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\section{Introduction}
Quanta image sensor (QIS) is a concept proposed by E. Fossum in 2005 aiming to overcome the limited sensor performance with the continuously shrinking pixel size~\cite{fossum200611,fossum2005gigapixel}. In a CMOS image sensor \cite{fossum1993active}, images are acquired by accumulating hundreds to thousands of photoelectrons in each pixel and converting the charges to voltages and then to digital signals. As pixel size shrinks, the amount of photons that fall on individual pixels drops. Under low-light (or high-speed imaging) conditions, photon shot noise dominates the acquisition process. As the photon becomes scarce, eventually the signal-to-noise ratio will drop to a point that no meaningful images can be generated. Small pixel also reduces the full-well capacity which limits the dynamic range. QIS overcomes the problem by spatial-temporally oversampling the scene. In a QIS, each ``image pixel'' is partitioned into a group of smaller pixels called jots, and each jot is sensitive enough for photon counting. Over the past decade, numerous studies have demonstrated the feasibility of QIS, with a few prototypes developed \cite{ma2015pump,ma2017photon,hynecek2001impactron,dutton2014320,bruschini2018monolithic}. On the image reconstruction side, various methods have also been proposed \cite{Chan14,Chan16,Elg18,yang2012bits}.
Despite the advances in QIS hardware and image reconstruction algorithms, one important open problem is the color imaging capability of QIS. Color imaging in low light is important for many computer vision applications such as night-vision. It is also important in biological studies, where using brighter light may lead to degradation of the biological matter. While active color imaging have been demonstrated using SPAD in \cite{Ren:18} passive color imaging using QIS is difficult for two reasons. First, the resolution of a typical single-photon detector array is not high, e.g., $320 \times 240$ pixels~\cite{dutton2016single}, or most recently $512 \times 512$ pixels ~ \cite{bruschini2018monolithic}. If a color filter array such as the standard Bayer pattern is implemented, the effective resolution will be halved to $160\times 120$ or $256\times256$. For detection/tracking applications \cite{gyongy2018single} this might be sufficient; but for photography the resolution could be low compared to the mainstream CMOS image sensors. Second, the color reconstruction of QIS data is different from CMOS image sensors because the QIS data is acquired under sparse-photon conditions with much lower signal-to-noise ratio. The demosaicing algorithm of QIS data needs to overcome the photon shot noise and accurately reproduce the color information. As we will discuss, QIS requires a joint reconstruction and demosaicing on top of the truncated Poisson statistics, a problem that has not been studied.
In this paper, we report the first demonstration of QIS color imaging. The QIS Pathfinder camera module developed at Gigajot Technology (as shown in Fig. \ref{fig: camera module and sensor}) was used in this demonstration. The camera sensor has the same core design as the sensor described in \cite{ma2017photon}, but has a Bayer pattern color filter array implemented on-chip. It has a spatial resolution of $1024\times1024$, single-bit and multi-bit photon-counting outputs, and up to $1040$ fps speed in single-bit mode. Figure \ref{fig:recSingleMulti} shows a snapshot of the work presented in this paper. The key innovation, besides the sensor, is a new customized image reconstruction algorithm which enables joint reconstruction and demosaicing of raw QIS data. Our method leverages a few recently developed techniques in the QIS literature, including the transform-denoise method \cite{Chan16} and the Plug-and-Play Alternating Direction Method of Multipliers (ADMM) \cite{chan2017plug}. We demonstrate both our new sensor and the new algorithm on real data, and make quantitative evaluation based on synthetic data.
\begin{figure}[t]
\centering
\begin{tabular}{cccc}
\includegraphics[width=0.23\linewidth]{Main_images/1Frame_Singlebit_q3p5.jpg}&
\hspace{-2.0ex}
\includegraphics[width=0.23\linewidth]{Main_images/ADMM_BM3D_lambda_0p150_rho_10p00_CCM_JJ1_35_1-bit_q_4p0_T_50_alpha_16.jpg}&
\hspace{-2.0ex}
\includegraphics[width=0.23\linewidth]{Main_images/1Frame_Multibit.jpg}&
\hspace{-2.0ex}\includegraphics[width=0.23\linewidth]{Main_images/ADMM_BM3D_lambda_0p0100_rho_1p00_gamma_1p00_WB_CCM_Linear_35_5-Bits_T5_demosaic.jpg}\\
\footnotesize (a) Raw input, 1-bit &
\hspace{-2.0ex}
\footnotesize (b) Processed from (a) &
\hspace{-2.0ex}\footnotesize (c) Raw input, 5-bit&
\hspace{-2.0ex}\footnotesize (d) Processed from (c) \\
\end{tabular}
\caption{(a) One 1-bit frame. (b) Reconstructed color image using \textcolor{black}{50 frames} of 1-bit input with threshold $q=4$. (c) One 5-bit frame. (d) Reconstructed color image using \textcolor{black}{10 frames} of 5-bit input. The average number of photons per frame is 5.}
\label{fig:recSingleMulti}
\vspace{-2.0ex}
\end{figure}
\section{Megapixel quanta image sensor}\label{sec:hardware}
When the concept of QIS was first proposed, the two mainstream design methodologies to realize single-photon detection were the electron-multiplying charge-coupled device (EMCCD) \cite{hynecek2001impactron} and the single-photon avalanche diode (SPAD) \cite{aull2002geiger,buchholz2018widefield}. Both devices amplify the signal from a single photoelectron with a physical phenomenon called electron \emph{avalanche multiplication}, where high electrical voltage (typically higher than 20V) is used to accelerate the photoelectron to produce more free electrons through collision. SPAD is more widely adopted in consumer applications because of its excellent low-noise performance and temporal resolution. The SwissSPAD \cite{charbon2008towards,charbon2007will,bruschini2018monolithic} and the SPAD developed at University of Edinburgh \cite{dutton2014320,dutton2016single} are two better known examples. However due to its fabrication technology SPAD often has to make compromise in at least one of the three factors - low dark current, high quantum efficiency, and small pixel pitch. The QIS we use in this paper is based on a new type of single-photon detector proposed by Ma \emph{et al.} \cite{ma2015pump,ma2015quanta,fossum2005gigapixel,ma2017photon,masoodian20171mjot}. The approach is to enhance the voltage signal generated by a single photoelectron by reducing the capacitance of the pixel output node so that the single-photon signal can overcome the background thermal noise.
\begin{figure*}[t]
\centering
\includegraphics[width=\linewidth]{Main_images/CMOSvsIphone.jpg}
\caption{Quanta Image Sensor (QIS) vs. CMOS Image Sensor (CIS). The top row shows a simulated CIS data at a photon level same as the QIS. The second row shows the real analog sensor data obtained from our prototype camera. For each photon flux level, we show both the raw input data and the denoised data. The CIS is assumed to have a read noise of $1.2 e^{-}$ r.m.s. , and we do not considered dark count. With dark count, the performance of CIS will deteriorate even more. }
\label{fig:IPhonevsQIS}
\end{figure*}
Table \ref{tab:Comparison} shows a comparison between a conventional CMOS image sensor, SPAD, and the prototype QIS used in this work. Among the listed different sensor parameters, the most noticeable difference between the QIS and SPADs is the small pixel pitch, where the QIS realized 1.1 $\mu$m pixel size, but SPADs typically have a pitch size around 5-20 $\mu$m. The smaller pixel size makes high spatial resolution (e.g. 10s or 100s of megapixel) possible with a small optical format (e.g. 1/4''). Another advantage of QIS over SPADs is the high quantum efficiency. Quantum efficiency measures the percentage of photons that are detectable by the sensor, which is critical for low-light imaging.
SPAD cameras generally have a low quantum efficiency (e.g. $<$50\%), because of low fill-factor and low photon detection capability normally called photon detection efficiency aka PDE. While low fill factor can be alleviated to an extent by using optical solutions such as microlens light-pipe. This reduces the photosensitive areal percentage in the SPAD pixels and limits the quantum efficiency. In contrast, since electron avalanche gain is not required in the QIS device, it has a quantum efficiency over 80\% within the visible wavelengths.
Figure \ref{fig:IPhonevsQIS} shows comparison between real QIS image and a simulted CIS image. We note that we assume that the CIS no dark current. Even without considering the advantages of a lower dark current, the performance of QIS in photon starved regime is much better than the CIS.
\begin{table*}
\footnotesize
\caption{\label{tab:Comparison}Comparison of the available image sensor technologies.}
\makebox[1 \textwidth][c]{
\begin{tabular}[b]{ c c c c c c}
\textbf{Camera} & \textbf{ Sony IMX253} & \textbf{ Edin. SPAD} & \textbf{SwissSPAD2 } & \textbf{QIS} & \textbf{sCMOS} \\
&CMOS&\cite{dutton2016spad}&\cite{bruschini2018monolithic}&(This work)&\cite{scmos}\\
\hline \hline
\textbf{Resolution} & 4096 $\times$ 3000 &320 $\times$ 240 & 512 $\times$ 512& 1024 $\times$ 1024 & 2048 $\times$ 2048 \\
\textbf{Mean Dark Count Rate} & 2.4 $e^{-}$/pix/s& 47 $e^{-}$/pix/s & 75 $e^{-}$/pix/s & 0.068 $e^{-}$/pix/s & 0.1 $e^{-}$/pix/s at $0^\circ C$\\
\textbf{Quantum Efficiency} & $72\%$ & $50\%$ & $39.5\%$ & $86\%$ & $82\%$ \\
& at 525 nm & at 520 nm & at 480 nm & at 480 nm & at 560 nm\\
\textbf{Pixel Pitch} &3.45$\mu$m & 8 $\mu$m& 16.4 $\mu$m & 1.1 $\mu$m & $6.5\mu m$ \\
\textbf{Frames / sec} & 30 & $16$k & 97.7k &1040 & 30\\
\textbf{Photon Counting} &No & Yes &Yes &Yes & No\\
\textbf{Color Capability} &Yes &No &No &Yes & Yes\\
\hline
\end{tabular}
}
[Note : Measurements under room temperature unless otherwise noted]
\end{table*}
\section{QIS color imaging model}
\label{sec:imagingModel}
The imaging model of a color QIS is illustrated in Fig. \ref{fig:Imgaing_model}. Relative to the previous QIS imaging models, e.g., \cite{yang2012bits,Chan16,Elg18}, the new model considers a color filter array which allows us to select wavelengths and multiple color channels. There are a few essential components in the model.
\begin{figure*}[!]
\centering
\includegraphics[width=\linewidth]{Main_images/image_formation.jpg}
\caption{QIS Imaging Model. When the scene image arrives at the sensor, the color filter array first selects the wavelength according to the colors. Each color pixel is then sensed using a photon-detector. In single-bit mode the detector reports a binary value, and in multi-bit mode the detector reports an integer up to the saturation limit. The measured data contains three subsampled sequences, each representing a measurement in the color channel.}
\label{fig:Imgaing_model}
\end{figure*}
\vspace{1ex}
\noindent\textbf{Spatial oversampling}. We represent the discretized scene image as a column vector $\vc = [\vc_r; \vc_g; \vc_b] \in [0,1]^{3N}$, where $N$ is the number of pixels, $\vc_r$, $\vc_g$, and $\vc_b$ are the light intensities of the red, green and blue channels, respectively. QIS was originally designed as a spatial-temporally oversampling device which samples each pixel using $K$ jots spatially. This means every pixel has $K$ measurements, and totally there are $M \bydef NK$ measurements. Thus, the model of this oversampling process is
\begin{equation}
\vv = [\vv_r; \vv_g; \vv_b] = \alpha \mG\vc \in \R^{3M},
\end{equation}
where $\mG \in \R^{3M \times 3N}$ represents the $K$-fold upsampling and interpolation filtering. The constant $\alpha$ is a gain factor. In our current prototype, the oversampling factor is $K=1$ as we want to maximize the spatial resolution. In this case, we have $\mG = \mI$, and $M=N$.
\vspace{1ex}
\noindent\textbf{Color filter array}. To obtain color information, a color filter is placed on the top of each jot to collect light for only one of the RGB colors, and filter out the remaining two colors. This process is represented by three mutually exclusive $M\times M$ diagonal matrices or masks: $\mS_r$, $\mS_g$ and $\mS_B$ for the red, green and blue channels, respectively. These matrices contain either zeros or ones on the diagonal, and $\mS_r + \mS_g + \mS_b = \mI$.
Incorporating the color filter array, the light exposure on the $M$ jots of QIS is represented by
\begin{equation}\label{eq:model}
\vtheta = \mS_r\vv_r+\mS_g\vv_g+\mS_b\vv_b \;\;\in \mathbb{R}^{M}.
\end{equation}
We call $\vtheta$ the mosaiced image. If we let $\mS \bydef [\mS_r, \; \mS_g, \; \mS_b] \in \R^{M \times 3M}$, then $\vtheta$ and $\vc$ are related by
\begin{equation}
\vtheta = \alpha\mS\mG\vc.
\end{equation}
\begin{figure}[!t]
\centering
\begin{tabular}{cc}
\includegraphics[width=0.35\linewidth]{Main_images/camera_1b.JPG}&
\includegraphics[width=0.35\linewidth]{Main_images/camera_1a.JPG}\\
(a) QIS sensor chip & (b) Prototype QIS camera module
\end{tabular}
\caption{Gigajot prototype QIS camera module and the QIS sensor chip. Note that there is no additional optics required besides a simple focusing lens. The camera is powered by standard 5V DC input, and has a USB3 data interface to transmit data to external storage.}
\label{fig: camera module and sensor}
\end{figure}
\vspace{1ex}
\noindent\textbf{Photon arrival}. The photon arrival of QIS is modeled as a Poisson process. Let $\mY \in \N^{M}$ be a non-negative random integer denoting the number of photons arrived at each jot during one integration period. Then, the probability of observing $\mY = \vy \in \N^{M}$ is
\begin{equation}\label{eq:prob_Ym}
\Pb(\mY = \vy)= \prod_{m=1}^{M} \frac{\theta_m ^{y_m} e^{-\theta_m}}{y_m!}.
\end{equation}
For single-bit QIS, we put a voltage comparator to threshold the arrived photon count into a binary decision. Thus, the read out of each jot is a binary random variable $\mB \in \{0,1\}^{M}$ with $B_m = 1$ if $Y_m \ge q$ and $B_m = 0$ if $Y_m < q$, where $q > 0$ is a threshold.
For multi-bit QIS, we use an
$L$-bit quantizer to quantize the measurement. In this case,
\begin{equation}
B_m =
\begin{cases}
Y_m,& Y_{m} < 2^{L}-1, \\
2^{L}-1,& Y_{m} \geq 2^{L}-1.
\end{cases}
\end{equation}
In other words, we count the photons until the number reaches a threshold. When the threshold is reached, we report the maximum number before saturation.
\begin{figure}[!t]
\centering
\includegraphics[width=0.6\linewidth]{Main_images/PCH.jpg}
\caption{Measured photo-electron count from a real sensor. The non-integer values of the count are due to the read noise of the readout circuit. However, with the ultra-low read noise provided by the QIS device, its negative effect on the photon-counting is negligible by setting appropriate thresholds.}
\label{fig:photon count histogram}
\end{figure}
\vspace{2ex}
\noindent\textbf{Assumptions}. While the above model is theoretically tractable, we made three assumptions. The first is regarding the read noise associated with $\mY$. If read noise is present, the measured photon count will become $\vy + \veta$ where $\veta$ is typically i.i.d. Gaussian. We assume that the read noise is negligible compared to the photon count, and can be eliminated by the thresholding process. We find that this is often the case with the QIS device used in this paper, which provides ultra-low read noise (0.24e- rms at room temperature) and accurate photon-counting capability. See Fig. \ref{fig:photon count histogram} for an actual measurement of the photon count.
The second assumption we make is that the multi-bit saturation does not happen. That is, during the multi-bit sensing, we will never reach the upper threshold $q$ and so $B_m = Y_m$, which is a standard Poisson random variable. In terms of hardware, this can be achieved by implementing an automatic exposure control scheme so that the threshold $q$ is increased when the scene is too bright. We will leave this to a follow up paper as it is outside the scope of this work.
The third assumption we make is that the camera is fast enough to catch up the scene movement, and the scene remains static during the multiple temporal samplings. This allows us to utilize multiple independent measurements over time to improve the statistics. In this case, the probability distribution of $\mY$ becomes
\begin{equation}\label{eq:prob_Ym 2}
\Pb(\mY = \vy)= \prod_{t=1}^T \prod_{m=1}^{M} \frac{\theta_m ^{y_{m,t}} e^{-\theta_m}}{y_{m,t}!},
\end{equation}
where $y_{m,t}$ is the Poisson count at pixel $m$ in time $t$.
\begin{figure*}[!]
\centering
\includegraphics[width = 1\linewidth]{Main_images/pipeline_2.jpg}
\caption{The image reconstruction pipeline consists of (i) a temporal binning step to sum the input frames, (ii) a variance stabilizing transform $\calT$ to transform the measurement so that the variance is stabilized, (iii) a joint reconstruction and demosaicing algorithm to recovery the color, (iv) an inverse transform to compensate the forward transform, and (v) a tone mapping operation to correct the contrast.}
\label{fig:color_image_pipeline}
\end{figure*}
\section{QIS color image reconstruction} \label{sec:pipeline}
The task of image reconstruction is to recover color scene $\vc$ from the measurements $\mB$. In the gray-scale setting, we can formulate the problem as maximum-likelihood and solve it using convex optimization tools \cite{yang2010optimal,yang2012bits,Chan14,Elg18}. We can also use learning-based methods, e.g., \cite{Rem16,choi2018image,chandramouli2018alittle,rojas2017learning} to reconstruct the signal. The method we present here is based on the transform-denoise approach by Chan \emph{et al.} \cite{Chan16}. We prefer transform-denoise because it is a physics-based approach and is robust to different sensor configurations. For example, in learning-based approaches, if we change the number of frames to sum, we need to train a different model or neural network.
\subsection{Reconstruction pipeline}
The pipeline of the proposed reconstruction algorithm is shown in Fig. \ref{fig:color_image_pipeline}. Given the random variable $\mB = \{B_{m,t}\}$ with realizations $\vb = \{b_{m,t}\}$, we first sum the frames to generate a single image $\mZ = \{Z_{m}\}$:
\begin{equation}
Z_m = \sum_{t=1}^T B_{m,t}.
\end{equation}
In the future, this step can be integrated into the hardware of the camera, so that the output of the camera will be the sum of multiple frames.
Depending on whether we are using the single-bit mode or the multi-bit mode, the statistics of $\{Z_m\}$ follows either a Binomial distribution or a Poisson distribution. In single-bit mode, the distribution of $\{Z_m\}$ is
\begin{equation}
\Pb[Z_m = k \;|\; \theta_m] = {T \choose k} \Psi_q(\theta_m)^{1-k}(1-\Psi_q(\theta_m))^{k},
\end{equation}
where $\Psi_q(\theta) \bydef \Pb[B = 0] = \sum_{j=0}^{q-1} \theta^j e^{-\theta}/j!$ is the probability of obtaining a zero in the binary variable $B$. As shown in \cite{Elg18}$, \Psi_q$ is the incomplete Gamma function. In multi-bit mode, since a sum of Poisson remains a Poisson, the distribution of $\{Z_m\}$ is
\begin{equation}
\Pb[Z_m = k \;|\; \theta_m] = \frac{(T\theta_m)^k e^{-T\theta_m}}{k!}.
\end{equation}
If we assume that there is no processing of the bits besides temporal summation, then a simple maximum-likelihood of the probability will return us the solution:
\begin{align}
\vtheta &= \argmax{\vtheta}\;\; \prod_{m=1}^M \Pb[Z_m = z_m \;|\; \theta_m] \notag \\
&\bydef \calM(\vec{z}) = \begin{cases}
\Psi_q^{-1}(1-\vec{z}/T), &\quad \mbox{Single-bit},\\
\vec{z}/T, &\quad \mbox{Multi-bit}.
\end{cases}
\end{align}
The operation $\calM$ can be interpreted as a tone-mapping.
\subsection{Variance stabilizing transform}
Now, we want to process the data besides just summing the frames. However, in either single-bit or multi-bit mode, the Binomial or the Poisson statistics are not easy as the variance changes with the mean. This prohibits the use any off-the-shelf algorithms that are based on i.i.d. Gaussian assumptions, e.g., denoising or inpainting.
The variance stabilizing transform (VST) is a statistical technique that tries to alleviate the changing variance problem \cite{azzari2017variance, azzari2016variance,foi2011noise}. Among the different types of VSTs, we are particularly interested in the two transforms originally proposed by F. Anscombe in 1948 \cite{Anscombe_1948}:
\begin{equation*}
\vbeta = \calT(\vz) \bydef
\begin{cases}
\sqrt{T+\frac{1}{2}}\sin^{-1}\sqrt{\frac{\vz+\frac{3}{8}}{T + \frac{3}{4}}}, &\quad\mbox{Single-bit},\\
\sqrt{\vz + \frac{3}{8}}, &\quad\mbox{Multi-bit}.
\end{cases}
\end{equation*}
Here, the first transform is called the Anscombe Binomial, and the second is called the Anscombe Poisson. Both transforms are pixel-wise, meaning that we can apply the transform to each pixel independently. As proved in \cite{Chan16} and \cite{Anscombe_1948}, the variance of $\calT(\vz)$ is 1/4 for both the single-bit and the multi-bit case. The inverse $\calT^{-1}$ is simply the algebraic inverse (or the unbiased inverse if we follow \cite{Makitalo_Foi_2011}).
The important question for color imaging is now that the photon count $\vz$ is the result of applying a color filter array matrix $\mS = [\mS_r,\mS_g,\mS_b]$. That is, instead of observing the full color $\vztilde \bydef [\vz_r;\vz_g;\vz_b]$, we only observe a subsampled version $\vz = \mS\vztilde$. However, since $\calT$ is point-wise and $\mS$ is a concatenation of three non-overlapping binary matrices, it holds that
\begin{equation}
\calT(\vz) = \calT(\mS\vztilde) = \mS\calT(\vztilde).
\end{equation}
Consequently, if we define $\calT(\vz)$ as the observed measurement (which is available from the sensor) and treat $\vx \bydef \calT(\vztilde)$ as the unknown quantity to be determined, then the problem can be formulated as
\begin{equation}
\vhat = \argmin{\vx}\;\; \|\mS\vx - \calT(\vz)\|^2 + \lambda g(\vx),
\label{eq: demosaic-denoise}
\end{equation}
where $g$ is some regularization function controlling the smoothness of $\vx$. Note that \eref{eq: demosaic-denoise} is a standard demosaicing-denoising problem assuming i.i.d. Gaussian noise.
\subsection{Joint reconstruction and demosaicing}
The optimization problem in \eref{eq: demosaic-denoise} is a standard least squares with regularization function $g$. Thus, most convex optimization algorithm can be used as long as $g$ is convex. In this paper, we adopt a variation of the alternating direction method of multiplier (ADMM) by replacing $g$ with an off-the-shelf image denoiser. Such algorithm is coined the name Plug-and-Play (PnP) \cite{venkatakrishnan2013plug} (and different versions thereafter \cite{chan2017plug}).
For the particular problem in \eref{eq: demosaic-denoise}, the PnP ADMM algorithm iteratively updates the following two steps:
\vspace{1ex}
\noindent\textbf{Demosaicing Module}:
\begin{equation}
\vx^{(k+1)} = (\mS^T\mS + \rho\mI)^{-1}(\mS^T\calT(\vz) + \rho(\vv^{(k)}-\vu^{(k)})),
\end{equation}
\noindent\textbf{Denoising Module}:
\begin{equation}
\vv^{(k+1)} = \calD_{\rho/\lambda}(\vx^{(k+1)}+\vu^{(k)}),
\end{equation}
and updates the Lagrange multiplier by $\vu^{(k+1)} = \vu^{(k)} - (\vx^{(k+1)}-\vv^{(k+1)})$. Readers interested in the detailed derivation can consult, e.g., \cite{chan2017plug}. Here, $\rho$ is an internal parameter that controls the convergence. The operator $\calD$ is an off-the-shelf image denoiser, e.g., Block-matching and 3D filtering (BM3D) or deep neural network denoisers. The subscript $\rho/\lambda$ denotes the denoising strength, i.e., the hypothesized ``noise variance''. Since $\mS^T\mS$ is a diagonal matrix, the inversion is pointwise.
\subsection{Non-iterative algorithm}
The $1.1\mu$m pixel pitch of the proposed QIS can potentially lead to a spatial resolution as high as or even higher than a conventional CMOS sensor. When this happens, in certain applications we can trade-off the color reconstruction efficiency and the resolution. For example, instead of using one jot for one pixel, we can use four jots for one pixel as shown in Fig. \ref{fig:2x and 1x}.
\begin{figure}[h]
\centering
\begin{tabular}{cc}
\includegraphics[width=0.35\linewidth]{Main_images/CFA_patterns_1}&
\includegraphics[width=0.35\linewidth]{Main_images/CFA_patterns_2}
\end{tabular}
\caption{In the future when QIS can achieve higher spatial resolution, we can use four color jots to reconstruct one pixel. In this case, we can bypass the iterative ADMM algorithm and use a one-shot denoising method.}
\label{fig:2x and 1x}
\end{figure}
Using four jots for one pixel allows us to bypass the iterative ADMM steps because there is no more missing pixel problem. In this case, the matrix $\mS \in \R^{M \times 3M}$ will become $\mS = \mbox{diag}\{\frac{1}{4}\mI, \frac{1}{2}\mI, \frac{1}{4}\mI\} \in \R^{3M \times 3M}$, and hence \eref{eq: demosaic-denoise} is simplified to a denoising problem with different noise levels for the three channels. In particular, the green channel has half of the variance of the red and the blue. For implementation, we can modify a denoiser, e.g., BM3D to accommodate the different noise variances. Since there is no more ADMM iteration, the algorithm is significantly faster. While we have used BM3D to demonstrate the results, any off the shelf denoiser which is used in CIS based cameras can be used to denoise the image for the four-jot to one-pixel method. We would also like to stress on the fact that both the Anscombe transform and the transform $\calM$ can be implemented as a look up table. So, this method can be as fast a current denoiser being used in a CIS based camera.
\subsection{Comparisons}
We compare the proposed method with several existing methods on a synthetic dataset shown in Fig. \ref{fig:demosaicing CMOS}. We simulate the raw color QIS data by assuming a Bayer pattern and using the image formation pipeline described in the previous section. We demosaic the images using: (a) a baseline method using MATLAB's demosaic preceded by gray-scale BM3D denoising of $R$, $G_1$, $G_2$ and $B$ channels and followed by color BM3D denoising; (b) Least-squares luma-chroma demultiplexing (LSLCD) method \cite{Jeon_Dubois_2013}, which has a built-in BM3D denoiser; (c) Hirakawa's PSDD method \cite{Korneliussen_Korneliussen_2014,Hirakawa_Meng_Wolfe_2007}, which does joint denoising and demosaicing for Poisson noise; and (d) the proposed method using BM3D with $(\lambda,\rho)=(0.001,5)$. We apply variance stabilizing transform, except for PSDD which is designed for Poisson noise. The results show that the proposed method has a significantly better performance, both in terms of Peak Signal to Noise Ratio (PSNR) and visual quality.
\begin{figure}[t]
\centering
\begin{tabular}{ccc}
\includegraphics[width=0.25\linewidth]{Main_images/GT.jpg} &
\includegraphics[width=0.25\linewidth]{Main_images/Raw.jpg}&
\includegraphics[width=0.25\linewidth]{Main_images/MATLAB.jpg} \\
(a) Ground truth & (b) Simulated Input & (c) MATLAB $30.91$dB\\
\includegraphics[width=0.25\linewidth]{Main_images/LSLCD.jpg} &
\includegraphics[width=0.25\linewidth]{Main_images/PSDD.jpg} &
\includegraphics[width=0.25\linewidth]{Main_images/ADMM.jpg}\\
(d) \cite{Jeon_Dubois_2013} $30.24$dB & (e) \cite{Hirakawa_Meng_Wolfe_2007} $29.57$dB & (f) Ours $31.51$dB
\end{tabular}
\vspace{0.5ex}
\caption{Simulated QIS experiment. The goal of this experiment is to compare the proposed iterative algorithm with existing methods. We assume the observed Bayer RGB image is from a 3-bit QIS sensor. (a) Ground Truth; (b) One 3-bit QIS frame; (c) MATLAB demosaic preceded and followed by BM3D; (d) LSLCD\cite{Jeon_Dubois_2013}; (e) Hirakawa's PSDD method \cite{Hirakawa_Meng_Wolfe_2007}, with a built-in wavelet shrinkage denoiser; (f) Proposed method with BM3D.}
\vspace{-2ex}
\label{fig:demosaicing CMOS}
\end{figure}
\section{Experimental results}
\subsection{Synthesized QIS data}
We conduct a synthetic experiment to provide a quantitative evaluation of the performance of the proposed algorithm. To this end, we simulate the image formation pipeline by passing through color images to generate the QIS raw input data, with different number of bits. Figure \ref{fig:recSingleMultiSynth} shows one example, and in the Supplementary Report we have additional examples.
In our simulation, we assume that the number of QIS frames is $T = 4$, and the average number of photon per pixel is $0.28$, $0.85$, $1.98$ and $4.23$ photons / frame for 1-bit, 2-bit, 3-bit and 4-bit QIS, respectively. On the measurement side, we generate single-bit and multi-bit data by thresholding the raw sensor output. To reconstruct the image, we use the proposed method with PnP and BM3D. The parameters are set as $\rho=1$ and $\lambda=0.007, 0.003, 0.002$ and $0.0007$ for 1-bit, 2-bit, 3-bit and 4-bit QIS, respectively. With as low as 1-bit, the reconstructed image in Fig. \ref{fig:recSingleMultiSynth} is already capturing most of the features. As the number of bits increases, the visual quality improves.
\begin{figure*}[!t]
\centering
\begin{tabular}{cccc}
\includegraphics[width=0.23\linewidth]{Main_images/Flower_1-bit_Raw.jpg}&
\hspace{-2.0ex}\includegraphics[width=0.23\linewidth]{Main_images/Flower_2-bit_Raw.jpg} &
\hspace{-2.0ex} \includegraphics[width=0.23\linewidth]{Main_images/Flower_3-bit_Raw.jpg} &
\hspace{-2.0ex} \includegraphics[width=0.23\linewidth]{Main_images/Flower_4-bit_Raw.jpg}\\
\includegraphics[width=0.23\linewidth]{Main_images/Flower_1-bit.jpg}&
\hspace{-2.0ex}\includegraphics[width=0.23\linewidth]{Main_images/Flower_2-bit.jpg} &
\hspace{-2.0ex} \includegraphics[width=0.23\linewidth]{Main_images/Flower_3-bit.jpg} &
\hspace{-2.0ex} \includegraphics[width=0.23\linewidth]{Main_images/Flower_4-bit.jpg}\\
\footnotesize 1-bit, $22.67$dB &
\footnotesize 2-bit, $23.86$dB &
\footnotesize 3-bit, $25.37$dB &
\footnotesize 4-bit, $26.73$dB \\
\end{tabular}
\caption{Synthetic experiment for quantitative evaluation. [Top row]: One frame of the QIS measurements using different number of bits. [Bottom row]: Reconstructed images using the proposed method with 20 frames of QIS data. The average photon counts per pixel are 0.25, 0.75, 1.75 and 3.75 for 1-bit, 2-bit, 3-bit and 4-bit QIS, respectively.}
\label{fig:recSingleMultiSynth}
\vspace{-1.0ex}
\end{figure*}
\subsection{Real QIS data}
We first show the reconstruction of an image of the Digital SG $\mathrm{ColorChecker}^{\footnotesize\textregistered}$ chart. We generate two sets of measurements: (a) a set of 50 one-bit frames, quantized with a threshold $q=4$, and (b) a set of 10 five-bit frames. We use PnP and BM3D with $(\lambda,\rho)=(0.15,10)$ and $(0.01,10)$ for the 1-bit and 5-bit data, respectively. After reconstruction, the results are multiplied by a $3\times 3$ color correction matrix to mitigate any color cross-talk. This matrix is generated by linear least square regression of the 24 Macbeth color patches that lies inside the SG ColorChecker chart. The results in Fig. \ref{fig:recSingleMulti} suggest that while 1-bit mode has color discrepancy with the ground truth, the 5-bit mode is producing a reasonably-high color accuracy.
Next, we show the result of imaging real scenes. See Fig. \ref{fig:results_reconstruction1}. In this experiment, the exposure time for each frame is set to 50$\mu$s. The average number of photons per pixel is approximately 4.2 photons for the ''QIS'' sign image, 3 photons for the Pathfinder image, 1.9 photons for the duck image, and 2.9 photons for the mushroom image. For all images, we collect the data using a 5-bit QIS.
We demonstrate two algorithms: (i) the proposed transform-denoise framework using PnP + BM3D, and (ii) assuming four-jots to one-pixel scenario by trading half of the spatial resolution. The typical runtime on an unoptimized MATLAB code is approximately 4 minutes for PnP, and 10 seconds for the four-jot to one-pixel method. An interesting observation is that even in the lower-resolution case, the details are not significantly deteriorated unless we zoom-in. However, the speed up we get is substantial.
\begin{figure*}[!]
\centering
\begin{tabular}{cccc}
\includegraphics[width=0.23\linewidth]{Main_images/qis_sign_one_5bitFrame1.jpg}&
\hspace{-2.0ex}\includegraphics[width=0.23\linewidth]{Main_images/model3_1frame.jpg} &
\hspace{-2.0ex}\includegraphics[width=0.23\linewidth]{Main_images/mushrromandduck3_1frame.jpg}&
\hspace{-2.0ex}\includegraphics[width=0.23\linewidth]{Main_images/dudeandmushroom_one_5bitFrame1.jpg}\\
\multicolumn{4}{c}{(a) Raw QIS 5-bit Data. 4 frames with exposure time of 50 $\mu$s were obtained for each scene.}\\
\includegraphics[width=0.23\linewidth]{Main_images/qis_sign_iterative.jpg}&
\hspace{-2.0ex}\includegraphics[width=0.23\linewidth]{Main_images/model3_2_iterative.jpg} &
\hspace{-2.0ex}\includegraphics[width=0.23\linewidth]{Main_images/mushroomandduck3_iterative.jpg} &
\hspace{-2.0ex}\includegraphics[width=0.23\linewidth]{Main_images/dudeandmushroom_iterative.jpg}\\
\multicolumn{4}{c}{(b) Reconstruction using the proposed method (iterative) $1024\times1024$}\\
\includegraphics[width=0.23\linewidth]{Main_images/qis_sign_noniterative.jpg}&
\hspace{-2.0ex}\includegraphics[width=0.23\linewidth]{Main_images/model3_2_noniterative.jpg} &
\hspace{-2.0ex}\includegraphics[width=0.23\linewidth]{Main_images/mushroomandduck3_noniterative.jpg} &
\hspace{-2.0ex}\includegraphics[width=0.23\linewidth]{Main_images/dudeandmushroom_noniterative.jpg}\\
\multicolumn{4}{c}{(c) Reconstruction using the proposed method (non-iterative) $512 \times 512$}
\end{tabular}
\vspace{-2ex}
\caption{ Real QIS image reconstruction. The exposure time for each frame is 50 $\mu$s. The average number of photons per frame is 4.2, 3.0, 1.9, and 2.9 for each image respectively. Both methods use 4 frames for reconstruction. The raw data has a resolution of $1024 \times 1024$ pixels. The ADMM method retains the resolution, whereas the non-iterative method reduces the resolution to $512 \times 512$. Reconstruction using both the methods are shown at the same size for easier visual comparison. Notice that the non-iterative algorithm is able to achieve a visual quality almost similar to the ADMM method. }
\label{fig:results_reconstruction1}
\vspace{-4ex}
\end{figure*}
\section{Conclusion and discussion}
We demonstrated the first color imaging for a mega-pixel Quanta Image Sensor (QIS). We designed a joint reconstruction-demosaicing algorithm for the quantized Poisson (and Poisson) statistics of the sensor data. Our solution involves a variance stabilizing transform and an iterative (and a non-iterative) algorithm. By integrating the sensor and the new algorithm, the sensor is able to acquire color images at a photon level as low as a few photons per pixel, a level that would be difficult for standard CMOS image sensors.
\section{Acknowledgement}
The authors thank Prof. Eric Fossum for many valuable feedback. The 1Mjot color Quanta Image Sensor used in the demonstration was designed by J. Ma, S. Masoodian, and D. Starkey at Dartmouth College Advanced Image Sensor Laboratory led by Dr. Eric Fossum. The sensor was manufactured by Taiwan Semiconductor Manufacturing Company (TSMC). The camera module developed at Gigajot Technology was partially supported by National Science Foundation SBIR program. A Gnanasambandam, O. Elgendy and S. H. Chan are supported, in part, by the U.S. National Science Foundation under Grant CCF-1718007.
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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Q: on hover/mouseover change color AND scale? I'm trying to create buttons that are small, and grey until moused over- then they become large and colored.
I've used Soh Tanaka's awesome greyscale tutorial-
but I can't figure out how to add the additional parameter of having the images also get bigger when moused over.
http://www.sohtanaka.com/web-design/examples/hover-over-trick/
any help is appreciated!!
A: Just animate width, height, top, left and margins. Here's a snippet from http://www.sohtanaka.com/web-design/fancy-thumbnail-hover-effect-w-jquery/
$("ul.thumb li").hover(function() {
$(this).css({'z-index' : '10'}); /*Add a higher z-index value so this image stays on top*/
$(this).find('img').addClass("hover").stop() /* Add class of "hover", then stop animation queue buildup*/
.animate({
marginTop: '-110px', /* The next 4 lines will vertically align this image */
marginLeft: '-110px',
top: '50%',
left: '50%',
width: '174px', /* Set new width */
height: '174px', /* Set new height */
padding: '20px'
}, 200); /* this value of "200" is the speed of how fast/slow this hover animates */
} , function() {
$(this).css({'z-index' : '0'}); /* Set z-index back to 0 */
$(this).find('img').removeClass("hover").stop() /* Remove the "hover" class , then stop animation queue buildup*/
.animate({
marginTop: '0', /* Set alignment back to default */
marginLeft: '0',
top: '0',
left: '0',
width: '100px', /* Set width back to default */
height: '100px', /* Set height back to default */
padding: '5px'
}, 400);
});
A: Try this one make Two class
.before_hover{
width:10px;
height:10px;
}
.on_hover{
width:10px !important;
height:10px !important;
}
Jquery
$(document).ready(function(){
$('ul li').hover(function(){
$(this).siblings().removeClass('on_hover');
$(this).addClass('on_hover');
},
function(){
});
});
you can make required changes in css.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
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Category Archive: Climate & Seasons
Home > Archive by category Climate & Seasons
The following airlines operate flights to/from Dushanbe (DYU)
AirBaltic (Riga)
Atlant-Soyuz Airlines (Krasnoyarsk, Moscow, Samara)
Avia Traffic Company (Bishkek)
China Southern Airlines (Urumqi)
Kam Air (Kabul)
Kyrgyzstan Airlines (Bishkek)
Iran Aseman Airlines (Tehran-Imam Khomeini, Mashhad)
Rossiya (St Petersburg)
S7 Airlines (Chelyabinsk, Novosibirsk, Perm)
Scat Air (Almaty)
Somon Air (Almaty, Dubai, Frankfurt, Istanbul-Atatürk, Moscow-Domodedovo,Urumqi)
Tajik Air (Almaty, Bishkek, Garm, Istanbul, Khujand, Khorog, Kulyab, Kurgan Tyube, Moscow, Novosibirsk, Samara, Tehran, Yekaterinburg)
Tatarstan Airlines (Kazan)
Dushanbe International Airport (DYU) is located 4 km (2 miles) southeast of the city. Airport facilities include left luggage office, exchange office, duty-free shops, restaurants, bars and open 24 hours. There is bus line #8 and trolleybus line #4 serve the airport as well as minibus shuttle service ##7, 8, 14, 16 to the city centre in daytime (travel time – 15 minutes). Taxis are readily available, travel time is 10-15 minutes.
Tajikistan borders with Kyrgyzstan, Uzbekistan, Afghanistan, and China. All of the borders have at least several designated land border crossings. Land borders working hours vary from place to place, some works round the clock, the others are not. It is especially important to check working hours if you are travelling around national holidays or important events. Force majeure like mudslides, avalanches or floods on border crossings based in the mountains may also be a subject to close the border.
Please note visa service is not available at the land borders, which means your Tajikistan visa must be prearranged prior arrival. Please check visa section for details.
Travelling thru Central Asia you can choose to cross at one of the following official land border crossings:
Over Tajikistan/Uzbekistan border: Tajikistan registered vehicles are not allowed to enter Uzbekistan, and Uzbekistan registered vehicles entering Tajikistan are subject to tax. So if you hire a taxi or share a seat, it is likely that you will only be taken as far as the border. You will have to cross the border on foot, pass thru formalities and get another vehicle on the other side. Taxis are readily available on both sides. Also "porterage" service available on both sides of the border. Paying little fee you can get your luggage loaded onto the curt or wheelbarrow and delivered straight to another taxi's trunk.
Climate & Seasons, Currency, History, Plants & Animals, Tajikistan in & out
Until the mid of the first millennium B.C., the territory of modern Tajikistan in the areas of Amudarya and Sirdarya were settled by Eastern Iranian tribes. Bactria and Sogdiana were the most ancient states, lying along the banks of the upper and middle Amudarya. In the 6th – 4th century B.C. these states were part of the Akhamenids Empire established by Persians. The agricultural area of Sogdiana, that included Fergana and Zerafshan valleys and reaching in the west the area of Bukhara, played an important role in international trade, as it was on trade routes that linked China and Central Asia. The Akhamenid State collapsed in 330 B.C. under pressure from Greek-Macedonian forces. Bactrians, Sogdians and other people of Central Asia were conquered, in spite of their heroic resistance to the army of Alexander the Great. About 200 years later, the population of Bactria and Sogdiana together with massage nomad tribes overthrew the Greek-Macedonian sway. A state was established in Bactria – Tokharistan, which later together with Sogdiana became part of the big Kushan kingdom. The Silk Road crossed via Tokharistan and silk was bought at the markets near the river Tarima and was delivered to the countries of Greek-Roman Empires of the west. In the reverse direction from Rome and Byzantine to China, glass products (such as crystal and thin multi-coloured glassware) were imported, from Central Asia – items such as adornments, gems and from India – paper, woollen fabrics and spices. In the 5th century, Tokharistan, Sogdiana and other areas of Central Asia were under the rule of Ephtalits, and later in the 6th century – nomad Turkic speaking tribes occupied the same region. Sogdians had a great influence on the nomads, who while settling mixed with the local population. During the 6th and 7th centuries in Tokharistan and Sogdiana there were many slaves and at the same time, the formation of a feudal system had already started. As a result, the economy and culture developed quickly. Iron, copper, lazurite and rubies were extracted at mines, irrigation channels were constructed, and arid places were irrigated. The importance of feudal cities started to grow, as well as craft and trade were developing. The main languages were Bactrian and Sogdian – which are Eastern Iranian languages. Numerous independent principalities emerged with the development of feudal relationships. But they were weak and could not resist the Arabs, who in the 7th & 8th centuries invaded Central Asia.
The population of Sogdiana and Tokharistan tirelessly struggled for their liberation. Because of continuous revolts it became difficult for the Arabs to keep control of Central Asia with the help of their governor-generals; therefore they more often resorted to the help of the service of their feudal partners. Their participation prepared the ground for releasing Central Asia from Arab rule. Hence, at the end of the 9th century independent from the Bagdad caliphate (successor of Muhammed), the Tajik state of Samanid formed. In the period of its most extensive growth Samanid stretched from the deserts of Central Asia to the Gulf and from the borders of India to Bagdad. During the Samanids Empire Tajik people and culture became widespread and Tajik language became dominant. The Samanids state lived in peace for more than 100 years which fostered the growth of cities , craft, development of farming and trade and mining. This was truly the era of Renaissance that produced some of the world's greatest humanitarians such as the founder of the Persian-Tajik poetry – Rudaki, creator of the immortal poem – 'Shahname' Firdawsi, and world renown scientist-Encyclopaedis Abu Ali ibn Sina (Avicenna). However internal conflicts and frequent raids of nomads undermined and weakened the Samanid state, which in 999 collapsed under the strike of the Turkic speaking tribes. The dynasties of Karakhanids (in the North) and Gaznavids (in the South) founded their power on the ruins of the Samanids state. In the beginning of the 13th century (1219-1221) Central Asia was invaded by Mongols, under the command of Chingizkhan. The country was completely devastated, cities destroyed, gardens and vineyards turned into pastures for horses of the invaders. Not long before his death Chingizkhan he divided the invaded lands between his sons. The main cultural areas of Central Asia went to ulus (independent principalities) of his second son Chagatai. People showed resistance. The biggest revolt broke out in 1238 in Bukhara, which was led by a craftsman Mahmud Tarabi. Another revolt in Samarkand was led by "sarbadars" – hanged men, who defeated the Mongol army in Bukhara, which brought to power a new ruler – Tamerlan. Timur's ascent to power in 1370 temporarily put an end to feudal factions. Timur, as a result of his numerous campaigns to conquer Eastern countries and unheard cruelty and genocide especially towards the Persian speaking population, built an enormous empire with its capital in Samarkand. The majority of today's Tajikistan was part of Timur's empire. During the reign of his son and especially his grandson Ulugbek– prominent scientist, astronomer and loyal ruler – the widely abandoned oasis of Central Asia was restored with the flourishing of astronomy, math, history, literature and art. But the inter-dynasty struggle and raids of nomads undermined this empire too. The leader of Uzbek nomad tribes Muhammad Shaibani khan, who lived in Ural, using hostility between the descendents of Timur, invaded Central Asia in 1500-1507. During his rule the state consisted of independent principalities with the biggest ones being Tashkent, Samarkand, Bukhara, and Balkh.
From 1557 to 1598 Central Asia was ruled by Shaibanid Abdulla Khan, founding Bukhara kingdom. In 1598 Abdulla Khan was killed and the power moved to the dynasty of Ashtarkhanids, being the rulers from a dynasty of Mongols. In this period downstream of Amudarya Khivin a principality was formed. Due to continuous wars and excessive taxes, the economy of this area declined. All the following enturies of this principality remained backwards and politically disconnected. What is known today in modern Tajikistan as Kulyab, Gissar, Karategin, Darvaz, Vakhan, and Shugnan principalities already existed. The majority of Tajiks of Central Asia lived in Bukhara and Kokand principalities, and minority in independent principalities. During the second half of the 14th century, areas of Central Asia were conquered by Russia and Turkestan and a general government was formed on its territory. The Northern areas of Tajikistan and the Pamirs were part of this new territory, and the central and Southern areas (called Eastern Bukhara) were left in the ownership of vassal of the Russian tsar – emir of Bukhara. At the end of the 14th century the tsarist government implemented agrarian reforms which meant that the settled population received irrigated lands, however, the majority of lands was withheld for the benefit of the government funds. Increases in taxes, numerous duties and illegality provoked in those areas of the emirate frequent disorders. Especially one of the biggest revolts was in 1888 in Baljuvanbekstvo, which was ruled by chum maker Vose. In 1900 peasants of Kelif bekstvo staged a rebellion, in 1901 – in Denau, in 1902 –in Kurgan Tube. All these outbursts of the peasant farmers were cruelly suppressed. World War I particularly intensified opposition and disintegration of society. In summer 1906 the biggest revolution broke out which was suppressed by the army of Emir. On 31 March 1917 the administrative centre of Russia influenced Central Asia and caused the Turkestan general government to be abolished. On October 6, 1920, the first all Bukhara national assembly was proclaimed and the Bukhara People Soviet Republic was established. Farmers were exempted from taxes, households that were affected by war were given loans, seeds and agricultural instruments. In 1924 a new state emerged in Central Asia: the Uzbek Soviet Socialist Republic, which also included the Tajik Autonomous Soviet Socialist Republic. In 1929 the Tajik ASSR was reformed into independent Tajik Soviet Socialist Republic. As a member of USSR, the Tajik Republic transformed into an agrarian-industrial country for a short period of time. Tajikistan became the main source of fine-fibre cotton for the USSR. A working class formed in the republic, as well as a national intelligentsia, and the first higher education institutions came into existence. It is also worth noting that Tajikistan's contribution in the fight against fascism in World War II was of some importance. More than 190,000 envoys fought in battlefronts of the war and more than 60,000 worked at military plants of Siberia and other cities of Russia. Fifty-four envoys of the republic were awarded the highest military award as Heros of the Soviet Union. After the war ended the economy of Tajikistan grew to a new level. Production of row cotton continued to increase. Tajikistan as part of USSR took the first place on the productivity of cotton and third on gross yield. The industry was well developed due to the construction of some big hydro-power stations among one of which was Nurek, which is the biggest in Central Asia. In September 1991 with the break-up of the Soviet Union, a new state emerged on the world map – the independent Republic of Tajikistan. In 1992 critical political conflict on the ground of regional-clan conflicts burst out which led to civil war. Protracted negotiations between Tajiks under the aegis of UN starting in April 1994 concluded by signing the Treaty for Peace and Reconciliation in Tajikistan on June 27, 1997, by the President of Tajikistan E.Rahmon and A. Nuri, the leader of the United Tajik Opposition in Moscow. Today Tajikistan is an independent state which is recognized by 117 countries worldwide. The country is a full member of the UN and other world international organizations.
You can make calls from hotels where you accommodate, the restaurants of big cities where have lunch. If you are travelling to remote regions you need to buy a SIM-card with balance and megabytes, to get in touch with friends or use the internet.
01 -Fire Department
02 -Police
03 -Ambulance
04 -Gas Department
07 -Operator assisted long-distance calls
09 -Information Department
235-45-45 -State traffic control (GAI) duty officer
There are 4 major mobile operators in Tajikistan: Tcell, Megafon, Beeline, and Babilon. Roaming agreements exist with most international mobile phone companies but check this with your domestic operator before arrival. If you plan extensive stay in the country it might be wise to purchase a SIM card from one of eight local operators to access local tariffs rather than use your phone in roaming. Coverage is limited to the main cities and its suburbs but this improving every year. In Gorny Badakhshan coverage is getting better and most of the main roads and villages are covered.
There are internet cafes in the main cities of Tajikistan. You can buy megabytes with the SIM-card and make a wide use while travelling in remote regions.
Tajikistan getting around
Tajik Air operates internal flights from Dushanbe to
Khorog – Travel time 1 hr
Khujand -Travel time 1hr
Kulob – Travel time 30 minutes
A flight to Khorog is one of the most technically demanding regularly scheduled flights in the world and is a subject to weather conditions of the region, therefore tickets cannot be reserved or purchased in advance and only sold on first come – first served basis an evening before the flight. Flight to be reconfirmed on the morning of departure day.
Internal flights are subject to frequent cancellations, delays, and overloading of passengers.
Tajikistan railroad network is limited by 480 km and does not cover Gorno Badakhshan which makes travel by train in Tajikistan it rather useless as you miss Pamirs – one of the most important country's highlights. Rail travel in Tajikistan is limited to only 3 directions:
Northern Tajikistan Line
Connecting to Tashkent and the Ferghana Valley
Nau – Proletarsk – Khujand – Kairakkum – Kairakchikum – Mahram – Madaniat- Kanibodom
Central Tajikistan Line
Dushanbe – Hanaka – Cheptura – Regar – Pakhtaabad
Southern Tajikistan Line
Connecting to Termez for international destinations and also connections to Dushanbe and north Tajikistan.
Khoshady – Shaartuz – Qurghonteppa – Dangara- Kulob
Long-distance carriages are usually offered private 4 berth compartments separated by sliding door from the corridor. You can lock the compartment from the inside though train conductors have their own key. Luggage trunks located under bed/seats and extra shelves placed over higher beds. Do not leave your luggage and compartment unattended.
Self-driving:
In Tajikistan, traffic drives on the right. The road network is wide and covers almost every corner of Tajikistan, though some parts may be seasonally impassable. Please note cars with driving wheel on the right side are forbidden and though it does not cover non-Tajikistan registered vehicles may be a subject for all sort of bribes and traffic police stops. Also, tinted windows are strictly forbidden.
Foreign drivers need the international driving license to drive in Tajikistan. Restricted areas or areas requiring additional permits are listed in Visa&Formalities section.
Traffic Police or The State Traffic Inspectorate (BDA), has checkpoints on the borders of city limits as well as in many cities and at regular intervals along highways outside the cities. Be ready for frequent stops for inspection of the vehicle and the driver's documents.
Bus: There are minibus services between the major towns when. In the south, buses go to Kurgan-Tyube and Kulyab and as far down as Pyanj and Ayvadaz. Buses to the east reach only around 100km (60 miles), as far as Komsomolabad. Information on timetables and fares can be found at the bus stations.
Taxi: Can be found in all major and minor towns. Many of them would be unlicensed and travelers should agree on a fare in advance. Officially marked taxis are safe, but sharing with strangers should be avoided.
OVIR registration
Tajikistan OVIR registration
Travellers staying in Tajikistan for the period of three days and longer are required to obtain OVIR registration stamps at the nearest Department of Visas and Registration of the Ministry of Internal Affairs (OVIR).
The holders of tourist visas are NO longer required to register with OVIR unless the entire stay in Tajikistan is over 30 days, in which case they should register with OVIR prior to the 30 day period.
Holders of all other visa categories are required to register with OVIR within three days of arrival in Tajikistan. Some hotels, however, are also allowed to register their guests on behalf of the visitor.
Please note: OVIR registration will be normally checked at departure by border control services, however, police can also check it any time. Lack of OVIR registration is a subject for at least 300 USD/person fine and several days of the proceeding. The departure of travellers failed to show OVIR registration stamp might be denied until after penalized travellers have paid a fine and obtained the registration stamps at OVIR.
Customs & Border Control
Tajikistan customs and border control
Arriving in Tajikistan whether over a land border or at the airport, you will be required to go thru customs control as well as the border guard. At customs fill out 2 customs declaration forms. Leave one with customs officer and get stamped another copy for yourself. Retain the declaration until your departure to prove that you are not leaving Tajikistan with more money than you actually brought into the country, otherwise, you may find yourselves answering questions on the course of the money. Forms are available in English and Russian. Make sure that you declare the full amount as well as other valuables on your customs form. The export of antiques and cultural valuables requires special permission. There are also currency restrictions. Please contact your nearest Tajikistan embassy for specific information on customs requirements.
Immigration cards
Before border control, you will be required to fill out immigration card, which will contain your personal information, arrival stamp and what's more important tourism as a reason for the visit. Travelling as a tourist you no longer required to register with local police within 30 days. Hence if you travel with the private visa you must register in nearest OVIR station within 3 days excluding Sunday and official holidays.
For longer stays exceeding 30 days travel visa period or travelling in Tajikistan with other than tourist types of visa, you must register with OVIR department of local police stations within 3 days following your arrival date. Not every police station has OVIR department but Murghob, Khorog, Kulob, Dushanbe, and Khujand. The serious penalty of minimum 300 USD per person follows failure on showing a registration ticket at the time of departure. This might result in your flight cancellation and unpleasant procedures which might take up to several days. For registration please provide 2 passport size photos, scans of your passport: photo page and Tajikistan entry stamp page, registration fee receipt paid in the local bank. The procedure might take up to 2 days but negotiate. Some small banknotes above the fee may seriously speed up the procedure.
Amounts not exceeding of 5000 USD do not need to be declared. Goods for personal use may be imported in reasonable quantities by persons of 18 years of age or older without incurring customs duty.
Allowed imports without incurring customs duty:
-Tobacco: 200 Cigarettes or 100 Cigars or 500g of Tobacco
-Alcohol: 2 litres of spirits
-Perfume: for personal use
Exports which requires additional permits from respective authorities:
-Firearms and weapons
-Pharmaceuticals
-Psychotropic drugs
-Equipment
-Live animals
-Radioactive substances
Invitation letter also known as visa support is given out by Tajik MFA by request of inviting party: a company or an individual. In order to receive visa support, an inviting party submits a request with Tajik MFA in advance of the desired travel date to Tajikistan.
Individuals travelling to Tajikistan by the invitation of a private Tajik resident (a friend or relative in Tajikistan) will be required to obtain a notification letter from OVIR before applying for visa support. Please plan it in advance as it might take something between 14 days to 45 days. Then Tajik MFA will issue Tajik visa support on the basis of the OVIR notification letter.
Applying for visa support letter you will be asked about the way of obtaining your visa: either thru the embassy abroad or on arrival to Dushanbe International Airport. State the embassy or consulate where you plan to obtain your Tajik visa as this is very important and normally will not be changed under any circumstance as visa support is finally will be sent to the embassy you have stated in your application.
Visa support letter usually contains the details of inviting party, traveller tails, the period of stay, reference number.
Please note the invitation letter would not work if you apply for your Tajikistan visa in other country or other Tajikistan embassy than stated in your Tajikistan visa support letter.
You normally should receive a copy of visa support from inviting party so you can provide it at the embassy or consulate in case they experience communication problems with their head office or claim they didn't get one for your name.
GBAO Permit
A trip to Gorno-Badakhshan Autonomous region of Tajikistan requires a special permit in addition to Tajikistan visa. GBAO permit can be obtained at Tajik embassies and consulates abroad at the time of applying for Tajikistan visa, or by applying to MFA or OVIR in Dushanbe. We can also pre-arrange it pre-arrange
GBAO permit may list the names of the settlements and cities as well as regions in Gorno-Badakhshan allowed you to visit: Kara-Kul, Murghab, Ishkashim, Vanj, Rushan, Darvaz, Khorog and Kalaikhum. But these days very often contains short "Allowed to entry and reside in GBAO".
There are some certain areas that require additional permits in addition to GBAO permit, they are Lake Sarez which requires additional permit from the Ministry of Emergency Situations, Lake Zorkul and some regions along Chinese or Afghan borders which requires additional permit from border guard authorities.
Tajikistan visa in passport
According to Tajikistan Ministry of Foreign Affairs the following nations can apply for a single-entry tourist visa valid for up to 45 days at Tajik embassies or consulates abroad on the basis of personal application without visa support letter: Australia, Austria, Algeria, Egypt, Argentina, Bulgaria, Luxemburg, Hungary, Vietnam, Brunei-Darussalam, Israel, Greece, China, India, Ireland, Iran, Pakistan, Iceland, Italy, Canada, Qatar, Cyprus, Bahrain, Belgium, Jordan, Denmark, Spain, Morocco, Netherlands, Norway, Saudi Arabia, Sweden, Kuwait, Latvia, Lithuania, Lebanon, Malaysia, Malta, Mexico, New Zealand, UAE, Oman, Poland, Portugal, Korea, Indonesia, Croatia, Romania, Singapore, Syria, Slovakia, Slovenia, Great Britain, USA, Thailand, Turkey, Tunis, Brazil, Germany, Philippines, Finland, France, Czech Republic, Switzerland, Estonia, SAR, Japan.
Travellers requiring the longer period of stay or several entries would need to arrange a visa support letter before applying for Tajik visa. Providing a visa support, travellers will be able to obtain multiple-entry visas valid for a longer time. Visa validity may also depend on the visa type: tourist visa -30 days, private visa -90 days, work or business visa -up to 1 year, student visa -up to 9 months extend your stay within Tajikistan you would need to apply for extension far in advance through Tajikistan Ministry of Foreign Affairs (work visa) or OVIR (tourist, work and business visa).
Please note: The government of Tajikistan requires all visitors who remain in the country for more than 90 days to submit to an HIV test or to present a medical certificate showing that they are HIV-free.
Entry /Exit
A valid passport and visa are required to enter and exit Tajikistan. Your Tajik visa should be valid for the entire period of your stay in Tajikistan. If the purpose of your stay is other than travel we strongly encourage you to request visa that allows for changing travel dates as it might be complicated to do within Tajikistan. Failed on the show a valid visa is subject to the penalty and immediate departure.
Obtaining Tajik Visa abroad
Travellers arriving in Tajikistan from countries where Tajikistan is represented by embassy or consulate must obtain Tajik visa abroad prior to their travel. Visa application forms (filled out in 2 copies) can be obtained in the embassy or downloaded here.
Please note your passport must be valid for at least six months following the duration of your planned stay in Tajikistan.
Please refer to the list of Tajikistan embassies abroad to locate nearest to you.
Obtaining Tajik Visa on arrival
This option is suitable for travellers arriving in Tajikistan from countries where no Tajik embassies or consulates represented. Please note you must obtain a visa support, in the form of a registered letter from the Tajik Ministry of Foreign Affairs confirming that request for the visa is approved and a visa may be issued at the consulate based in arrival lounge of Dushanbe International Airport on arrival.
At this moment visas on arrival could only be obtained arriving at Dushanbe International Airport and does not apply to any other Tajik airports or land borders.
Visa application forms (fill out 2 copies) can be obtained from the consulate at the arrival lounge of Dushanbe International Airport or downloaded here.
Travellers are also required to provide two passport-size photos.
Visas issued on arrival at Dushanbe International Airport are by default valid for 45 days.
Receiving a visa at the airport may also entail some waiting.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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As with the majority of treatments for kidney disease and failure, patients have the option to include home hemodialysis as part of their daily regimen. The ability to maintain a life outside of a dialysis center can almost inspiring for some patients, especially those that feel as though their independence is being hindered by their treatments.
However, because hemodialysis is going to be taking place in someone's home, the patient and his/her caregiver (if applicable) will be responsible for maintaining the treatment. This is no small task, and while it does grant a patient a major boost in confidence because they are a part of the process, it does ask a lot of them as well.
1. The history of hemodialysis does lend itself to having a place at a patient's home. Within ten years of hemodialysis being used as a treatment, nearly half of these patients received treatment at home. For a patient considering home treatment, this is good to hear.
2. There will be an initially large amount of anxiety for involved in the process. Remember, many of the patients taking on their hemodialysis at home do not have a background in medicine. Their caregiver, who is usually a spouse, also may not have medical experience, so both of you will experience some level of hesitation and worry about making sure everything goes right so as to not make the patient vulnerable.
3. There is less social interaction experienced by patients who manage their hemodialysis at home. Dialysis centers are set up for the comfort of patients who, at times, must be at the center multiple times a week for hours at a time. It becomes a new normal, however, and patients sitting next to one another become close because they have a common bond. Patients receiving treatment at home may either interact with their caregiver or, if they live alone, possibly no one unless they receive visitors.
4. Per the National Institute of Diabetes and Digestive and Kidney Diseases (NIDDK), "Hemodialysis is a treatment for kidney failure that uses a machine to filter your blood outside your body." In short, this treatment is necessary for your kidneys to do their job, which is essential to you staying alive. You and/or your caregiver will also need to undergo weeks of training to make sure you can handle treatments at home. There is a major 'if' attached to that because if you and/or your caregiver cannot handle the steps in treatment, then a safe environment cannot guaranteed.
5. The law requires that patients must be notified that they have a choice regarding at-home hemodialysis treatment. They must receive information regarding where they can go to receive training on the process, as well as where they can receive general education on treatment options.
Home hemodialysis can be a fitting way for a patient becoming accustomed to a new way of life to feel as though they still have independence and are not being hindered by their kidney failure. However, this independence does have quite a large amount of responsibility, and it is the job of all parts of the healthcare team, including the patient, to understand this responsibility and choose the treatment option that is optimal for their health.
Why Do People Want to Make Money?
Is Your Marketing on the Right Or the Left?
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{
"redpajama_set_name": "RedPajamaC4"
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/ Haiti Police Hold President's Palace Security Chief
Haiti Police Hold President's Palace Security Chief
Radio Eko Lagras 15 July 2021
PORT-AU-PRINCE, Haiti — The head of palace security for President Jovenel Moïse, who was assassinated last week at his home by a team of armed men in an attack that has shaken an already fragile nation, was taken into police custody on Thursday.
The detention of the security chief for the presidential palace, Dimitri Hérard, was confirmed by Marie Michelle Verrier, a spokeswoman for Haiti's National Police.
Mr. Hérard was one of four members of the president's security personnel whom the state prosecutor was planning to call in for questioning earlier this week, as questions remained over how the attackers managed to enter the heavily guarded home.
The head of the presidential guard and two other top bodyguards had also been called in for questioning, Bedford Claude, the chief public prosecutor in Port-au-Prince, the Haitian capital, said earlier this week. He had issued summons for the four to appear as part of the investigation into the assassination.
Mr. Hérard's detention comes as questions have begun to be raised about the response of security guards after two dozen armed mercenaries opened fire after driving up to the president's home in several vehicles, and entered the residence after being met with little resistance.
Last week, Mr. Hérard declined to respond to questions from The New York Times, texting: "Unfortunately, after consulting with my lawyer, I am not in a position to comment on this at the very moment as this is an open investigation and a matter of national security."
When asked, he did not provide the name of his lawyer.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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Razi ibn Abu Turab Mashadi (known as Danesh) (died 1665) was Iranian Poet laureate of the Mughal Empire between 1655 and 1662.
Life
He was born in Isfahan. His father was a poet who immigrated to India and died in 1650 in Heydarabad. Danesh went to India before the death of his father. Danesh sang a Qasida in the court of Shah Jahan and gained 2000 Rupees. Also, he served Shah Jahan and Shah Shoja Mozaffari.
Sources
People from Mashhad
1665 deaths
Iranian male poets
17th-century poets
Iranian emigrants to the Mughal Empire
Mughal Empire poets
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{
"redpajama_set_name": "RedPajamaWikipedia"
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I've tried to create report with nested loops, but have some problems. Customer care department was unable to help me, so i have to ask you.
There is link below, which contains all report files and test qpr file. Also I attached word file where explained what i wanted to do and how I tried to complete my task.
It would be grate if you help me, thank you.
Sorry that your problem was not solved in QPR CC.
Do you have the original QPRXXXXXX number from the mails that you sent to QPR CustomerCare?
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{
"redpajama_set_name": "RedPajamaC4"
}
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\section{Introduction}
Remote sensing of spherical objects with electromagnetic beams has numerous applications in non-destructive evaluation (NDE) \cite{NDE}, biology \cite{BIO}, and medicine \cite{MED}. One promising medical application is corneal water content sensing for detection and management of Fuchs's dystrophy and assessment of corneal grafts. Fuchs' dystrophy causes corneal hyperhydration through degradation of the cornea's endothelium and its hydration maintenance functions. Corneal graft rejection is typically preceded by an inflammatory response characterized by increased corneal tissue water content (CTWC) \cite{Zach}. In both cases, excess corneal water content is usually detected via visual inspection of diffuse light scattering arising from swollen collagen fibers. In both cases, detection by visually apparent corneal edema is too late in the disease/rejection process to salvage cornea tissue.
Current diagnostic methods restrict \textit{in vivo} CTWC quantification to pachymetry, which infers tissue hydration from thickness quantified via ultrasonic or optical backscatter measurements. However, the mapping from the pachymetry-measured central corneal thickness (CCT) to corneal tissue hydration does not account for physiologic variation and returns significant abnormal hydration for corneal thicknesses off the nominal 580 \(\mu\)m \cite{vika}.
Recent research suggests that terahertz (THz) systems are suitable for mapping human cornea water content by coupling to the cornea's lossy longitudinal modes \cite{Cornea1,Cornea2}. When referenced to THz band wavelengths, the cornea presents as an optically smooth, spherical shell structure with radially varying dielectric properties arising from the water content gradient in the axial dimension \cite{Zach}. Furthermore, natural physiologic variation in radius of curvature, when referenced to THz band wavelengths, is negligible thus curvature can be assumed and fixed when designing optical systems.
In many reports describing illumination of the cornea, the incident field is modeled as a Gaussian beam in which the radius of curvature (RoC) of the wavefront matches the spherical surface at the optical axis \cite{aleksi,eucap}. As the focused spot size of the incident field is less than the radius of the cornea, the electromagnetic properties of the cornea can be modeled with stratified media theory (SMT) \cite{strat1} to capture the dielectric gradient, and effective media theories (EMT) \cite{strat2} to compute the effective permittivity at a given depth. Application of SMT and EMT implies that a plane wave on a planar film atop an aqueous half-space is a sufficient analogue for spherical cornea under the right illumination. This matching criterion is called the plane-wave condition. Optimization routines can then be applied to estimate corneal water content and thickness from backscattered THz illumination.
However, as the Gaussian beam is observed further from the optical axis, a mismatch between the wavefront and spherical surface increases between the field wavefront and the spherical surface \cite{aleksi}, see Figure \ref{fig:phase}. The phase deviation between the incident wavefront and the desired wavefront matching with the surface of the sphere reduces the theoretical CTWC measurement accuracy when the plane-wave condition and corresponding EMT and SMT are applied. For improved agreement between THz frequency corneal backscatter and that predicted by plane wave based analysis, there is a need to synthesize and simulate a field whose wavefront matches the spherical corneal shape at any point on the surface. In other words, a method that can create the required incident field from an arbitrary, complex electric field distribution initiated at the spherical surface. Furthermore, the method should also be able to model electromagnetic wave interactions between the incident beam and the cornea interior modeled as a multilayered spherical structure that generates complex internal and scattered electric fields from the initial surface distribution. This leads to an inverse imaging problem; first, the optimal wavefront distribution should be created at the cornea's surface, and then the incident beam should be synthesized with the ability to satisfy this surface criterion.
Internal and scattered fields arising from a homogenous sphere with arbitrary, given electric field distributions on the spherical surface can be computed with conventional methods including full-wave simulations, geometrical optics (GO) \cite{geo}, or physical optics (PO) \cite{PO,Sanford}. Conventional methods are also sufficient for calculating propagating fields from source distributions defined on a sphere. These techniques are well studied and accurate given the model fidelity and a suitable wavelength range. However, they cannot assess the internal and scattered electric fields from multilayered spherical objects without considerable computational effort. This problem occurs in full-wave simulation as the volume of the multilayered sphere may require discretization on the order of \(10^9\) cells, which would exceed available memory on most modern computers. The same discretization and computer memory constraint occurs with the boundary element method, as each layer boundary should be discretized, to, again, on the order of \(10^9\) boundary elements. Likewise, geometrical optics is limited to spheres much larger than the illumination wavelength, and ray splitting to accommodate a sufficient number of layers may also exceed available memory. Furthermore, obtaining the steady-state results with physical optics would need tracking of local reflected and transmitted fields from each boundary leading to an immense number of data points and equally infeasible for same computer memory limitations.
The ratio of the corneal radius of curvature (\(\sim\) 7.8 mm) and THz band wavelengths (\(\sim\) 3 mm - 0.3 mm) places the problem in the Mie scattering region thus supporting analysis as a pure scattering problem. Mie theory and the generalized Lorentz-Mie theory are powerful methods to evaluate the internal and scattered fields from the multilayered spheres \cite{GOUESBET20111}. First, the known incident field is presented in vector spherical harmonics (VSH) expansion which is defined by VSH coefficients \cite{bohrn}. However, obtaining the coefficients for arbitrarily shaped incident fields is difficult \cite{GOUESBET20111}. These VSH coefficients for the incident field can be obtained by the angular spectrum (AS) method from the planar surface distributions \cite{goodman}. The AS method presents the incident field in the angular spectrum domain as the sum of differently oriented plane waves. When this angular summation of the plane waves is presented in a direction cosine coordinate system, it can be used directly to compute the incident VSH coefficients. Then the incident field is expanded to the VSH presentation. Additionally, the internal and scattered fields from a multilayered dielectric sphere can be mapped with the T-matrix method in the Mie scattering region \cite{khaled}. The limitation of this AS method is that it is nominally compatible with planar interfaces. In \cite{StepAS}, the planar AS method was expanded to approximate diffractive fields from 2D curved surfaces. This was obtained by dividing the curved 2D surface into the step-wise subregions windowed by the Gaussian function. This approach was later expanded to the 3D surfaces in \cite{Ebers:20}, where the authors used the same planar step-wise subsections with Gaussian distribution. Both methods approximate well the diffraction, reflections, and transmission from the curved single-layer boundaries, when the RoC is much larger than the wavelength \cite{Worku:18}. Thus these methods are not suitable for cornea analysis in the THz region, where the RoC is comparable to the wavelength.
\begin{figure}[!h]
\centering
\includegraphics[width=0.75\linewidth]{Fig/PhaseError.png}
\caption{\small The incident beam propagating towards the sphere, and the mismatch between the beam wavefront and the spherical surface is proportional to the distance from the optical axis.}
\label{fig:phase}
\end{figure}\FloatBarrier
In this study, we start with the same approach of dividing the spherical surface into planar subregions. These subregions are presented with equal amplitude, without Gaussian distribution or windowing. Then these subregions are shrunk into infinitesimally small areas, which approach the Dirac's delta function at the limit. This approach gives exact results of the diffractive field from the spherical surface on the limit of infinitesimally small partition. Thus, this study proposes an accurate 3D AS method for the beam synthesis from the spherical surfaces, which can be expanded to arbitrary surfaces taking into account surface structures comparable to the wavelength. Also, with the proposed method the radius of curvature of the surface can be as small as couple of wavelengths. Moreover, this result is equivalent to the angular spectrum presentation of the Stratton-Chu method with magnetic dipole sources \cite{6710963}. The obtained theory is then utilized to create an incident electric field with the ideal spherical wavefront \cite{Mooses}, and the scattered field is calculated from the cornea model with the layered spherical structure of 50 layers.
\section{Theory}
This work presents the synthesis of the incident electromagnetic field from the arbitrary electric field distribution positioned on the spherical surface with the modified AS method. The method's main idea is that Riemann's surface integral combines differential surface elements, which can be approximated as locally planar elements. Then the radiated field from each element is created with the planar AS method. The total electromagnetic field is the sum of the fields created from the differential sources by the superposition principle \cite{goodman}. In this approach, we make a physical assumption that when the areas of the elements are small enough, the created field from the differential elements approaches the field from the original surface. This approximation works well for a smooth surface with radius of curvature that is larger than the wavelength.
The derivation starts on an arbitrary point on the unit-sphere, where the spherical components \(r,\theta,\phi\) define the surface. These orthogonal spherical components \((\mathbf{e}_n,\, \mathbf{e}_{\theta},\mathbf{e}_{\phi})\) can be used as base vectors for the local Cartesian coordinate system, whose origin is positioned at the same observation point on the spherical surface. Figure \ref{fig:diff} presents the situation, where the spherical components on the surface element are used as base vectors for the local coordinate system.
The main steps of the upcoming derivation follow the following structure;
\begin{itemize}
\item The relation between the local Cartesian coordinates of each surface element and the global spherical coordinates is derived from the unit-sphere base vectors \((\mathbf{e}_n,\, \mathbf{e}_{\theta},\mathbf{e}_{\phi})\).
\item Each differential element is considered as a field source and the radiated field is presented in the angular domain by the Fourier transform of the electric field distribution.
\item The total field is obtained by summing together the fields from the inverse Fourier transform of angular domain presentation.
\item Then the total field with polarization vector parallel to the \(\mathbf{e}_{\theta}\) is expanded into the VSH presentation.
\item Each local coordinate system is presented in the global coordinate system with the origin shift for satisfying the VSH symmetry.
\item Finally, the equations of the incident field synthesized from the spherical surface are presented.
\end{itemize}
After the derivation, the obtained result is proved to be mathematically exact in Appendix A and the result can be generalized to the spheres with arbitrary radii by taking account of the wavelength. Furthermore, the scattered field equations are presented in Appendix B.
\subsection{Incident field synthesis}
Let us start by defining the base vectors and the relation between coordinate systems. Position vector in standard form is
\begin{equation}\label{eq1}
\begin{array}{l}
\mathbf{r}=x\mathbf{e}_x+y\mathbf{e}_y+z\mathbf{e}_z,
\end{array}
\end{equation}
where \(\mathbf{e}_x,\mathbf{e}_y\) and \(\mathbf{e}_z\) are orthonormal Cartesian base-vectors. Position at unit-sphere is
\begin{equation}\label{eq2}
\begin{array}{l}
\mathbf{n}=\mathbf{n}(\theta,\phi)=n_x\mathbf{e}_x+n_y\mathbf{e}_y+n_z\mathbf{e}_z,
\end{array}
\end{equation}
where \(|\mathbf{n}|=1\), and in spherical coordinates \(n_x=\sin{\theta}\cos{\phi},n_y=\sin{\theta}\sin{\phi}\) and \(n_z=\cos{\theta}\).
\begin{figure}[!ht]
\centering
\includegraphics[width=0.63\linewidth]{Fig/Figure2.png}
\caption{\small Global spherical unit-vectors as local Cartesian base-vectors.}
\label{fig:diff}
\end{figure}\FloatBarrier
The differential surface element on the unit-sphere in spherical coordinates is \(d\Omega=\sin{\theta}d\theta d\phi\). Let the electric field on the segment of the sphere \(\Omega\) be a continuous function \(E_0=E_0 (\theta,\phi),\{0\leq\theta\leq\theta_0,0\leq\phi\leq2\pi\)\}. Let the position \(\mathbf{r}\) be fixed and let's observe the electric field at \(\mathbf{n}(\theta,\phi)\). The differential field element is considered to be on the plane defined by the unit vectors \(\mathbf{e}_{\theta}\) and \(-\mathbf{e}_{\phi}\). Together with their normal \(\mathbf{e}_n=-\mathbf{n}\), these determine the right handed orthonormal base \((\mathbf{e}_{\theta},\mathbf{-e}_{\phi},\mathbf{e}_n)\). Fixing origin at \(\mathbf{n}\) this base defines a local coordinate system, where coordinates are denoted by upper-bar like \((\bar{x}, \bar{y},\bar{z})\). A well known relation between spherical and Cartesian coordinates gives
\begin{equation}\label{eq3}
\begin{array}{l}
\begin{aligned}
\mathbf{e}_\theta&=\frac{d\mathbf{n}}{d\theta}=\cos{\theta}\cos{\phi}\mathbf{e}_x+\cos{\theta}\sin{\phi}\mathbf{e}_y-\sin{\theta}\mathbf{e}_z\\
-\mathbf{e}_\phi&=\frac{-1}{\sin{\theta}}\frac{d\mathbf{n}}{d\phi}=\sin{\phi}\mathbf{e}_x-\cos{\phi}\mathbf{e}_y.
\end{aligned}
\end{array}
\end{equation}
The transformation matrix between the coordinates at the base \((\mathbf{e}_{\theta},\mathbf{-e}_{\phi,}\mathbf{e}_n)\) and the coordinates at the standard base \((\mathbf{e}_x,\mathbf{e}_y,\mathbf{e}_z)\), is defined as
\begin{equation}\label{eq4}
\begin{array}{l}
\Theta(\theta,\phi) = \left[\mathbf{e}_{\theta}\,\,\mathbf{-e}_{\phi}\,\,\mathbf{e}_n\right],
\end{array}
\end{equation}
where the columns are the vectors \(\mathbf{e}_{\theta},\mathbf{-e}_{\phi}\) and \(\mathbf{e}_n\). It is an orthonormal matrix, thus \(\Theta^{T}\Theta=I\).
The position \(\mathbf{\bar{r}}\) in global coordinate system is
\begin{equation}\label{eq5}
\begin{array}{l}
\mathbf{\bar{r}}=\mathbf{r}-\mathbf{n}=(x-\sin{\theta}\cos{\phi})\mathbf{e}_x+(y-\sin{\theta}\sin{\phi})\mathbf{e}_y+(z-\cos{\theta})\mathbf{e}_z.
\end{array}
\end{equation}
Let's change (\ref{eq5}) expressed with respect to the base \((\mathbf{e}_{\theta},\mathbf{-e}_{\phi},\mathbf{e}_n):\, \mathbf{r}-\mathbf{n}=\bar{x}\mathbf{e}_{\theta}+\bar{y}(\mathbf{-e}_{\phi})+\bar{z}\mathbf{e}_n\). By (\ref{eq4}) we have position \(\mathbf{\bar{r}}\) in local coordinate system as
\begin{equation}\label{eq6}
\begin{array}{l}
\left[\bar{x}\,\,\bar{y}\,\,\bar{z}\right]^{T}=\Theta^{T}(\mathbf{r}-\mathbf{n})=\Theta^{T}[x-\sin{\theta}\cos{\phi}\,\,\,\,\,\,y-\sin{\theta}\sin{\phi}\,\,\,\,\,\,z-\cos{\theta}]^{T}.
\end{array}
\end{equation}
Let us divide the segment of the sphere \(\Omega\) containing the electric field distribution to finite partition of small separate sets \(\cup_c\Omega_c=\Omega\). Select the point \(\mathbf{\bar{n}}=\mathbf{\bar{n}}(\bar{\theta},\bar{\phi})\) on differential element \(\Omega_c\), the vectors associated with it are \(\mathbf{e}_\theta\) and \(\mathbf{e}_\phi\). Area \(\Omega_c\) is projected radially to the plane spanned by the vectors \(\mathbf{e}_\theta\) and \(\mathbf{e}_\phi\). The resulting area at the plane is marked as \(\Omega_t\). Likewise, function \(E_0\) is projected radially into that plane, limited to set \(\Omega_t\) and zero elsewhere. This geometry is presented in Figure \ref{fig:projection}. The resulting restriction is marked as \(E_0^t\). In the local coordinate system it holds \(\bar{z}=0\) on the piece \(\Omega_t\), because this is located on the \(\bar{x}\bar{y}-\)plane. \(E_0^t\) is valid when \(\bar{x}\approx0\) and \(\bar{y}\approx0\). Let the Fourier transform of the function \(E_0^t\) on the \(\bar{x}\bar{y}-\)plane be
\begin{equation}\label{eq7}
\begin{array}{l}
A(k_{\bar{x}},k_{\bar{y}};\bar{0})=\mathcal{F}\left\{E_0^t\right\}(k_{\bar{x}},k_{\bar{y}})=\iint_{\mathbb{R}^2}E_0^t e^{-i(k_{\bar{x}}\bar{x}+ k_{\bar{y}}\bar{y})}d\bar{x}d\bar{y},
\end{array}
\end{equation}
where \(k=|\mathbf{k}|=\sqrt{k_x^2+k_y^2+k_z^2}=2\pi/\lambda\). The inverse Fourier transform \(\mathcal{F}\left\{A(k_{\bar{x}},k_{\bar{y}};\bar{0})\right\}\) gives back \(E^t_0(\bar{x},\bar{y})=E(\bar{x},\bar{y},\bar{0})\).
Analogously the angular spectrum \(A(k_{\bar{x}},k_{\bar{y}};\bar{z})\) gives back the desired field \(E(\bar{x},\bar{y},\bar{z})\) \cite{khaled}, thus
\begin{equation}\label{eq8}
\begin{array}{l}
E(\bar{x},\bar{y},\bar{z})=\frac{1}{4\pi^2}\iint_{\mathbb{R}^2} A(k_{\bar{x}},k_{\bar{y}},\bar{z})e^{i(k_{\bar{x}}\bar{x}+ k_{\bar{y}}\bar{y})}dk_{\bar{x}}dk_{\bar{y}}.
\end{array}
\end{equation}
The relation between \(A(k_{\bar{x}},k_{\bar{y}};\bar{0})\) and \(A(k_{\bar{x}},k_{\bar{y}};\bar{z})\) can be obtained by operating the Helmholtz equation \(\nabla^2E+k^2E=0\) to the function (\ref{eq8}) \cite{goodman}. The obtained relation is
\begin{equation}\label{eq9}
\begin{array}{l}
A(k_{\bar{x}},k_{\bar{y}},\bar{z})=A(k_{\bar{x}},k_{\bar{y}};\bar{0})e^{ik_{\bar{z}}|\bar{z}|}=\mathcal{F}\big\{E_0^t\big\}(k_{\bar{x}},k_{\bar{y}})e^{i|\bar{z}|\sqrt{k^2-k_{\bar{x}}^2-k_{\bar{y}}^2}},
\end{array}
\end{equation}
where the solution \(A(k_{\bar{x}},k_{\bar{y}};\bar{z})=C_\pm e^{\pm\ ik_{\bar{z}}\bar{z}}\) allows both propagation directions \(\pm z\) as \(e^{ik_{\bar{z}}|\bar{z}|}\).
Above the angular spectrum theory is locally applied on the local coordinate system to function \(E_0^t\) valid on piece \(\Omega_t\). Let's observe the field it creates at point \(\mathbf{\bar{r}}=\bar{x}\mathbf{e}_{\theta}+\bar{y}(\mathbf{-e}_{\phi})+\bar{z}\mathbf{e}_n\), where the coordinates are as in (\ref{eq6}). As shown above, it holds at a fixed point \(\mathbf{r}\)
\begin{equation}\label{eq10}
\begin{array}{l}
E_t(\mathbf{r})=\frac{1}{4\pi^2}\iint_{\mathbb{R}^2}\mathcal{F}\big\{E_0^t\big\}(k_{\bar{x}},k_{\bar{y}})e^{i(k_{\bar{x}}\bar{x}+ k_{\bar{y}}\bar{y})}e^{i|\bar{z}|\sqrt{k^2-k_{\bar{x}}^2-k_{\bar{y}}^2}}dk_{\bar{x}}dk_{\bar{y}}.
\end{array}
\end{equation}
The total field from the spherical surface is obtained by summing the fields from the differential sources by superposition principle as
\begin{equation}\label{eq11}
\begin{array}{l}
\begin{aligned}
\mathbf{E_{\theta}}(\mathbf{r})&\approx\sum_t E_t(\mathbf{r})\mathbf{e}_{\theta}=\sum_t\mathbf{e}_{\theta}\frac{1}{4\pi^2}\iint_{\mathbb{R}^2}\mathcal{F}\big\{E_0^t\big\}(k_{\bar{x}},k_{\bar{y}})e^{i(k_{\bar{x}}\bar{x}+k_{\bar{y}}\bar{y})}\\&\times e^{i|\bar{z}|\sqrt{k^2-k_{\bar{x}}^2-k_{\bar{y}}^2}}dk_{\bar{x}}dk_{\bar{y}},
\end{aligned}
\end{array}
\end{equation}
where local coordinates \((\bar{x},\bar{y},\bar{z})\) as in (\ref{eq6}) and the total field is polarized along the global \(\mathbf{e}_{\theta}\) unit-vector. Because the operator \(\Delta=\nabla^2\) is linear, the field (\ref{eq11}) also satisfies the Helmholtz equation as unit vector \(\mathbf{e}_{\theta}\) is locally constant. In order to be precise, if the limit with respect to the areas \(|\Omega_t|\) of the sets \(\Omega_t\) exists, we define mathematically precisely
\begin{equation}\label{eq12}
\begin{array}{l}
\mathbf{E}_{\theta}(\mathbf{r})=\mathbf{E}_{\theta}(x,y,z)=lim_{|\Omega_t|\to 0} \sum_t E_t(\mathbf{r})\,\mathbf{e}_{\theta}.
\end{array}
\end{equation}
As pointed out at the beginning of the Theory section, this limit also works physically when considering smooth surfaces, and is even more precise when the wavelength decreases compared to the radius of the sphere. Finally, a further examination is made to obtain a closed-form for the clause (\ref{eq12}); we postpone an exact proof at the Appendix A.
Heuristically, when \(|\Omega_t|\) is getting smaller, the function \(E_0^t/E_0^t(\bar{0},\bar{0})|\Omega_t|\), normed in volume, is approaching Dirac delta \(\delta_{(0,0)}\) in the sense of the distribution theory; \(E_0^t/E_0^t(\bar{0},\bar{0})|\Omega_t|\rightarrow\delta_{(0,0)}\) thus \(E_0^t\rightarrow E_0^t(\bar{0},\bar{0})|\Omega_t|\delta_{(0,0)}\). As known, the Fourier transform of Dirac delta is a constant and equals to 1. Thus, by small \(|\Omega_t|\)
\begin{equation}\label{eq13}
\begin{array}{l}
\begin{aligned}
\mathcal{F}\big\{E_0^t\big\}(k_{\bar{x}},k_{\bar{y}})&=\iint_{\mathbb{R}^2} E_0^t e^{-i(k_{\bar{x}}\bar{x}+ k_{\bar{y}}\bar{y})}d\bar{x}d\bar{y}\\
&\approx\iint_{\mathbb{R}^2} E_0^t(\bar{0},\bar{0})|\Omega_t|\delta_{(0,0)} e^{-i(k_{\bar{x}}\bar{x}+ k_{\bar{y}}\bar{y})}d\bar{x}d\bar{y}\\
&=E_0(\theta,\phi)|\Omega|\mathcal{F}\big\{\delta_{(0,0)}\big\}=E_0(\theta,\phi)|\Omega_t|,
\end{aligned}
\end{array}
\end{equation}
since \(E_0(\theta,\phi)=E_0^t(\bar{0},\bar{0})\) when fixed \(\mathbf{n}\). Then (\ref{eq11}) and (\ref{eq13}) yield first the form of a Riemann sum, and then an integral as a limit
\begin{equation}\label{eq14}
\begin{array}{l}
\begin{aligned}
\mathbf{E}_{\theta}(\mathbf{r})&=\mathbf{E}_{\theta}(x,y,z)=lim_{|\Omega_t|\to 0} \sum_t\mathbf{e}_{\theta}\frac{1}{4\pi^2} E_0(\theta,\phi)|\Omega_t|\\&\times\iint_{\mathbb{R}^2}e^{i(k_{\bar{x}}\bar{x}+ k_{\bar{y}}\bar{y})}e^{i|\bar{z}|\sqrt{k^2-k_{\bar{x}}^2-k_{\bar{y}}^2}}dk_{\bar{x}}dk_{\bar{y}}\\&=\frac{1}{4\pi^2}\iint_{S}E_0(\theta,\phi)\mathbf{e}_{\theta}(\theta,\phi)\iint_{\mathbb{R}^2}e^{i(k_{\bar{x}}\bar{x}+k_{\bar{y}}\bar{y})}e^{i|\bar{z}|\sqrt{k^2-k_{\bar{x}}^2-k_{\bar{y}}^2}}dk_{\bar{x}}dk_{\bar{y}}d\Omega.
\end{aligned}
\end{array}
\end{equation}
The final form can be written as
\begin{equation}\label{eq15}
\begin{array}{l}
\mathbf{E}_{\theta}(\mathbf{r})=\frac{1}{4\pi^2}\iint_{\Omega}E_0(\theta,\phi)\mathbf{E}_t(\mathbf{r};\theta,\phi)\sin{\theta}d\theta d\phi,
\end{array}
\end{equation}
where
\begin{equation}\label{eq16}
\begin{array}{l}
\mathbf{E}_t(\mathbf{r};\theta,\phi)=\mathbf{e}_{\theta}(\theta,\phi)\iint_{\mathbb{R}^2}e^{i(k_{\bar{x}}\bar{x}+k_{\bar{y}}\bar{y})}e^{i|\bar{z}|\sqrt{k^2-k_{\bar{x}}^2-k_{\bar{y}}^2}}dk_{\bar{x}}dk_{\bar{y}}\\
\end{array}
\end{equation}
and \(\bar{x},\bar{y}\) and \(\bar{z}\) are as in (\ref{eq6}). It is worthwhile to mention that equation (\ref{eq16}) is similar to the \(\mathbf{E}_t=-\nabla\times(\mathbf{M}G)\) after replacing the scalar Green's function \(G\) by its angular spectrum given by Weyl identity, where \(\mathbf{M}=-2\mathbf{n}\times\mathbf{e}_{\theta}\) is a small magnetic dipole \cite{Chew}. This leads equation (\ref{eq15}) to be a particular case of the Stratton-Chu equation for electric field in which only the magnetic sources \(\mathbf{M}=-\mathbf{n}\times\mathbf{E}_{surf}\) are presented and the electric ones \(\mathbf{J}=\mathbf{n}\times\mathbf{H}_{surf}\) have been removed \cite{Stratton}.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.61\linewidth]{Fig/Figure3.png}
\caption{\small Spherical segment \(\Omega_c\) projected into a plane \(\Omega_t\).}
\label{fig:projection}
\end{figure}\FloatBarrier
\subsection{Polarization}
In the derived equation (\ref{eq15}), the electric field component is along \(\mathbf{e}_{\theta}\) at the spherical surface, creating a globally continuous polarization. The polarization of each differential source is selected on \(\bar{x}\bar{z}-\)plane at source's local coordinate system. Let us consider function (\ref{eq16}), with the electric field along \(\mathbf{e}_{\theta}\)
\begin{equation}\label{eq17}
\begin{array}{l}
\mathbf{E}_t(\bar{x},\bar{y},\bar{z})=\mathbf{e}_{\theta}\iint_{\mathbb{R}^2}e^{i\mathbf{\bar{k}}\cdot\mathbf{\bar{r}}_{||}}d k_{\bar{x}}d k_{\bar{y}},
\end{array}
\end{equation}
where \(\mathbf{\bar{r}}_{||}=(\bar{x},\bar{y},|\bar{z}|)\). Equation (\ref{eq17}) will give us the local \(\bar{x}\)-component of the electric field. The local \(\bar{z}-\)component for the polarization is obtained from \(\mathbf{E}_t\cdot\mathbf{\bar{k}}=E_{\bar{x}}k_{\bar{x}}+E_{\bar{y}}k_{\bar{y}}+E_{\bar{z}}k_{\bar{z}}=0\) for each plane wave \cite{esam94}. Because of polarization on the \(\bar{x}\bar{z}-\)plane as mentioned above, we have \(E_y=0\) and
\begin{equation}\label{eq18}
\begin{array}{l}
E_{\bar{z}}=-\Big(\frac{k_{\bar{x}}}{k_{\bar{z}}}\Big)E_{\bar{x}}.
\end{array}
\end{equation}
The final form of (\ref{eq17}) is obtained as
\begin{equation}\label{eq19}
\begin{array}{l}
\mathbf{E}_t(\bar{x},\bar{y},\bar{z})=\iint_{\mathbb{R}^2}e^{i\mathbf{\bar{k}}\cdot\mathbf{\bar{r}}_{||}}\Big[\mathbf{e}_{\bar{x}}-\Big(\frac{k_{\bar{x}}}{k_{\bar{z}}}\Big)\mathbf{e}_{\bar{z}}\Big]d k_{\bar{x}}d k_{\bar{y}},
\end{array}
\end{equation}
where \(\mathbf{e}_{\bar{x}}=\mathbf{e}_{\theta},\ \mathbf{e}_{\bar{y}}=-\mathbf{e}_\phi\) and \(\mathbf{e}_{\bar{z}}=\mathbf{e}_n=-\mathbf{n}\) as defined in (\ref{eq2}) and (\ref{eq3}). The total polarization at the spherical surface is thus in line with the orthodrome lines between the poles of the sphere, and creates a desired polarization when the middle point of the segment \(\Omega\) is located at the \(\theta=\pi/2\). The function (\ref{eq19}) is integrated over the real disk \(k_{\bar{x}}^2+k_{\bar{y}}^2\leq k^2\) for obtaining only propagating waves and those waves outside of the disk are evanescent. Using direction cosine coordinate system, the above equation can be written as \cite{khaled}
\begin{equation}\label{eq20}
\begin{array}{l}
\mathbf{E}_t(\bar{x},\bar{y},\bar{z})=k^2\int_{0}^{\pi}\sin{\xi}\Big\{\int_{0}^{\pi}\mathbf{e}_0(\xi,\zeta)e^{ik(\mathbf{\bar{s}}\cdot\mathbf{\bar{r}}_{||})}\sin{\zeta}d\zeta\Big\} d\xi,
\end{array}
\end{equation}
where \(\xi\) and \(\zeta\) are direction cosines related to spherical coordinates as \(\xi=\cos^{-1}{(\sin{\theta}\cos{\phi})}\) and \(\zeta=\cos^{-1}{(\sin{\theta}\sin{\phi})}\). The polarization \(\mathbf{e}_0\) in terms of direction cosines is given as
\begin{equation}\label{eq21}
\begin{array}{l}
\mathbf{e}_0(\xi,\zeta)=\mathbf{e}_{\bar{x}}-\frac{\cos{\xi}}{(1-s^2)^{1/2}}\mathbf{e}_{\bar{z}}.
\end{array}
\end{equation}
Additionally \(s^2=\cos^2{\xi}+\cos^2{\zeta}\), and the vector \(\mathbf{\bar{s}}\) is
\begin{equation}\label{eq22}
\begin{array}{l}
\mathbf{\bar{s}}=\cos{\xi}\mathbf{e}_{\bar{x}}+\cos{\zeta}\mathbf{e}_{\bar{y}}+(1-s^2)^{1/2}\mathbf{e}_{\bar{z}}.
\end{array}
\end{equation}
\subsection{VSH expansion and origin shift}
The total incident electric field from the spherical surface distribution is obtained from the formula (\ref{eq15}). When (\ref{eq16}) is expanded to the vector spherical harmonics, the origin of the local coordinate system needs to be shifted to the global origin due to the spherical symmetry of the VSH presentation. This means that the dielectric spheres in each local coordinate system will be centered on the local origin, i.e., at the surface. Thus each local spherical object needs to be shifted to the global origin while the position of the surface source is kept fixed. This is done by first shifting the local origins to the global origins and then shifting the positions of the differential source elements back to the original places. After this procedure, each local source element is presented in the global coordinate system with the ability to add the dielectric sphere to the right location. To do so, equation (\ref{eq20}) is evaluated by numerical integration over a disk of radius \(k\) using the trapezoidal rule with uniform-width \(l\) \cite{khaled}
\begin{equation}\label{eq23}
\begin{array}{l}
\mathbf{E}_{t_{ij}}\approx p^2\sum_i\sin{\xi_i}\sum_j\sin{\zeta_j}\mathbf{e}_{0_{ij}}(\xi_{t_i},\zeta_{t_j})e^{ik(\mathbf{\bar{s_{ij}}}\cdot\mathbf{\bar{r}}_{||})},
\end{array}
\end{equation}
where the indices \(i\) and \(j\) refers to plane waves with propagation constants \(k_x^i\) and \(k_y^j\) respectively to \(\bar{x}\) and \(\bar{y}\) directions. \(p=\xi_{t+1}-\xi_t=\zeta_{t+1}-\zeta_t\)
is the step-size of the numerical integration.
Integration limit of radius \(k\) is selected, because the electric field outside that area is rapidly decreasing. The trapezoidal rule method is highly accurate on periodic functions \cite{trap}, such as equation (\ref{eq20}). The transformation matrix from (\(\bar{x}\bar{y}\bar{z}\)) to (\(xyz\)) is \(\Theta(\theta,\phi)\) as in (\ref{eq4}).
We want to express the dot product \(\mathbf{\bar{s}}_{ij}\cdot\mathbf{\bar{r}}_{||}\) in the global coordinates. Let's write \(\mathbf{\bar{r}}=(\bar{x},\bar{y},\bar{z})\). We have a simple connection \(\mathbf{r}-\mathbf{n}=\Theta\,\mathbf{\bar{r}}\) as in (\ref{eq6}); that will be used. So we define \(\mathbf{\bar{s}}_{ij}^{||}=\mathbf{\bar{s}}_{ij}\), when \(\bar{z}>0\), but, when \(\bar{z}<0\), the sign of \(\bar{z}-\)coordinates in \(\mathbf{\bar{s}}_{ij}\) is exchanged. Consequently, we can write \(\mathbf{\bar{s}}_{ij}\cdot\mathbf{\bar{r}}_{||}=\mathbf{\bar{s}}_{ij}^{||}\cdot\mathbf{\bar{r}}\). Let us define
\begin{equation}\label{eq24}
\begin{array}{l}
\mathbf{s}_{ij}=\Theta\,\mathbf{\bar{s}}_{ij}^{||}.
\end{array}
\end{equation}
As a result we get
\begin{equation}\label{eq25}
\begin{array}{l}
\mathbf{\bar{s}}_{ij}\cdot\mathbf{\bar{r}}_{||}=\mathbf{\bar{s}}_{ij}^{||}\cdot\mathbf{\bar{r}}=\mathbf{\bar{s}}_{ij}^{||}\cdot\Theta^{T}(\mathbf{r}-\mathbf{n})=\Theta\,\mathbf{\bar{s}}_{ij}^{||}\cdot(\mathbf{r}-\mathbf{n})\\=\mathbf{s}_{ij}\cdot(\mathbf{r}-\mathbf{n})=\mathbf{s}_{ij}\cdot\mathbf{r}-\mathbf{s}_{ij}\cdot\mathbf{n}.
\end{array}
\end{equation}
Based on (\ref{eq25}), the term \(e^{ik(\mathbf{\bar{s_{ij}}}\cdot\mathbf{\bar{r}}_{||})}\), which is in the local coordinate system, can be presented in the global coordinate system as
\begin{equation}\label{eq26}
\begin{array}{l}
e^{ik(\mathbf{\bar{s_{ij}}}\cdot\mathbf{\bar{r}}_{||})}=e^{ik(\mathbf{s_{ij}}\cdot\mathbf{r})}e^{-ik(\mathbf{s_{ij}}\cdot\mathbf{n})},
\end{array}
\end{equation}
where \(e^{-ik(\mathbf{s_{ij}}\cdot\mathbf{n})}\) presents the shift of beam waist of the differential source from global origin to the unit-sphere surface.
Now each plane wave in the angular spectrum created from the differential source is presented in a global coordinate system. The total incident electric field is obtained as the sum of the plane waves in the angular spectrum, and at an arbitrary point \(\mathbf{r}\) in space expressed by VSHs can be written as \cite{khaled}
\begin{equation}\label{eq27}
\begin{array}{l}
\begin{aligned}
\mathbf{E}_t(k\mathbf{r})&=p^2\sum_m\sum_n D_{mn}\Big[a^t_{\text{emn}}\mathbf{M}^1_{\text{emn}}(k\mathbf{\bar{r}})+a^t_{\text{omn}}\mathbf{M}^1_{\text{omn}}(k\mathbf{\bar{r}})\\&+b^t_{\text{emn}}\mathbf{N}^1_{\text{emn}}(k\mathbf{\bar{r}})+b^t_{\text{omn}}\mathbf{N}^1_{\text{omn}}(k\mathbf{\bar{r}})\Big],
\end{aligned}
\end{array}
\end{equation}
where \(D_{mn}=\frac{\epsilon_m(2n+1)(n-m)!}{4n(n+1)(n+m)!}\) is a normalization factor. \(\epsilon_m\) is equal to 1 for \(m=0\) and is equal to 2 for \(m>0\). \(\mathbf{M}^1_{\text{emn}},\mathbf{M}^1_{\text{omn}},\mathbf{N}^1_{\text{emn}}\) and \(\mathbf{N}^1_{\text{omn}}\) are the VSH of the first kind and \(a^t_{\text{emn}},a^t_{\text{omn}},b^t_{\text{emn}}\) and \(b^t_{\text{omn}}\) are incident beam expansion coefficients given as
\begin{equation}\label{eq29}
\begin{array}{l}
a^t_{\text{emn}}=\sum_i\sin{\xi_i}\sum_j\sin{\zeta_j}a^t_{{\text{emn}}_{ij}}e^{-ik(\mathbf{s_{ij}}\cdot\mathbf{n})},\\
a^t_{\text{omn}}=\sum_i\sin{\xi_i}\sum_j\sin{\zeta_j}a^t_{{\text{omn}}_{ij}}e^{-ik(\mathbf{s_{ij}}\cdot\mathbf{n})},\\
b^t_{\text{emn}}=\sum_i\sin{\xi_i}\sum_j\sin{\zeta_j}b^t_{{\text{emn}}_{ij}}e^{-ik(\mathbf{s_{ij}}\cdot\mathbf{n})},\\
b^t_{\text{omn}}=\sum_i\sin{\xi_i}\sum_j\sin{\zeta_j}b^t_{{\text{omn}}_{ij}}e^{-ik(\mathbf{s_{ij}}\cdot\mathbf{n})}.
\end{array}
\end{equation}
The VSH coefficients for each \(ij-\) plane wave \(a^t_{{\text{emn}}_{ij}}, a^t_{{\text{omn}}_{ij}}, b^t_{{\text{emn}}_{ij}}\) and \(b^t_{{\text{omn}}_{ij}}\) are defined in Appendix B.
The equation for the incident field (\ref{eq15}), adapted by polarization (\ref{eq19}), can also be written in vector spherical expansion (\ref{eq27}) as
\begin{equation}\label{INC}
\begin{array}{l}
\mathbf{E}(\mathbf{r})_{inc}=\frac{1}{4\pi^2}\iint_{\Omega}E_0(\theta,\phi)\mathbf{E}_t(k\mathbf{r};\theta,\phi)\sin{\theta}d\theta d\phi.
\end{array}
\end{equation}
A similar scattered field presentation mapped with the T-matrix method is presented as
\begin{equation}\label{SCAT}
\begin{array}{l}
\mathbf{E}(\mathbf{r})_{sca}=\frac{1}{4\pi^2}\iint_{\Omega}E_0(\theta,\phi)\mathbf{E}_{scat}(k\mathbf{r};\theta,\phi)\sin{\theta}d\theta d\phi,
\end{array}
\end{equation}
where \(\mathbf{E}_{scat}\) is introduced in Appendix B.
\section{Results}
Now we present three different simulation scenarios, and they are: 1) Incident field simulations and comparison to similar physical optics simulations, to validate the accuracy of the field synthesis. 2) The spherical bandstop filter simulations to verify the benefits of the proposed method. 3) The cornea simulations to confirm the increased accuracy of cornea´s thickness and water content analysis.
\subsection{Incident field simulations}
First, the incident electric field was synthesized from the top-hat field distribution positioned on the spherical surface with Fibonacci sampling using a derived equation (\ref{INC}). This field has a matching wavefront at the spherical surface, with the main polarization along the \(\mathbf{e}_{\theta}\) direction. The incident field was evaluated at the plane transverse to the propagation x-axis at 40 mm from the origin of the sphere. The position of the evaluation plane along the propagation axis can be changed without restraints. In our case, the evaluation plane location is selected to match the approximated location of the lens in a quasioptical system. At this distance, the converging beam can be focused with reasonably small focusing elements. Also, this location is identical to the lens location in a future quasioptical system, designed to create the desired field distribution matching the spherical surface. All the following simulations were realized with a constant electric field within subtended angle of 30{$^\circ$} on the spherical surface and the transverse evaluation plane at the yz-plane, see Figure \ref{fig:sphearrangemet}. The only exception is shown in Figure \ref{fig:Etot1}, where the evaluation plane is along the xy-plane.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.7\linewidth]{Fig/Arrangement.png}
\caption{\small The simulation arrangement of beam propagation between the spherical surface and evaluating plane, where evaluating plane is marked as blue dots. This plane has 100 \(\times\) 100 mm\(^2\) area, located at x = 40 mm from the origin of the sphere. The source grid with 681 Fibonacci distributed differential sources is marked as red dots at the spherical surface and at the enlargement of the surface grid. The subtended angle of the spherical cap enclosing the differential source grid is 30{$^\circ$}.}
\label{fig:sphearrangemet}
\end{figure}\FloatBarrier
Then the Cartesian components of the incident field (\(E_{3D\,ASM}\)) calculated by the proposed method were compared to the identical physical-optics simulation (\(E_{PO}\)) at 175 GHz frequency. The comparizon provides precise matching down to less than a \(-41\) dB average difference in amplitude and less than 1{$^\circ$} average difference in phase, see Figure \ref{Diff}. This difference was calculated for each component as \(||E_{3D\,ASM}-E_{PO}||_{ave}\) for the amplitude and \(||\angle E_{3D\,ASM}-\angle E_{PO}||_{ave}\) for the phase. The magnitude and phase components of the electric field computed by the physical-optics are visually identical to the \(E_{3D\,ASM}\) results, thus they are not plotted.
\begin{figure*}[!ht]
\centering
\begin{subfigure}[h]{1\textwidth}
\centering \includegraphics[width=0.97\textwidth]{Fig/Difference1.png}
\end{subfigure}\vskip 0.1ex
\begin{subfigure}[h]{0.955\textwidth}
\centering \includegraphics[width=1\textwidth]{Fig/Difference2.png}
\end{subfigure}\vskip 0.1ex
\caption{\small Subfigures \(\mathbf{a)}\)-\(\mathbf{c)}\) are the magnitudes of the electric field components computed by the proposed 3D ASM method, and \(\mathbf{d)}\)-\(\mathbf{f)}\) are the magnitude differences between the 3D ASM and PO method at 175 GHz.
Subfigures \(\mathbf{g)}\)-\(\mathbf{i)}\) are the phases of the electric field components computed by the proposed 3D ASM method, and \(\mathbf{j)}\)-\(\mathbf{l)}\) are the phase differences between the 3D ASM and PO method at 175 GHz.}
\label{Diff}
\end{figure*}\FloatBarrier
\subsection{Banstop filter backscatter simulations}
After the simulation verification of the incident field, let's investigate the analogy of the spherical layered structure compared to the planar layered structure, see Figure \ref{Structure}. To compare these structures properly, we must take into account two different scenarios. When the spherical layered structure has a lossless core, it creates a cavity with multiple internal reflections inside the core. Then the analog planar structure is presented in subfigure \ref{Structure} b). If the spherical layered structure has a lossy core to prevent the internal reflection in the core, the analog planar structure is presented in subfigure \ref{Structure} c).
\begin{figure}[!ht]
\centering
\includegraphics[width=1\linewidth]{Fig/Structures.png}
\caption{\small Subfigure \(\mathbf{a)}\) is stratified spherical shell structure, subfigure \(\mathbf{b)}\) presents the analog planar structure when the spherical structure have a lossless core, and subfigure \(\mathbf{c)}\) presents the analog planar structure when the spherical structure have a lossy core.}
\label{Structure}
\end{figure}\FloatBarrier
Then, the benefits of illuminating spherical layered structures with the beam with a spherically matching wavefront compared to the Gaussian beam are presented. This is done by reproducing a planar 27-layer bandstop filter to an analog spherical structure. The original filter is 3627 nm thick with alternating layers of \(TiO_2\) and \(SiO_2\), with refractive indexes of 2.5 and 1.45 respectively \cite{Filter}. A spherically analog structure was created with a 4000 nm radius sphere with a lossless and lossy (\(\epsilon = 1-0.5i\)) core to prevent the internal reflections. Then spherical bandpass filter was illuminated in the 450 - 700 nm wavelength range by a normal and tapered top-hat beam created by the proposed 3D ASM method and by Gaussian beam with different beam size and position scenarios presented in \cite{FaezNew,Fa2022}. These results are compared to the plane wave illumination of the similar planar filter structures in Figure \ref{filter}.
\begin{figure}[!h]
\centering
\includegraphics[width=1\linewidth]{Fig/FilterCoupling.png}
\caption{\small Coupling coefficients from the spherical layered filter model, where \(\mathbf{a)}\) presents the lossy core with \(\epsilon=1-0.5i\) permittivity and \(\mathbf{b)}\) presents the lossless core scenario.}
\label{filter}
\end{figure}
In this simulation example, the spherical layers extend the entire radius of the sphere from the surface to the core, allowing the beam to disperse as it propagates to the deepest layers. This phenomenon produces a small difference between the planar stratified medium theory model with PW illumination and the model of the spherical structure with converging beam illumination. Nevertheless, the simulation shows an increased and relatively well-matched coupling coefficient (\(K\)) between the incident and scattered fields with normal and tapered top-hat beams compared to the superconfocal Gaussian beam illumination. The formula for computing \(K\) is presented in Appendix B.
\subsection{Cornea backscatter simulations}
The cornea is the transparent front part of the eye covering the iris, pupil, and anterior chamber. The water percentage of the cornea increases gradually from the anterior surface to the posterior surface. Thus, the permittivity of the cornea differs radially and can be modeled by the spherical multilayered structure. The cornea was modeled as a 7.8 mm radius sphere with a 580 \(\mu\)m thick multilayered spherical shell structure with a pure-water core. The permittivity gradient of the cornea was modeled with 50 layers. Each layer's permittivity was obtained with the effective medium theory via the Bruggeman model \cite{Brugg}, and the permittivity of the pure-water core was calculated by the double-Debye model \cite{Debye}. The corneal shell consisted of 40 \% water on the anterior surface and 70 \% water on the posterior surface. The cornea is modeled as relatively thin layered structure compared to the radius of the sphere. Due to this, the propagating beam does not disperse as much as in the bandpass filter and the difference between the scattering from planar and spherical structure is decreased. First, the total field \(\mathbf{E}_{tot}=\mathbf{E}_{inc}+\mathbf{E}_{scat}\), including the scattered field from the 50-layer cornea model was simulated in the plane along the propagation axis at 175 GHz frequency. This is done for illustrating the beam behavior and the equal phase wavefront on the spherical surface, where the internal field \(\mathbf{E}_{int}\) is not plotted, see Figure \ref{fig:Etot1}. The scattered field was calculated by the formula (\ref{eqB4}) from Appendix B. It's worthwhile to mention that the 3D ASM method has no singularities at the source points due to its eigenfunction nature, and the incident and scattered fields can be evaluated at the source points as well. The source points are included in Figure \ref{fig:Etot1} at the spherical surface.
\begin{figure}[!ht]
\centering
\includegraphics[width=1\linewidth]{Fig/Etot}
\caption{\small $\mathbf{a)}$ presents the radially changing permittivity of the cornea shell, $\mathbf{b)}$ is amplitude (dB) and $\mathbf{c)}$ is phase (rad) of the total electric field outside the cornea at xy-plane at 175 GHz. The cornea is represented with a red circle. The top part in $\mathbf{b)}$ shows a zoom-in plot in the vicinity of the cornea marked with the rectangle in the lower part of the red circle.}
\label{fig:Etot1}
\end{figure}\FloatBarrier
Each Cartesian component of the scattered field from the same cornea model was computed at the transverse evaluation plane arrangement, introduced in the Figure \ref{fig:sphearrangemet}. The components are normalized by the main polarization \(E_z\)-component of the incident field from the Figure \ref{Diff} (c).
\begin{figure}[!h]
\centering
\includegraphics[width=1\linewidth]{Fig/ScatteredField.png}
\caption{\small The scattering simulation from the multilayered spherical structure with 3D ASM method at 175 GHz. Subfigures \(\mathbf{a)}\)-\(\mathbf{c)}\) presents the magnitudes, and \(\mathbf{d)}\)-\(\mathbf{f)}\) present the phases of the Cartesian components at the transverse plane at 40 mm from the origin of the cornea. The cornea was modeled as 50-layer structure.}.
\label{fig:scat}
\end{figure}
The change in amplitude distribution of the scattered field compared to the incident field is in an excellent match down to less than \(-38\) dB difference to the change calculated from a similar multilayered stratified medium structure, i.e., 50 layered planar surfaces illuminated with the plane wave. This comparison was made as \({E_{3D\,ASM}}_{scat.}/{E_{3D\,ASM}}_{inc.}-{E_{PW}}_{scat.}/{E_{PW}}_{inc.}\). The plane-wave illumination to the planar surface gives the reference scattering, which is used to evaluate the thickness and CTWC of the cornea.
The coupling coefficient \(K\) between the incident and scattered fields plays a critical role in the cornea's CTWC and thickness analysis. In this analysis, the obtained \textit{K} in the observed frequency range is fitted to match the cornea's CTWC- and thickness-dependent reflectivity curve from the planar reference structure. In this paper, the \textit{K} from the cornea model was calculated with two different illumination scenarios in the 200 GHz - 400 GHz band; by the field synthesized from the spherical top-hat field distribution and by the Gaussian beam, whose frequency-dependent waist was located at the origin of the cornea \cite{aleksi}. Obtained values were compared to the reference values, i.e, plane-wave illumination of the planar layered structure with the same layer thickness and permittivity values, see Figure \ref{fig:Coupling}.
The \textit{K} of the equal phase top-hat distribution indicates a high match with amplitude less than 0.8\% and tapered top-hat less than 0.7\% average difference compared to the reference plane wave scenario. Typical Gaussian-beam illumination produces a 4-5\% amplitude difference. This improved matching in \textit{K} cut down the fitting errors in the cornea's CTWC and thickness analysis. It is also worthwhile to mention that the total reflected energy of the beam with the spherical tapered and untapered top-hat distribution matches with less than 0.07\% average difference compared to the plane wave scenario, thus we can conclude that the small difference in the \textit{K} magnitude of the top-hat beam and reference values are due to the differences in the diffraction patterns between the incident and scattered fields.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.7\linewidth]{Fig/Coupling}
\caption{\small The magnitude of the coupling coefficient from 200 to 400 GHz frequency range in four scenarios. The blue line presents the reference value from plane wave illumination into the planar structure, the red line presents the top-hat illumination, the yellow line presents tapered top-hat illumination, and the purple line presents the typical Gaussian beam illumination for spherical structure.}
\label{fig:Coupling}
\end{figure}\FloatBarrier
\section{Conclusions}
This paper proposes a novel method for expanding the nominally planar AS method to synthesize incident electromagnetic beams from electric field distributions on spherical surfaces. The derivation of the method was presented, and the resulting equations were proved mathematically in Appendix A. The proposed method can be expanded to synthesize beams from the arbitrary smooth surfaces taking the radii of curvatures and frequency range into account. The minimum radius of curvature of the surface compared to the wavelength is \(RoC>2\lambda\). The benefit of this method is the ability to present the synthesized fields in the angular spectrum domain, which can be used to compute the VSH coefficients of the incident fields. Thus the incident field can be presented with VSH expansion and scattered fields from multilayered spherical structures, of an arbitrarily large number of layers, can be obtained utilizing the Mie theory. The computational effort of presented 50-layer spherical scattering simulations requires much less memory and fewer arithmetical operations than, for example, similar full-wave computation.
First, the advantages of the beams with spherically matching wavefronts compared to the Gaussian beam illumination for spherical objects were presented. This was done by comparing the scattering behavior of the top-hat, tapered top-hat and Gaussian beam illumination from the spherical 27-layer bandpass filter. These results were compared to the reference scattering from interaction with plane wave and planar 27-layer bandpass filter. The beams with spherically matching wavefront had significantly increased coupling coefficient compared to the traditional Gaussian beam illumination. Next, the Cartesian components of the incident field were evaluated at the transverse plane at 40 mm from the origin of the cornea. These results were compared to the physical optics simulations and demonstrated good agreement. Then the gradient permittivity of the cornea was modeled as a multilayered spherical structure of 50 layers, and the scattered beam was evaluated from the ideal incident field with a spherical top-hat and tapered top-hat wavefront by the T-matrix method. The coupling coefficients between the incident and scattered fields of the spherical top-hat distributions were evaluated in the 200 GHz to 400 GHz region and compared to Gaussian-beam illumination and the reference plane-wave illumination on an analog planar surface. The coupling coefficient amplitude of the spherical tapered top-hat illumination was closest to the reference amplitude with less than 0.7\% average difference. The good agreement between the coupling coefficient and the reference validates the proposed method. The results indicate that the incident beam with a matching spherical wavefront interacts with the multilayered sphere similar to the plane wave interacting with the planar structures. Thus, the developed method enables the synthesis of beams whose interaction with the cornea can be more accurately predicted/analyzed with standard EMT and SMT and improves the accuracy of CTWC and thickness measurement in THz band.
The presented method will enable the computation of electric field propagation between different surfaces. The desired incident field can be synthesized from the cornea's surface distribution as presented in this paper and computed at the optical elements. This electric field distribution at the optical element can then be used as a source distribution and propagated back to the cornea. Through this methodology, the minimum optical element apertures can be evaluated by comparing the surface field distribution from the propagated field to the original surface distribution.
\section{Appendix A}
\begin{theorem}
Let the function \(E_0\) be continuous in a closed segment of sphere \(\Omega\) and let \(\mathbf{r}\in\mathbb{R}^3\) be such that \(\mathbf{r}\cdot\mathbf{n}\neq1\) for all \(\mathbf{n}\in\Omega\). The field \(E\) created by function \(E_0\) is presented by (\ref{eq15}) and (\ref{eq16}) at point \(\mathbf{r}\), where \(\bar{x},\bar{y}\) and \(\bar{z}\) are as in (\ref{eq6}).
\end{theorem}
\begin{proof}
Let us define \(\epsilon>0\). Let us show first, that when the partition {\(\Omega_t\)} is small enough, in other words, the areas \(|\Omega_t|\) are small enough regardless of t, by replacing the function \(E_0^t\) supported by each piece \(\Omega_t\) with a constant \(E_0^t(\bar{0},\bar{0})=E_0(\theta,\phi)\), the error made in formula (\ref{eq10}) is smaller than \(\epsilon/4\). Function \(E_0\) is uniformly continuous on the compact segment of sphere \(\Omega\), and based on this, with sufficiently small \(|\Omega_t|\) it holds
\begin{equation}\label{eq33}
\begin{array}{l}
|E^t_0(\bar{x},\bar{y})-E_0(\theta,\phi)|\leq\frac{\epsilon}{4I|\Omega|}
\end{array}
\end{equation}
for all \(\bar{x},\bar{y}\in\Omega_t\) and \(t\), where the constant \(I\) will be defined later on. In the Fourier transform (\ref{eq7}) on the plane holds
\begin{equation}\label{eq34}
\begin{array}{l}
|\mathcal{F}\left\{E_0^t\right\}-\mathcal{F}\left\{E_0(\theta,\phi)\right\}|
\leq\iint_{\Omega_t}|E^t_0(\bar{x},\bar{y})-E_0(\theta,\phi)||e^{-i(k_{\bar{x}}\bar{x}+ k_{\bar{y}}\bar{y})}|d\bar{x}d\bar{y}\\\leq\iint_{\Omega_t}\frac{\epsilon}{4I\Omega}|e^{-i(k_{\bar{x}}\bar{x}+ k_{\bar{y}}\bar{y})}|d\bar{x}d\bar{y}
\leq\iint_{\Omega_t}\frac{\epsilon}{4I|\Omega|}d\bar{x}d\bar{y}=\frac{\epsilon|\Omega_t|}{4I|\Omega|},
\end{array}
\end{equation}
for all \((k_{\bar{x}},k_{\bar{y}})\in\mathbb{R}^2\). Function \(e^{i|\bar{z}|\sqrt{k^2-k_{\bar{x}}^2-k_{\bar{y}}^2}}\) is integrable on \((k_{\bar{x}},k_{\bar{y}})\) plane, because outside of the circle \(k_{\bar{x}}^2+k_{\bar{y}}^2=k^2\), the exponential becomes real and negative. Let \(I\) be at first a continuous elementary function
\begin{equation}\label{eq35}
\begin{array}{l}
I(|\bar{z}|)=\iint_{\mathbb{R}^2}|e^{i|\bar{z}|\sqrt{k^2-k_{\bar{x}}^2-k_{\bar{y}}^2}}|dk_{\bar{x}}dk_{\bar{y}},
\end{array}
\end{equation}
where \(\bar{z}=\bar{z}(\theta,\phi)\); see (\ref{eq6}). A continuous function \(|\bar{z}(\theta,\phi)|\) gets its minimum value \(\bar{z}_{min}>0\) on the compact set \(\left\{(\theta,\phi)|\theta\in[0,\theta_0],\phi\in[0,2\pi]\right\}\). Especially \(|\bar{z}|\geq \bar{z}_{min}\) for all \(t\) regardless of partitioning. Finally, the constant \(I\) is defined
\begin{equation}\label{eq36}
\begin{array}{l}
I=I(z_{min})<\infty.
\end{array}
\end{equation}
Due the monotone of integral, \(I(|\bar{z}(\theta,\phi)|)\leq I\) on all \(t\) regardless of partitions. The upper limit for the error due \(E_0(\theta,\phi)\) on the formula (\ref{eq12}) is obtained as
\begin{equation}\label{eq37}
\begin{array}{l}
\frac{1}{4\pi^2}\iint_{\mathbb{R}^2}|\mathcal{F}\left\{E_0^t\right\}-\mathcal{F}\left\{E_0(\theta,\phi)\right\}|(k_{\bar{x}},k_{\bar{y}})|e^{i|\bar{z}|\sqrt{k^2-k_{\bar{x}}^2-k_{\bar{y}}^2}}|dk_{\bar{x}}dk_{\bar{y}}\\
\leq\frac{1}{4\pi^2}\frac{\epsilon|\Omega_t|}{4I|\Omega|}\iint_{\mathbb{R}^2}|e^{i|\bar{z}|\sqrt{k^2-k_{\bar{x}}^2-k_{\bar{y}}^2}}|dk_{\bar{x}}dk_{\bar{y}}\\\leq\frac{1}{4\pi^2}\frac{\epsilon|\Omega_t|}{4I|\Omega|}I\leq\frac{\epsilon|\Omega_t|}{4|\Omega|},
\end{array}
\end{equation}
where (\ref{eq33}) and (\ref{eq36}) have been used. Because of \(||\mathbf{e}_{\theta}||=1\) and the triangle inequality the error in formula (\ref{eq13}) is
\begin{equation}\label{eq38}
\begin{array}{l}
err_1\leq\sum_t\frac{\epsilon|\Omega_t|}{4|\Omega|}=\frac{\epsilon}{4}.
\end{array}
\end{equation}
In the above approximation, a constant \(E_0(\theta,\phi)\) is used on piece \(\Omega\); as a function it is zero on the plane outside of the piece, let us denote this function simply as \(E_0(\theta,\phi)\). Next, we consider the Fourier transform of function \(E_0(\theta,\phi)\). Because function \(e^{i|\bar{z}|\sqrt{k^2-k_{\bar{x}}^2-k_{\bar{y}}^2}}\) is integrable, there can be found an origin centered closed disk \(\bar{B}\) such that
\begin{equation}\label{eq39}
\begin{array}{l}
\left[max_\Omega|E_0|\right]\iint_{\mathbb{R}^2/\bar{B}}|e^{i|\bar{z}|\sqrt{k^2-k_{\bar{x}}^2-k_{\bar{y}}^2}}|dk_{\bar{x}}dk_{\bar{y}}<\frac{\epsilon}{4|\Omega|}
\end{array}
\end{equation}
for all \(t\) because \(|\bar{z}|\geq \bar{z}_{min}\). For all \((k_{\bar{x}},k_{\bar{y}})\in\mathbb{R}^2\) it holds
\begin{equation}\label{eq40}
\begin{array}{l}
|\mathcal{F}\left\{E_0(\theta,\phi)\right\}(k_{\bar{x}},k_{\bar{y}})|\\\leq \left[max_\Omega|E_0|\right]\iint_{\Omega_t}1dk_{\bar{x}}dk_{\bar{y}}=max|E_0||\Omega_t|.
\end{array}
\end{equation}
Let us now consider the error (\(err_2\)) made in integral (\ref{eq12}) when \(\mathcal{F}\left\{E_0(\theta,\phi)\right\}\) is replaced by the constant \(E_0(\theta_t,\phi_t)|\Omega_t|\). Under estimates (\ref{eq39}) and (\ref{eq40}) it follows
\begin{equation}\label{eq41}
\begin{array}{l}
\iint_{\mathbb{R}^2/\bar{B}}|\mathcal{F}\left\{E_0(\theta,\phi)\right\}-E_0(\theta,\phi)|\Omega_t|||e^{i|\bar{z}|\sqrt{k^2-k_{\bar{x}}^2-k_{\bar{y}}^2}}|dk_{\bar{x}}dk_{\bar{y}}\\
\leq\iint_{\mathbb{R}^2/\bar{B}}2\left[max_\Omega|E_0|\right]|\Omega_t||e^{i|\bar{z}|\sqrt{k^2-k_{\bar{x}}^2-k_{\bar{y}}^2}}|dk_{\bar{x}}dk_{\bar{y}}\\
\leq 2|\Omega_t|\frac{\epsilon}{4|\Omega|}=\frac{\epsilon|\Omega_t|}{2|\Omega|}.
\end{array}
\end{equation}
For all \((k_{\bar{x}},k_{\bar{y}})\in\bar{B}\) and (\(\bar{x},\bar{y})\in\Omega_t\), we can estimate by continuity of the function
\begin{equation}\label{eq42}
\begin{array}{l}
\left[max_\Omega|E_0|\right]|e^{-i(k_{\bar{x}}\bar{x}+k_{\bar{y}}\bar{y})}-1|\leq\frac{\epsilon}{4|\bar{B}|\Omega|}
\end{array}
\end{equation}
always when \(|\Omega_t|\) is small enough, because then \(\bar{x}\approx0, \bar{y}\approx0\) and hence \(k_{\bar{x}}\bar{x}+k_{\bar{y}}\bar{y}\approx0\ -\) that is the assignment of \(\bar{B}\). Then in the Fourier transform at point \((k_{\bar{x}},k_{\bar{y}})\in\bar{B}\) it holds
\begin{equation}\label{eq43}
\begin{array}{l}
|\mathcal{F}\left\{E_0(\theta,\phi)\right\}(k_{\bar{x}},k_{\bar{y}})-E_0(\theta,\phi)|\Omega_t||\\
=|\iint_{\Omega_t}E_0(\theta,\phi)e^{-i(k_{\bar{x}}\bar{x}+k_{\bar{y}}\bar{y})}d\bar{x}d\bar{y}-E_0(\theta,\phi)1d\bar{x}d\bar{y}|\\
\leq|E_0(\theta,\phi)|\iint_{\Omega_t}|e^{-i(k_{\bar{x}}\bar{x}+k_{\bar{y}}\bar{y})}-1|d\bar{x}d\bar{y}\\
\leq\iint_{\Omega_t}\frac{\epsilon}{4|\bar{B}||\Omega|}d\bar{x}d\bar{y}=\frac{\epsilon|\Omega_t|}{4|\bar{B}||\Omega|}.
\end{array}
\end{equation}
Replacing by \(\mathcal{F}\left\{E_0(\theta,\phi)\right\}\) with the constant \(E_0(\theta,\phi)|\Omega_t|\) in formula (\ref{eq12}) we make an error
\begin{equation}\label{eq44}
\begin{array}{l}
\frac{1}{4\pi^2}\Big(\iint_{\bar{B}}+\iint_{\mathbb{R}^2/\bar{B}}\Big)|\mathcal{F}\left\{E_0(\theta,\phi)\right\}-E_0(\theta,\phi)|\Omega_t||e^{i|\bar{z}|\sqrt{k^2-k_{\bar{x}}^2-k_{\bar{y}}^2}}|dk_{\bar{x}}dk_{\bar{y}}\\
\leq\frac{\epsilon|\Omega_t|}{4|\bar{B}||\Omega_t|}\iint_{\bar{B}}1dk_{\bar{x}}dk_{\bar{y}}+\frac{\epsilon|\Omega_t|}{2|\Omega|}=\frac{\epsilon|\Omega_t|}{4|\Omega|}+\frac{\epsilon|\Omega_t|}{2|\Omega|}=\frac{3\epsilon|\Omega_t|}{4|\Omega|},
\end{array}
\end{equation}
where the estimates (\ref{eq41}) and (\ref{eq43}) have been used. The error for sum (\ref{eq13}) is obtained as
\begin{equation}\label{eq45}
\begin{array}{l}
err_{2}\leq\sum_t\frac{3\epsilon|\Omega_t|}{4|\Omega|}=\frac{3\epsilon}{4}.
\end{array}
\end{equation}
Thus, when the partition {\(\Omega_t\)} is dense enough and \(\mathbf{r}\in\mathbb{R}^3\) be such that \(\mathbf{r}\cdot\mathbf{n}\neq1\) for all \(\mathbf{n}\in\Omega\), we get by (\ref{eq38}) and (\ref{eq45})
\begin{equation}\label{eq46}
\begin{array}{l}
\Big|\Big|\sum_tE_t(\mathbf{r})\mathbf{e}_{\theta}-\frac{1}{4\pi^2}\sum_tE_0(\theta,\phi)\mathbf{e}_{\theta}(\theta,\phi)|\Omega_t|\iint_\mathbb{R}e^{i(k_{\bar{x}}\bar{x}+ k_{\bar{y}}\bar{y})}e^{i|\bar{z}|\sqrt{k^2-k_{\bar{x}}^2-k_{\bar{y}}^2}}dk_{\bar{x}}dk_{\bar{y}}\Big|\Big|\\
\leq err_1+err_{2}
\leq\frac{\epsilon}{4}+\frac{3\epsilon}{4}=\epsilon.
\end{array}
\end{equation}
Finally, by continuity of the integrand, the integral in (\ref{eq15}) exists, and, because of the estimate (\ref{eq46}), for the points \(\mathbf{r}\in\mathbb{R}^3\) like in Theorem there also exists
\begin{equation}\label{eq47}
\begin{array}{l}
\mathbf{E}_{\theta}(\mathbf{r})=\mathbf{E}_{\theta}(x,y,z)=lim_{|\Omega_t|\to 0}\sum_tE_t(\mathbf{r})\mathbf{e}_{\theta}\\
=\frac{1}{4\pi^2}\iint_{\Omega}E_0(\theta,\phi)\mathbf{E}_t(\mathbf{r};\theta,\phi)\sin{\theta}d\theta d\phi.
\end{array}
\end{equation}
\end{proof}
The proof above is also valid for the sum of the second vector component of (\ref{eq19}). Only a few modifications are needed.
\section{Appendix B}
The VSH beam shape coefficients for each \(ij-\) plane wave of the incident field is defined as \cite{barber}
\begin{equation}\label{eqB1}
\begin{array}{l}
a^t_{{\text{emn}}_{ij}}=4i^n\mathbf{e}_{o_{ij}}\cdot\big[-\mathbf{e}_{\theta}\sin{(m\phi)}\frac{m}{\sin{\theta}}P^m_n(\cos{\theta})-\mathbf{e}_{\phi}\cos{(m\phi)}\frac{d}{d\theta}P^m_n(\cos{\theta})\big],\\
a^t_{{\text{omn}}_{ij}}=4i^n\mathbf{e}_{o_{ij}}\cdot\big[\mathbf{e}_{\theta}\cos{(m\phi)}\frac{m}{\sin{\theta}}P^m_n(\cos{\theta})-\mathbf{e}_{\phi}\sin{(m\phi)}\frac{d}{d\theta}P^m_n(\cos{\theta})\big],\\
b^t_{{\text{emn}}_{ij}}=-4i^{n+1}\mathbf{e}_{o_{ij}}\cdot\big[\mathbf{e}_{\theta}\cos{(m\phi)}\frac{m}{\sin{\theta}}P^m_n(\cos{\theta})-\mathbf{e}_{\phi}\sin{(m\phi)}\frac{d}{d\theta}P^m_n(\cos{\theta})\big],\\
b^t_{{\text{omn}}_{ij}}=-4i^{n+1}\mathbf{e}_{o_{ij}}\cdot\big[\mathbf{e}_{\theta}\sin{(m\phi)}\frac{m}{\sin{\theta}}P^m_n(\cos{\theta})+\mathbf{e}_{\phi}\cos{(m\phi)}\frac{d}{d\theta}P^m_n(\cos{\theta})\big],
\end{array}
\end{equation}
where \(P^m_n\) is the associated Legendre function of the first kind of degree \(n\) and order \(m\).
The scattered field from differential element in VSH presentation is obtained as
\begin{equation}\label{eqB2}
\begin{array}{l}
\begin{aligned}
\mathbf{E}_{scat}(k\mathbf{r})&=p^2\sum_m\sum_n D_{mn}\Big[f_{\text{emn}}\mathbf{M}^{(3)}_{\text{emn}}(k\mathbf{\bar{r}})+f_{\text{omn}}\mathbf{M}^{(3)}_{\text{omn}}(k\mathbf{\bar{r}})\\&+g_{\text{emn}}\mathbf{N}^{(3)}_{\text{emn}}(k\mathbf{\bar{r}})+g_{\text{omn}}\mathbf{N}^{(3)}_{\text{omn}}(k\mathbf{\bar{r}})\Big],
\end{aligned}
\end{array}
\end{equation}
where the superscripts (3) presents the vectors spherical harmonics (outgoing wave) with spherical Hankel function of the first kind \(h_{\text{n}}^{(1)}(kr)\). The \(f_{\text{emn}},f_{\text{omn}},g_{\text{emn}}\) and \(g_{\text{omn}}\) are vector spherical harmonic coefficient for the scattered field calculated from the T-matrix method as
\begin{equation}\label{eqB3}
\begin{array}{l}
\begin{bmatrix}f_{\text{emn}}\\f_{omn}\\g_{\text{emn}}\\g_{omn}\end{bmatrix}=\begin{bmatrix}T_{11} & 0 & 0 & 0\\
0 & T_{22} & 0 & 0\\0 & 0 & T_{33} & 0\\0 & 0 & 0 & T_{44}\end{bmatrix}\begin{bmatrix}a_{\text{emn}}\\a_{omn}\\b_{\text{emn}}\\b_{omn}\end{bmatrix},
\end{array}
\end{equation}
where the T-matrix elements for the multilayered sphere is obtained by the algorithm defined in \cite{pena}. The coupling coefficient between the incident and scattered field can be obtained as
\begin{equation}\label{eqB4}
K=\frac{\int\int\big[\mathbf{E}_{inc}\cdot\mathbf{E}_{sca}\big]dxdy}{\int\int\big[\mathbf{E}_{inc}\cdot{\mathbf{E}_{inc}}^{*}\big]dxdy},
\end{equation}
where \(\mathbf{E}_{inc}\) and \(\mathbf{E}_{sca}\) are computed by Eq. (\ref{INC}) and Eq. (\ref{SCAT}), and $^*$ denotes the complex conjugate.
\begin{backmatter}
\bmsection{Funding}
This research was supported by Academy of Finland $\#$327640: Assessment of the Graft Rejection Using Millimeter Waves (AGRUM) and the Agencia Estatal Investigación PID2019-107885GB-C31/AEI/10.13039,
PRE2018-084326.
\bmsection{Disclosures}
The authors declare no conflicts of interest.
\bmsection{Data availability}
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
\end{backmatter}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,914
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Q: Can you have more than one production database in Heroku? In Heroku, is it possible to have more than one "production" Postgres database? In other words, let's say that (for whatever reason), I want to have both a "crane" instance and a "kappa" instance. It sure seems like Heroku would be happy to let me give them the money :) but, it doesn't want to seem to let me do this. Am I missing something? Is it the new dashboard/UI that's not letting me? Do I have to create a separate app for the new db?
A: Yes, you should be able to provision multiple production databases, however they are then differentiated by the COLOR in their config name. In order to add two you would:
heroku addons:add heroku-postgresql:crane
heroku addons:add heroku-postgresql:kappa
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,226
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Émile Thienpondt (ur. 1904, zm. ?) – belgijski pływak, uczestnik Letnich Igrzysk 1924 w Paryżu.
Wystartował w sztafecie 4 × 200 metrów stylem dowolnym podczas igrzysk olimpijskich w 1924 roku. Razem z zespołem odpadli w eliminacjach z czasem 11:14,8.
Linki zewnętrzne
Belgijscy olimpijczycy
Belgijscy pływacy
Uczestnicy Letnich Igrzysk Olimpijskich 1924
Urodzeni w 1904
Nieznana data śmierci
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 7,237
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At a time when we can't go ten minutes without hearing someone declare how they're going to turn everything around and make 2017 their best year yet, it's a radical notion to propose that we simply continue on the course we're on.
We live in a culture that's obsessed with instant results.
New Years comes around and we think the reason we didn't get what we wanted in 2016 was because we didn't have the right goals, the right intention or the right vision board, but is that really true? Couldn't it also be true that we didn't stick with something long enough?
The downside to over-New-Years-Resolution-ing is we are constantly changing directions in a way that has us never actually arrive anywhere. Our addiction to visioning becomes the very thing that keeps us stuck, because as soon as we don't get the fast, instant success we were promised by Tai Lopez, we go back to the drawing board and plot a new course, forgetting that success takes time, commitment and years of hard work.
One of the most life-changing articles I've ever read was called 5-Year Commitments by Steve Pavlina. It's premise is summed up beautifully in the first line.
People commonly overestimate how far they can get in a year, but grossly underestimate how far they can get in 5 years.
Isn't that true? Look at the things in your life you've accomplished, look at the skills you've built and the businesses you've been a part of or started. Can you see how they all took far longer than a year to make successful? Can you see how mastery took years and years of dedication and practice?
Maybe instead of making new resolutions in 2017 we should go back and look at our resolutions from 2016, or 2015 and continue to work on those. Instead of changing directions again because it's fun to make New Years resolutions with our friends we could just keep on the path we're on, because we may be a lot closer to success than we think.
Or maybe you don't want to succeed on the path you're on, and that's ok too.
Dabbling is an important part of your life, especially in your twenties and thirties. If that's true for you, why not really let yourself dabble? If you're twenty-four years old, give yourself permission to start three different businesses this year, or work five different jobs, or take the whole year off and travel. Hell — we should give ourselves permission to do this at any age, if it feels appropriate.
To really commit to something for five years, you need to be sure you want to do it, and the only way to know that is to have done the thing, so don't be afraid to try something, knowing it may not be your end game.
I spent so much of my twenties dabbling, then beating myself up because I thought I had to "find my purpose" at age twenty-five. What a ridiculous sight it is watching a twenty-something beat themselves up for not knowing what they want to do with their life.
Nope. I started that day, and I published two weeks in a row so when 2015 came, I was already on my way to writing once a week.
If you don't like what you're working on, quit.
If you aren't successful yet, consider that maybe you haven't given it enough time, and sticking with things the way they are for another year, or two years, or five years might just be the best resolution you could ever make.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,004
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The Ferreira Rooftop Is Home to 2 Beehives!
I was very interested in working with Alvéole, a Montreal company that guides individuals and organizations in beekeeping and honey production. Thanks to our collaboration, Ferreira has had 2 beehives on its roof since spring.
To me, the decline in the Quebec bee population is very concerning. These foragers are a vital link, since they pollinate 70% of all the flowering plants in the world, thereby contributing to the survival of many plant species.
I feel that it's our duty as participants in the restaurant industry to help bees survive and to raise awareness of the issue among Montrealers.
I hope the Ferreira bees stay with us for at least the next 20 years.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,077
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Kändler, Kaendler:
Johann Joachim Kändler (1706–1775), the most important modelleur of the Meissen porcelain manufacture
5195 Kaendler (3289 T-1), a main-belt asteroid discovered in 1971
Tiit Kändler (born 1948), Estonian humorist, publicist and science journalist
German-language surnames
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,914
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\section{Introduction}
Density functional theory (DFT) has been widely used to study the electronic structure of various types of materials from atoms and molecules to
nanostructures to periodic materials.
The popularity of DFT stems from its low computational expense combined with relatively good accuracy.
The self-interaction error (SIE) that arises from the density functional approximations (DFAs) of the exchange-correlation
functional is well-documented \cite{PhysRevB.23.5048}. This error arises since the self-Coulomb energy is not completely canceled by
the self-exchange energy when the exact, but unknown, exchange-correlation functional is approximated.
This leads to a number of problems. For example, the one-electron potential in DFA does not have the correct asymptotic behavior due to the presence of the SIE, leaving the highest occupied orbitals in stable anions unbound as a result.
The Perdew-Zunger self-interaction correction formalism (PZ-SIC) is a one-electron self-interaction-free
approximation where an orbital by orbital correction is applied to the DFA total energy \cite{PhysRevB.23.5048}. A number of implementations of SIC to DFT
exist \cite{doi:10.1063/1.481421, doi:10.1063/1.1327269, doi:10.1063/1.1370527, doi:10.1063/1.1468640, doi:10.1021/jp014184v,PhysRevA.55.1765,doi:10.1080/00268970110111788, Polo2003, doi:10.1063/1.1630017, B311840A,doi:10.1063/1.1794633, doi:10.1063/1.1897378, doi:10.1063/1.2176608, doi:10.1063/1.2204599,doi:10.1002/jcc.10279,PhysRevA.45.101, PhysRevA.46.5453, PhysRevA.47.165}, including a
recent implementation by J\'onsson \textit{et al.} using complex orbitals that has shown promising results \cite{doi:10.1021/acs.jctc.6b00347}.
The PZ-SIC formalism corrects SIE, but it also leads to an orbital-dependent theory since the orbital-dependent total energy is not invariant under a unitary transformation of the
occupied
orbitals. The set of orbitals that yields the minimum self-interaction corrected total energy therefore must be found. Pederson \textit{et al.} showed that these minimum-energy local orbitals
satisfy additional pairwise conditions known as the localization equations (LE) \cite{doi:10.1063/1.446959, doi:10.1063/1.448266}.
Varying the N$^2$ elements of a unitary transformation to find local orbitals that satisfy the LE is a process that scales poorly with increasing numbers of orbitals, making the solution of
the LE computationally challenging. Another problem with traditional PZ-SIC is that it is not formally size-extensive.
The canonical Kohn-Sham (KS) orbitals tend to delocalize with increasing system size. In the limit of very large sizes and very delocalized orbitals, the correction terms in PZ-SIC tend to zero\cite{Perdew1990113}. This leads to a breakdown of size extensivity when the lowest-energy correction for a single atom is positive.
An alternative approach to solving the LE in PZ-SIC was introduced by Pederson, Perdew, and Ruzsinszky through the use of Fermi-L\"owdin orbitals (FLO)
\cite{doi:10.1063/1.4869581} to evaluate the PZ-SIC total energy. (The resulting method is known as FLOSIC.)
The FLOs are orthonormal local orbitals that are a linear combination of Fermi orbitals (FO). The FOs
depend on the density matrix and spin density at certain points in space called Fermi orbital descriptors (FODs).
The FOs are obtained from the KS orbitals as
\begin{equation}\label{eq:fod}
\phi_{i\sigma}^{FO}(r) =\frac{\sum_j^{N_\sigma}\psi_{j\sigma}^* ({\bf a}_{i\sigma}) \psi_{j\sigma}(\vec{r})}{\sqrt{\rho_\sigma({\bf a}_{i\sigma})}}
\end{equation}
where $\psi_{j\sigma}$, $\rho_\sigma$, $\bf{a}_{i\sigma}$, $N_\sigma$ denote KS orbital,
total electron density, FOD, and number of occupied orbitals of spin $\sigma$, respectively. The FO transformation is unitarily invariant, \textit{i.e.} the same set of FO's is produced by any orthonormal set of orbitals spanning the occupied space. The total energy in FLOSIC therefore depends on the FOD positions and the LE do not need to be applied. In addition, because the FLOs are localized, the FLOSIC method restores size extensivity \cite{doi:10.1063/1.4869581}.
The FO are determined by the positions of
the FODs; therefore, only 3N variables are needed to determine
the optimal set of local orbitals, compared to N$^2$ coefficients of a unitary transformation needed in traditional PZ-SIC. Thus, in principle, FLOSIC provides a computationally simpler way to incorporate the self-interaction correction.
In practical FLOSIC calculations, optimal FOD positions are found using gradients of the energy with respect to FOD positions, in a procedure analogous to molecular geometry optimizations
\cite{doi:10.1063/1.4907592,PEDERSON2015153}. A number of studies have been conducted using the FLOSIC method \cite{doi:10.1063/1.4869581,doi:10.1063/1.4996498,magnetochemistry3040031,doi:10.1080/00268976.2016.1225992,doi:10.1063/1.4947042,doi:10.1021/acs.jpca.8b09940, doi:10.1063/1.5050809, PhysRevA.100.012505,doi:10.1002/jcc.26008}.
To date, FLOSIC has been applied mostly to the LDA level of theory where nearly all properties of atoms and molecules are significantly improved
\cite{doi:10.1063/1.4996498,PhysRevA.95.052505,doi:10.1063/1.4907592}.
On the other hand, SIC-based improvements are known to be less uniform with semilocal generalized gradient approximations (GGA) and meta-GGAs \cite{PhysRevA.84.050501,doi:10.1063/1.1794633,doi:10.1063/1.5087065}.
Recently, Perdew and coworkers have provided insight into this problem \cite{doi:10.1063/1.5087065},
showing that the lobed one electron densities needed for applying SIC are problematic for semilocal functionals such as the Perdew, Burke, and Ernzerhof (PBE) \cite{PhysRevLett.77.3865,PhysRevLett.78.1396} GGA and the strongly constrained and appropriately normed (SCAN) \cite{PhysRevB.54.16533, *PhysRevB.57.14999, PhysRevLett.115.036402} meta-GGA.
While the use of complex orbitals can lessen the problem, it does not
eliminate it \cite{doi:10.1063/1.5087065}. In related work, Santra and Perdew showed applying SIC to a semilocal functional causes appropriate norms that are built in to the functional to be violated \cite{doi:10.1063/1.5090534}.
Because these recent developments may lead to new approaches to implementing SIC and because SCAN is the most successful nonempirical
semilocal functional for predicting the properties of atoms, molecules,
and solids, it is important to thoroughly benchmark the performance of SCAN when used with the existing FLOSIC methodology. We note that although some initial applications of FLOSIC-SCAN were included in the recent publications \cite{doi:10.1063/1.5087065,doi:10.1063/1.5090534}, this article presents the details of the FLOSIC-SCAN implementation for the first time, including a description of refinements to the numerical integration grid that are necessary to insure accurate results. It also gives
a full account of how FLOSIC-SCAN performs for a number of properties such as atomic energies, $\Delta$-SCF ionization potentials and electron affinities, ionization potential estimates from the HOMO energies of atoms and molecules, dissociation energies using benchmark sets that are known to be sensitive to SIEs, and atomization energies. In all cases, we compare the performance of FLOSIC-SCAN to that of FLOSIC-LDA and FLOSIC-PBE and the uncorrected SCAN functional.
We also examine the effectiveness of using FODs
optimized at the FLOSIC-LDA level
in FLOSIC-SCAN calculations.
Finally, we also investigate the quality of the self-consistent FLOSIC-DFA electron density
by using it in place of the corresponding self-consistent DFA density in the parent DFA functional. Since the FLOSIC method restores the correct asymptotic behavior to the DFA potential for a localized system, it is expected to improve the quality of the density in the asymptotic region. Hence, the more physically correct electron density from FLOSIC,
when combined with an accurate
functional such as SCAN, may lead to improved estimates of total energies
by removing density driven errors \cite{VERMA201210, PhysRevLett.111.073003}. Our results show that using the FLOSIC density in the parent functional often leads to electronic properties near equilibrium that are improved over those of the parent functional.
This article is organized as follows.
In Sec. \ref{cpsetup}, we present our computational method and also discuss the
implementation of SCAN in the FLOSIC code.
Calculated data for the atoms and their ionization potentials and electron affinities using the FLOSIC method are discussed
in Sec. \ref{atomandip}.
FLOSIC total and atomization energies of selected molecules are presented and discussed in
Sec. \ref{atomizationenergymol}.
FLOSIC dissociation energies are presented in Sec. \ref{sec:sie11}.
Finally in Sec. \ref{evaluehoorbitals}, we discuss the eigenvalues of the highest occupied molecular orbitals using FLOSIC.
\section{Computational Method}\label{cpsetup}
All of the results presented in this manuscript are calculated with the FLOSIC code, which is based on the UTEP version of the NRLMOL code \cite{FLOSICcode}, a Gaussian orbital-based electronic structure code \cite{PhysRevB.41.7453, PhysRevB.42.3276, doi:10.1002/(SICI)1521-3951(200001)217:1<197::AID-PSSB197>3.0.CO;2-B}. Among the features included in this version is an interface to the exchange-correlation library called LIBXC. The latter provides access to a large number exchange-correlation functionals \cite{LEHTOLA20181, MARQUES20122272}.
The FLOSIC code inherits the optimized Gaussian basis sets of NRLMOL \cite{PhysRevA.60.2840} and an accurate numerical integration grid scheme \cite{PhysRevB.41.7453}.
In all of our calculations, the default NRLMOL basis sets are used.
A recent study which studied ionization potentials and enthalpies of formation using FLOSIC approach, the default NRLMOL basis
set was found to provide results comparable to the cc-pVQZ basis set \cite{doi:10.1002/jcc.25586}.
The SIC calculations require finer mesh as orbital densities are involved in calculation of orbital depedent
potentials. A default NRLMOL mesh for FLOSIC calculation, on average, has 25000 grid points per atom. This results in integration of charge density that is accurate to the order of $10^{-8} e$.
The exchange-correlation (XC) functionals used in this study are the LSDA implementation of Perdew and Wang (LDA) \cite{PhysRevB.45.13244}, Perdew, Burke \& Ernzerhof (PBE) \cite{PhysRevLett.77.3865, PhysRevLett.78.1396}, and SCAN \cite{PhysRevLett.115.036402}.
FLOSIC calculations require an initial set of trial FOD positions. Whenever they are available, previously reported FOD positions are used as starting points. In other cases, FODs are generated from scratch and further optimized using a conjugate gradient algorithm. We use the convergence criteria of $10^{-6}$ Ha on the FLOSIC total energy for these optimizations. We find that FLOSIC-LDA optimized FOD positions are typically a good starting point for FLOSIC-PBE and FLOSIC-SCAN calculations. For example, the FOD positions for neutral atoms shifted an average of only 0.073 Bohr after optimization with FLOSIC-SCAN, while keeping similar overall arrangements.
Meta-GGA functionals, including SCAN, are sensitive to the numerical details of a calculation, and this sensitivity extends to FLOSIC-SCAN calculations.
The standard variational integration mesh method \cite{PhysRevB.41.7453} employed in the FLOSIC code provides good accuracy for the LSDA and PBE functionals, but not for SCAN calculations.
Semilocal meta-GGA functionals use
a dimensionless variable defined as
\begin{equation}
\alpha = \frac {\tau - \tau^W}{\tau^\text{unif}} > 0
\end{equation}
where $\tau$ is the kinetic energy density,
$\tau^W=|\vec{\nabla} \rho|^2/8\rho$ is the Weizs\"acker kinetic energy density, and $\tau^\text{unif}=(3/10)(3\pi^2)^{2/3} \rho^{5/3}$ is the kinetic energy density at the uniform-density limit.
The numerical challenges of using SCAN are related to changes in $\alpha$.
Recently, Bart\'ok and Yates showed that the numerical instabilities arising from switching function in SCAN can be eliminated by modifying the switching function \cite{doi:10.1063/1.5094646}; however, such modification results in violation of some exact constraints.
The exchange enhancement factor of SCAN has a mathematical form given as
\begin{equation}
F_x(s,\alpha)= \{ h_x^1(s,\alpha)+ f_x(\alpha) [ h_x^0 - h_x^1(s,\alpha) ] \} g_x(s),
\end{equation}
\begin{equation}\label{eqn_fx}
f_x(\alpha) = \exp \left[- \frac{c_{1x} \alpha}{ 1-\alpha}\right] \theta(1-\alpha) -d_x \exp \left[\frac{c_{2x}}{1-\alpha} \right] \theta(\alpha - 1),
\end{equation}
\begin{equation}
s=\frac{|\vec{\nabla} \rho|}{2(3\pi^2)^{1/3}\rho^{4/3}}
\end{equation}
where $h_x^1 (s,\alpha)$ is a function of $s$ and $\alpha$, $g_x(s)$ is a function of $s$, $h_x^0 = 1.174$, $s$ is dimensionless density gradient,
$c_{1x}$, $c_{2x}$, $d_x$ are interpolation parameters, and $\theta(x)$ is a step function of $x$ \cite{PhysRevLett.115.036402}. Figure \ref{scanmesh} shows $f_x(\alpha)$ (Eq. (\ref{eqn_fx})) and its derivative, $\frac {df_x(\alpha)}{d\alpha}$, as functions of $\alpha$. A large oscillation of $\frac {df_x(\alpha)}{d\alpha}$ is seen near $\alpha = 1$.
A high density of grid points is needed in the areas where the $\frac {df_x(\alpha)}{d\alpha}$ term changes rapidly in space, and similarly for the $\frac {df_c(\alpha)}{d\alpha}$ function used in the correlation term.
The enhanced mesh used in the FLOSIC code was designed to provide this.
To obtain numerically converged results, following procedure was adopted. We begin by adding radial points with uniform increments until the integrals are converged.
This is a brute force approach of mesh generation.
This is done to eliminate any assumption about the problematic ($\alpha \approx 1$) region. We then decrease the number of radial grid points in the region farther from the nuclei by maintaining the same grid density in the problematic ($\alpha \approx 1$) region. It is ensured that the integrals accuracy remains
same ($10^{-8}$ Ha for exchange-correlation energy) while
reducing the grid density. This approach has worked well but still results in a numerical mesh that is approximately three to six times larger than the default variational mesh.
The SCAN mesh used in this work is roughly 140000 grid points per atom. This results in integration of charge density which is accurate in the order of $10^{-10} e$.
Further improvement of
the numerical grid to reduce the need of such dense grid is being explored and will be reported in future.
\subsection{Meta-GGA implementation}
The meta-GGA exchange-correlation energy has the form given as
\begin{equation}\label{ExcMGGA}
E_{XC}[\rho_\uparrow,\rho_\downarrow] = \int e_{XC}(\rho_\uparrow, \rho_\downarrow, \vec{\nabla}\rho_\uparrow, \vec{\nabla}\rho_\downarrow ,\tau_\uparrow, \tau_\downarrow)d \vec{r}
\end{equation}
where $e_{XC}$ is the exchange-correlation energy density function, $\rho_\uparrow$ and $\rho_\downarrow$ are electron spin densities, and $\tau_\uparrow$ and $\tau_\downarrow$ are kinetic energy density. The kinetic energy density is calculated from the KS orbitals $\psi_i$ as
\begin{equation}
\tau(\vec{r}) = \frac 1 2 \sum_i \vec{\nabla} \psi_i (\vec{r}) \cdot \vec{\nabla} \psi_i (\vec{r}).
\end{equation}
To obtain the exchange-correlation potential, functional derivatives of $E_{XC}$ are required.
In the case of Eq. (\ref{ExcMGGA}), the functional derivative of exchange-correlation energy with respect to density is
\begin{equation}\label{dExcdrho}
\begin{aligned}
\frac {\delta E_{XC}[\rho]} {\delta\rho(\vec{r})} =& \frac {\partial e_{XC}(\rho(\vec{r}),\vec{\nabla} \rho(\vec{r}), \tau(\vec{r}))} {\partial \rho(\vec{r})}
- \vec{\nabla} \frac {\partial e_{XC}(\rho(\vec{r}),\vec{\nabla} \rho(\vec{r}), \tau(\vec{r}))} {\partial \vec{\nabla} \rho(\vec{r})}\\
+& \int \frac {\partial e_{XC}(\rho(\vec{r'}),\vec{\nabla} \rho(\vec{r'}), \tau(\vec{r'}))} {\partial \tau(\vec{r'})}
\frac {\delta \tau [\rho] (\vec{r'})}{\delta \rho(\vec{r})} d\vec{r'}
\end{aligned}
\end{equation}
where the third term is obtained with the functional derivative chain rules.
Typically, an exchange-correlation functional is implemented in quantum chemistry software in such a way that $\frac {\partial e_{XC}(\rho,\vec{\nabla} \rho, \tau)} {\partial \rho}$, $\frac {\partial e_{XC}(\rho,\vec{\nabla} \rho, \tau)} {\partial \vec{\nabla} \rho}$, and $ \frac {\partial e_{XC}(\rho,\vec{\nabla} \rho, \tau)} {\partial \tau}$ are returned from
subroutines.
The $\frac{\delta \tau[\rho](\vec{r})}{\delta \rho}$ in Eq. (\ref{dExcdrho}) can be calculated as $\frac{\delta \tau}{\delta \psi} \frac{\delta \psi}{\delta \rho}$; however, computing $\frac{\delta \psi[\rho](\vec{r})}{\delta \rho}$ is difficult.
It was suggested by Zahariev \textit{et al.} \cite{doi:10.1063/1.4811270} and Yang \textit{et al.} \cite{PhysRevB.93.205205} that the Hamiltonian matrix elements of the pure meta-GGA exchange-correlation potential can be written as follows, using integrations-by-parts:
\begin{equation}
\begin{aligned}
\int & \psi_i (\vec{r}) \frac{\delta E_{XC}[\tau[\rho]]}{\delta \rho(\vec{r})} \psi_j(\vec{r}) d\vec{r} \\
\approx & \frac 1 2 \int \frac{\delta E_{XC}[\tau]}{\delta\tau(\vec{r})}\vec{\nabla} \psi_i(\vec{r})\cdot \vec{\nabla}\psi_j(\vec{r})d\vec{r}.
\end{aligned}
\end{equation}
This approach of computing the Hamiltonian matrix elements is used for the meta-GGA implementation in the FLOSIC code.
\vfill
\subsection{FLOSIC}
FLOSIC uses the PZ-SIC total energy expression that removes the self-interaction of the occupied orbitals on an orbital by orbital basis:
\begin{equation}\label{eqnESIC}
E_{}^{SIC}[\rho_\uparrow,\rho_\downarrow] = E_{}[\rho_\uparrow,\rho_\downarrow]-\sum_\sigma \sum_i^{N_\sigma}\Big(U[\rho_{i\sigma}]+E_{XC}[\rho_{i\sigma},0]\Big)
\end{equation}
where $\sigma$ is the spin index, $i$ is the orbital index, and $N_{i\sigma}$ is the number of orbitals for spin $\sigma$. $\rho_\uparrow$ and $\rho_\downarrow$ denote spin up and spin down electron densities. $\rho_{i\sigma} = |\phi_{i\sigma}|^2$, where the $\phi_{i\sigma}$ are the Fermi-L\"{o}wdin orbitals (FLO). The FO are constructed from a transformation on the KS orbitals using Eq. (\ref{eq:fod}). These are normalized, but not mutually orthogonal. L\"{o}wdin orthogonalization yields the FLOs.
The DFA-SIC single particle equations are
\begin{equation}
(H_\sigma^{DFA} + V_{i\sigma}^{SIC})\phi_{i\sigma} = \sum_j^{N_\sigma} \lambda_{ji\sigma} \phi_{j\sigma}.
\end{equation}
These are satisfied self-consistently for a given choice of the FODs, following the approach of Ref. \cite{PhysRevA.95.052505}. We use an SCF convergence tolerance of $10^{-6}$ Ha.
\section{Results and discussion}
\subsection{Atoms: total energies, ionization energies, and electron affinities}\label{atomandip}
The focus of this work is to give a comprehensive assessment of the results of FLOSIC-SCAN calculations. To do that we compare these to corresponding results for FLOSIC-LDA, FLOSIC-PBE, and for the corresponding uncorrected DFA's.
The FLOSIC energies for atoms from H--Ar ($Z=1-18$) can be compared against accurate non-relativistic total energies reported
by Chakravorty \textit{et al.} \cite{PhysRevA.47.3649}.
The deviation of the calculated total energies are given on a per electron basis as $(E - E_\text{Ref})/N_e$, where $E$ is the
FLOSIC energy, $E_\text{Ref}$ is the reference energy,
and $N_e$ is the number of electrons in the given system.
The results are shown in Fig. \ref{ldas}--\ref{scans},
and the numerical errors of FLOSIC energies with respect to $E_\text{Ref}$ are presented
in Table \ref{flosicstable}.
As noted in earlier works \cite{doi:10.1063/1.4869581, PEDERSON2015153, doi:10.1063/1.4996498}, we find that the total energies with
LSDA improve
within the FLOSIC method (shown in Fig. \ref{ldas})
with a decrease in mean absolute error (MAE) from $0.73$ Ha (LSDA) to $0.38$ Ha (FLOSIC-LSDA).
On the other hand, both PBE and SCAN total energies show a larger deviation when
corrected for self-interaction using FLOSIC as shown in Figures \ref{pbes} and \ref{scans}.
The MAEs for total energy with PBE and FLOSIC-PBE are $0.083$
and $0.159$ Ha respectively; for SCAN and FLOSIC-SCAN the MAE's are $0.019$ and $0.15$ Ha. Thus, FLOSIC-PBE and FLOSIC-SCAN perform better than LSDA and FLOSIC-LSDA, but not as well as PBE and SCAN.
DFT calculation using accurate electron densities can eliminate density driven errors and give better energies \cite{VERMA201210,wasser-burke}.
Since SIC restores the correct asymptotic behavior of the potential and one-electron self-interaction freedom
\cite{doi:10.1063/1.5090534}, it can provide a more physically reasonable density than a DFA calculation.
It is therefore of interest to calculate the total energies using the
self-consistent FLOSIC density in the standard GGA (PBE)
and meta-GGA (SCAN) functionals. We denote these results as DFA@FLOSIC-DFA. For example,
the SCAN@FLOSIC-SCAN is the result obtained by using
the self-consistent FLOSIC-SCAN electron density to evaluate the SCAN total energy.
The DFA@FLOSIC-DFA with LDA, PBE, and SCAN produces atomic total energies that are very close to the self-consistent total energies of the respective DFA
as shown in Figs. \ref{ldas}--\ref{scans}.
For completeness, we also tested the FLOSIC-LDA and FLOSIC-PBE densities in SCAN. The
SCAN@FLOSIC-SCAN, SCAN@FLOSIC-PBE and SCAN@FLOSIC-LDA energies
are very close, indicating that the respective FLOSIC densities are similar.
Note that these DFA@FLOSIC-DFA results are obtained at no additional computational cost
beyond that of the FLOSIC calculations.
We also calculated the ionization potentials (IPs) for H--Kr atoms with FLOSIC applied to the LDA, PBE, and SCAN functionals.
The FOD optimization of cations is performed independently, and the resulting cation total energy $E_\text{cat}$ is then used to calculate the IP as
\begin{equation}
E_\text{IP} = E_\text{cat} - E_\text{neut}.
\end{equation}
The results from
FLOSIC-LDA, FLOSIC-PBE, and FLOSIC-SCAN calculations
are summarized in Table \ref{errordeltascf}, and the energy differences from corresponding experimental energies \cite{NIST_ASD} are shown in Fig. \ref{figureip}.
FLOSIC-LDA tends to overestimate the IPs with a few exceptions.
On the other hand, FLOSIC-PBE and FLOSIC-SCAN energies underestimate the experimental values.
The mean absolute percentage errors (MAPE) in ionization energies are $7.68$, $5.13$, and $5.18$ \%
for LDA, PBE, and SCAN, respectively. The MAPE values in IP are $5.01$, $5.04$, and $3.30$ \%
for FLOSIC-LDA, FLOSIC-PBE, and FLOSIC-SCAN respectively. For all three functionals, the values of IP are reduced overall with SIC compared to without. The IPs of PBE and SCAN are over-corrected with SIC.
This is seen in the sign of mean errors (ME); with SIC, the ME in IP changes from $0.342$ to $-0.230$ eV for PBE and from $0.277$ to $-0.278$ eV for SCAN. In terms of mean absolute errors (MAE), FLOSIC improves the MAE for LSDA, from $0.619$ to $0.402$ eV, but increases it for PBE, from $0.397$ to $0.468$ eV. The MAE is improved from $0.398$ to $0.299$ eV for SCAN and FLOSIC-SCAN. The results for LSDA and PBE are consistent with those of Vydrov and Scuseria \cite{doi:10.1063/1.1897378}. By comparing the SIC energy corrections for the neutral atoms and their cations, we observe that
the overcorrection of IPs with semilocal functionals occurs because the neutrals have a larger positive correction than the cations, in most cases.
We point out that the optimization of the FOD at the level of the meta-GGA is important. We compared our FLOSIC-SCAN results with those calculated using descriptors optimized with FLOSIC-LDA. We find that the IPs with FLOSIC-SCAN
show a sizable improvement after performing FOD optimization. The MAE using FLOSIC-LDA optimized FODs is $0.448$ eV; this decreases to
$0.299$ eV upon FLOSIC-SCAN optimization. This reduction comes about in part by improving the Co IP. Using FLOSIC-LDA FODs, the error for Co is $-5.082$ eV; using FLOSIC-SCAN FODs, the error drops to $-0.137$ eV.
This points to the importance of optimizing the FODs with a consistent functional.
Similarly to what we have done for the total energy of atoms, we performed DFA@FLOSIC-DFA calculations for the IP.
PBE@FLOSIC-PBE gives MAPE of $4.91$ \%, which is a smaller error than both PBE and FLOSIC-PBE. For SCAN@FLOSIC-SCAN (MAPE = $5.28$\%), we do not see a performance improvement compared to SCAN ($5.18$\%) or FLOSIC-SCAN ($3.30$\%.)
Finally, the electron affinities (EA) of the atoms were computed by taking the difference $EA = E_\text{neut} - E_\text{anion}$.
For the anion calculations, we added additional single Gaussian orbitals (s, p, and d-type) to the default NRLMOL basis set to account for the more diffuse nature of the anion wave functions. These extra orbitals share the same Gaussian exponents that are obtained using the relation $\beta(N+1) = \beta(N)^2/\beta(N-1)$ where $\beta(N)$ is the N-th Gaussian exponent in the basis.
We computed EAs for H, Li, B, C, O, F, Na, Al, Si, P, S, Cl, K, Ti, Cu, Ga, Ge, As, Se, and Br, for which experimental EA values are available in Ref. \cite{NIST_CCCBD}.
In all the DFA anion calculations, the orbital eigenvalue of the highest occupied orbital becomes positive due to SIE \cite{PhysRevB.23.5048}, implying that the fully charged anions are not truly bound in DFA. Despite this, we adopt the common practice of computing EA values by taking total energy difference of an atom and its anion via $\Delta$-SCF. These are listed in Table \ref{tabea} and are comparable to
those reported by Vydrov and Scuseria \cite{doi:10.1063/1.2176608}. The application of SIC results in negative HOMO orbital energies, due to the improved description of the exchange potential in the asymptotic region. FLOSIC-PBE and FLOSIC-SCAN generally underestimate the EAs as seen from ME and MAE as well as in Fig. \ref{fig:flosicsa}. Overall, the performance of FLOSIC-LSDA is the
best among the three FLOSIC-DFAs.
DFA@FLOSIC-DFA calculations were also performed for EA similarly to the IP calculations.
For all three functionals, the errors with respect to experimental values are noticeably reduced compared to the pure DFA calculations (cf. Table III). This suggests that density-driven errors may be particularly important in describing the EA.
\subsection{Atomization energies}\label{atomizationenergymol}
FLOSIC-LDA, -PBE, and -SCAN are also used to calculate the total and atomization energies (AE) of a set of
$37$ molecules. This supplements the FLOSIC-SCAN results that appeared recently \cite{doi:10.1063/1.5087065}.
Most of the molecules are taken from the G2/97 test set \cite{doi:10.1063/1.460205}; in addition, we include the six molecules from the AE6 test set \cite{doi:10.1021/jp035287b}, as well as HBr, LiBr, NaBr, FBr, Br$_2$, and cyclopentadienyl. Most of the geometries for these molecules were optimized using B3LYP with the 6-31G(2df,p) basis \cite{NIST_CCCBD}.
The geometries for
O$_2$, CO, CO$_2$, C$_2$H$_2$, Li$_2$, CH$_4$, NH$_3$, and H$_2$O
were optimized using the PBE functional and the default NRLMOL basis set.
FOD positions were initially optimized using FLOSIC-LDA and further optimized for FLOSIC-SCAN.
The atomization energy of a molecule is defined as
\begin{equation}\label{eqnatomization}
E_\text{a}= \sum_i^{N_\text{atom}} E_i - E_\text{mol} > 0
\end{equation}
where $E_i$ is the energy of individual atoms, $N_\text{atom}$ is the number of atoms in the given molecule, and $E_\text{mol}$ is the total energy of a molecule.
Table \ref{errorAtomization} summarizes the errors in calculated AEs for DFA only, FLOSIC-DFA, and DFA@FLOSIC-DFA calculations.
The experimental energies are taken from Ref. \cite{NIST_CCCBD}.
The MAEs are $99.0$, $65.7$, $196.0$, $84.3$, and $73.7$ kJ/mol
for PBE, SCAN, FLOSIC-LDA, FLOSIC-PBE, and FLOSIC-SCAN respectively.
At the DFA level, SCAN performs much better than PBE resulting in the smallest MAE of 65.7 kJ/mol and MAPE of 5.22 \% among all five cases.
On the other hand, FLOSIC-PBE and FLOSIC-SCAN results are generally worse than those of their parent functionals.
We find that FLOSIC-LDA performs the worst of the above five cases with overestimated AE for many systems and especially for Br$_2$ for which the MAPE is
13.42 \%.
FLOSIC-PBE and FLOSIC-SCAN atomization energies have similar MAEs and MAPEs.
It is interesting to note that for FLOSIC-SCAN, the
MAE is 94.5 kJ/mol using LDA-optimized FODs and it improves to 73.7 kJ/mol after FOD optimization in FLOSIC-SCAN, indicating again that it is important to optimize FODs at a consistent level of theory.
Application of SIC generally results in an underestimation of the AEs compared to uncorrected DFA calculations (see Fig. \ref{figatomization}). This is similar to results seen previously for semilocal functionals \cite{doi:10.1063/1.2176608}. In the FLOSIC calculations with semilocal functionals, we observe that SIC treatment raises the total energies of the molecules more than it raises the combined total energies of separated atoms with a few exceptions.
This observation was also noted by Shahi \textit{et al.} for real localized SIC orbitals \cite{doi:10.1063/1.5087065}. Consequently, the SIC treatment lowers atomization energies according to Eq. (\ref{eqnatomization}).
We find that DFA@FLOSIC-DFA improves atomization energies with respect to \textit{both} the parent DFA and FLOSIC-DFA calculations.
The MAPE in AE for PBE is $8.64\%$ while that for FLOSIC-PBE is $9.67\%$. The MAPE for PBE@FLOSIC-PBE, on the other hand, is considerably smaller $7.72\%$. Similar improvement is also observed for SCAN. The MAPE of SCAN@FLOSIC-SCAN (5.05\%) is smaller than both the FLOSIC-SCAN ($10.24$ \%) and SCAN ($5.22$ \%).
\subsection{Dissociation energies}\label{sec:sie11}
We use SIE11 and SIE4$\times$4 test sets \cite{doi:10.1021/ct900489g,C7CP04913G}, sets of benchmark reactions that are known to be sensitive to self-interaction errors,
to investigate the performance of FLOSIC-SCAN on the dissociation energy calculations. The SIE11 test set consists of 11 systems that are directly affected from SIE.
The SIE4$\times$4 set consists of 4 positively charged dimers
(H$_2^+$, He$_2^+$, (NH$_3$)$_2^+$, and (H$_2$O)$_2^+$) separated at four different distances $R$ from the equilibrium distances $R_e$ ($R/R_e$=1.0, 1.25, 1.5, and 1.75); this set is designed to capture the effects of pure one-electron SIE.
Previously, Sharkas \textit{et al.} studied both SIE11 and SIE4$\times$4 with FLOSIC-LDA and FLOSIC-PBE and found that removal of self-interaction improves the performance in both case \cite{doi:10.1021/acs.jpca.8b09940}.
The dissociation energy is given as the difference of the complex total energy $E(X)$ and the fragments $E(X^+)$ and $E(X_2^+)$ as
\begin{equation}
E_D = E(X) + E(X^+) - E(X_2^+).
\end{equation}
The results are compared against the reference values in Ref. \cite{doi:10.1021/ct900489g} and are shown in Table \ref{tableSIE}.
For LDA and PBE, we find MAE decreases from DFA to FLOSIC. The DFA calculations overestimate the total energies of both complexes and fragments, and it leads to large errors in the dissociation energies. FLOSIC is able to correct the total energies and improves errors in dissociation. This is expected since a removal of SIE should improve the results.
SCAN has relatively small self-interaction compared to other functionals, and DFA-SCAN shows smaller MAE in SIE11 ($10.4$ kcal/mol) than that for FLOSIC-LDA ($11.7$ kcal/mol).
In those data sets, the SIC treatment improves the performance of SCAN.
We find that FLOSIC-SCAN (MAE $= 5.7$ kcal/mol for SIE11 and $2.2$ kcal/mol for SIE4$\times$4) performs very well among the three functionals under both DFA and FLOSIC.
The SIE11 set is divided into five positively charged cationic and six neutral systems.
DFA@FLOSIC-DFA calculations improve the errors for the neutral systems. This implies that those neutral systems are susceptible to density driven errors.
For the SIE11 cationic systems, on the other hand, the MAEs of DFA@FLOSIC-DFA fall between FLOSIC and DFA indicating that full SIC treatment is needed.
We observed the similar results for SIE4$\times$4 where full SIC is required as this dataset contains strechted bonds.
\subsection{Eigenvalues of the highest occupied orbitals}\label{evaluehoorbitals}
In exact DFT, the negative of
the highest occupied eigenvalue equals the first ionization energy of the system
\cite{PhysRevA.30.2745, PhysRevB.60.4545}.
This property has been widely used to adjust the magnitude of the exchange
potential or exact exchange potentials in practical DFT calculations \cite{PhysRevLett.105.266802, doi:10.1021/ct5000617}.
In Fig. \ref{figureeho} we compare the SCAN and FLOSIC-SCAN HOMO orbital eigenvalues $\varepsilon_{HO}$ of atoms $Z=1-36$ against experimental electron removal energies. We also include the corresponding results for LDA, PBE, FLOSIC-LDA, and FLOSIC-PBE for comparison.
Table \ref{errorEhomo} shows that the MAEs of DFA orbital eigenvalues are $4.06$, $4.15$, and $3.88$ eV
for LDA, PBE, and SCAN respectively, and MAE of FLOSIC-DFA eigenvalues are $0.67$, $0.59$, and $0.61$ eV
in the same order.
Although the size of the errors of the HOMO eigenvalues is similar to the errors in IP calculated using total energy differences, the corrections to
the HOMO
eigenvalues are much larger.
The DFA HOMO eigenvalues significantly underestimate the electron removal energies for all three functionals. FLOSIC corrects
this and reduces the MAE by a factor of 6 to 7.
Similar improvement in the eigenvalues of the HOMO is also seen for the set of molecules studied here (Fig. \ref{figureehomol}). As with the atoms, the HOMOs for the molecules are too high, understimating electron removal energies. In all cases, the HOMO eigenvalues are significantly lowered resulting in overestimated ionization potentials with FLOSIC.
Eliminating self-interaction error
improves the description of the potential seen by the electrons in the asymptotic region. This accounts for the significant improvement
in the eigenvalue of the highest occupied orbitals as can be seen from Tables \ref{errorEhomo} and \ref{errorEhomoMol}.
As HOMO eigenvalue is related to the asymptotic decay of the electron density
\cite{PhysRevA.32.2010, PhysRevA.30.2745},
it is reasonable to expect that the FLOSIC electron density is more accurate in the valence region than the corresponding uncorrected DFA density.
\section{Conclusion}
We implemented meta-GGA functionals in the FLOSIC code and compared the performance of FLOSIC-SCAN to that of FLOSIC-LDA and FLOSIC-PBE calculations for a variety of properties. Total energies of atoms from H--Kr are obtained.
We find that SCAN performs well in the total energy calculations, however, correcting for self-interaction errors using
FLOSIC worsens the total energies. As also has been noted in a few earlier PZ-SIC works,
the application of the FLOSIC method deteriorates the total energies and atomization energies where self-interaction
errors are small. Only in the case of LDA, the removal of self-interaction errors improves the results
over the parent DFA functional. For ionization potentials, FLOSIC improves ionization potentials
for LDA but worsens them for PBE and SCAN.
A pragmatic solution to obtain meaningful estimates of the atomization and total energies is to
compute these quantities using
the self-consistent self-interaction corrected electron density and Kohn-Sham orbitals in the parent functional.
This peturbative procedure
does not require any additional computational effort beyond the FLOSIC calculation.
Our results show that the
total energies, atomization energies, electron affinities and ionization energies (using $\Delta$-SCF) obtained using
such a procedure are of comparable quality as of their parent functionals while keeping the benefits from SIC such as physically accurate electron densities and improved occupied orbital eigenvalues.
For the SCAN functional, we saw some improvement over DFA-SCAN in total and atomization energies as judged from
MAEs of these quantities.
The procedure adopted here
is similar to that used in removing delocalization errors (density driven errors) in
the literature \cite{VERMA201210} and is expected to be more accurate for ionization potentials and electron
affinities for larger systems. The present work shows that FLOSIC calculations can provide accurate estimates of the near equilibirum properties (e.g. total and atomic energies)
where SIE are small by employing DFA@FLOSIC-DFA approach while providing
accurate description of properties like dissociation energies (using full FLOSIC-DFA)
where SIC errors are large. Alternative approaches to rectify the overcorrection of the PZ-SIC/FLOSIC methods are being pursued in our laboratory.
\newpage
\section*{Figures}
\begin{figure}[!htb]
\centering
\includegraphics[width=\columnwidth]{Fig1.eps}
\caption{A plot of $f_x (\alpha)$ (Eq. \ref{eqn_fx}) and ${df_x(\alpha)}/{d\alpha}$ used in the SCAN exchange enhancement factor. A large oscillation of ${df_x(\alpha)}/{d\alpha}$ is seen near $\alpha=1$. \label{scanmesh}}
\end{figure}
\newpage
\newpage
\begin{figure}[!htb]
\centering
\includegraphics[width=\columnwidth]{Fig2.eps}
\caption{Atomic total energies (in Ha) for LDA (black circles), FLOSIC-LDA (red squares), and LDA@FLOSIC-LDA (blue diamonds), compared against the reference values of Ref. \cite{PhysRevA.47.3649}. $(E - E_\text{Ref})/N_{e}$ is shown, where $N_e$ is the number of electrons.
\label{ldas}}
\end{figure}
\newpage
\begin{figure}[!htb]
\centering
\includegraphics[width=\columnwidth]{Fig3.eps}
\caption{Atomic total energies (in Ha) for PBE (black circles), FLOSIC-PBE (red squares), and PBE@FLOSIC-PBE (blue diamonds), compared against the reference values of Ref. \cite{PhysRevA.47.3649}. $(E - E_\text{Ref})/N_{e}$ is shown, where $N_e$ is the number of electrons.
\label{pbes}}
\end{figure}
\newpage
\begin{figure}[!htb]
\centering
\includegraphics[width=\columnwidth]{Fig4.eps}
\caption{Atomic total energies (in Ha) for SCAN (black circles), FLOSIC-SCAN (red squares), and SCAN@FLOSIC-SCAN (blue diamonds) compared against the reference values of Ref. \cite{PhysRevA.47.3649}. $(E - E_\text{Ref})/N_{e}$ is shown, where $N_e$ is the number of electrons.
\label{scans}}
\end{figure}
\newpage
\begin{figure}[!htb]
\centering
\includegraphics[width=\columnwidth]{Fig5.eps}
\caption{Ionization energies (in eV) of atoms computed using FLOSIC-LDA (blue circles), FLOSIC-PBE (red squares), and FLOSIC-SCAN (green diamonds).
The energies are obtained by $\Delta$-SCF and compared against the experimental values of Ref. \cite{NIST_ASD}.}
\label{figureip}
\end{figure}
\newpage
\begin{figure}[!htb]
\centering
\includegraphics[width=\columnwidth]{Fig6.eps}
\caption{\label{fig:flosicsa} Electron affinities (in eV) of 20 atoms computed using LDA (blue circles), PBE (red squares), SCAN(green diamonds), FLOSIC-LDA (black triangles), FLOSIC-PBE (magenta crosses), and FLOSIC-SCAN (green xs). The energies are obtained by $\Delta$-SCF and compared against the experimental values of Ref. \cite{NIST_CCCBD}.}
\end{figure}
\newpage
\begin{figure}[!htb]
\centering
\includegraphics[width=0.7\columnwidth]{Fig7-1.eps}\\
\vspace{10mm}
\includegraphics[width=0.7\columnwidth]{Fig7-2.eps}\\
\vspace{10mm}
\includegraphics[width=0.7\columnwidth]{Fig7-3.eps}\\
\caption{Atomization energies of molecules compared against reference experimental values found in Ref. \cite{NIST_CCCBD}. $(E - E_\text{Ref})/E_\text{Ref}$ is shown:
(a) DFA, (b) FLOSIC, and (c) DFA@FLOSIC-DFA.
\label{figatomization}}
\end{figure}
\newpage
\begin{figure}[!htb]
\centering
\includegraphics[width=\columnwidth]{Fig8.eps}
\caption{Deviation of $-\varepsilon_{HO}$ from the corresponding experimental ionization potential\cite{NIST_ASD} (in eV) for atoms with $Z = 1 - 36$. LDA (blue circles), PBE (red squares), SCAN (green diamonds), FLOSIC-LDA (filled blue circles), FLOSIC-PBE (filled red squares), and FLOSIC-SCAN (filled green diamonds) values are shown.}
\label{figureeho}
\end{figure}
\newpage
\begin{figure}[!htb]
\centering
\includegraphics[width=\columnwidth]{Fig9.eps}
\caption{Deviation of $-\varepsilon_{HO}$ from the corresponding experimental ionization potential (in eV) for a test set of molecules. The experimental values are from Ref. \cite{NIST_webbook} and Ref. \cite{HuberHerzberg1979}.
PBE (red squares), SCAN (green diamonds), FLOSIC-LDA (filled blue circles), FLOSIC-PBE (filled red squares), and FLOSIC-SCAN (filled green diamonds) values are shown.}
\label{figureehomol}
\end{figure}
\newpage
\section*{Tables}
\begin{table}[!htb]
\begin{center}
\caption{Mean absolute error (MAE in Ha) of the total energies of atoms with $Z=1-18$ calculated with various methods when compared against reference values given in Ref. \cite{PhysRevA.47.3649}.}
\label{flosicstable}
\begin{ruledtabular}
\begin{tabular}{|c|c|}
Method & MAE (Ha) \\\hline
LDA & 0.726125 \\
FLOSIC-LDA & 0.380502 \\
LDA@FLOSIC-LDA & 0.734249 \\\hline
PBE & 0.082958 \\
FLOSIC-PBE & 0.159131 \\
PBE@FLOSIC-PBE & 0.089404 \\\hline
SCAN & 0.019197 \\
FLOSIC-SCAN & 0.147113 \\
SCAN@FLOSIC-SCAN & 0.017547 \\
\end{tabular}
\end{ruledtabular}
\end{center}
\end{table}
\begin{table}[!htb]
\begin{center}
\caption{Deviation of calculated ($\Delta$-SCF) ionization potentials from experimental values for atoms $Z=2-36$ for several methods. Mean errors (ME, in eV), mean absolute errors (MAE, in eV), and mean absolute percentage errors (MAPE) are shown.
}
\label{errordeltascf}
\begin{ruledtabular}
\begin{tabular}{|c|c|c|c|}
Method & ME & MAE & MAPE \\
& (eV) & (eV) & (\%) \\\hline
LDA & 0.586 & 0.619 & 7.68 \\
FLOSIC-LDA & 0.214 & 0.402 & 5.01 \\
LDA@FLOSIC-LDA & 0.482 & 0.521 & 6.45 \\\hline
PBE & 0.342 & 0.397 & 5.13 \\
FLOSIC-PBE & -0.230 & 0.468 & 5.04 \\
PBE@FLOSIC-PBE & 0.272 & 0.372 & 4.91 \\\hline
SCAN & 0.277 & 0.398 & 5.18 \\
FLOSIC-SCAN (LDA FOD) & -0.278 & 0.448 & 5.17 \\
FLOSIC-SCAN (Optimized FOD)& -0.123 & 0.299 & 3.30 \\
SCAN@FLOSIC-SCAN (LDA FOD) & 0.244 & 0.402 & 5.28 \\%0.40237 \\
SCAN@FLOSIC-SCAN (Optimized FOD)& 0.241 & 0.402 & 5.28 \\%0.40206 \\
\end{tabular}
\end{ruledtabular}
\end{center}
\end{table}
\newpage
\begin{table}[!htb]
\begin{center}
\caption{\label{tabea}%
Electron affinities of 20 atoms calculated with various methods and compared to experimental values \cite{NIST_CCCBD}. Mean error (ME) and mean absolute error (MAE) are shown, both in eV.}
\begin{ruledtabular}
\begin{tabular}{|l|c|c|}
\textrm{Method}&
\textrm{ME }&
\textrm{MAE}\\
\colrule
LDA & 0.359 & 0.362 \\%& 85.66 \\
FLOSIC-LDA & -0.133 & 0.189 \\%& 89.36 \\
LDA@FLOSIC-LDA & 0.227 & 0.231 \\ \hline
PBE & 0.159 & 0.172 \\%& 59.01 \\
FLOSIC-PBE & -0.531 & 0.531 \\%& 158.13 \\
PBE@FLOSIC-PBE & 0.038 & 0.080 \\ \hline
SCAN & 0.093 & 0.148 \\%& 61.01 \\
FLOSIC-SCAN & -0.341 & 0.341 \\%& \\
SCAN@FLOSIC-SCAN & 0.031 & 0.126 \\
\end{tabular}
\end{ruledtabular}
\end{center}
\end{table}
\begin{table}[!htb]
\begin{center}
\caption{Atomization energies for the test set of molecules featured in Fig. \ref{figatomization}. Mean absolute errors (MAE, in kJ/mol), mean percentage errors (MPE), mean absolute percentage errors (MAPE), and root mean square errors (RMS, in kJ/mol) are shown.
}
\label{errorAtomization}
\begin{ruledtabular}
\begin{tabular}{|c|c|c|c|c|}
Method & MAE & MPE & MAPE & RMS \\
& (kJ/mol) & (\%) & (\%) & (kJ/mol)\\\hline
FLOSIC-LDA & 195.95 & 11.93 & 13.42 & 321.16 \\
LDA@FLOSIC-LDA & 267.41 & 22.78 & 23.00 & 381.49 \\\hline
PBE & 98.99 & 7.24 & 8.64 & 146.48 \\
FLOSIC-PBE & 84.30 & -4.81 & 9.67 & 114.21 \\
PBE@FLOSIC-PBE & 88.85 & 5.93 & 7.72 & 133.27 \\\hline
SCAN & 65.69 & 3.01 & 5.22 & 102.42 \\
FLOSIC-SCAN (LDA FOD) & 94.50 & -4.84 & 10.45 & 131.78 \\
FLOSIC-SCAN (Optimized FOD) & 73.72 & -6.78 & 10.24 & 97.83 \\
SCAN@FLOSIC-SCAN (LDA FOD) & 63.38 & 2.31 & 5.10 & 98.82 \\
SCAN@FLOSIC-SCAN (Optimized FOD)& 62.84 & 2.35 & 5.05 & 97.87 \\
\end{tabular}
\end{ruledtabular}
\end{center}
\end{table}
\newpage
\begin{table}[!htb]
\begin{center}
\caption{\label{tableSIE}SIE11 and SIE4$\times$4 dissociation energies calculated by various methods and compared to reference values from Ref. \cite{doi:10.1021/ct900489g}. Mean absolute errors (MAE, in kcal/mol) of SIE11 (5 cationic, 6 neutral, and 11 combined systems) and SIE4$\times$4 are shown.}
\begin{ruledtabular}
\begin{tabular}{|l|c|c|c|c|}
Method & SIE11, 5 cationic & SIE11, 6 neutral & SIE11 & SIE4$\times$4 \\\hline
LDA & 22.9 & 13.4 & 17.8 & 27.5 \\
FLOSIC-LDA & 14.8 & 9.0 & 11.7 & 3.0 \\
LDA@FLOSIC-LDA & 20.1 & 8.9 & 14.1 & 21.2 \\ \hline
PBE & 12.7 & 10.9 & 12.1 & 23.3 \\
FLOSIC-PBE & 8.9 & 6.4 & 7.5 & 3.4 \\
PBE@FLOSIC-PBE & 9.6 & 4.5 & 7.2 & 15.1 \\ \hline
SCAN & 10.4 & 9.9 & 10.4 & 17.9 \\
FLOSIC-SCAN & 5.1 & 6.2 & 5.7 & 2.2 \\
SCAN@FLOSIC-SCAN & 8.8 & 4.9 & 6.9 & 12.4 \\
\end{tabular}
\end{ruledtabular}
\end{center}
\end{table}
\begin{table}[!htb]
\begin{center}
\caption{Deviation of $-\varepsilon_{HO}$ from the corresponding experimental ionization potential for atoms with $Z = 1-36$. Mean errors (ME) and mean absolute errors (MAE) are given in eV.}
\label{errorEhomo}
\begin{ruledtabular}
\begin{tabular}{|c|c|c|}
Method & ME & MAE \\\hline
LDA & -4.059 & 4.059 \\
PBE & -4.150 & 4.150 \\
SCAN & -3.880 & 3.880 \\\hline
FLOSIC-LDA & 0.494 & 0.672 \\
FLOSIC-PBE & 0.189 & 0.590 \\
FLOSIC-SCAN (LDA FODs) & 0.314 & 0.622 \\
FLOSIC-SCAN (FOD optimized)& 0.318 & 0.606 \\
\end{tabular}
\end{ruledtabular}
\end{center}
\end{table}
\begin{table}[!htb]
\begin{center}
\caption{Deviation of $-\varepsilon_{HO}$ from the corresponding experimental ionization potential for the set of molecules featured in Fig. \ref{figureehomol}. Mean errors (ME) and mean absolute errors (MAE) are given in eV.}
\label{errorEhomoMol}
\begin{ruledtabular}
\begin{tabular}{|c|c|c|}
Method & ME & MAE \\\hline
PBE & -4.023 & 4.023 \\
SCAN & -3.699 & 3.699 \\\hline
FLOSIC-LDA & 2.104 & 2.104 \\
FLOSIC-PBE & 1.658 & 1.667 \\
FLOSIC-SCAN (LDA FODs) & 1.790 & 1.790 \\
FLOSIC-SCAN (Optimized FOD)& 1.762 & 1.762 \\
\end{tabular}
\end{ruledtabular}
\end{center}
\end{table}
\newpage
\vfill
\section*{Supplementary material}
See supplementary material for detailed results of the total energies, IP, EA, Atomization energies, and $-\varepsilon_{HO}$ for the systems studied in this manuscript and detailed results for SIE11, SIE4$\times$4, and BH6 molecular test sets.
\begin{acknowledgments}
The authors gratefully acknowledge discussions with Profs. Mark R. Pederson, John Perdew, Jianwei Sun, and Dr. Jorge Vargas.
The initial phase of the work (implementation of meta-GGAs) was supported by the Office of Basic Energy Sciences, U.S. Department of Energy DE-SC0002168 and DE-SC0006818 while the applications using FLOSIC are supported by DE-SC0018331 as a part of the Computational Chemical Sciences program.
This research used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility operated under Contract No. DE-AC02-05CH11231.
\end{acknowledgments}
\clearpage
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,260
|
\section{Approximation lemmas}\label{aplem}
In the proofs of our recursive formulas we use the following lemmas. For ease of reference, we give two separate formulations, but clearly the first lemma follows from the second.
\begin{lemma}\label{approx1}
Suppose that $f$ and $g$ are two holomorphic functions on a smooth compact analytic surface $S$ with boundary. Suppose that they have the same divisor $D$, which is transverse to the boundary. (We do not assume that $D$ is reduced.) Suppose that the unit $u=f/g$ always has positive real part. Then, for sufficiently small $\sigma$, the fibration over the circle $|\epsilon|=\sigma$ with fibers $f=\epsilon$ is smoothly isotopic to the fibration with fibers $g=\epsilon$.
\end{lemma}
\begin{lemma}\label{approx2}
Over a circle $|x|=\rho$, let $S$ be the total space of a continuous family of smooth compact analytic surfaces $S_x$ with boundary. Suppose that $f$ and $g$ are two continuous functions such that, for each $x$, their restrictions $f_x$ and $g_x$ are holomorphic functions on $S_x$ having the same divisor $D_x$. Suppose that each $D_x$ is transverse to the boundary. Suppose that the unit $u=f/g$ always has positive real part. Then, for sufficiently small $\sigma$, the fibration over the torus $|x|=\rho, |\epsilon|=\sigma$ with fibers $f_x=\epsilon$ is isotopic to the fibration with fibers $g_x=\epsilon$.
\end{lemma}
\begin{proof}
Let $D$ be the union of the divisors $D_x$. We argue that in a punctured neighborhood of $D$, the interpolation $F_t = tf + (1-t)g$ (with $0 \leq t \leq 1$) has a non-vanishing gradient (as does its restriction to the boundary). Then by the Ehresmann fibration theorem, $F_t$ provides a locally trivial fibration.
\par
There is a neighborhood of $D$ on which, away from $D$ itself, the relative gradient $\nabla g$ does not vanish. Indeed, let $V$ be the variety on which $\nabla g$ vanishes. Then $g$ must be constant on each component of $V$, and each such component either misses $D$ or is completely contained within it. Similarly, we claim that there is a (punctured) neighborhood of $D$ on which $\nabla f$ is never a negative multiple of $\nabla g$. To see this, consider the variety $V$ on which the two gradients are linearly dependent; note that $D$ is contained in $V$. Then the quotient $\lambda = \nabla f/ \nabla g$ is a well-defined analytic function on $V$ at least away from $D$. Suppose we have a map $\gamma: (C, p) \rightarrow V$ from a nonsingular curve germ, with $\gamma (p) \in D$. Then on $C$ we have
$$\lambda = f'/g' = u + \frac{g}{g'}u'.$$
The quotient $g/g'$ has a removable singularity at $p$ and vanishes there. Thus $\lambda(p) = u(p)$. Since the curve $C$ is arbitrary, this shows that $\lambda$ is well-defined on $D$ and agrees with $u$ there. Thus there is a neighborhood of $V$ in which the real part of $\lambda$ cannot be negative; in the punctured neighborhood $\nabla F_t$ does not vanish.
\par
Finally, since each $D_x$ is transverse to the boundary, we can find a local trivialization of a neighborhood of $D_x \cap \partial S$ in $\partial S$, with fibers isomorphic to the complex disk. Then a similar argument as above applies to $f$ and $g$ restricted to the boundary.
\end{proof}
\section{Plane curves}\label{plcur}
Consider a germ at the origin of an irreducible analytic plane curve defined by $f(y,z)=0$; we will simply call it a ``curve.'' (For basic notions and facts about singular plane curves see \cite{BK} or \cite{Wall}.)
The {\em Milnor fiber} $F$ is the set of points $(y,z)$ obtained by the following process:
\begin{itemize}
\item[(1)]
requiring that $\|(y,z)\|\leq\delta$, a sufficiently small radius,
\item[(2)]
then requiring that $f(y,z)=\epsilon$, a number sufficiently close to zero.
\end{itemize}
The boundary of the Milnor fiber is a link in the sphere.
Letting $\epsilon$ vary over a circle centered at 0 we obtain the {\em Milnor fibration} (which we will also call the {\em horizontal fibration}).
Let $h_q:H_q(F;\mathbf{Q}) \to H_q(F;\mathbf{Q})$ be the monodromy operator.
The graded characteristic function
$$
\mathbf H(t)=\frac{\det(tI-h_0)}{\det(tI-h_1)}
$$
is called the {\em horizontal monodromy}. (In the literature it is sometimes called a {\em zeta function}.) Taking its degree computes the Euler characteristic $\chi$ of $F$.
\par
Assuming that the curve is not the axis $y=0$, there is a parametrization
$$
y=t^d, \quad z=\sum_{j} c_{j}t^{j},
$$
where the exponents are positive integers and all coefficients are nonzero. The integer $d$ (which we call the {\em degree}) is the number of sheets for the projection $\pi: (y,z) \mapsto y$, and over a slitted neighborhood of $0$ we may parametrize each sheet by
$$
\zeta=\sum_{j} c_{j}y^{j/d},
$$
having chosen one of the $d$ possible roots. We prefer to write this as follows:
\begin{equation}\label{puiseux}
\zeta=\sum c_{\mu}y^{\mu},
\end{equation}
where the sum is now over certain positive rational numbers with common denominator $d$ (arranged in increasing order); this is called the {\em Puiseux series} of the curve. One can recover $f$ by forming a product over all conjugates:
$$f(y,z) = \prod^d(z-\zeta).$$
(Note our notation for recording the number of conjugates.)
\par
An exponent of the Puiseux series is called {\em essential} (or {\em characteristic}) if its denominator does not divide the common denominator of the previous exponents. In particular (by the convention that the least common multiple of the empty set is 1) all integer exponents are inessential, but the first noninteger exponent is essential. Clearly there are only finitely many essential exponents $\mu_1 < \mu_2 < \dots < \mu_e$. The sum
\begin{equation}\label{prototype}
\sum_{i=1}^{e} y^{\mu_i}
\end{equation}
parametrizes the $d$ sheets of a singular curve which we call the {\em prototype}.
\begin{theorem} \label{protosame}
A curve and its prototype have the same horizontal monodromy.
\end{theorem}
For example, if there are no essential exponents then the curve is nonsingular at the origin, its prototype is $z=0$, and the horizontal monodromy is $t-1$.
We will prove Theorem \ref{protosame} by induction on $e$, at the same time that we prove a set of recursive formulas.
To this end, we define the {\em truncation} of a singular curve with prototype
$$\sum_{i=1}^{e}y^{\mu_i}$$
to be the curve with Puiseux series
$$\zeta_1=y^{\mu_1}=y^{n/m}$$
(where the second equation defines the relatively prime integers $m$ and $n$).
Its {\em derived curve} is the curve with Puiseux series
$$\zeta'=\sum_{i=1}^{e-1}y^{\mu'_i},$$
with the new exponents computed by
$$\mu'_i=m(\mu_{i+1}-\mu_1+n).$$
(An example is worked out at the end of this section.)
Let $d_1$ and $d'$ denote the degrees of the truncation and the derived curve, respectively. Similarly, let $\chi_1$ and $\chi'$ denote the Euler characteristics of their Milnor fibers; let $\mathbf H_1$ and $\mathbf H'$ denote their horizontal monodromies.
\begin{theorem}\label{curverecursion}
The degree, Euler characteristic, and horizontal monodromy are determined by these formulas.
\begin{enumerate}
\item
$d_1=m$
\item
$d=d_1 d'$
\item
$
\chi_1=m+n-mn
$
\item
$
\chi=d'(\chi_1-1)+\chi'
$
\item
$$
\mathbf H_1(t)=
\frac{(t^{m}-1)(t^{n}-1)}
{t^{mn}-1}
$$
\item
$$\mathbf H(t)=
\frac{\mathbf H_1(t^{d'})\cdot \mathbf H'(t)}
{t^{d'}-1}$$
\end{enumerate}
\end{theorem}
Before embarking on the proof, we describe its key idea. As is well known, one may obtain an embedded resolution of a curve singularity by a resolution process whose steps are dictated by the Puiseux exponents, and from such a resolution one can compute the monodromy by invoking a formula of A'Campo \cite{AC}. Our proof does not use this full process of resolution, but just the first step of it: the toric transformation prescribed by the leading exponent. In general the strict transform that we obtain is still highly singular. We strip away all of the exceptional divisors except for the sole divisor meeting the strict transform, called the ``rupture component.'' We then observe that the remaining configuration, consisting of the strict transform together with the rupture component, can be blown down in a certain way so as to obtain a new singular curve. This is the derived curve. Other authors have also used this idea of partial resolution, e.g. \cite{GLM}.
\begin{proof} As indicated, we will simultaneously provide an inductive proof of Theorem \ref{protosame} (inducting on the number of essential exponents) and a recursive proof of Theorem \ref{curverecursion}.
\par
The Milnor fiber of the truncation, which is defined by $z^m-y^n=\epsilon$, is projected by $\pi$ onto a neighborhood of $0$ on the $y$-line, with total ramification above the $n$th roots of $-\epsilon$. This neighborhood can be retracted onto the union $L$ of line segments from $0$ to these points, in such a way that there is a compatible retraction of the Milnor fiber onto $\pi^{-1}L$, which is the complete bigraph on the $n$ points $((-\epsilon)^{1/n},0)$ and the $m$ points $(0,\epsilon^{1/m})$. As $\epsilon$ goes around a circle, each set of points is cyclically permuted. Since $m$ and $n$ are relatively prime, the $mn$ edges of the graph are likewise cyclically permuted. Thus the odd-numbered formulas are confirmed.
\par
To verify the recursive formulas and to handle the inductive step in the proof of Theorem \ref{protosame}, suppose we are given a curve with Puiseux series (\ref{puiseux}) and prototype (\ref{prototype}).
We first replace
$$
\frac{z-\sum_{\mu\in\mathbf Z} c_{\mu}y^{\mu}}{c_{\mu_1}}.
$$
by $z$.
In the new coordinate system, the curve is defined by the vanishing of
$$
f=\prod^d\left(z-\left[y^{n/m}+\sum_{\mu>n/m} c_{\mu}y^{\mu}\right]\right),
$$
(where for simplicity the coefficients have been renamed).
The truncation is defined by the vanishing of
$$
f_1= \prod^m(z-y^{n/m})=z^m-y^n.
$$
Note that $m$ divides $d$, and that, as we vary the $d$th root of $y$, each value of $y^{1/m}$ occurs $d/m$ times.
Thus
\begin{equation}\label{compare}
\frac{f}{f_1^{d/m}}= \prod^d\left(1-\frac{\sum_{\mu>n/m} c_{\mu}y^{\mu}}{z-y^{n/m}}\right).
\end{equation}
\par
One can obtain an embedded resolution of the truncation by a sequence of blowups dictated by its exponent $\mu_1=n/m$ and the Euclidean algorithm. The total transform will consist of a chain of exceptional divisors occurring with certain multiplicities, together with a strict transform meeting just one such exceptional divisor, which we call the {\em rupture component}. Along this chain the function $z^m/y^n$ has no indeterminacy, and in fact except along the rupture component its value is either $0$ or $\infty$. In either case one immediately verifies that the value of (\ref{compare}) is 1.
\par
To work in a chart containing the rupture component, we use substitutions dictated by the matrix
$$ \left[ \begin{array}{cc}
m & n \\
r & s
\end{array} \right],$$
where $r$ and $s$ are the smallest positive integers for which the determinant is 1,
namely
\begin{gather*}
y=u^{m} v^{r} \\
z=u^{n} v^{s}.
\end{gather*}
We find that in this chart the total transform of the truncation is defined by the vanishing of
$$
f_1=u^{mn}v^{rn}(v-1),
$$
and its strict transform is defined by the vanishing of the last factor. Note that it meets the $v$-axis at the point
$(u,v)=(0,1)$.
The total transform of the given curve is defined by the vanishing of
$$
f=\prod^d\left(u^{n}v^{s}-\left[u^{n}v^{rn/m}+\sum_{\mu>n/m} c_{\mu}u^{m\mu}v^{r\mu}\right]\right)
$$
which may be rewritten as
\begin{equation}\label{rewritten}
f=u^{nd}v^{rnd/m}\prod^d\left(v^{1/m}-\left[1+\sum_{\mu>n/m} c_{\mu}u^{m\mu-n}v^{r(m\mu-n)/m}\right]\right).
\end{equation}
The strict transform is defined by the vanishing of the last $d$ factors, and again it meets the $v$-axis at $(0,1)$.
Note that
$$
\frac{f}{f_1^{d/m}}= \prod^d\left(1-\frac{\sum_{\mu>n/m} c_{\mu}u^{m\mu-n}v^{r(m\mu-n)/m}}{v^{1/m}-1}\right),
$$
which is indeterminate at $(0,1)$ but whose value elsewhere on the rupture component is 1.
\par
Introducing two new variables $y'$ and $w$, let $B$ denote a small ball $\|(y',w)\|\leq\delta'$ centered at the origin, and map it to a neighborhood $N$ of $(u,v)=(0,1)$ by letting $u=\frac{y'}{(w+1)^r}$ and $v=(w+1)^m$. Note that this map is nonsingular at the origin. When pulled back via this map, just one of the values $v^{1/m}$ becomes $w+1$. Thus $d/m$ of the factors at the end of (\ref{rewritten}) become
$$
w-\sum_{\mu>n/m} c_{\mu}(y')^{m\mu-n},
$$
whereas the remaining $d-d/m$ factors become units.
\par
We can regard the Milnor fiber of our original curve as a subset of the surface obtained by the sequence of blowups. Let us assume that the choices of $\delta$ and $\epsilon$ made in defining the Milnor fiber are made subsequent to the choice of $\delta'$. We claim that by choosing $\delta$ sufficiently small we can guarantee that the strict transform of the original curve germ lies entirely within $N$. Indeed,
we note that on the strict transform
\begin{equation*}
v^{1/m}=1+\sum_{\mu>n/m} c_{\mu}y^{\mu-n/m}
\end{equation*}
(for some choice of conjugate). Thus we can force $v$ to be arbitrarily close to 1 by choosing $\delta$ sufficiently small, and since $u^m=y/v^r$ we can likewise force $u$ arbitrarily close to 0.
Then by appropriate choice of $\epsilon$ we can arrange that the Milnor fiber of our curve is transverse to the boundary of $N$, and that its boundary lies completely within $N$. Our Milnor fiber is thus divided into two pieces. (See Figure \ref{twopieces}.)
\par
\begin{figure}
\scalebox{0.25}
{\includegraphics{twopieces.png}}
\caption{The Milnor fiber (the thickened curve) is divided into two pieces by the boundary of $N$ (indicated by a circle). The rupture component is horizontal, and another exceptional divisor is shown vertically. The strict transform enters from above.}
\label{twopieces}
\end{figure}
\par
Consider first the piece of the Milnor fiber lying outside of $N$. Having excluded the points of indeterminacy of $f/f_1^{d/m}$, we may apply the approximation lemma \ref{approx1} to conclude that the monodromy of $f$ is the same as the monodromy of $f_1^{d/m}$.
The Milnor fiber has $d/m$ connected components corresponding to all possible values of $\epsilon^{m/d}$, and each one is a copy of the Milnor fiber for $f_1$. Fixing one such value $\eta$, we see as above that the corresponding component can be contracted onto the complete bigraph on the $n$ points $((-\eta)^{1/n},0)$ and the $m$ points $(0,\eta^{1/m})$. As $\epsilon$ goes around a circle the values of $\epsilon^{m/d}$ are cyclically permuted; thus the components are likewise permuted. As $\epsilon$ goes around this circle $d/m$ times, however, each $\eta$ goes once around a circle. Thus the monodromy of this piece is $\mathbf H_1(t^{d/m})$.
\par
Now consider the piece of the Milnor fiber lying inside $N$. Note that it has two sorts of boundary components: the components of the original link $L$ and those components created by its intersection with the boundary sphere of $N$. To analyze it, we look at its inverse image in the ball $B$.
By the approximation lemma \ref{approx1}, we may ignore all unit factors in $f$. Thus we may assume that the function defining this piece of the Milnor fiber is
$$
(y')^{nd}\prod^{d/m}\left(w-\sum_{\mu>n/m} c_{\mu}(y')^{m\mu-n}\right).
$$
The map $(y',w) \mapsto (y',(y')^{nm}w)$ takes this piece to the Milnor fiber of the curve with Puiseux series
\begin{equation}\label{derivedseries}
\sum_{\mu>n/m} c_{\mu}(y')^{m\mu-n+nm},
\end{equation}
but it misses disks centered at the $d/m$ points $(0,\epsilon^{m/d})$. Note that these disks are cyclically permuted by the monodromy. In (\ref{derivedseries}) there are $e-1$ essential terms, whereas our original Puiseux series had $e$ essential terms. By the inductive hypothesis, the monodromy of this curve is the same as that of its prototype, which has Puiseux series
$$
\sum_{i=2}^{e}(y')^{m(\mu_{i}-\mu_1+n)};
$$
by reindexing we obtain the Puiseux series of the derived curve. Thus $d'=d/m$, confirming formula (2) of the theorem, and the monodromy of this piece of the Milnor fiber is
$$
\frac{\mathbf H'(t)}
{t^{d'}-1}.
$$
Combining this with our conclusion about the monodromy of the first piece, we obtain formula (6). Finally we obtain formula (4) by computing the degree of both sides of (6).
\end{proof}
Here is an example. Suppose we begin with the curve whose Puiseux series is
$$
\zeta=y^{3/2}+y^{7/4}+y^{11/6}.
$$
Then its truncation is parametrized by $\zeta_1=y^{3/2}$, and its derived curve is parametrized by
$$
\zeta'=y^{13/2}+y^{20/3}.
$$
Repeating this process, we obtain truncation $\zeta'_1=y^{13/2}$ and second derived curve
$$
\zeta''=y^{79/3}.
$$
By repeated use of the first two formulas in Theorem \ref{curverecursion}, we have $d=2d'=4d''=12$.
By formulas (3) and (4), the Euler characteristic of the Milnor fiber is
$$
\chi=d'(\chi_1-1)+d''(\chi'_1-1)+\chi''=6(-2)+3(-12)+(-155)=-203.
$$
By formulas (5) and (6), the horizontal monodromy is
$$
\mathbf H(t)=
\frac{\mathbf H_1(t^{d'})}{t^{d'}-1}
\cdot \frac{\mathbf H_1(t^{d''})}{t^{d''}-1}
\cdot \mathbf H''(t)
=\frac{(t^{12}-1)(t^{18}-1)(t^{39}-1)(t^{79}-1)}{(t^{36}-1)(t^{78}-1)(t^{237}-1)}.
$$
\section{Quasi-ordinary surfaces}\label{qos}
We now turn to quasi-ordinary surfaces, beginning with a compressed account of the essential facts and definitions. A reader seeking more information should consult \cite {Li0, Li1, Li2}.
\par
We suppose that $S$ is a germ at the origin of an irreducible analytic surface defined by the vanishing of a function $f(x,y,z)$.
The quasi-ordinary condition means that we can arrange a projection $\pi: (x,y,z) \mapsto (x,y)$ so that $\pi|_S$ has discriminant locus contained in the coordinate axes $xy = 0$. In particular $\pi|_S$ is a finite covering space map on the complement of the axes. It is known that $S$ has many curve-like properties. Foremost among them is the existence of a fractional-exponent power series
\begin{equation}\label{powseries}
\zeta(x,y) = \sum c_{\lambda\mu} x^{\lambda} y^{\mu}
\end{equation}
which parametrizes $S$ via $(x,y) \mapsto (x,y, \zeta(x,y))$, where we vary the conjugate of $\zeta$ so as to obtain the various sheets of the cover. The exponents can all be taken to have a common denominator, and we write only those terms in which $c_{\lambda\mu}\neq 0$. One can recover $f$ by forming a product over all conjugates:
$$
f(x,y,z) = \prod^d(z-\zeta(x,y)).
$$
(Here $d$ denotes the number of conjugates and thus the number of sheets.)
\par
Define an ordering on pairs of exponents as follows: we say that $(\lambda,\mu)<(\lambda^{*},\mu^{*})$ if $\lambda\leq\lambda^{*}$, $\mu\leq\mu^{*}$, and they are not the same pair. The restriction on the discriminant locus implies that among the exponent pairs of (\ref{powseries}) we may find a finite sequence of {\em characteristic pairs}
\begin{equation}\label{cpairs}
(\lambda_1,\mu_1)<(\lambda_2,\mu_2)<\cdots<(\lambda_e,\mu_e)
\end{equation}
with these properties:
\begin{enumerate}
\item
Each $(\lambda_i,\mu_i)$ is not contained in the subgroup of $\mathbf Q\times\mathbf Q$ generated by $\mathbf Z\times\mathbf Z$ and by the previous characteristic pairs.
\item
If $(\lambda,\mu)$ is a noncharacteristic pair, then it is contained in the subgroup generated by those characteristic pairs for which $(\lambda_i,\mu_i)<(\lambda,\mu)$.
\end{enumerate}
\par
We say that $S$ is {\em reduced} (as a quasi-ordinary surface) if $\mu_1 \neq 0$. In this case, one immediately verifies that the singular locus of $S$ is contained in the pair of coordinate axes in the $x$-$y$ plane. For such a surface we define the {\em Milnor fiber of a transverse slice} to be the set of points $(x,y,z)$ obtained by the following process:
\begin{itemize}
\item[(1)]
requiring that $\|(x,y,z)\|\leq\delta$, a sufficiently small radius,
\item[(2)]
then requiring that $x$ be a fixed number sufficiently close to zero,
\item[(3)]
then requiring that $f(x,y,z)=\epsilon$, a number sufficiently close to zero.
\end{itemize}
Denote this transverse Milnor fiber by $F$ and its Euler characteristic by $\chi$.
We should point out a subtlety in the definition: the tranverse slice (obtained by the first two steps but then staying on the surface $f=0$) may be a plane curve with several branches. For example, the transverse slice of $z^2=x^{3}y^{2}$ is a pair of lines, and thus its transverse Milnor fiber has two boundary components.
\par
By keeping $x$ fixed but letting $\epsilon$ vary over a circle centered at 0, we obtain the {\em horizontal fibration}.
Keeping $\epsilon$ fixed but letting $x$ vary over a circle centered at 0, we obtain the {\em vertical fibration}.
Thus we have a fibration over a torus.
Let $h_q:H_q(F;\mathbf{Q}) \to H_q(F;\mathbf{Q})$
and $v_q:H_q(F;\mathbf{Q}) \to H_q(F;\mathbf{Q})$
be the respective monodromy operators. The graded characteristic functions
$$
\mathbf H(t)=\frac{\det(tI-h_0)}{\det(tI-h_1)}
\qquad
\text{and}
\qquad
\mathbf V(t)=\frac{\det(tI-v_0)}{\det(tI-v_1)}
$$
are called the {\em horizontal monodromy} and {\em vertical monodromy}.
\par
For a non-reduced quasi-ordinary surface, the definitions of horizontal and vertical monodromy need to be formulated in a slightly different way. We discuss this case in the last section of the paper. In all circumstances our definitions agree with those of Kulikov \cite{Ku}, p. 137
(except in those cases where the surface is not singular along or above the $x$-axis, in which case our formulas yield trivial monodromy).
\section{Recursive formulas for horizontal and vertical monodromy}
\label{rfhvm}
Suppose we begin with a series (\ref{powseries}) defining the germ at the origin of an irreducible quasi-ordinary surface $S$.
As in the case of plane curves, we create a new series using just the characteristic pairs,
\begin{equation}\label{prototype2}
\sum_{i=1}^{e} x^{\lambda_i} y^{\mu_i},
\end{equation}
and call the corresponding surface the {\em prototype}.
\begin{theorem}\label{protosame2}
A reduced quasi-ordinary surface and its prototype have the same horizontal monodromy and the same vertical monodromy.
\end{theorem}
We will establish this as in the case of plane curves: by induction on $e$, while simultaneously proving a set of recursive formulas. The case $e=0$ is trivial, and henceforth we assume that $e>0$.
We define the {\em truncation} to be the surface $S_1$ determined by
$$
\zeta_1=x^{\lambda_1}y^{\mu_1}=x^{\frac{a}{mb}}y^{\frac{n}{m}},
$$
where $n$ and $m$ are relatively prime, as are $a$ and $b$.
\par
As before, let $r$ and $s$ be the smallest nonnegative integers so that
$$ \left[ \begin{array}{cc}
m & n \\
r & s
\end{array} \right]$$
has determinant 1.
The {\em derived surface} is the surface $S'$ determined by
$$\zeta'=\sum_{i=1}^{e-1}x^{\lambda'_i}y^{\mu'_i},$$ where the new exponents are computed by these formulas:
\begin{align*}
\mu_i' &= m(\mu_{i + 1} - \mu_1 + mb\mu_1) \\
\lambda_i' &= b(\lambda_{i + 1} - \lambda_1 + mb \lambda_1 + r \mu_i' \lambda_1).
\end{align*}
(An example is worked out at the end of this section.)
\par
For the truncation, let $d_1$, $\chi_1$, $\mathbf H_1$, and $\mathbf V_1$ denote its degree, the Euler characteristic of its transverse Milnor fiber, and its horizontal and vertical monodromies. Let $d'$, $\chi'$, $\mathbf H'$, and $\mathbf V'$ denote the same things for the derived surface. Let $(n,a)$ denote the greatest common divisor.
\begin{theorem}\label{surfacerecursion}
For a reduced quasi-ordinary surface germ, its degree, the Euler characteristic of its transverse Milnor fiber, its horizontal monodromy, and its vertical monodromy are determined by these formulas.
\begin{enumerate}
\item
$d_1=mb$
\item
$d=d_1 d'$
\item
$
\chi_1=mb+nb-mnb^2
$
\item
$
\chi=d'(\chi_1-b)+b\chi'=d'\chi_1+b(\chi'-d')
$
\item
$$
\mathbf H_1(t)=\frac {(t^{mb}-1)(t^{nb}-1)} {(t^{mnb}-1)^b}
$$
\item
$$
\mathbf H(t)=\frac {\mathbf H_1(t^{d'})(\mathbf H'(t))^b} {(t^{d'}-1)^b}
$$
\item
$$
\mathbf V_1(t)=\frac {(t-1)^{mb}} {(t^{nb/(n,a)}-1)^{(n,a)(mb -1)}}
$$
\item
$$
\mathbf V(t)=\frac{(\mathbf V_1(t))^{d'}\mathbf V'(t^b)} {(t^b-1)^{d'}}
$$
\end{enumerate}
\end{theorem}
\begin{proof}
As indicated, we will simultaneously provide an inductive proof of Theorem \ref{protosame2} (inducting on the number of characteristic pairs) and a recursive proof of Theorem \ref{surfacerecursion}.
\par
Fixing a value of $x$, consider the transverse Milnor fiber of the truncation, defined by $z^{mb}-x^{a}y^{nb}=\epsilon$, and its image under the projection $\pi$. There is total ramification above the $(nb)$th roots of $(-\epsilon/x^a)$. We can retract a neighborhood of 0 onto the union $L_x$ of line segments from $0$ to these points, in such a way that there is a compatible retraction of the Milnor fiber onto $\pi^{-1}L_x$, which is the complete bigraph on the $nb$ points
\begin{equation}\label{set1}
\left(\sqrt[nb]{-\epsilon/x^a},0\right)
\end{equation}
and the $mb$ points
\begin{equation}\label{set2}
\left(0,\sqrt[mb]{\epsilon}\right).
\end{equation}
As $\epsilon$ goes around a circle, each set of points is cyclically permuted. Since $m$ and $n$ are relatively prime, the $mnb^2$ edges of the graph fall into $b$ orbits of length $mnb$. This confirms formula (5).
If $\epsilon$ is fixed but $x$ varies, the retractions of the Milnor fibers fit together continuously. The points (\ref{set2}) are fixed but the points (\ref{set1}) fall into $(n,a)$ orbits each of size $nb/(n,a)$. For the edges of the graph the orbits likewise have this size, and there are $(n,a)mb$ such orbits. This confirms formula (7). Formula (3) follows by taking the degree, and formula (1) is trivial.
\par
To verify the recursive formulas and to handle the inductive step in the proof of Theorem \ref{protosame2},
suppose we are given a curve with series (\ref{powseries}) and prototype (\ref{prototype2}).
We first replace
$$
\frac{z-\sum_{(\lambda,\mu)\in\mathbf Z\times\mathbf Z} c_{\lambda\mu} x^{\lambda} y^{\mu}}{c_{\lambda_1\mu_1}}.
$$
by $z$.
In the new coordinate system, the surface is defined by the vanishing of
\begin{equation}
\label{fproduct}
f=\prod^d\left(z-\left[x^{\frac{a}{mb}}y^{\frac{n}{m}}
+\sum_{(\lambda,\mu)>\left(\frac{a}{mb},\frac{n}{m}\right)} c_{\lambda\mu}x^{\lambda}y^{\mu}\right]\right),
\end{equation}
(where for simplicity the coefficients have been renamed).
The truncation is defined by the vanishing of
\begin{equation}
\label{f1product}
f_1= \prod^{mb}(z-x^{\frac{a}{mb}}y^{\frac{n}{m}})=z^{mb}-x^{a}y^{nb}.
\end{equation}
\par
Dividing (\ref{fproduct})
by a power of (\ref{f1product}), we claim that
\begin{equation}\label{compare2}
\frac{f}{f_1^{d/(mb)}}= \prod^d\left(1-\frac{\sum_{(\lambda,\mu)>\left(\frac{a}{mb},\frac{n}{m}\right)} c_{\lambda\mu}x^{\lambda}y^{\mu}}{z-x^{\frac{a}{mb}}y^{\frac{n}{m}}}\right).
\end{equation}
To justify this we argue as follows. Let $(x,y)$ be a point close to the origin but not lying on the $x$- or $y$-axis. Let $d_x$ be the common denominator of all $x$-exponents appearing in (\ref{fproduct}); similarly let $d_y$ be the common denominator of all $y$-exponents. Fix a value $\bar{x}=x^{1/d_x}$ and similarly a value $\bar{y}=y^{1/d_y}$.
Then there is a map from the product of two groups of roots of unity:
$$
\mu_{d_x} \times \mu_{d_y} \to \text{points on the surface projecting to $(x,y)$}
$$
whose last coordinate is given by
\begin{equation}
\label{deck}
(\alpha,\beta) \mapsto (\alpha\bar{x})^{ad_x/(mb)}(\beta\bar{y})^{nd_y/m}
+\sum_{(\lambda,\mu)>\left(\frac{a}{mb},\frac{n}{m}\right)} c_{\lambda\mu}(\alpha\bar{x})^{\lambda d_x}(\beta\bar{y})^{\mu d_y}.
\end{equation}
(Note that all exponents are integers.) This map factors through the quotient $(\mu_{d_x} \times \mu_{d_y})/K$, where $K$ consists of all elements determining the same point as $(1,1)$. This quotient group has order $d$.
Similarly there is a map
$$
(\alpha,\beta) \mapsto (\alpha\bar{x})^{ad_x/(mb)}(\beta\bar{y})^{nd_y/m}
$$
onto the points of the truncation surface, with kernel $K_1$ and with quotient group $(\mu_{d_x} \times \mu_{d_y})/K_1$ of order $mb$. A fiber of the homomorphism
$$
(\mu_{d_x} \times \mu_{d_y})/K \to (\mu_{d_x} \times \mu_{d_y})/K_1
$$
(i.e, a coset of the kernel $K_1/K$) corresponds to all distinct series in (\ref{deck}) compatible with a specified first term. Since these fibers all have the same cardinality $d/(mb)$, the calculation leading to (\ref{compare2}) is justified.
\par
Now we suppose that $x$ moves on the circle of radius $\rho$. All of our constructions will be done equivariantly, i.e., by doing the same thing simultaneously to all transverse slices. First, in each transverse slice, we perform the series of blowups dictated by $\mu_1=n/m$ and the Euclidean algorithm. Doing this for the truncation, we obtain (for each transverse slice) a total transform consisting of certain exceptional divisors occurring with certain multiplicities, together with a strict transform meeting just one exceptional divisor, which we call the {\em rupture component}. Along this chain the function $z^m/y^n$ has no indeterminacy, and in fact except along the rupture component its value is either $0$ or $\infty$.
\par
If all of the exponents $\mu$ appearing in (\ref{compare2}) were strictly greater than $n/m$, then we could argue, as in the earlier proof of Theorem \ref{curverecursion}, that the value of (\ref{compare2}) along a non-rupture exceptional divisor is 1. But since there may be a repetition of exponents (even in the characteristic pairs) we need to be more careful. If $z^m/y^n=0$, then
\begin{equation*}
\frac{f}{f_1^{d/(mb)}}
=
\prod^d\left(1+
\sum_{(\lambda,\mu)>\left(\frac{a}{mb},\frac{n}{m}\right)} c_{\lambda\mu}x^{\lambda-a/(mb)}y^{\mu-n/m}
\right),
\end{equation*}
and since $y$ vanishes everywhere along the exceptional divisors we find that
\begin{equation*}
\frac{f}{f_1^{d/(mb)}}
=
\prod^d\left(1+
\sum_{\lambda>\frac{a}{mb}} c_{\lambda\mu_1}x^{\lambda-a/(mb)}
\right).
\end{equation*}
Note that by choosing $x$ sufficiently close to 0 we can guarantee that this value has positive real part. If $z^m/y^n=\infty$, i.e. $y^n/z^m=0$, then a similar calculation shows that the value of (\ref{compare2}) is 1.
\par
To work in a chart containing the rupture component, we use substitutions dictated by the matrix
$$ \left[ \begin{array}{cc}
m & n \\
r & s
\end{array} \right],$$
where $r$ and $s$ are the smallest positive integers for which the determinant is 1,
namely
\begin{gather*}
y=u^{m} v^{r} \\
z=u^{n} v^{s}.
\end{gather*}
We find that in this chart the total transform of the truncation is defined by the vanishing of
$$
f_1=u^{mnb}v^{rnb}(v^{b}-x^{a}),
$$
and its strict transform is defined by the vanishing of the last factor. Note that it meets the $v$-axis in $b$ points, and that as $x$ travels around a small circle these points trace out the torus knot $v^b=x^a$.
The total transform of the given surface is defined by the vanishing of
$$
f=\prod^d\left(u^{n}v^{s}-\left[x^{\frac{a}{mb}}u^{n}v^{rn/m}+\sum_{(\lambda,\mu)>\left(\frac{a}{mb},\frac{n}{m}\right)} c_{\lambda\mu}x^{\lambda}u^{m\mu}v^{r\mu}\right]\right)
$$
which may be rewritten as
\begin{equation}\label{expanded}
\begin{split}
f=&u^{nd}v^{rnd/m}x^{ad/(mb)} \\
&\prod^d\left(
\left(\frac{v}{x^{a/b}}
\right)^{1/m}
-\left[
1
+\sum_{(\lambda,\mu)>\left(\frac{a}{mb},\frac{n}{m}\right)}
c_{\lambda\mu}x^{\lambda-a/(mb)}u^{m\mu-n}v^{r(m\mu-n)/m}
\right]
\right).
\end{split}
\end{equation}
Again if all the values of $\mu$ appearing in (\ref{expanded}) are strictly greater than $n/m$, then we can assert that the strict transform meets the $v$-axis in the same set of $b$ points, but if there is a repetition of exponents then we find that the strict transform meets this axis at all points at which (for some choice of conjugate)
\begin{equation}\label{clustered}
v^b=\left(1+
\sum_{\lambda>\frac{a}{mb}} c_{\lambda\mu_1}x^{\lambda-a/(mb)}
\right)^{mb}
x^a.
\end{equation}
\par
We also note that
$$
\frac{f}{f_1^{d/(mb)}}= \prod^d\left(1-\frac{\sum_{(\lambda,\mu)>\left(\frac{a}{mb},\frac{n}{m}\right)}
c_{\lambda\mu}x^{\lambda-a/(mb)}u^{m\mu-n}v^{r(m\mu-n)/m}}{\left(\frac{v}{x^{a/b}}
\right)^{1/m}-1}\right),
$$
and that its restriction to the rupture component is
\begin{equation}\label{restriction}
\prod^d\left(1-\frac{\sum_{\lambda>\frac{a}{mb}}
c_{\lambda\mu_1}x^{\lambda-a/(mb)}}{\left(\frac{v}{x^{a/b}}
\right)^{1/m}-1}\right).
\end{equation}
\par
Introducing three new variables $x'$, $y'$, and $w$, let $B$ denote the product of the circle $\| x' \| =\rho^{1/b}$ and the ball $\|(y',w)\|\leq\delta'$.
Map this product to a neighborhood $N$ of the torus knot as follows:
\begin{gather*}
x=(x')^{b} \\
u = \frac{y'}{(w+1)^{r}\rho^{ar/(mb)}} \\
v = (w+1)^{m} (x')^{a}
\end{gather*}
(See Figure \ref{fancymap}.)
Note that the circle $(y',w)=(0,0)$ is mapped onto the knot. We claim that if $\delta'$ is sufficiently small then the map is injective (regardless of the value of $\rho$).
Indeed, suppose that $(x'_1,y'_1,w_1)$ and $(x'_2,y'_2,w_2)$ are two points whose images agree. Then
$$
\left(\frac{w_2+1}{w_1+1}\right)^m=\left(\frac{x'_1}{x'_2}\right)^a,
$$
where the quantity on the right is a $b$th root of 1. If $w_1$ and $w_2$ are sufficiently close to 0 then this root must be 1 itself. Since $a$ and $b$ are relatively prime, this implies that $x'_1/x'_2=1$. Since the map $w \mapsto (w+1)^m$ is injective near 0, we see that $w_1=w_2$ and then that $y'_1=y'_2$.
\par
Thus $N$ is a tubular neighborhood of the torus knot: its intersection with each transverse plane consists of $b$ disjoint topological disks, each of which encloses one of the points where the torus knot meets the plane.
\par
\begin{figure}
\[
\xy
(0,0)*
{\xy
{\ar(0,0)*{};(40,0)*+{x'}
\ar(0,0)*{};(0,40)*+{y',w}
\ar@{.. }(0,3)*{};(36,3)*{}
\ar@{.. }(0,-3)*{};(36,-3)*{}
}
\endxy};
(35,0)*
{\xy
{\ar(0,0)*{};(10,0){}}
\endxy};
(70,0)*
{\xy
{\ar(0,0)*{};(40,0)*+{x}
\ar(0,0)*{};(0,40)*+{u,v}
\ar@{ }(0,0)*{};(36,24)*{}
\ar@{ }(0,12)*{};(36,36)*{}
\ar@{ }(0,24)*{};(18,36)*{}
\ar@{ }(18,0)*{};(36,12)*{}
\ar@{.. }(0,2)*{};(36,26)*{}
\ar@{.. }(3,0)*{};(36,22)*{}
\ar@{.. }(0,34)*{};(3,36)*{}
\ar@{.. }(0,14)*{};(33,36)*{}
\ar@{.. }(0,10)*{};(36,34)*{}
\ar@{.. }(33,0)*{};(36,2)*{}
\ar@{.. }(0,26)*{};(15,36)*{}
\ar@{.. }(0,22)*{};(21,36)*{}
\ar@{.. }(15,0)*{};(36,14)*{}
\ar@{.. }(21,0)*{};(36,10)*{}
\ar@{ }(28,-2)*{};(28,43)*+{\text{transverse slice}}
}
\endxy};
\endxy
\]
\caption{A tubular neighborhood $B$ of the circle $\| x' \| =\rho^{1/b}$ is mapped onto a tubular neighborhood $N$ of the torus knot $v^b=x^a$ (where $u=0$, and $x$ moves on the circle of radius $\rho$). Each transverse slice $x=\text{constant}$ meets $N$ in $b$ disjoint topological balls. In this example, $a=2$ and $b=3$.}
\label{fancymap}
\end{figure}
\par
We can regard each transverse Milnor fiber as a subset of the surface obtained from the transverse plane $x=\text{constant}$ by the sequence of blowups. Let us assume that the choices of $\delta$, $x$, and $\epsilon$ which determine the transverse Milnor fiber are made subsequent to the choice of $\delta'$. We claim that we can make these choices so as to guarantee that the strict transform of the surface lies entirely within $N$. Indeed, we note that on the strict transform
$$
w=
\sum_{(\lambda,\mu)>\left(\frac{a}{mb},\frac{n}{m}\right)} c_{\lambda\mu}x^{\lambda-a/(mb)}y^{\mu-n/m},
$$
where in each term at least one of the exponents is positive.
Thus by choosing $\delta$ and $\|x\|$ sufficiently small we may force $w$ arbitrarily close to 0. Now observe that
$$
(y')^m=y \left( \frac{x'}{\rho^{1/b}} \right)^{-ar}
$$
and that $\|x'/\rho^{1/b}\|=1$. Thus we may also force $\|y'\|$ to be arbitrarily small. Note in particular that $N$ will contain the points where the strict transform meets the $v$-axis (as determined by equation (\ref{clustered})); Figure \ref{braiding} shows an example.
\par
Looking at formula (\ref{restriction}), we note that outside of $N$ the value of $\left(\frac{v}{x^{a/b}}\right)^{1/m}$ along the rupture component is bounded away from 1, with the bound being independent of the choice of $x$; thus by choosing $x$ sufficiently close to 0 we can guarantee that the value of (\ref{restriction}) has positive real part. Finally by choosing $\epsilon$ sufficiently close to 0, we can guarantee that the Milnor fiber is transverse to the boundary of $N$ and that its boundary lies entirely within $N$. Our transverse Milnor fiber is thus divided into two pieces. (See Figure \ref{twopieces2}.)
\par
\begin{figure}
\scalebox{0.40}
{\includegraphics{braid.png}}
\caption{The strict transform of a transverse slice of the quasi-ordinary surface $\zeta=x^{1/2}y^{4/3}+x^{2/3}y^{4/3}+x^{11/12}y^{4/3}$ meets the (complex) $v$-axis in $12$ points, which are clustered around the two points where the torus knot $v^2=x^3$ pierces the axis. The tubular neighborhood $N$ meets the axis in two topological disks.}
\label{braiding}
\end{figure}
\par
\begin{figure}
\scalebox{0.25}
{\includegraphics{twopieces2.png}}
\caption{The transverse Milnor fiber is divided into two pieces by the boundary of $N$ (indicated by two circles). The rupture component is horizontal, and another exceptional divisor is shown vertically. The strict transform enters from above.}
\label{twopieces2}
\end{figure}
\par
Consider first the piece of the Milnor fiber lying outside of $N$. By the approximation lemma \ref{approx2}, for this piece the monodromy of $f$ is the same as the monodromy of $f_1^{d/(mb)}$. The Milnor fiber has $d/(mb)$ connected components corresponding to all possible values of
$\eta=\epsilon^{mb/d}$, and each one is a copy of the Milnor fiber for $f_1$.
As $\epsilon$ goes around a circle, these copies are cyclically permuted. As
$\epsilon$ goes around this circle $d/(mb)$ times, however, each $\eta$ goes once around a circle. Thus the horizontal monodromy of this piece is $\mathbf H_1(t^{d/(mb)})$.
But if $\epsilon$ is fixed and $x$ varies, then each copy is individually acted upon by the vertical monodromy, so that the contribution from this piece is $(\mathbf V_1(t))^{d/(mb)}$.
\par
Now consider the piece of the Milnor fiber lying inside $N$. Note that it has two sorts of boundary components: the components of the original link and those components created by its intersection with the boundary sphere of $N$. To analyze it, we look at its inverse image in $B$, which is contained in the $b$ disjoint balls centered at the points $(x',y',w)=(x^{1/b},0,0)$ (allowing all possible roots).
\par
When pulled back to $B$, most of the $d$ factors at the end of (\ref{expanded}) become units. To see this, first observe that we can force the value in square brackets to be arbitrarily close to 1 by choosing sufficiently small radii
$\delta'$ and $\rho$. To obtain a non-unit, we must therefore pick the ``principal value'' of $x^{1/b}$ for which it equals $x'$ and then similarly pick the appropriate $m$th root of $v/(x')^a$ so that
$$
\left(\frac{v}{(x')^{a}}
\right)^{1/m}=w+1;
$$
note that these choices can be made uniformly throughout $B$.
Thus $d/(mb)$ of the factors at the end of (\ref{expanded}) become
$$
w-\sum_{(\lambda,\mu)>\left(\frac{a}{mb},\frac{n}{m}\right)} c'_{\lambda\mu}(x')^{b\lambda-a/m+ar(m\mu-n)/m}(y')^{m\mu-n}
$$
(where $c'_{\lambda\mu}=c_{\lambda\mu}\rho^{-ar(m\mu-n)/(mb)}$), whereas the remaining $d-d/(mb)$ factors become units. Each such unit takes its values in an arbitrarily small neighborhood of some $e-1$, where $e$ is a nontrivial $(mb)$th root of unity. Thus by the approximation lemma \ref{approx2}, we may ignore all unit factors in $f$.
Thus we may assume that the function defining this piece of the Milnor fiber is
$$
(x')^{ads}(y')^{nd}\prod^{d/(mb)}\left(w-\sum_{(\lambda,\mu)>\left(\frac{a}{mb},\frac{n}{m}\right)} c'_{\lambda\mu}(x')^{b\lambda-a/m+ar(m\mu-n)/m}(y')^{m\mu-n}\right).
$$
\par
The map $(x',y',w) \mapsto (x',y',(x')^{asmb}(y')^{nmb}w)$ takes this piece to the transverse Milnor fiber of the quasi-ordinary surface with series
\begin{equation}\label{derivedseries2}
\sum_{(\lambda,\mu)>\left(\frac{a}{mb},\frac{n}{m}\right)} c'_{\lambda\mu}(x')^{b\lambda-a/m+ar(m\mu-n)/m+ambs}(y')^{m\mu-n+nmb},
\end{equation}
but it misses disks centered at the $d/(mb)$ points
\begin{equation}\label{centers}
(x',0,\epsilon^{d/(mb)}).
\end{equation}
(Note that all of the exponents on $y'$ in (\ref{derivedseries2}) are positive; thus we are still in the reduced case.)
The horizontal monodromy permutes these disks. In (\ref{derivedseries2}) there are $e-1$ characteristic pairs, whereas our original series had $e$ characteristic pairs. By the inductive hypothesis, the horizontal monodromy of this curve is the same as that of its prototype, which has series
$$
\sum_{i=2}^{e}
(x')^{b[\lambda_i-\lambda_1+mb\lambda_1+rm(\mu_i-\mu_1+mb\mu_1)\lambda_1]}
(y')^{m(\mu_i-\mu_1+mb\mu_1)}.
$$
(In calculating the first exponent we have used $ms=rn+1$.)
By reindexing we obtain the series of the derived surface.
Thus $d'=d/(mb)$, confirming formula (2) of the theorem. Since there are $b$ copies of this situation (one for each $b$th root of $x$),
the monodromy of this piece of the transverse Milnor fiber is
$$
\left(\frac{\mathbf H'(t)}
{t^{d'}-1}\right)^{b}.
$$
Combining this with our conclusion about the monodromy of the first piece, we obtain formula (6). Then we obtain formula (4) by computing the degree of both sides of (6).
\par
Turning to the vertical monodromy, we remark that it cyclically permutes the individual pieces of the Milnor fiber cut out by the $b$ disjoint balls. Its $b$th power acts on each such piece by the vertical monodromy of the derived surface, in such a way that the disks of (\ref{centers}) are cyclically permuted. Thus the contribution to the vertical monodromy of our original surface is
$$
\frac{\mathbf V'(T)} {(T-1)^{d'}}
$$
where $T=t^b$.
Combining this with our conclusion about the vertical monodromy of the first piece, we obtain formula (8).
\end{proof}
Here is an example. If we begin with the surface parametrized by
$$
\zeta=x^{1/2}y^{3/2}+x^{1/2}y^{7/4}+x^{2/3}y^{11/6},
$$
then its truncation and derived surface are parametrized by
$$
\zeta_1=x^{1/2}y^{3/2}\qquad
\text{and}
\qquad
\zeta'=x^{17/4}y^{13/2}+x^{9/2}y^{20/3}.
$$
Repeating the process, the new truncation and the second derived surface are parametrized by
$$
\zeta'_1=x^{17/4}y^{13/2}\qquad
\text{and}
\qquad
\zeta''=x^{1438/3}y^{157/3}.
$$
By repeated use of the first two formulas in Theorem \ref{surfacerecursion},
we find that the degree of the quasi-ordinary surface is
$$
d=d_1 d'_1 d''=2\cdot4\cdot3=24.
$$
By formulas (3) and (4), the Euler characteristic of the transverse Milnor fiber is
$$
\chi=d'(\chi_1-b)+d''(\chi'_1-b')+b'\chi''=12(-1-1)+3(-74-2)+2(-311)=-874.
$$
By formulas (5) and (6), the horizontal monodromy is
\begin{equation}
\begin{split}
\mathbf H(t)&=
\frac{\mathbf H_1(t^{d'})}{(t^{d'}-1)^{b}}
\left[ \frac{\mathbf H'_1(t^{d''})}{(t^{d''}-1)^{b'}} \right]^{b}
\left[ \mathbf H''(t) \right]^{bb'} \\
&=
\frac{(t^{24}-1)(t^{36}-1)}{(t^{72}-1)(t^{12}-1)}
\left[ \frac{(t^{12}-1)(t^{78}-1)}{(t^{156}-1)^{2}(t^{3}-1)^{2}} \right]^{1}
\left[ \frac{(t^{3}-1)(t^{157}-1)}{t^{471}-1} \right]^{2}.
\end{split}
\end{equation}
By formulas (7) and (8), the vertical monodromy is
\begin{equation}
\begin{split}
\mathbf V(t)&=
\left[ \frac{\mathbf V_1(t)}{t^b-1} \right]^{d'}
\left[ \frac{\mathbf V'_1(t^{b})}{(t^{bb'}-1)} \right]^{d''}
\cdot \mathbf V''(t^{bb'}) \\
&=
\left[ \frac{(t-1)^2}{(t^3-1)(t-1)} \right]^{12}
\left[ \frac{(t-1)^4}{(t^{26}-1)^{2}(t^2-1)} \right]^{3}
\cdot \frac{(t^2-1)^3}{(t^{314}-1)^2}.
\end{split}
\end{equation}
\section{Non-reduced quasi-ordinary surfaces}
\label{nrqos}
We now consider the non-reduced case. Suppose that in (\ref{cpairs})
we have $\mu_i = 0$ for $1 \leq i \leq s < e$. Then the singular locus of $S$ may contain a curve which does not lie in the $x$-$y$ plane, namely the intersection of $S$ with the plane $y=0$. This curve projects to the $x$-axis, and if we restrict our attention to those points lying over a small circle we see an $N$-sheeted covering $C \to S^1$, where $N$ is the least common denominator of $\{ \lambda_i \}_{i = 1}^s$. The transverse slice of $S$ (as defined in section \ref{qos}) will then be a curve with $N$ singularities.
For example, on the surface parametrized by $\zeta=x^{3/2}+x^{2}y^{3/2}$ the curve $z^2=x^3$ is a component of the singular locus. A transverse slice is shown in Figure \ref{multi}.
\par
In this case, the correct definitions of the horizontal and vertical fibrations use Milnor fibers at the points of $C$. Such a Milnor fiber consists of those points within a transverse slice, within a sufficiently small neighborhood of the specified point of $C$, and satisfying $f=\epsilon$ (for sufficiently small $\epsilon$). Each transverse slice will contain $N$ such Milnor fibers, and they form the fibers of a fibration over $C \times S^1$ (the latter factor consisting of all $\epsilon$ on a small circle). One obtains the horizontal or vertical fibration by fixing (respectively) the point of C or the value of $\epsilon$.
\par
\begin{figure}
\scalebox{0.75}
{\includegraphics{multising.png}}
\caption{The real points of the transverse slice of the quasi-ordinary surface parametrized by $\zeta=x^{3/2}+x^{2}y^{3/2}$. Here $N=2$.}
\label{multi}
\end{figure}
\par
Lipman \cite{Li2} (p. 65 ff.) shows that we can replace $S$ by a reduced quasi-ordinary surface $S'$ with characteristic pairs $\{ (\lambda_i', \mu_i') = (N\lambda_{i+s}, \mu_{i+s}) \}$, $1 \leq i \leq e - s$, so that the horizontal and vertical fibrations of $S$ (as just defined) are the same as those of $S'$ (as defined in section \ref{qos}). Thus the characteristic pairs $\{ (\lambda_i, 0) \} _{i = 1}^s$ are invisible in these monodromies, but they are precisely what is recovered by the topological zeta function of the two-dimensional singularity; see \cite{MN} and \cite{McE}.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,455
|
\section{Introduction}
In \citet{CMPer-Silling} and \cite{States} a self consistent non-local continuum mechanics is proposed. This formulation known as peridynamics has been employed in the computational reproduction of dynamic fracture as well as offering dynamically based explanations for features observed in fracture, see e.g.,\cite{CMPer-Silling5,CMPer-Silling7,CMPer-Lipton2,BobaruHu,HaBobaru,SillBob,CMPer-Agwai,CMPer-Ghajari}. These references are by no means complete and a recent review of this approach together with further references to the literature can be found in \cite{Handbook}.
The peridynamic formulation expresses internal forces as functions of displacement differences as opposed to displacement gradients. This generalization allows for an extended kinematics and provides a unified treatment of differentiable and non-differentiable displacements.
The motion of a point $\bolds{x}$ is influenced by its neighbors through non-local forces. In its simplest formulation forces act within a horizon and only neighbors confined to a ball of radius $\epsilon$ surrounding $\bolds{x}$ can influence the motion of $\bolds{x}$. The radius $\epsilon$ is referred to as the peridynamic horizon.
When the forces are linear in the strain and when length scale of nonlocality $\epsilon$ tends to zero the peridynamic models converge to the linear elastic model \cite{CMPer-Emmrich,CMPer-Silling4,AksoyluUnlu,CMPer-Mengesha2}. If one considers non-linear forces associated with two point interactions that are initially elastic and then soften after a critical strain, then the dynamic evolutions are found to converge to a different ``limiting'' dynamics associated with a crack set and a displacement that satisfies the balance of linear momentum away from the crack set and has bounded elastic energy and Griffith surface energy, see \cite{CMPer-Lipton,CMPer-Lipton3} and \cite{CMPer-JhaLipton}. A numerical analysis of this two-point interaction or bond based peridynamic model is carried out in \cite{CMPer-JhaLipton,CMPer-JhaLipton3}. In these works the a-priori convergence rates for finite difference and finite element methods together with different time stepping schemes are reported.
This article focuses on the numerical analysis of a state based peridynamic fracture model governed by forces that are initially elastic and then soften for sufficiently large tensile and hydrostatic strains. Attention is given to the prototypical state-based peridynamic model proposed in \cite{CMPer-Lipton4}. The analysis performed here provides a-priori upper bounds on the convergence rate for a numerical scheme that applies the finite difference approximation in space and the forward Euler discretization scheme in time. The state based peridynamic model treated here has two components of non-local force acting on a point. The first force is due to tensile strains acting on $\bolds{x}$ by its neighbors $\bolds{y}$, while the second force is due to the net hydrostatic strain on $\bolds{x}$ associated with the change in volume about $\bolds{x}$. In this article we analyze the convergence of the numerical scheme for two different cases of constitutive law relating non-local force to strain. For the first case we take both tensile and hydrostatic forces to be initially linear and increasing with the strain and then after reaching critical values of tensile and hydrostatic strain respectively the forces decrease to zero with strain, see figures \ref{ConvexConcave}(b) and \ref{ConvexConcaveFunctionG}(b).
For the second case we choose the hydrostatic force to be a linear function of the hydrostatic strain (see dashed line \ref{ConvexConcaveFunctionG}(b)) while the tensile force is initially linear and then decreases to zero after a critical tensile strain is reached, see \autoref{ConvexConcave}(b). The choice of the two constitutive models studied here is motivated by the prospect of simulating materials that exhibit failure due to extreme local tensile stress or strain or materials that fail due to extreme local hydrostatic stress or strain.
Here the quadratic potential function for the dilatational strain can be associated with materials that fail under extreme local tensile loads while the convex-concave dilatational potential function can be associated with materials in which fail under extreme local hydrostatic loads.
The primary new contribution of this paper is that a-priori convergence rates are established for numerical schemes used for simulation using these prototypical state based peridynamic models. As mentioned earlier the constitutive behavior is non-linear, non-convex and material properties can degrade during the course of the evolution.
We consider the class of H\"older continuous displacement fields and show the existence of a unique H\"older continuous evolution for a prescribed H\"older continuous initial condition and body force, see \autoref{thm:existence over finite time domain}.
To obtain a-priori bounds on the error, we develop an $L^2$ approximation theory for the finite difference approximation in the spatial variables and the forward Euler approximation in time, see \autoref{s:finite difference}.
We show that discrete approximations converge to the exact H\"older continuous solution uniformly over finite time intervals with respect to the $L^2$ norm. The a-priori rate of convergence in the $L^2$ norm is given by $(C_t\Delta t + C_s h^\gamma/\epsilon^2)$, where $\Delta t$ is the size of the time step, $h$ is the size of spatial mesh discretization, $\gamma\in (0,1]$ is the H\"older exponent, and $\epsilon$ is the length scale of nonlocal interaction relative to the size of the domain, see \autoref{thm:convergence} The constant $C_t$ depends on the $L^2$ norm of the time derivatives of the solution, $C_s$ depends on the H\"older norm of the solution and the Lipschitz constant of peridynamic force.
We point out that the convergence results derived here can be extended to general single step time discretization using arguments provided in \cite{CMPer-JhaLipton}. Although the constitutive law relating force to strain is nonlinear we are still able to establish stability for the semi-discrete approximation and it is shown that the energy at any given time $t$ is bounded above by the energy of the initial conditions and the total work done by the body force up to time $t$, see \autoref{thm:stab semi}.
Our numerical simulations support the theoretical studies, see \autoref{s:numerical}. In the simulations we introduce a straight crack and it propagates in response to applied boundary conditions. For these simulations we use piecewise constant interpolants and record the rate of convergence with respect to mesh size while keeping the horizon fixed. Our results show that convergence rate remains above the a-priori estimated rate of $1$ during the simulation. For illustration we also present numerical simulations for a pre-cracked samples subject to a bending load.
It is pointed out that there is now a significant number of investigations examining the numerical approximation of singular kernels for non-local problems with applications to nonlocal diffusion, advection, and continuum mechanics. Numerical formulations and convergence theory for nonlocal $p$-Laplacian formulations are developed in \cite{DeEllaGunzberger}, \cite{Nochetto1}. Numerical analysis of nonlocal steady state diffusion is presented in \cite{CMPer-Du2} and \cite{CMPer-Du3}, and \cite{CMPer-Chen}. The use of fractional Sobolev spaces for nonlocal problems is investigated and developed in \cite{CMPer-Du1}. Quadrature approximations and stability conditions for linear peridynamics are analyzed in \cite{CMPer-Weckner} and \cite{CMPer-Silling8}. The interplay between nonlocal interaction length and grid refinement for linear peridynamic models is presented in \cite{CMPer-Bobaru}. Analysis of adaptive refinement and domain decomposition for the linearized peridynamics are provided in \cite{AksoyluParks}, \cite{LindParks}, and \cite{AksMen}. This list is by no means complete and the literature continues to grow rapidly.
The paper is organized as follows. In \autoref{s:nonlocal dynamics}, we describe the nonlocal model and state the peridynamic equation of motion. The Lipschitz continuity of the peridynamic force and global existence of unique solutions are presented in \autoref{ss:existence holder}.
The finite difference discretization is introduced in \autoref{s:finite difference}. We demonstrate the energy stability of the semi-discrete approximation in \autoref{semidiscrete}. In \autoref{time discrete} we give the a-priori bound on the error for the fully discrete approximation, see \autoref{thm:convergence}. The numerical simulations are described and presented in \autoref{s:numerical}.
The Lipschitz continuity of the peridynamic force and stability of the semi-discrete approximation are proved in \autoref{s:proofs} and \autoref{ss:stab proof}.
In \autoref{s:conclusions} we summarize our results.
\section{Nonlocal Dynamics}\label{s:nonlocal dynamics}
We now formulate the nonlocal dynamics. Let $D \subset \mathbb{R}^d$ denote the material domain of dimension $d=2,3$ and let the horizon be given by $\epsilon >0$. We make the assumption of small (infinitesimal) deformations so that the displacement field $\bolds{u}: [0,T]\times D \to \mathbb{R}^d$ is small compared to the size of $D$ and the deformed configuration is the same as the reference configuration. We have $\bolds{u}=\bolds{u}(t,\bolds{x})$ as a function of space and time but will suppress the $\bolds{x}$ dependence when convenient and write $\bolds{u}(t)$. The tensile strain $S$ between two points $\bolds{x},\bolds{y} \in D$ along the direction $\bolds{e}_{\bolds{y}-\bolds{x}}$is defined as
\begin{align}\label{strain}
S(\bolds{y},\bolds{x},\bolds{u}(t))=\frac{\bolds{u}(t,\bolds{y})-\bolds{u}(t,\bolds{x})}{|\bolds{y}-\bolds{x}|}\cdot \bolds{e}_{\bolds{y}-\bolds{x}},
\end{align}
where $ \bolds{e}_{\bolds{y}-\bolds{x}}=\frac{\bolds{y}-\bolds{x}}{|\bolds{y}-\bolds{x}|}$ is a unit vector and ``$\cdot$'' is the dot product. The influence function $J^\epsilon(|\bolds{y}-\bolds{x}|)$ is a measure of the influence that the point $\bolds{y}$ has on $\bolds{x}$. Only points inside the horizon can influence $x$ so $J^\epsilon(|\bolds{y}-\bolds{x}|)$ nonzero for $|\bolds{y} - \bolds{x}| < \epsilon$ and zero otherwise. We take $J^\epsilon$ to be of the form: $J^\epsilon(|\bolds{y} - \bolds{x}|) = J(\frac{|\bolds{y} - \bolds{x}|}{\epsilon})$ with $J(r) = 0$ for $r\geq 1$ and $0\leq J(r)\leq M < \infty$ for $r<1$. We also introduce the boundary function $\omega(\bolds{x})$ providing the influence of the boundary on the non-local force. Here $\omega(\bolds{x})$ takes the value $1$, for all $\bolds{x} \in D$, an $\epsilon$ distance away from $\partial D$. As $\bolds{x} $ approaches $\partial D$ from the interior, $\omega(\bolds{x})$ smoothly decays from $1$ to $0$ on $\partial D$ and is extended by zero outside $D$.
The spherical or hydrostatic strain at $\bolds{x}$ is a measure of the volume change about $\bolds{x}$ and is given by
\begin{align}\label{sphericalstrain}
\theta(\bolds{x},\bolds{u}(t))=\frac{1}{\epsilon^d \omega_d}\int_{H_\epsilon(\bolds{x})} \omega(\bolds{y}) J^\epsilon(|\bolds{y}-\bolds{x}|)S(\bolds{y},\bolds{x},\bolds{u}(t)){|\bolds{y}-\bolds{x}|}\,d\bolds{y},
\end{align}
where $\omega_d$ is the volume of the unit ball in dimension $d=2,3$, and $H_\epsilon(\bolds{x})$ denotes the ball of radius $\epsilon$ centered at $\bolds{x}$.
\subsection{The class of nonlocal potentials}
Motivated by potentials of Lennard-Jones type, the force potential for tensile strain is defined by
\begin{equation}\label{tensilepot}
\mathcal{W}^\epsilon (S(\bolds{y},\bolds{x},\bolds{u}(t)))=\omega(\bolds{x}) \omega(\bolds{y}) J^\epsilon(|\bolds{y}-\bolds{x}|)\frac{1}{\epsilon|\bolds{y}-\bolds{x}|}f(\sqrt{|\bolds{y}-\bolds{x}|}S(\bolds{y},\bolds{x},\bolds{u}(t)))
\end{equation}
and the potential for hydrostatic strain is defined as
\begin{equation}\label{hydropot}
\mathcal{V}^\epsilon(\theta(\bolds{x},\bolds{u}(t)))= \omega(\bolds{x})\frac{g(\theta (\bolds{x},\bolds{u}(t)))}{\epsilon^2}\\
\end{equation}
where $\mathcal{W}^\epsilon (S(\bolds{y},\bolds{x},\bolds{u}(t)))$ is the pairwise force potential per unit length between two points $\bolds{x}$ and $\bolds{y}$ and $\mathcal{V}^\epsilon(\theta(\bolds{x},\bolds{u}(t)))$ is the hydrostatic force potential density at $\bolds{x}$. They are described in terms of their potential functions $f$ and $g$, see \autoref{ConvexConcave} and \autoref{ConvexConcaveFunctionG}.
The potential function $f$ represents a convex-concave potential such that the associated force acting between material points $\bolds{x}$ and $\bolds{y}$ are initially elastic and then soften and decay to zero as the strain between points increases, see \autoref{ConvexConcave}. The first well for $\mathcal{W}^\epsilon (S(\bolds{y},\bolds{x},\bolds{u}(t)))$ is at zero tensile strain and the potential function satisfies
\begin{align}
\label{choice at oregon}
f(0)=f'(0) = 0.
\end{align}
The behavior for infinite tensile strain is characterized by the horizontal asymptotes $\lim_{S\rightarrow \infty} f(S)=C^+$ and $\lim_{S\rightarrow -\infty} f(S)=C^-$ respectively, see \autoref{ConvexConcave}. The critical tensile strain $S^+_c>0$ for which the force begins to soften is given by the inflection point $r^+>0$ of $f$ and is
\begin{equation}
S^+_c=\frac{r^+}{\sqrt{|y-x|}}.
\label{crittensileplus}
\end{equation}
The critical negative tensile strain is chosen much larger in magnitude than $S_c^+$ and is
\begin{equation}
S^-_c=\frac{r^-}{\sqrt{|y-x|}},
\label{crittensileminus}
\end{equation}
with $r^-<0$ and $r^+<<|r^-|$.
We assume here that the all the potential functions are bounded and have bounded derivatives up to order $3$, We denote the $i^{th}$ derivative of the function $f$ by $f^{(i)}$, $i=1,2,3$. Let $C^f_i$ for $i=0,1,2,3$ denote the bounds on the functions and derivatives given by
\begin{align}\label{eq:def Cfi}
C^f_0 := \sup_r |f(r)|, \qquad C^f_i := \sup_r |f^{(i)}(r)| \quad \text{for } i = 1,2,3,
\end{align}
and $C^f_i < \infty$ for $i=0,1,2,3$.
We will consider two types of potentials associated with hydrostatic strain. The first potential we consider is a quadratic potential characterized by a quadratic potential function $g$ with a minimum at zero strain. The second potential we consider is characterized by a convex-concave potential function $g$, see \autoref{ConvexConcaveFunctionG} . If $g$ is assumed to be quadratic then the force due to spherical strain is linear and there is no softening of the material. However, if $g$ is convex-concave the force internal to the material is initially linear and increasing but then becomes decreasing with strain as the hydrostatic strain exceeds a critical value. For the convex-concave $g$, the critical values $0<\theta_c^+$ and $\theta^-_c< 0$ beyond which the force begins to soften is related to the inflection point $r^+_\ast$ and $r_\ast^-$ of $g$ as follows
\begin{equation}
\theta^+_c={r^+_\ast}, \qquad \theta^-_c={r_\ast^-}.
\label{crittensileplus theta}
\end{equation}
The critical compressive hydrostatic strain where the force begins to soften for negative hydrostatic strain is chosen much larger in magnitude than $\theta_c^+$, i.e. $\theta^+_c << |\theta_c^-|$. When $g$ is convex-concave we assume it is bounded and has bounded derivatives up to order three. These bounds are denoted by $C^g_i< \infty$ for $i=0,1,2,3$ and,
\begin{align}\label{eq:def Cgi}
C^g_0 := \sup_r |g(r)|, \qquad C^g_i := \sup_r |g^{(i)}(r)| \quad \text{for } i = 1,2,3.
\end{align}
\begin{figure}
\centering
\begin{subfigure}{.45\linewidth}
\begin{tikzpicture}[xscale=0.75,yscale=0.75]
\draw [<->,thick] (0,5) -- (0,0) -- (3.0,0);
\draw [-,thick] (0,0) -- (-3.5,0);
\draw [-,thin] (0,2.15) -- (2.5,2.15);
\draw [-, thin] (-3.5,4.15) -- (0,4.15);
\draw [-,thick] (0,0) to [out=0,in=-175] (2.5,2);
\draw [-,thick] (-3.5,4) to [out=-5,in=180] (0,0);
\draw (1.5,-0.2) -- (1.5, 0.2);
\node [below] at (1.5,-0.2) {${r}^+$};
\draw (-2.25,-0.2) -- (-2.25, 0.2);
\node [below] at (-2.0,-0.2) {${r}^-$};
\node [right] at (3,0) {$r$};
\node [left] at (0,2.250) {$C^+$};
\node [left] at (0.9,4.15) {$C^-$};
\node [above] at (-2.5,2.20) {$f(r)$};
\end{tikzpicture}
\caption{}
\label{ConvexConcavea}
\end{subfigure}
\hskip2em
\begin{subfigure}{.45\linewidth}
\begin{tikzpicture}[xscale=0.6,yscale=0.6]
\draw [<-,thick] (0,3) -- (0,-3);
\draw [->,thick] (-5,0) -- (3.5,0);
\draw [-,thick] (0,0) to [out=65,in=180] (1.5,1.5) to [out=0,in=165] (3,0.25);
\draw [-,thick] (-4.6,-0.5) to [out=-20,in=130] (-3.0,-2.0) to [out=-50, in=245] (0,0);
\draw (1.5,-0.2) -- (1.5, 0.2);
\draw (-2.25,-0.2) -- (-2.25, 0.2);
\node [below] at (1.5,-0.2) {${r}^+$};
\node [below] at (-2.0,-0.2) {${r}^-$};
\node [right] at (3.5,0) {${r}$};
\node [right] at (0,2.2) {$f'(r)$};
\end{tikzpicture}
\caption{}
\label{ConvexConcaveb}
\end{subfigure}
\caption{(a) The potential function $f(r)$ for tensile force. Here $C^+$ and $C^-$ are the two asymptotic values of $f$. (b) Cohesive tensile force.}\label{ConvexConcave}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}{.45\linewidth}
\begin{tikzpicture}[xscale=0.75,yscale=0.75]
\draw [<->,thick] (0,6.5) -- (0,0) -- (2.5,0);
\draw [-,thick] (0,0) -- (-2.5,0);
\draw [-] (0,2.15) -- (2.5,2.15);
\draw [-] (0,6.15) -- (-3.5,6.15);
\draw [-,thick] (0,0) to [out=0,in=-175] (2.5,2);
\draw [-,thick] (-3.5,6) to [out=-5,in=180] (0,0);
\draw [-,thick, dashed] (0,0) parabola (2.5,6);
\draw [-,thick, dashed] (0,0) parabola (-2.5,6);
\draw (1,-0.2) -- (1, 0.2);
\node [below] at (1,-0.2) {${r}_\ast^+$};
\draw (-1.5,-0.2) -- (-1.5, 0.2);
\node [below] at (-1.5,-0.2) {${r}_\ast^-$};
\node [right] at (2.5,0) {$r$};
\node [left] at (0,2.250) {$C^+_\ast$};
\node [left] at (0.9,6.1) {$C^-_\ast$};
\end{tikzpicture}
\caption{}
\label{ConvexConcaveFunctionGa}
\end{subfigure}
\hskip2em
\begin{subfigure}{.45\linewidth}
\begin{tikzpicture}[xscale=0.4,yscale=0.4]
\draw [<-,thick] (0,3) -- (0,-10);
\draw [->,thick] (-6.5,0) -- (4,0);
\draw [-,thick] (-5.5,-10.25) to [out=-30,in=180] (-4.5,-10.5) to [out=0, in=245 ] (0,0) to [out=65,in=180] (1.5,1.5)
to [out=0,in=165] (3,0.25);
\draw [-,thick,dashed] (-3,-9) -- (1,3);
\draw (1.5,-0.2) -- (1.5, 0.2);
\draw (-4.5,-0.2) -- (-4.5, 0.2);
\node [below] at (1.5,-0.2) {${r}_\ast^+$};
\node [below] at (-4.5,-0.2) {${r}_\ast^-$};
\node [right] at (4,0) {${r}$};
\end{tikzpicture}
\caption{}
\label{ConvexConcaveFunctionGb}
\end{subfigure}
\caption{(a) Two types of potential function $g(r)$ for hydrostatic force. The dashed line corresponds to the quadratic potential $g(r) = \beta r^2/2$. The solid line corresponds to the convex-concave type potential $g(r)$. For the convex-concave type potential, there are two special points $r^-_\ast$ and $r^+_\ast$ at which material points start to soften. $C^+_\ast$ and $C^-_\ast$ are two extreme values. (b) Hydrostatic forces. The dashed line corresponds to the quadratic potential and solid line corresponds to the convex-concave potential.}
\label{ConvexConcaveFunctionG}
\end{figure}
\subsection{Peridynamic equation of motion}
The potential energy of the motion is given by
\begin{equation}\label{new peri}
\begin{aligned}
PD^\epsilon(\bolds{u})=\frac{1}{\epsilon^d \omega_d}\int_D \int_{H_\epsilon(\bolds{x})} |\bolds{y}-\bolds{x}|\mathcal{W}^\epsilon(S(\bolds{y},\bolds{x},\bolds{u}(t)))\,d\bolds{y} d\bolds{x}\\
+\int_D \mathcal{V}^\epsilon(\theta(\bolds{x},\bolds{u}(t)))\,d\bolds{x}.
\end{aligned}
\end{equation}
In this treatment the material is assumed homogeneous and the density $\rho$ is constant. We denote the body force by $\bolds{b}(\bolds{x},t)$ and define the {Lagrangian} $${\rm{L}}(\bolds{u},\partial_t \bolds{u},t)=\frac{\rho}{2}||\dot \bolds{u}||^2 _{L^2 (D;\mathbb{R}^d)}-PD^\epsilon(\bolds{u})+\int_D \bolds{b}\cdot \bolds{u} d\bolds{x},$$
where $\dot \bolds{u}=\frac{\partial \bolds{u}}{\partial t}$ is the velocity and $\Vert \dot \bolds{u}\Vert_{L^2(D;\mathbb{R}^d)}$ denotes the $L^2$ norm of the vector field $\dot \bolds{u}: D\rightarrow \mathbb{R}^d$.
Applying the {principal of least action} gives the nonlocal dynamics
\begin{equation}\label{energy based model2}
\begin{aligned}
\rho \ddot{\bolds{u}}(\bolds{x},t)=\mathcal{L}^\epsilon(\bolds{u})(\bolds{x},t)+\bolds{b}(\bolds{x},t),\hbox{ for $\bolds{x}\in D$},
\end{aligned}
\end{equation}
where
\begin{align}\label{eq:total peri force}
\mathcal{L}^\epsilon(\bolds{u})(\bolds{x},t) = \mathcal{L}^\epsilon_T(\bolds{u})(\bolds{x},t) + \mathcal{L}^\epsilon_D(\bolds{u})(\bolds{x},t).
\end{align}
Here $\mathcal{L}^\epsilon_T(\bolds{u})$ is the peridynamic force due to the tensile strain and is given by
\begin{align}\label{nonlocforcetensite}
&\mathcal{L}^\epsilon_T(\bolds{u})(\bolds{x},t) \notag \\
&=\frac{2}{\epsilon^d \omega_d}\int_{H_\epsilon(\bolds{x})} \omega(\bolds{x}) \omega(\bolds{y}) \frac{J^\epsilon(|\bolds{y}-\bolds{x}|)}{\epsilon|\bolds{y}-\bolds{x}|}\partial_S f(\sqrt{|\bolds{y}-\bolds{x}|}S(\bolds{y},\bolds{x},\bolds{u}(t)))\bolds{e}_{\bolds{y}-\bolds{x}}\,d\bolds{y},
\end{align}
and $\mathcal{L}^\epsilon_D(\bolds{u})$ is the peridynamic force due to the hydrostatic strain and is given by
\begin{align}\label{nonlocforcedevia}
&\mathcal{L}^\epsilon_D(\bolds{u})(\bolds{x},t)\notag \\
&=\frac{1}{\epsilon^d \omega_d}\int_{H_\epsilon(\bolds{x})} \omega(\bolds{x}) \omega(\bolds{y}) \frac{J^\epsilon(|\bolds{y}-\bolds{x}|)}{\epsilon^2}\left[\partial_\theta g(\theta(\bolds{y},\bolds{u}(t)))+\partial_\theta g(\theta(\bolds{x},\bolds{u}(t)))\right]\bolds{e}_{\bolds{y}-\bolds{x}}\,d\bolds{y}.
\end{align}
The dynamics is complemented with the initial data
\begin{align}\label{idata}
\bolds{u}(\bolds{x},0)=\bolds{u}_0(\bolds{x}), \qquad \partial_t \bolds{u}(\bolds{x},0)=\bolds{v}_0(\bolds{x}),
\end{align}
and we prescribe zero Dirichlet boundary condition on the boundary $\partial D$
\begin{align}\label{eq:bc}
\bolds{u}(\bolds{x}) = \mathbf{0} \qquad \forall \bolds{x} \in \partial D.
\end{align}
The zero boundary value is extended outside $D$ by zero to $\mathbb{R}^d$. Last we note that since the material is homogeneous we will divide both sides of the equation of motion by $\rho$ and assume, without loss of generality, that $\rho = 1$.
\section{Existence of solutions}\label{ss:existence holder}
Let $\Cholder{D;\bbR^{d}}$ be the H\"{o}lder space with exponent $\gamma \in (0,1]$. We introduce $C_0^{0,\gamma}(D)=C^{0,\gamma}(D)\cap C_0(D)$ where $C_0(D)$ is the closure of continuous functions with compact support on $D$ in the supremum norm. Functions in $C_0(D)$ are uniquely extended to $\overline{D}$ and take zero values on $\partial D$, see \cite{MA-Driver}. In this paper we extend all functions in $C_0^{0,\gamma}(D)$ by zero outside $D$. The norm of $\bolds{u} \in \Cholderz{D;\bbR^{d}}$ is given by
\begin{align*}
\Choldernorm{\bolds{u}}{D;\bbR^{d}} &:= \sup_{\bolds{x} \in D} \abs{\bolds{u}(\bolds{x})} + \left[\bolds{u} \right]_{\Cholder{D;\bbR^{d}}},
\end{align*}
where $\left[\bolds{u} \right]_{\Cholder{D;\bbR^{d}}}$ is the H\"{o}lder semi norm and given by
\begin{align*}
\left[\bolds{u} \right]_{\Cholder{D;\bbR^{d}}} &:= \sup_{\substack{\bolds{x}\neq \bolds{y},\\
\bolds{x},\bolds{y} \in D}} \dfrac{\abs{\bolds{u}(\bolds{x})-\bolds{u}(\bolds{y})}}{\abs{\bolds{x} - \bolds{y}}^\gamma},
\end{align*}
and $\Cholderz{D;\bbR^{d}}$ is a Banach space with this norm. Here we make the hypothesis that the domain function $\omega$ belongs to $C^{0,\gamma}_0(D;[0,1])$.
We consider the first order system of equations equivalent to \autoref{energy based model2}. Let $y_1(t) = \bolds{u}(t)$, $y_2(t) = {\bolds{v}}(t)$ with $\bolds{v}(t) = \dot{\bolds{u}}(t)$. We form the vector $y = (y_1,y_2)^T$ where $y_1,y_2 \in \Cholderz{D;\bbR^{d}}$ and let $F^{\epsilon}(y,t) = (F^{\epsilon}_1(y,t), F^{\epsilon}_2(y,t))^T$ with
\begin{align}
F^\epsilon_1(y,t) &:= y_2 \label{eq:per first order eqn 1} \\
F^\epsilon_2(y, t) &:= \mathcal{L}^\epsilon(y_1(t)) + \bolds{b}(t). \label{eq:per first order eqn 2}
\end{align}
The initial boundary value associated with the evolution \autoref{energy based model2} is equivalent to the initial boundary value problem for the first order system given by
\begin{align}\label{eq:per first order}
\dfrac{d}{dt}y = F^{\epsilon}(y,t),
\end{align}
with initial condition given by $y(0) = (\bolds{u}_0, \bolds{v}_0)^T \in \Cholderz{D;\bbR^{d}}\times\Cholderz{D;\bbR^{d}}$.
We next show that $F^\epsilon(y,t)$ is Lipschitz continuous.
{\vskip 2mm}
\begin{proposition}\label{prop:lipschitz}
\textbf{Lipschitz continuity and bound}\\
Let $X = \Cholderz{D;\bbR^{d}} \times \Cholderz{D;\bbR^{d}}$. We suppose that the boundary function $\omega$ belongs to $C^{0,\gamma}_0(D;[0,1])$. Let $f$ be a convex-concave potential function satisfying \autoref{eq:def Cfi} and let the potential function $g$ either be a quadratic function or be a convex-concave function satisfying \autoref{eq:def Cgi}, then the function $F^\epsilon(y,t) = (F^\epsilon_1, F^\epsilon_2)^T$, as defined in \autoref{eq:per first order eqn 1} and \autoref{eq:per first order eqn 2}, is Lipschitz continuous in any bounded subset of $X$. We have, for any $y,z \in X$ and $t> 0$,
\begin{align}\label{eq:lipschitz property of F}
&\normX{F^{\epsilon}(y,t) - F^{\epsilon}(z,t)}{X} \notag \\
&\leq \frac{L_1 (1+||\omega||_{\Cholder{}}) (1+ ||y||_X + ||z||_X)}{\epsilon^{2+\alpha(\gamma)}} ||y - z||_X.
\end{align}
where $L_1$ is independent of $\bolds{u},\bolds{v}$ and $\epsilon$, and depends on $f$, $J$, and $g$. The exponent $\alpha(\gamma)$ is $0$ if $\gamma \geq 1/2$ and is $1/2 - \gamma$ if $\gamma \leq 1/2$.
Furthermore, for any $y \in X$ and any $t\in [0,T]$, we have the bound
\begin{align}\label{eq:bound on F}
\normX{F^\epsilon(y,t)}{X} &\leq \frac{L_2(1+||\omega||_{\Cholder{}}) (1+||y||_{X})}{\epsilon^2} + b,
\end{align}
where $b = \sup_{t} \Choldernorm{\bolds{b}(t)}{D;\bbR^{d}}$ and $L_2$ is independent of $y$.
\end{proposition}
{\vskip 2mm}
We easily see that on choosing $z=0$ in \autoref{eq:lipschitz property of F} that $\mathcal{L}^\epsilon(\bolds{u}) $ is in $C^{0,\gamma}(D;\mathbb{R}^d)$ provided that $\bolds{u}$ belongs to $C^{0,\gamma}(D;\mathbb{R}^3)$. Moreover since $\mathcal{L}^\epsilon(\bolds{u})$ takes the value $\mathbf{0}$ on $\partial D$ we can conclude that $\mathcal{L}^\epsilon(\bolds{u})$ also belongs to $C^{0,\gamma}_0(D;\mathbb{R}^d)$.
The following theorem gives the existence and uniqueness of solution in any given time domain $I_0 = (-T, T)$.
{\vskip 2mm}
\begin{theorem}\label{thm:existence over finite time domain}
\textbf{Existence and uniqueness of H\"older solutions over finite time intervals}\\
Let $f$ be a convex-concave function satisfying \autoref{eq:def Cfi} and let $g$ either be a quadratic function or a convex-concave function satisfying \autoref{eq:def Cgi}. For any initial condition $x_0\in X ={ \Cholderz{D;\bbR^{d}} \times \Cholderz{D;\bbR^{d}}}$, time interval $I_0=(-T,T)$, and right hand side $\bolds{b}(t)$ continuous in time for $t\in I_0$ such that $\bolds{b}(t)$ satisfies $\sup_{t\in I_0} {||\bolds{b}(t)||_{\Cholder{}}}<\infty$, there is a unique solution $y(t)\in C^1(I_0;X)$ of
\begin{equation}
y(t)=x_0+\int_0^tF^\epsilon(y(\tau),\tau)\,d\tau,
\label{10}
\end{equation}
or equivalently
\begin{equation}
y'(t)=F^\epsilon(y(t),t),\hbox{with $y(0)=x_0$},
\label{11}
\end{equation}
where $y(t)$ and $y'(t)$ are Lipschitz continuous in time for $t\in I_0$.
\end{theorem}
{\vskip 2mm}
The proof of this theorem follows directly from \sautoref{Proposition}{prop:lipschitz} and is established along the same lines as the existence proof for H\"older continuous solutions of bond based peridynamics given in [Theorem 2,\cite{CMPer-JhaLipton}].
We conclude this section by stating following result which shows the Lipschitz bound of peridynamic force in $L^2$ norm for functions in $L^2_0(D; \mathbb{R}^d)$. Here $L^2_0(D;\mathbb{R}^d)$ denotes the space of functions $\bolds{u} \in L^2(D;\mathbb{R}^d)$ such that $\bolds{u} = \mathbf{0}$ on $\partial D$. We assume that functions in $L^2_0(D;\mathbb{R}^d)$ are extended to $\mathbb{R}^d$ by zero.
{\vskip 2mm}
\begin{proposition}\label{prop:lipschitz L2}
\textbf{Lipschitz continuity of peridynamic force in $L^2$}\\
Let $f$ and $g$ satisfy the hypothesis of Proposition \sautoref{Proposition}{prop:lipschitz}, then for any $\bolds{u},\bolds{v} \in L^2_0(D;\mathbb{R}^d)$ we have
\begin{align}\label{eq:lipschitz L2}
||\mathcal{L}^\epsilon(\bolds{u}) - \mathcal{L}^\epsilon(\bolds{v})||_{L^2(D;\mathbb{R}^d)} &\leq \dfrac{L_3}{\epsilon^2} ||\bolds{u} - \bolds{v}||_{L^2(D;\mathbb{R}^d)},
\end{align}
where the constants $L_3$ and $L_4$ are independent of $\epsilon$, $\bolds{u}$ and $\bolds{v}$. Here $L_3 = 4 (C^f_1 \bar{J}_1 + C^g_2 \bar{J}_0^2)$, for convex-concave $g$, and $L_3 = 4 (C^f_1 \bar{J}_1 + g''(0) \bar{J}_0^2)$, for quadratic $g$. Here $\bar{J}_\alpha = \frac{1}{\omega_d} \int_{H_1(\mathbf{0})} J(|{\boldsymbol{\xi}}|) |{\boldsymbol{\xi}}|^{-\alpha} d{\boldsymbol{\xi}}$.
\end{proposition}
{\vskip 2mm}
The proofs of \sautoref{Proposition}{prop:lipschitz} and \sautoref{Proposition}{prop:lipschitz L2} are provided in \autoref{s:proofs}. We now describe the finite difference scheme and analyze the rate of convergence to H\"older continuous solutions of the peridynamic equation of motion.
\section{Finite difference approximation}
\label{s:finite difference}
In this section we consider the discrete approximation to the dynamics given by finite differences in space and the forward Euler discretization in time.
\begin{figure}[h]
\centering
\includegraphics[scale=0.5]{mesh_peridynamic.png}
\caption{(a) Typical mesh of size $h$. (b) Unit cell $U_i$ corresponding to material point $\bolds{x}_i$.}\label{fig:peridynamic mesh}
\end{figure} Let $h$ denote the mesh size and $D_h = D\cap(h \mathbb{Z})^d$ be the associated discretization of the material domain $D$. In this paper we will keep the horizon length scale $\epsilon$ fixed and assume that the spatial discretization length satisfies $h< \epsilon<1$. Let $i\in \mathbb{Z}^d$ be the index such that $\bolds{x}_i = hi \in D$, see \autoref{fig:peridynamic mesh}. Let $U_i$ be a the cell of volume $h^d$ corresponding to the grid point $\bolds{x}_i$. The exact solution evaluated at grid points is denoted by $(\bolds{u}_i(t),\bolds{v}_i(t))$. Given any discrete set $\{ \hat{\bolds{u}}_i \}_{i, \bolds{x}_i\in D}$, where $i$ is index representing grid point of mesh, we define its piecewise constant extension as
\begin{align}\label{eq:const extension u}
\hat{\bolds{u}}(\bolds{x}) := \sum_{i, \bolds{x}_i \in D} \hat{\bolds{u}}_i \chi_{U_i}(\bolds{x}).
\end{align}
In this way we have representation of the discrete set as a piecewise constant function.
We now describe the $L^2$-projection of the function $\bolds{u} : D\to \mathbb{R}^d$ onto the space of piecewise constant functions defined over the cells $U_i$. We denote the average of $\bolds{u}$ over the unit cell $U_i$ as $\tilde{\bolds{u}}_i$ and
\begin{align}
\tilde{\bolds{u}}_i &:= \frac{1}{h^d} \int_{U_i} \bolds{u}(\bolds{x}) d\bolds{x}
\end{align}
and the $L^2$ projection of $\bolds{u}$ onto piecewise constant functions is $\tilde{\bolds{u}}$ given by
\begin{align}\label{eq:projn u}
\tilde{\bolds{u}}(\bolds{x}) &:= \sum_{i,\bolds{x}_i \in D} \tilde{\bolds{u}}_i \chi_{U_i}(\bolds{x}).
\end{align}
\begin{lemma}\label{lem:proj error}
Let $\bolds{u} \in C^{0,\gamma}_0(D;\mathbb{R}^d)$ and let $\tilde{\bolds{u}}$ be its $L^2$ projection defined in \autoref{eq:projn u}, then we have
\begin{align}\label{eq:proj error}
|\tilde{\bolds{u}}(\bolds{x}) - \bolds{u}(\bolds{x})| &\leq \left[c^\gamma ||\bolds{u}||_{\Cholder{}}\right] h^\gamma, \forall \bolds{x} \in D, \notag \\
||\tilde{\bolds{u}}(\bolds{x}) - \bolds{u}(\bolds{x})||_{L^2} &\leq \left[c^\gamma \sqrt{|D|} ||\bolds{u}||_{\Cholder{}} \right] h^\gamma,
\end{align}
where $c = \sqrt{2}$ for $d=2$ and $c = \sqrt{3}$ for $d=3$.
\end{lemma}
This lemma can be demonstrated easily by substituting \autoref{eq:projn u} for $\tilde{\bolds{u}}$ and using the fact that $\bolds{u} \in \Cholderz{D;\bbR^{d}}$. We also note that first line of \autoref{eq:proj error} remains valid of $\bolds{x}$ in a layer of thickness $2\epsilon$ surrounding $D$.
\subsection{Stability of the semi-discrete approximation}
\label{semidiscrete}
We first introduce the semi-discrete boundary condition by setting $\hat{\bolds{u}}_i(t) = \mathbf{0}$ for all $t$ and for all $\bolds{x}_i \notin D$.
Let $\{\hat{\bolds{u}}_i(t)\}_{i,\bolds{x}_i\in D}$ denote the semi-discrete approximate solution which satisfies the following, for all $t\in [0,T]$ and $i$ such that $\bolds{x}_i\in D$,
\begin{align}\label{eq:fd semi discrete}
\ddot{\hat{\bolds{u}}}_i(t) = \mathcal{L}^\epsilon(\hat{\bolds{u}}(t))(\bolds{x}_i) + \bolds{b}(\bolds{x}_i,t),
\end{align}
where $\hat{\bolds{u}}(t)$ is the piecewise constant extension of discrete set $\{\hat{\bolds{u}}_i(t) \}_i$ and is defined as
\begin{equation}
\label{eq:def piecewise ext}
\hat{\bolds{u}}(\bolds{x},t) =
\begin{cases}
\sum_{i, \bolds{x}_i \in D} \hat{\bolds{u}}_i(t) \chi_{U_i}(\bolds{x}),\\
\mathbf{0},\,\,\,i,\hbox{ such that }\bolds{x}_i\not\in D.
\end{cases}
\end{equation}
The scheme is complemented with the discretized initial conditions $\hat{\bolds{u}}_i(0) = \bolds{u}_0(\bolds{x}_i)$ and $\hat{\bolds{v}}_i(0) =\bolds{v}_0(\bolds{x}_i)$.
The total kinetic and potential energy is given by
\begin{align*}
\mathcal{E}^\epsilon(\bolds{u})(t) = \frac{1}{2} ||\dot{\bolds{u}}(t)||^2_{L^2} + PD^\epsilon(\bolds{u}(t)),
\end{align*}
and we introduce the augmented energy given by
\begin{align}\label{eq:per energy total new}
\bar{\mathcal{E}}^\epsilon(\bolds{u})(t) := \mathcal{E}^\epsilon(\bolds{u})(t) + \frac{1}{2}||\bolds{u}(t)||^2_{L^2}.
\end{align}
We have the stability of the semi-discrete evolution.
{\vskip 2mm}
\begin{theorem}\label{thm:stab semi}
\textbf{Energy stability of the semi-discrete approximation}\\
Let $\{\hat{\bolds{u}}_i(t) \}_{i,\bolds{x}_i\in D}$ be the solution to the semidiscrete initial boundary value problem \autoref{eq:fd semi discrete} and $\hat{\bolds{u}}(t)$ denote its piecewise constant extension. Similarly let $\hat{\bolds{b}}(t,\bolds{x})$ denote the piecewise constant extension of $\{ \bolds{b}(t,\bolds{x}_i)\}_{i,\bolds{x}_i\in D}$.
If $f$ and $g$ are convex-concave type functions satisfying \autoref{eq:def Cfi} and \autoref{eq:def Cgi}, then the total energy $\mathcal{E}^\epsilon(\hat{\bu})(t)$ satisfies,
\begin{align}\label{eq:inequal energy doublewell g}
\mathcal{E}^\epsilon(\hat{\bolds{u}})(t) &\leq \left( \sqrt{\mathcal{E}^\epsilon(\hat{\bolds{u}})(0)} + \dfrac{t C}{\epsilon^2} + \int_0^t ||\hat{\bolds{b}}(s)||_{L^2} ds \right)^2, \hbox{ $\forall t \in [0,T]$,}
\end{align}
and the constant $C$ is independent of $\epsilon$ and $h$.
If $f$ is a convex-concave type function satisfying \autoref{eq:def Cfi} and $g$ is quadratic then the augmented energy $\bar{\mathcal{E}}^\epsilon(\hat{\bu})(t)$ satisfies,
\begin{align}\label{eq:inequal energy quadratic g}
\bar{\mathcal{E}}^\epsilon(\hat{\bolds{u}})(t) &\leq \exp[3(C_2/\epsilon^2 +1) t] \bigg( \bar{\mathcal{E}}^\epsilon(\hat{\bolds{u}})(0) \notag \\
&\quad \quad + \int_0^T (\frac{C_1^2}{\epsilon^4} + ||\hat{\bolds{b}}(s)||^2_{L^2}) \exp[-3(C_2/\epsilon^2 +1) s] ds \bigg),\hbox{ $\forall t \in [0,T]$},
\end{align}
where the constants $C_1$ and $C_2$ are independent of $\epsilon$ and $h$.
\end{theorem}
{\vskip 2mm}
We provide proof of \autoref{thm:stab semi} in \autoref{ss:stab proof}. We now discuss the fully discrete scheme.
\subsection{Time discretization}
\label{time discrete}
Let $\Delta t$ be the size of the time step and $[0,T] \cap (\Delta t \mathbb{Z})$ be the discretization of the time domain. We denote the fully discrete solution at $(t^k = k\Delta t, \bolds{x}_i = ih)$ as $(\hat{\bolds{u}}^k_{i}, \hat{\bolds{v}}^k_i)$ and the exact solution as $(\bolds{u}^k_i,\bolds{v}^k_i)$. We enforce the boundary condition $\hat{\bolds{u}}^k_i = \mathbf{0}$ for all $\bolds{x}_i \notin D$ and for all $k$. The piecewise constant extension of $\{\hat{\bolds{u}}^k_i\}_{i\in \mathbb{Z}^d}$ and $\{\hat{\bolds{v}}^k_i\}_{i\in \mathbb{Z}^d}$ are denoted by $\hat{\bolds{u}}^k$ and $\hat{\bolds{v}}^k$ respectively. The $L^2$-projection of $\bolds{u}^k$ and $\bolds{v}^k$ onto piecewise constant functions are denoted by $\tilde{\bolds{u}}^k$ and $\tilde{\bolds{v}}^k$ respectively.
The forward Euler time discretization, with respect to velocity, and the finite difference scheme for $(\hat{\bolds{u}}^k_{i}, \hat{\bolds{v}}^k_i)$ is written
\begin{align}
\dfrac{\hat{\bu}^{k+1}_i - \hat{\bu}^k_i}{\Delta t} &= \hat{\bv}^{k+1}_i \label{eq:finite diff eqn u} \\
\dfrac{\hat{\bv}^{k+1}_i - \hat{\bv}^k_i}{\Delta t} &= \mathcal{L}^\epsilon(\hat{\bolds{u}}^k)(\bolds{x}_i) + \bolds{b}^k_i. \label{eq:finite diff eqn v}
\end{align}
The initial condition is enforced by setting $\hat{\bu}^{0}_i =(\hat{\bu}_0)_i$ and $\hat{\bv}^{0}_i =(\hat{\bv}_0)_i$. We note that the forward difference scheme for the system reduces to the central difference scheme for the second order differential equation \autoref{energy based model2} on substitution of \autoref{eq:finite diff eqn u} into \autoref{eq:finite diff eqn v}.
\subsubsection{Convergence of approximation}
In this section we provide an upper bound on the convergence rate of the fully discrete approximation to the H\"older continuous solution as measured by the $L^2$ norm.
The $L^2$ approximation error $E^k$ at time $t^k$, for $0< t^k \leq T$, given by
\begin{align*}
E^k &:= \Ltwonorm{\hat{\bu}^k - \bolds{u}^k}{D;\bbR^{d}} + \Ltwonorm{\hat{\bv}^k- \bolds{v}^k}{D;\bbR^{d}} .
\end{align*}
The following theorem gives an explicit a-priori upper bound on the convergence rate.
{\vskip 2mm}
\begin{theorem}\label{thm:convergence}
\textbf{Convergence of finite difference approximation (forward Euler time discretization)}\\
Let $\epsilon>0$ be fixed. Let $(\bolds{u}, \bolds{v})$ be the solution of peridynamic equation \autoref{eq:per first order}. We assume $\bolds{u}, \bolds{v} \in \Ctwointime{[0,T]}{\Cholderz{D;\bbR^{d}}}$. Then the finite difference scheme given by \autoref{eq:finite diff eqn u} and \autoref{eq:finite diff eqn v} is consistent in both time and spatial discretization and converges to the exact solution uniformly in time with respect to the $\Ltwo{D;\bbR^{d}}$ norm. If we assume the error at the initial step is zero then the error $E^k$ at time $t^k$ is bounded and satisfies
\begin{align}\label{eq: first est}
\sup_{0\leq k \leq T/\Delta t} E^k\leq O\left( C_t\Delta t + C_s\dfrac{h^\gamma}{\epsilon^2} \right),
\end{align}
where constant $C_s$ and $C_t$ are independent of $h$ and $\Delta t$ and $C_s$ depends on the H\"older norm of the solution and $C_t$ depends on the $L^2$ norms of time derivatives of the solution.
\end{theorem}
{\vskip 2mm}
Here we have assumed the initial error is zero for ease of exposition only.
We remark that the explicit constants leading to \autoref{eq: first est} can be large. The inequality that delivers \autoref{eq: first est} is given by
\begin{align}\label{eq: fund est initial}
\sup_{0\leq k \leq T/\Delta t} E^k\leq \exp \left[T (1 + L_3/\epsilon^2) \right] T \left[ C_t \Delta t + (C_s/\epsilon^2) h^\gamma \right],
\end{align}
where the constants $L_3$, $C_t$ and $C_s$ are given by \autoref{eq:def L3}, \autoref{eq:const Ct}, and \autoref{eq:const Cs}. The explicit constant $C_t$ depends on the spatial $L^2$ norm of the time derivatives of the solution and $C_s$ depends on the spatial H\"older continuity of the solution and the constant $L_3$. The constant $L_3$ is bounded independently of horizon $\epsilon$. Although the constants are necessarily pessimistic they deliver a-priori error estimates. We carry out numerical simulations for different values of the horizon $\epsilon$ in \autoref{s:numerical}. We find that the convergence rate for piecewise constant finite difference interpolation functions is greater than or equal to $\gamma=1$ for simulations lasting in the tens of microseconds. These results are consistent with the a-priori estimates given in \autoref{thm:convergence} above.
\subsubsection{Error analysis}\label{ss:error analysis1}
We split the error between $(\hat{\bu}^k, \hat{\bv}^k)^T$ and $(\bolds{u}^k,\bolds{v}^k)^T$ in two parts as follows
\begin{align}
E^k &= ||\hat{\bu}^k - \bolds{u}^k||_{L^2} + ||\hat{\bv}^k- \bolds{v}^k||_{L^2} \notag \\
&\leq \left[ ||\tilde{\bu}^k - \bolds{u}^k||_{L^2} + ||\tilde{\bv}^k- \bolds{v}^k||_{L^2} \right] + \left[ ||\hat{\bu}^k - \tilde{\bu}^k||_{L^2} + ||\hat{\bv}^k- \tilde{\bv}^k||_{L^2} \right].
\end{align}
In section \autoref{ss:error analysis} we will show that the error between the $L^2$ projections of the actual solution and the discrete approximation for both forward Euler and implicit one step methods decay according to
\begin{align}\label{eq:write estimate error ek}
\sup_{0\leq k \leq T/\Delta t} \left( ||\hat{\bu}^k - \tilde{\bu}^k||_{L^2} + ||\hat{\bv}^k- \tilde{\bv}^k||_{L^2} \right) &= O\left( \Delta t + \dfrac{h^\gamma}{\epsilon^2} \right).
\end{align}
And using \sautoref{Lemma}{lem:proj error} we have
\begin{align*}
& \sup_{k} \left( ||\tilde{\bu}^k - \bolds{u}^k||_{L^2} + ||\tilde{\bv}^k- \bolds{v}^k||_{L^2} \right) \notag \\
&= c^\gamma \sqrt{|D|} \left[\sup_{t\in [0,T]} ||\bolds{u}(t)||_{\Cholder{}} + \sup_{t\in [0,T]} ||\bolds{v}(t)||_{\Cholder{}} \right] h^\gamma.
\end{align*}
We now study the difference $\hat{\bu}^k - \tilde{\bu}^k$ and $\hat{\bv}^k - \tilde{\bv}^k$.
\subsubsection{Error analysis for approximation of $L^2$ projection of the exact solution}\label{ss:error analysis}
Let the differences be denoted by $\bolds{e}^k(u) := \hat{\bu}^k - \tilde{\bu}^k$ and $\bolds{e}^k(v ):= \hat{\bv}^k - \tilde{\bv}^k$ and their evaluation at grid points are $\bolds{e}^k_i(u) := \hat{\bu}^k_i - \tilde{\bu}^k_i$ and $\bolds{e}^k_i(v) := \hat{\bv}^k_i - \tilde{\bv}^k_i$. We have the following lemma for the evolution of the differences in the discrete dynamics.
\begin{lemma}
\label{lem: discretedifferences}
The differences $\bolds{e}^k_i(u)$ and $\bolds{e}^k_i(v)$ discretely evolve according to the equations:
\begin{align}\label{eq:error eqn in u}
\bolds{e}^{k+1}_i(u) &= \bolds{e}^k_i(u) + \Delta t \bolds{e}^{k+1}_i(v) + \Delta t\tau^{k}_i(u)
\end{align}
and
\begin{align}\label{eq:error eqn in v}
\bolds{e}^{k+1}_i(v) &= \bolds{e}^k_i(v) + \Delta t \left(\tau^k_i(v) + \sigma^k_i(u) + \sigma^k_i(v) \right) \notag \\
&\quad + \Delta t \left( \mathcal{L}^{\epsilon}(\hat{\bu}^k)(\bolds{x}_i) - \mathcal{L}^{\epsilon}(\tilde{\bu}^k)(\bolds{x}_i) \right).
\end{align}
Here $\tau^k_i(u),\tau^k_i(v)$ and $\sigma^k_i(u),\sigma^k_i(v)$ are consistency error terms and are defined as
\begin{align}\label{eq:def consistency error}
\tau^k_i(u) &:= \dparder{\tilde{\bu}^k_i}{t} - \dfrac{\tilde{\bu}^{k+1}_i - \tilde{\bu}^k_i}{\Delta t}, \notag \\
\tau^k_i(v) &:= \dparder{\tilde{\bv}^k_i}{t} - \dfrac{\tilde{\bv}^{k+1}_i - \tilde{\bv}^k_i}{\Delta t} , \notag \\
\sigma^k_i(u) &:= \left( \mathcal{L}^{\epsilon}(\tilde{\bu}^k)(\bolds{x}_i) - \mathcal{L}^{\epsilon}(\bolds{u}^k)(\bolds{x}_i) \right) \notag \\
\sigma^k_i(v) &:= \dparder{\bolds{v}^k_i}{t} - \dparder{\tilde{\bv}^k_i}{t}.
\end{align}
\end{lemma}
To prove this we start by subtracting $(\tilde{\bu}^{k+1}_i - \tilde{\bu}^k_i)/\Delta t$ from \autoref{eq:finite diff eqn u} to get
\begin{align*}
& \dfrac{\hat{\bu}^{k+1}_i - \hat{\bu}^k_i}{\Delta t} - \dfrac{\tilde{\bu}^{k+1}_i - \tilde{\bu}^k_i}{\Delta t} \\
&= \hat{\bv}^{k+1}_i - \dfrac{\tilde{\bu}^{k+1}_i - \tilde{\bu}^k_i}{\Delta t} \notag \\
&= \hat{\bv}^{k+1}_i - \tilde{\bv}^{k+1}_i + \left( \tilde{\bv}^{k+1}_i - \dparder{\tilde{\bu}^{k+1}_i}{t} \right) + \left( \dparder{\tilde{\bu}^{k+1}_i}{t} - \dfrac{\tilde{\bu}^{k+1}_i - \tilde{\bu}^k_i}{\Delta t} \right).
\end{align*}
Taking the average over unit cell $U_i$ of the exact peridynamic equation \autoref{eq:per first order} at time $t^k$, we will get $\tilde{\bv}^{k+1}_i - \dparder{\tilde{\bu}^{k+1}_i}{t} = 0$ and we recover \autoref{eq:error eqn in u}.
Next, we subtract $(\tilde{\bv}^{k+1}_i - \tilde{\bv}^k_i)/\Delta t$ from \autoref{eq:finite diff eqn v} and add and subtract terms to get
\begin{align}\label{eq:error in v 1}
\dfrac{\hat{\bv}^{k+1}_i - \hat{\bv}^k_i}{\Delta t} - \dfrac{\tilde{\bv}^{k+1}_i - \tilde{\bv}^k_i}{\Delta t} &= \mathcal{L}^\epsilon(\hat{\bolds{u}}^k)(\bolds{x}_i) + \bolds{b}^k_i - \dparder{\bolds{v}^k_i}{t} + \left( \dparder{\bolds{v}^k_i}{t} - \dfrac{\tilde{\bv}^{k+1}_i - \tilde{\bv}^k_i}{\Delta t}\right) \notag \\
&= \mathcal{L}^\epsilon(\hat{\bolds{u}}^k)(\bolds{x}_i) + \bolds{b}^k_i - \dparder{\bolds{v}^k_i}{t} \notag \\
&\quad + \left( \dparder{\tilde{\bv}^k_i}{t} - \dfrac{\tilde{\bv}^{k+1}_i - \tilde{\bv}^k_i}{\Delta t}\right) + \left( \dparder{\bolds{v}^k_i}{t} - \dparder{\tilde{\bv}^k_i}{t}\right),
\end{align}
where from \autoref{eq:def consistency error}
\begin{align}\label{eq:consistency error in v}
\tau^k_i(v) &:= \dparder{\tilde{\bv}^k_i}{t} - \dfrac{\tilde{\bv}^{k+1}_i - \tilde{\bv}^k_i}{\Delta t}.
\end{align}
Note that from the exact peridynamic equation, we have
\begin{align}\label{eq:exact per eqn v 1}
\bolds{b}^k_i - \dparder{\bolds{v}^k_i}{t} =-\mathcal{L}^\epsilon({\bolds{u}}^k)(\bolds{x}_i) .
\end{align}
Combining \autoref{eq:error in v 1}, \autoref{eq:consistency error in v}, and \autoref{eq:exact per eqn v 1}, gives
\begin{align*}
\bolds{e}^{k+1}_i(v) &= \bolds{e}^k_i(v) + \Delta t \tau^k_i(v) + \Delta t \left( \dparder{\bolds{v}^k_i}{t} - \dparder{\tilde{\bv}^k_i}{t}\right) \notag \\
&\quad + \Delta t \left( \mathcal{L}^\epsilon(\hat{\bolds{u}}^k)(\bolds{x}_i) -\mathcal{L}^\epsilon({\bolds{u}}^k)(\bolds{x}_i) \right) \notag \\
&= \bolds{e}^k_i(v) + \Delta t \tau^k_i(v) + \Delta t \left( \dparder{\bolds{v}^k_i}{t} - \dparder{\tilde{\bv}^k_i}{t}\right) \notag \\
&\quad + \Delta t \left( \mathcal{L}^\epsilon(\hat{\bolds{u}}^k)(\bolds{x}_i) -\mathcal{L}^\epsilon(\tilde{\bolds{u}}^k)(\bolds{x}_i) \right) \notag \\
&\quad + \Delta t \left( \mathcal{L}^\epsilon(\tilde{\bolds{u}}^k)(\bolds{x}_i) - \mathcal{L}^\epsilon({\bolds{u}}^k)(\bolds{x}_i) \right).
\end{align*}
and the lemma follows on applying the definitions given in \autoref{eq:def consistency error}.
\subsubsection{Consistency}\label{sss:consistency}
In this section we provide upper bounds on the consistency errors. This error is measured in the $L^2$ norm. Here the upper bound on the consistency error with respect to time follows using Taylor's series expansion. The upper bound on the spatial consistency error is established using the H\"older continuity of nonlocal forces.
\textbf{Time discretization: } We apply a Taylor series expansion in time to estimate $\tau^k_i(u)$ as follows
\begin{align*}
\tau^k_i(u) &= \dfrac{1}{h^d} \int_{U_i} \left( \dparder{\bolds{u}^k(\bolds{x})}{t} - \dfrac{\bolds{u}^{k+1}(\bolds{x}) - \bolds{u}^k(\bolds{x})}{\Delta t} \right) d\bolds{x} \\
&= \dfrac{1}{h^d} \int_{U_i} \left( -\dfrac{1}{2} \dsecder{\bolds{u}^k(\bolds{x})}{t} \Delta t + O((\Delta t)^2) \right) d\bolds{x} .
\end{align*}
We form the $\Ltwo{}$ norm of $\tau^k_i(u)$ and apply Jensen's inequality to get
\begin{align*}
\Ltwonorm{\tau^k(u)}{} &\leq \frac{\Delta t}{2} \Ltwonorm{\dsecder{\bolds{u}^k}{t}}{} + O((\Delta t)^2) \notag \\
&\leq \frac{\Delta t}{2} \sup_{t} \Ltwonorm{\dsecder{\bolds{u}(t)}{t}}{} + O((\Delta t)^2).
\end{align*}
A similar argument gives
\begin{align*}
\Ltwonorm{\tau^k(v)}{} = \frac{\Delta t}{2} \sup_{t} \Ltwonorm{\dsecder{\bolds{v}(t)}{t}}{} + O((\Delta t)^2).
\end{align*}
\
\textbf{Spatial discretization: }From \autoref{eq:def consistency error} one can write
\begin{align*}
\sigma^k_i(v) &= \frac{\partial \bolds{v}^k_i}{\partial t} - \frac{\partial \tilde{\bv}^k_i}{\partial t} = \frac{\partial \bolds{v}^k(\bolds{x}_i)}{\partial t} - \frac{\partial \tilde{\bv}^k(\bolds{x}_i)}{\partial t}.
\end{align*}
Applying \sautoref{Lemma}{lem:proj error} gives
\begin{align*}
|\sigma^k_i(v)| &\leq c^\gamma h^\gamma \Choldernorm{\dparder{\bolds{v}^k}{t}}{} \leq c^\gamma h^\gamma \sup_t \Choldernorm{\dparder{\bolds{v}(t)}{t}}{}.
\end{align*}
Taking the $L^2$ norm and using the estimates given above yields the inequality
\begin{align*}
\Ltwonorm{\sigma^k(v)}{} &\leq h^{\gamma} c^{\gamma} \sqrt{ \abs{D} } \sup_{t} \Choldernorm{\dparder{\bolds{v}(t)}{t}}{}.
\end{align*}
We now estimate $\abs{\sigma^k_i(u)}$. Since $\mathcal{L}^\epsilon = \mathcal{L}^\epsilon_T + \mathcal{L}^\epsilon_D$, we have from \autoref{eq:def consistency error}
\begin{align}\label{eq:estimate sigma u 1}
|\sigma^k_i(u)| &\leq |\mathcal{L}^\epsilon_T(\tilde{\bu}^k)(\bolds{x}_i) - \mathcal{L}^\epsilon_T(\bolds{u}^k)(\bolds{x}_i)| + |\mathcal{L}^\epsilon_D(\tilde{\bu}^k)(\bolds{x}_i) - \mathcal{L}^\epsilon_D(\bolds{u}^k)(\bolds{x}_i)| \notag \\
&= I_1 + I_2
\end{align}
To expedite the calculations we employ the following notation for ${\boldsymbol{\xi}} \in H_1(\mathbf{0})$,
\begin{align}\label{eq:notations consist}
s_{\boldsymbol{\xi}} &:= \epsilon |{\boldsymbol{\xi}}|, \: \bolds{e}_{\boldsymbol{\xi}} := \frac{{\boldsymbol{\xi}}}{|{\boldsymbol{\xi}}|}, \notag \\
\omega_{\boldsymbol{\xi}}(\bolds{x}) &:= \omega(\bolds{x} + \epsilon {\boldsymbol{\xi}}) \omega(\bolds{x}), \notag \\
\bar{\bu}_{\boldsymbol{\xi}}(\bolds{x}) &:= \bolds{u}(\bolds{x}+ \epsilon {\boldsymbol{\xi}}) - \bolds{u}(\bolds{x}) .
\end{align}
We also write hydrostatic strain (see \autoref{sphericalstrain}) as follows
\begin{align}\label{eq:hydro strain}
\theta(\bolds{x};\bolds{u}) = \frac{1}{\omega_d} \int_{H_1(\mathbf{0})}\omega(\bolds{x} + \epsilon{\boldsymbol{\xi}}) J(|{\boldsymbol{\xi}}|)\bar{\bu}_{\boldsymbol{\xi}}(\bolds{x})\cdot \bolds{e}_{\boldsymbol{\xi}} d{\boldsymbol{\xi}}.
\end{align}
In our calculations we will also encounter various moments of influence function $J$ therefore we define following term
\begin{align}\label{eq:def bar J}
\bar{J}_\alpha := \frac{1}{\omega_d} \int_{H_1(\mathbf{0})} J(|{\boldsymbol{\xi}}|) |{\boldsymbol{\xi}}|^{-\alpha} d{\boldsymbol{\xi}}, \qquad \text{ for } \alpha \in \mathbb{R}.
\end{align}
Recall that $J(|{\boldsymbol{\xi}}|) = 0$ for ${\boldsymbol{\xi}} \notin H_1(\mathbf{0})$ and $0\leq J(|{\boldsymbol{\xi}}|) \leq M$ for ${\boldsymbol{\xi}}\in H_1(\mathbf{0})$.
Applying the notation $\mathcal{L}^\epsilon_T$ becomes
\begin{align}\label{eq:per bondbased simple force expression}
\mathcal{L}^\epsilon_T(\bolds{u})(\bolds{x}) &= \frac{2}{\epsilon \omega_d} \int_{H_1(\mathbf{0})} \omega_{\boldsymbol{\xi}}(\bolds{x}) \frac{J(|{\boldsymbol{\xi}}|)}{\sqrt{s_{\boldsymbol{\xi}}}} f'(\bar{\bu}_{\boldsymbol{\xi}}(\bolds{x})\cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}}) \bolds{e}_{{\boldsymbol{\xi}}} d{\boldsymbol{\xi}}.
\end{align}
On choosing $\bolds{u} = \bolds{u}^k$ and $\bolds{u}= \tilde{\bu}^k$ in $\mathcal{L}^\epsilon_T$ given by \autoref{eq:per bondbased simple force expression} we get
\begin{align}\label{eq:consist est 0}
I_1 &\leq \frac{2}{\epsilon \omega_d} \int_{H_1(\mathbf{0})} \omega_{\boldsymbol{\xi}}(\bolds{x}_i) \frac{J(|{\boldsymbol{\xi}}|)}{\sqrt{s_{\boldsymbol{\xi}}}} |f'(\bar{\tilde{\bu}}^k_{\boldsymbol{\xi}}(\bolds{x}_i) \cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}}) - f'(\bar{\bolds{u}}^k_{\boldsymbol{\xi}}(\bolds{x}_i) \cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}})| d{\boldsymbol{\xi}} \notag \\
&\leq \frac{2C^f_2}{\epsilon \omega_d} \int_{H_1(\mathbf{0})} \frac{J(|{\boldsymbol{\xi}}|)}{s_{\boldsymbol{\xi}}} |\bar{\tilde{\bu}}^k_{\boldsymbol{\xi}}(\bolds{x}_i)- \bar{\bolds{u}}^k_{\boldsymbol{\xi}}(\bolds{x}_i)| d{\boldsymbol{\xi}},
\end{align}
where we have applied \autoref{eq:def Cfi} and used the fact that $|f'(r_1) - f'(r_2)| \leq C^f_2 |r_1 - r_2|$ and $0\leq \omega(\bolds{x}) \leq 1$. We use \sautoref{Lemma}{lem:proj error} to estimate $|\bar{\tilde{\bu}}^k_{\boldsymbol{\xi}}(\bolds{x}_i)- \bar{\bolds{u}}^k_{\boldsymbol{\xi}}(\bolds{x}_i)|$ as follows
\begin{align}
|\bar{\tilde{\bu}}^k_{\boldsymbol{\xi}}(\bolds{x}_i)- \bar{\bolds{u}}^k_{\boldsymbol{\xi}}(\bolds{x}_i)| &\leq |\tilde{\bu}^k(\bolds{x}_i + \epsilon {\boldsymbol{\xi}})- \bolds{u}^k(\bolds{x}_i+ \epsilon{\boldsymbol{\xi}})| + |\tilde{\bu}^k(\bolds{x}_i )- \bolds{u}^k(\bolds{x}_i)| \notag \\
&\leq 2 c^\gamma ||\bolds{u}(t^k)||_{\Cholder{}} h^\gamma \leq 2 c^\gamma \sup_t ||\bolds{u}(t)||_{\Cholder{}} h^\gamma.
\end{align}
From this we get
\begin{align}
I_1 &\leq \left[\frac{4C^f_2 c^\gamma \bar{J}_1}{\epsilon^2} \sup_t ||\bolds{u}(t)||_{\Cholder{}} \right] h^\gamma,
\end{align}
where $\bar{J}_\alpha$ for $\alpha \in \mathbb{R}$ is defined in \autoref{eq:def bar J}. Clearly,
\begin{align}\label{eq:consist est 1}
\sum_{i,\bolds{x}_i \in D} h^d I_1^2 &\leq \left[\frac{4C^f_2 c^\gamma \bar{J}_1 \sqrt{|D|}}{\epsilon^2} \sup_t ||\bolds{u}(t)||_{\Cholder{}} \right]^2 h^{2\gamma} .
\end{align}
We now estimate $I_2$ in \autoref{eq:estimate sigma u 1}. We will consider $g$ of convex-concave type satisfying $C^g_i < \infty$ for $i=0,1,2,3$ where $C^g_0 = \sup |g(r)|$ and $C^g_i = \sup |g^{(i)}(r)|$ for $i=1,2,3$. It is noted that the upper bound for the choice of quadratic $g$ is also found using the steps presented here. We can write $\mathcal{L}^\epsilon_D(\bolds{u})(\bolds{x})$ (see \autoref{nonlocforcedevia}) as follows
\begin{align}\label{eq:per statebased simple force expression}
\mathcal{L}^\epsilon_D(\bolds{u})(\bolds{x}) &= \frac{1}{\epsilon^2 \omega_d} \int_{H_1(\mathbf{0})} \omega_{\boldsymbol{\xi}}(\bolds{x}) J(|{\boldsymbol{\xi}}|) [g'(\theta(\bolds{x}+\epsilon{\boldsymbol{\xi}};\bolds{u})) + g'(\theta(\bolds{x};\bolds{u}))] \bolds{e}_{\boldsymbol{\xi}} d{\boldsymbol{\xi}}.
\end{align}
Using this expression we have the upper bound
\begin{align}\label{eq:consist est 2}
I_2 &= \bigg\vert \frac{1}{\epsilon^2 \omega_d} \int_{H_1(\mathbf{0})} \omega_{\boldsymbol{\xi}}(\bolds{x}_i) J(|{\boldsymbol{\xi}}|) [g'(\theta(\bolds{x}_i+\epsilon{\boldsymbol{\xi}};\tilde{\bu}^k)) + g'(\theta(\bolds{x}_i;\tilde{\bu}^k)) \notag \\
&\quad \quad - g'(\theta(\bolds{x}_i+\epsilon{\boldsymbol{\xi}};\bolds{u}^k)) + g'(\theta(\bolds{x}_i;\bolds{u}^k))] \bolds{e}_{\boldsymbol{\xi}} d{\boldsymbol{\xi}} \bigg\vert \notag \\
&\leq \frac{1}{\epsilon^2 \omega_d} \int_{H_1(\mathbf{0})} J(|{\boldsymbol{\xi}}|) (|g'(\theta(\bolds{x}_i+\epsilon{\boldsymbol{\xi}};\tilde{\bu}^k)) - g'(\theta(\bolds{x}_i+\epsilon {\boldsymbol{\xi}};\bolds{u}^k))| \notag \\
&\quad \quad + |g'(\theta(\bolds{x}_i;\tilde{\bu}^k)) - g'(\theta(\bolds{x}_i;\bolds{u}^k))|) d{\boldsymbol{\xi}} \notag \\
&\leq \frac{C^g_2}{\epsilon^2 \omega_d} \int_{H_1(\mathbf{0})} J(|{\boldsymbol{\xi}}|) (|\theta(\bolds{x}_i+\epsilon{\boldsymbol{\xi}};\tilde{\bu}^k) - \theta(\bolds{x}_i+\epsilon {\boldsymbol{\xi}};\bolds{u}^k)| \notag \\
&\quad \quad + |\theta(\bolds{x}_i;\tilde{\bu}^k) - \theta(\bolds{x}_i;\bolds{u}^k)|) d{\boldsymbol{\xi}} .
\end{align}
We proceed further as follows using expression of $\theta$ in \autoref{eq:hydro strain}
\begin{align}
& |\theta(\bolds{x}_i+\epsilon{\boldsymbol{\xi}};\tilde{\bu}^k) - \theta(\bolds{x}_i+\epsilon {\boldsymbol{\xi}};\bolds{u}^k)| \notag \\
&\leq \bigg\vert \frac{1}{\omega_d} \int_{H_1(\mathbf{0})} \omega(\bolds{x}_i + \epsilon {\boldsymbol{\xi}} + \epsilon \boldsymbol{\eta}) J(|\boldsymbol{\eta}|) (\tilde{\bu}^k(\bolds{x}_i + \epsilon {\boldsymbol{\xi}} + \epsilon \boldsymbol{\eta}) \notag \\
&\quad\quad - \bolds{u}^k(\bolds{x}_i + \epsilon {\boldsymbol{\xi}} + \epsilon \boldsymbol{\eta}) - \tilde{\bu}^k(\bolds{x}_i + \epsilon {\boldsymbol{\xi}}) + \bolds{u}^k(\bolds{x}_i + \epsilon {\boldsymbol{\xi}}))\cdot \bolds{e}_{\boldsymbol{\eta}} d\boldsymbol{\eta} \bigg\vert \notag \\
&\leq \frac{1}{\omega_d} \int_{H_1(\mathbf{0})} J(|\boldsymbol{\eta}|) (|\tilde{\bu}^k(\bolds{x}_i + \epsilon {\boldsymbol{\xi}} + \epsilon \boldsymbol{\eta}) - \bolds{u}^k(\bolds{x}_i + \epsilon {\boldsymbol{\xi}} + \epsilon \boldsymbol{\eta})| \notag \\
&\quad \quad + |\tilde{\bu}^k(\bolds{x}_i + \epsilon {\boldsymbol{\xi}}) - \bolds{u}^k(\bolds{x}_i + \epsilon {\boldsymbol{\xi}})|)d\boldsymbol{\eta} \notag \\
&\leq 2 c^\gamma h^\gamma \sup_t ||\bolds{u}(t)||_{\Cholder{}} \bar{J}_0
\end{align}
where we used \sautoref{Lemma}{lem:proj error} in last step. We combine above estimate in \autoref{eq:consist est 2} to get
\begin{align}
I_2 &\leq \left[ \frac{4C^g_2c^\gamma \bar{J}_0^2 }{\epsilon^2} \sup_t ||\bolds{u}(t)||_{\Cholder{}} \right] h^\gamma
\end{align}
and
\begin{align}\label{eq:consist est 3}
\sum_{i,\bolds{x}_i \in D} h^d I_2^2 &\leq \left[ \frac{4C^g_2c^\gamma \bar{J}_0^2 \sqrt{|D|} }{\epsilon^2} \sup_t ||\bolds{u}(t)||_{\Cholder{}} \right]^2 h^{2\gamma} .
\end{align}
Applying \autoref{eq:consist est 1}, \autoref{eq:consist est 3} and \autoref{eq:estimate sigma u 1} gives
\begin{align*}
||\sigma^k(u)||_{L^2} &\leq \sqrt{\sum_{i,\bolds{x}_i \in D} h^d I_1^2} + \sqrt{\sum_{i,\bolds{x}_i \in D} h^d I_2^2} \notag \\
&\leq \left[ \frac{4(C^g_2 \bar{J}^2_0 + C^f_2 \bar{J}_1)c^\gamma \sqrt{|D|} }{\epsilon^2} \sup_t ||\bolds{u}(t)||_{\Cholder{}} \right] h^\gamma.
\end{align*}
Here we define the constant
\begin{equation}\label{eq:def L3}
L_3 = \left\{\begin{array}{l l}4 (C^f_1 \bar{J}_1 + C^g_2 \bar{J}^2_0),& \hbox{ for $g$ convex-concave}\\4 (C^f_1 \bar{J}_1 + g''(0) \bar{J}^2_0),& \hbox {for $g$ quadratic} \end{array} \right.
\end{equation}
this is also the Lipschitz constant related to Lipschitz continuity of peridynamic force in $L^2$, see \sautoref{Proposition}{prop:lipschitz L2}. Thus, we have shown for $g$ convex-concave that
\begin{align}
||\sigma^k(u)||_{L^2} &\leq \left[ \frac{L_3 c^\gamma \sqrt{|D|} }{\epsilon^2} \sup_t ||\bolds{u}(t)||_{\Cholder{}} \right] h^\gamma.
\end{align}
The same arguments show that an identical inequality holds for quadratic $g$ using the other definition of $L_3$ and this completes the estimation of the consistency errors.
\subsubsection{Stability}\label{sss:stability}
In this subsection we establish estimates that ensure stability of the evolution and apply the consistency estimates of the previous subsection to establish \autoref{thm:convergence}.
Let $e^k$ be the total error at the $k^{\text{th}}$ time step. It is defined as
\begin{align*}
e^k &:= \Ltwonorm{\bolds{e}^k(u)}{} + \Ltwonorm{\bolds{e}^k(v)}{}.
\end{align*}
To simplify the calculations, we collect all the consistency errors and write them as
\begin{align*}
\tau &:= \sup_t \left(\Ltwonorm{\tau^k(u)}{} + \Ltwonorm{\tau^k(v)}{} + \Ltwonorm{\sigma^k(u)}{} + \Ltwonorm{\sigma^k(v)}{}\right),
\end{align*}
and from our consistency analysis, we know that to leading order in $\Delta t$ and $h$ that
\begin{align}\label{eq:estimate tau}
\tau &\leq C_t \Delta t + \dfrac{C_s}{\epsilon^2} h^\gamma
\end{align}
where,
\begin{align}
C_t &:= \frac{1}{2} \sup_{t} \Ltwonorm{\dsecder{\bolds{u}(t)}{t}}{} + \frac{1}{2} \sup_{t} \Ltwonorm{\dfrac{\partial^3 \bolds{u}(t)}{\partial t^3}}{}, \label{eq:const Ct} \\
C_s &:= c^\gamma \sqrt{|D|} \left[ \epsilon^2 \sup_{t} \Choldernorm{\dfrac{\partial^2 \bolds{u}(t)}{\partial t^2}}{} + L_3 \sup_t \Choldernorm{\bolds{u}(t)}{} \right]. \label{eq:const Cs}
\end{align}
We take the $\Ltwo{}$ norm of \autoref{eq:error eqn in u} and \autoref{eq:error eqn in v} and add them. Using the definition of $\tau$ we get
\begin{align}\label{eq:error k ineq 1}
e^{k+1} &\leq e^k + \Delta t \Ltwonorm{\bolds{e}^{k+1}(v)}{D;\bbR^{d}} + \Delta t \tau \notag \\
&\quad + \Delta t \sqrt{ \sum_{i, \bolds{x}_i \in D} h^d \abs{ \mathcal{L}^{\epsilon}(\hat{\bu}^k)(\bolds{x}_i) - \mathcal{L}^{\epsilon}(\tilde{\bu}^k)(\bolds{x}_i)}^2 }.
\end{align}
It now remains to estimate the last term in the above equation. We illustrate the calculations for convex-concave $g$ noting the identical steps apply to quadratic $g$ as well. Let
\begin{align}\label{eq:consist est 3.1}
H &:= \sqrt{ \sum_{i, \bolds{x}_i \in D} h^d \abs{ \mathcal{L}^{\epsilon}(\hat{\bu}^k)(\bolds{x}_i) - \mathcal{L}^{\epsilon}(\tilde{\bu}^k)(\bolds{x}_i)}^2} \notag \\
&\leq \sqrt{ \sum_{i, \bolds{x}_i \in D} h^d \abs{ \mathcal{L}^{\epsilon}_T(\hat{\bu}^k)(\bolds{x}_i) - \mathcal{L}^{\epsilon}_T(\tilde{\bu}^k)(\bolds{x}_i)}^2 } \notag \\
&\quad+ \sqrt{ \sum_{i, \bolds{x}_i \in D} h^d \abs{ \mathcal{L}^{\epsilon}_D(\hat{\bu}^k)(\bolds{x}_i) - \mathcal{L}^{\epsilon}_D(\tilde{\bu}^k)(\bolds{x}_i)}^2 } \notag \\
&=: H_1 + H_2.
\end{align}
Choosing $\bolds{u} = \hat{\bu}^k$ and $\bolds{u}= \tilde{\bu}^k$ with $\mathcal{L}^\epsilon_T$ given by \autoref{eq:per bondbased simple force expression} we get
\begin{align}\label{eq:consist est 4}
H_1^2 &\leq \sum_{i, \bolds{x}_i \in D} h^d \bigg\vert \frac{2C^f_2}{\epsilon^2 \omega_d} \int_{H_1(\mathbf{0})} \frac{J(|{\boldsymbol{\xi}}|)}{|{\boldsymbol{\xi}}|} |\bar{\hat{\bu}}^k_{\boldsymbol{\xi}}(\bolds{x}_i)- \bar{\tilde{\bu}}^k_{\boldsymbol{\xi}}(\bolds{x}_i)| d{\boldsymbol{\xi}} \bigg\vert^2 ,
\end{align}
where $\bar{\hat{\bu}}^k_{\boldsymbol{\xi}}(\bolds{x}) = \hat{\bu}^k(\bolds{x} + \epsilon {\boldsymbol{\xi}}) - \hat{\bu}^k(\bolds{x} )$.
We will make use of the following inequality in the sequel. Let $p({\boldsymbol{\xi}})$ be a scalar valued function of ${\boldsymbol{\xi}}$ and $\alpha \in \mathbb{R}$ then
\begin{align}\label{eq:ineq symm square}
&\bigg\vert \frac{C}{\omega_d} \int_{H_1(\mathbf{0})} \frac{J(|{\boldsymbol{\xi}}|)}{|{\boldsymbol{\xi}}|^\alpha} p({\boldsymbol{\xi}}) d{\boldsymbol{\xi}} \bigg\vert^2 \notag \\
&\leq \left(\frac{C}{\omega_d} \right)^2 \int_{H_1(\mathbf{0})} \int_{H_1(\mathbf{0})} \frac{J(|{\boldsymbol{\xi}}|)}{|{\boldsymbol{\xi}}|^\alpha} \frac{J(|\boldsymbol{\eta}|)}{|\boldsymbol{\eta}|^\alpha} p({\boldsymbol{\xi}}) p(\boldsymbol{\eta}) d{\boldsymbol{\xi}} d\boldsymbol{\eta} \notag \\
&\leq \left(\frac{C}{\omega_d} \right)^2 \int_{H_1(\mathbf{0})} \int_{H_1(\mathbf{0})} \frac{J(|{\boldsymbol{\xi}}|)}{|{\boldsymbol{\xi}}|^\alpha} \frac{J(|\boldsymbol{\eta}|)}{|\boldsymbol{\eta}|^\alpha} \frac{p({\boldsymbol{\xi}})^2 + p(\boldsymbol{\eta})^2}{2} d{\boldsymbol{\xi}} d\boldsymbol{\eta} \notag \\
&= C^2 \frac{\bar{J}_\alpha}{\omega_d } \int_{H_1(\mathbf{0})} \frac{J(|{\boldsymbol{\xi}}|)}{|{\boldsymbol{\xi}}|^\alpha} p({\boldsymbol{\xi}})^2 d{\boldsymbol{\xi}}.
\end{align}
On applying \autoref{eq:ineq symm square} in \autoref{eq:consist est 4} with $C = \frac{2C^f_2}{\epsilon^2}$, $\alpha = 1$, $p(|{\boldsymbol{\xi}}|) = |\bar{\hat{\bu}}^k_{\boldsymbol{\xi}}(\bolds{x}_i)- \bar{\tilde{\bu}}^k_{\boldsymbol{\xi}}(\bolds{x}_i)|$ we get
\begin{align}
H_1^2 &\leq \sum_{i, \bolds{x}_i \in D} h^d \left(\frac{2C^f_2}{\epsilon^2}\right)^2 \frac{\bar{J}_1}{\omega_d} \int_{H_1(\mathbf{0})} \frac{J(|{\boldsymbol{\xi}}|)}{|{\boldsymbol{\xi}}|} |\bar{\hat{\bu}}^k_{\boldsymbol{\xi}}(\bolds{x}_i)- \bar{\tilde{\bu}}^k_{\boldsymbol{\xi}}(\bolds{x}_i)|^2 d{\boldsymbol{\xi}} \notag \\
&\leq \left(\frac{2C^f_2}{\epsilon^2}\right)^2 \frac{\bar{J}_1}{\omega_d} \int_{H_1(\mathbf{0})} \frac{J(|{\boldsymbol{\xi}}|)}{|{\boldsymbol{\xi}}|} \notag \\
&\quad \left[ \sum_{i, \bolds{x}_i \in D} h^d 2(|\hat{\bu}^k(\bolds{x}_i + \epsilon {\boldsymbol{\xi}}) - \tilde{\bu}^k(\bolds{x}_i + \epsilon {\boldsymbol{\xi}})|^2 + |\hat{\bu}^k(\bolds{x}_i) - \tilde{\bu}^k(\bolds{x}_i)|^2) \right] d{\boldsymbol{\xi}} \notag \\
&\leq \left(\frac{2C^f_2}{\epsilon^2}\right)^2 \frac{\bar{J}_1}{\omega_d} \int_{H_1(\mathbf{0})} \frac{J(|{\boldsymbol{\xi}}|)}{|{\boldsymbol{\xi}}|} \notag \\
&\quad \left[ \sum_{i, \bolds{x}_i \in D} h^d 2(|\bolds{e}^k(u)(\bolds{x}_i + \epsilon {\boldsymbol{\xi}})|^2 + |\bolds{e}^k(u)(\bolds{x}_i)|^2) \right] d{\boldsymbol{\xi}},
\end{align}
where we substituted definition of $\bar{\hat{\bu}}^k_{\boldsymbol{\xi}}$ and $\bar{\tilde{\bu}}^k_{\boldsymbol{\xi}}$ and used inequality $(a+b)^2 \leq 2 a^2 + 2 b^2$ in third step, and identified terms as $\bolds{e}^k(u)$ in last step. Since $\bolds{e}^k(u)(\bolds{x}) = \sum_{i,\bolds{x}_i \in D} \bolds{e}^k_i(u) \chi_{U_i}(\bolds{x})$, we have
\begin{align}\label{eq:consist est 5}
H_1^2 &\leq \left(\frac{2C^f_2}{\epsilon^2}\right)^2 \frac{\bar{J}_1}{\omega_d} \int_{H_1(\mathbf{0})} \frac{J(|{\boldsymbol{\xi}}|)}{|{\boldsymbol{\xi}}|} 4 ||\bolds{e}^k(u)||_{L^2}^2 d{\boldsymbol{\xi}}, \notag
\end{align}
so
\begin{equation}\label{eq:consist est for H1}
H_1 \leq \frac{4C^f_2 \bar{J}_1}{\epsilon^2} ||\bolds{e}^k(u)||_{L^2} .
\end{equation}
We now estimate $H_2$. Note that for $I_2 = |\mathcal{L}^\epsilon_D(\tilde{\bu}^k)(\bolds{x}_i) - \mathcal{L}^\epsilon_D(\bolds{u}^k)(\bolds{x}_i)|$, we have the inequality given by \autoref{eq:consist est 2}. We now use \autoref{eq:consist est 2} but with $\tilde{\bu}^k$ replaced by $\hat{\bu}^k$ and $\bolds{u}^k$ replaced by $\tilde{\bu}^k$ together with the identity $\theta(\bolds{x};\hat{\bu}^k) - \theta(\bolds{x}; \tilde{\bu}^k) = \theta(\bolds{x};\hat{\bu}^k - \tilde{\bu}^k) = \theta(\bolds{x}; \bolds{e}^k(u))$, to see that
\begin{align}
H_2^2 &\leq \sum_{i, \bolds{x}_i \in D} h^d \bigg( \frac{C^g_2}{\epsilon^2 \omega_d} \int_{H_1(\mathbf{0})} J(|{\boldsymbol{\xi}}|) (|\theta(\bolds{x}_i+\epsilon{\boldsymbol{\xi}};\bolds{e}^k(u))|+ |\theta(\bolds{x}_i;\bolds{e}^k(u))|) d{\boldsymbol{\xi}} \bigg)^2.
\end{align}
We use inequality \autoref{eq:ineq symm square} with $C = C^g_2/\epsilon^2$, $\alpha = 0$, and $p({\boldsymbol{\xi}} ) = |\theta(\bolds{x}_i+\epsilon{\boldsymbol{\xi}};\bolds{e}^k(u))|+ |\theta(\bolds{x}_i;\bolds{e}^k(u))|$ to get
\begin{align}\label{eq:consist est 6}
H_2^2 &\leq \sum_{i, \bolds{x}_i \in D} h^d \left( \frac{C^g_2}{\epsilon^2}\right)^2 \frac{\bar{J}_0}{\omega_d } \int_{H_1(\mathbf{0})} J(|{\boldsymbol{\xi}}|) ( |\theta(\bolds{x}_i+\epsilon{\boldsymbol{\xi}};\bolds{e}^k(u))|+ |\theta(\bolds{x}_i;\bolds{e}^k(u))| )^2 d{\boldsymbol{\xi}} \notag \\
&\leq \left( \frac{C^g_2}{\epsilon^2}\right)^2 \frac{\bar{J}_0}{\omega_d } \int_{H_1(\mathbf{0})} J(|{\boldsymbol{\xi}}|) \notag \\
& \quad \left[ \sum_{i, \bolds{x}_i \in D} h^d 2 (|\theta(\bolds{x}_i+\epsilon{\boldsymbol{\xi}};\bolds{e}^k(u))|^2+ |\theta(\bolds{x}_i;\bolds{e}^k(u))|^2 ) \right] d{\boldsymbol{\xi}},
\end{align}
where we used inequality $(a+b)^2 \leq 2a^2 + 2b^2$ in the second step. We now proceed to estimate the first sum in the last line of \autoref{eq:consist est 6},
\begin{align}
& \sum_{i, \bolds{x}_i \in D} h^d |\theta(\bolds{x}_i+\epsilon{\boldsymbol{\xi}};\bolds{e}^k(u))|^2 \notag \\
&\leq \sum_{i, \bolds{x}_i \in D} h^d \bigg\vert \frac{1}{\omega_d }\int_{H_1(\mathbf{0})} J(|\boldsymbol{\eta}|) \omega(\bolds{x}_i + \epsilon {\boldsymbol{\xi}} + \epsilon\boldsymbol{\eta}) \notag \\
& \qquad ( \bolds{e}^k(u)(\bolds{x}_i + \epsilon {\boldsymbol{\xi}} + \epsilon\boldsymbol{\eta}) - \bolds{e}^k(u) (\bolds{x}_i + \epsilon {\boldsymbol{\xi}})) \cdot \bolds{e}_{\boldsymbol{\eta}} d\boldsymbol{\eta} \bigg\vert^2 \notag \\
&\leq \sum_{i, \bolds{x}_i \in D} h^d \bigg( \frac{1}{\omega_d }\int_{H_1(\mathbf{0})} J(|\boldsymbol{\eta}|) (|\bolds{e}^k(u)(\bolds{x}_i + \epsilon {\boldsymbol{\xi}} + \epsilon\boldsymbol{\eta})| + |\bolds{e}^k(u)(\bolds{x}_i + \epsilon {\boldsymbol{\xi}})|) d\boldsymbol{\eta} \bigg)^2,
\end{align}
where we used expression of $\theta$ from \autoref{eq:hydro strain} in first step, and used $0\leq \omega(\bolds{x}) \leq 1$ in the second step. The second summation on the last line of \autoref{eq:consist est 6} is also bounded above the same way. We apply inequality \autoref{eq:ineq symm square} with $C =1$, $\alpha = 0$, and $p(\boldsymbol{\eta}) = |\bolds{e}^k(u)(\bolds{x}_i + \epsilon {\boldsymbol{\xi}} + \epsilon\boldsymbol{\eta})| + |\bolds{e}^k(u)(\bolds{x}_i + \epsilon {\boldsymbol{\xi}})|$ to get
\begin{align}\label{eq:consist est 7}
& \sum_{i, \bolds{x}_i \in D} h^d |\theta(\bolds{x}_i+\epsilon{\boldsymbol{\xi}};\bolds{e}^k(u))|^2 \notag \\
&\leq \sum_{i, \bolds{x}_i \in D} h^d \frac{\bar{J}_0}{\omega_d} \int_{H_1(\mathbf{0})} J(|\boldsymbol{\eta}|) (|\bolds{e}^k(u)(\bolds{x}_i + \epsilon {\boldsymbol{\xi}} + \epsilon\boldsymbol{\eta})| + |\bolds{e}^k(u)(\bolds{x}_i + \epsilon {\boldsymbol{\xi}})|)^2 d \boldsymbol{\eta} \notag \\
&\leq \frac{\bar{J}_0}{\omega_d} \int_{H_1(\mathbf{0})} J(|\boldsymbol{\eta}|) 2 \sum_{i, \bolds{x}_i \in D} h^d(|\bolds{e}^k(u)(\bolds{x}_i + \epsilon {\boldsymbol{\xi}} + \epsilon\boldsymbol{\eta})|^2 + |\bolds{e}^k(u)(\bolds{x}_i + \epsilon {\boldsymbol{\xi}})|^2) d \boldsymbol{\eta} \notag \\
&\leq \frac{\bar{J}_0}{\omega_d} \int_{H_1(\mathbf{0})} J(|\boldsymbol{\eta}|) 4 ||\bolds{e}^k(u)||^2_{L^2} d \boldsymbol{\eta} \notag \\
& = 4 \bar{J}^2_0 ||\bolds{e}^k(u)||^2_{L^2},
\end{align}
where as before we have used the Cauchy inequality. We next apply the estimate \autoref{eq:consist est 7} to \autoref{eq:consist est 6} to see that
\begin{align}\label{eq:consist est 8}
&H_2^2 \leq 16 \bar{J}^2_0 ||\bolds{e}^k(u)||^2_{L^2} \left( \frac{C^g_2}{\epsilon^2}\right)^2 \frac{\bar{J}_0}{\omega_d } \int_{H_1(\mathbf{0})} J(|{\boldsymbol{\xi}}|) d{\boldsymbol{\xi}}, \notag
\end{align}
so
\begin{equation}\label{eq:consist est H2}
H_2 \leq \frac{4 C^g_2 \bar{J}^2_0}{\epsilon^2} ||\bolds{e}^k(u)||_{L^2} .
\end{equation}
Finally, we apply the inequalities given by \autoref{eq:consist est for H1} and \autoref{eq:consist est H2} to \autoref{eq:consist est 3.1} and obtain
\begin{align}\label{eq:consist est 9}
H &= \sqrt{\sum_{i, \bolds{x}_i \in D} h^d \abs{ \mathcal{L}^{\epsilon}(\hat{\bu}^k)(\bolds{x}_i) - \mathcal{L}^{\epsilon}(\tilde{\bu}^k)(\bolds{x}_i)}^2 } \notag \\
&\leq H_1 + H_2 \notag \\
&\leq \frac{4(C^f_2 \bar{J}_1 + C^g_2 \bar{J}^2_0)}{\epsilon^2 } ||\bolds{e}^k(u)||_{L^2} \notag \\
&\leq \left[ \frac{L_3}{\epsilon^2 } ||\bolds{e}^k(u)||_{L^2} \right]^2,
\end{align}
where $L_3=4(C^f_2 \bar{J}_1 + C^g_2 \bar{J}^2_0)$ for convex-concave $g$. For the case of quadratic $g$ we have the same inequality but with $L_3=4(C^f_2 \bar{J}_1 + g''(0) \bar{J}^2_0)$
Applying the inequality given by \autoref{eq:consist est 9} to \autoref{eq:error k ineq 1} gives
\begin{align*}
e^{k+1} &\leq e^k + \Delta t \Ltwonorm{\bolds{e}^{k+1}(v)}{D;\bbR^{d}} + \Delta t \tau + \Delta t \dfrac{L_3}{\epsilon^2}\Ltwonorm{\bolds{e}^k(u)}{D;\bbR^{d}}
\end{align*}
We now add $\Delta t ||e^{k+1}(u)||_{L^2(D;\mathbb{R}^d)} + \Delta t \dfrac{L_3}{\epsilon^2} ||\bolds{e}^k(v)||_{L^2(D;\mathbb{R}^d)} $ to the right side of the equation above to get
\begin{align*}
&e^{k+1} \leq ( 1 + \Delta t \dfrac{L_3}{\epsilon^2}) e^k + \Delta t e^{k+1} + \Delta t \tau \\
\Rightarrow & e^{k+1} \leq \dfrac{( 1 + \Delta t {L_3}/{\epsilon^2})}{1 - \Delta t} e^{k} + \dfrac{\Delta t }{1 - \Delta t} \tau.
\end{align*}
We now recursively substitute $e^{j}$ as follows
\begin{align}
e^{k+1} &\leq \dfrac{( 1 + \Delta t {L_3}/\epsilon^2)}{1 - \Delta t} e^k + \dfrac{\Delta t }{1 - \Delta t} \tau \notag \\
&\leq \left(\dfrac{( 1 + \Delta t {L_3}/\epsilon^2)}{1 - \Delta t} \right)^2 e^{k-1} + \dfrac{\Delta t }{1 - \Delta t} \tau \left(1 + \dfrac{( 1 + \Delta t {L_3}/\epsilon^2)}{1 - \Delta t}\right) \notag \\
&\leq ...\notag \\
&\leq \left(\dfrac{( 1 + \Delta t {L_3}/\epsilon^2)}{1 - \Delta t} \right)^{k+1} e^0 + \dfrac{\Delta t }{1 - \Delta t} \tau \sum_{j=0}^k \left(\dfrac{( 1 + \Delta t {L_3}/\epsilon^2)}{1 - \Delta t} \right)^{k-j}. \label{eq:ek estimnate}
\end{align}
Since $1/(1-\Delta t)= 1 + \Delta t + \Delta t^2 + O(\Delta t^3)$, we have
\begin{align*}
\dfrac{( 1 + \Delta t {L_3}/\epsilon^2)}{1 - \Delta t} &\leq 1 + (1 + {L_3}/\epsilon^2) \Delta t + (1 + {L_3}/\epsilon^2) \Delta t^2 + O({L_3}/\epsilon^2) O(\Delta t^3).
\end{align*}
Now, for any $k \leq T /\Delta t$ and using the identity $(1+ a)^k \leq \exp [ka]$ for $a\leq 0$, we have
\begin{align*}
&\left( \dfrac{ 1 + \Delta t {L_3}/\epsilon^2}{1 - \Delta t} \right)^k \\
&\leq \exp \left[k (1 + {L_3}/\epsilon^2) \Delta t + k(1 + {L_3}/\epsilon^2) \Delta t^2 + k O(L_3/\epsilon^2) O(\Delta t^3) \right] \\
&\leq \exp \left[T (1 + {L_3}/\epsilon^2) + T (1 + {L_3}/\epsilon^2) \Delta t + O(T{L_3}/\epsilon^2) O(\Delta t^2) \right].
\end{align*}
We write above equation in more compact form as follows
\begin{align*}
&\left( \dfrac{ 1 + \Delta t {L_3}/\epsilon^2}{1 - \Delta t} \right)^k \\
&\leq \exp \left[T (1 + {L_3}/\epsilon^2) (1 + \Delta t + O(\Delta t^2)) \right].
\end{align*}
We use above estimate in \autoref{eq:ek estimnate} and get following inequality for $e^k$
\begin{align*}
e^{k+1} &\leq \exp \left[T (1 + {L_3}/\epsilon^2) (1 + \Delta t + O(\Delta t^2)) \right] \left( e^0 + (k+1) \tau \Delta t/(1- \Delta t) \right) \notag \\
&\leq \exp \left[T (1 + {L_3}/\epsilon^2) (1 + \Delta t + O(\Delta t^2)) \right] \left( e^0 + T\tau (1 + \Delta t + O(\Delta t^2) \right).
\end{align*}
where we used the fact that $1/(1-\Delta t) = 1+ \Delta t + O(\Delta t^2)$.
Assuming the error in initial data is zero, i.e. $e^0= 0$, and noting the estimate of $\tau$ in \autoref{eq:estimate tau}, we have
\begin{align*}
&\sup_k e^k \leq \exp \left[T (1 + L_3/\epsilon^2) \right] T \tau
\end{align*}
and we conclude to leading order that
\begin{align}\label{eq: fund est}
\sup_k e^k \leq \exp \left[T (1 + L_3/\epsilon^2) \right] T \left[ C_t \Delta t + (C_s/\epsilon^2) h^\gamma \right],
\end{align}
Here the constants $C_t$ and $C_s$ are given by \autoref{eq:const Ct} and \autoref{eq:const Cs}.
This shows the stability of the numerical scheme. We note that constant $L_3 = 4 (C^f_1 \bar{J}_1 + C^g_2 \bar{J}_0^2)$, where $C^f_2 = \sup |f''(r)|, C^g_2 = \sup |g''(r)|$, corresponds to the case when $g$ is convex-concave type. For quadratic $g$ the constant is given by $L_3 = 4 (C^f_1 \bar{J}_1 + g''(0) \bar{J}_0^2)$.
\section{Numerical results}\label{s:numerical}
In this section, we present numerical simulations that support the theoretical upper bound on the convergence rate and to illustrate the displacement field and fracture set under different loading conditions.
We specify the density $\rho = 1200 \,kg/m^3$, bulk modulus $K = 25 \, GPa$, and critical energy release rate $G_c = 500 \,Jm^{-2}$. The pairwise interaction and the hydrostatic interaction are characterized by potentials $f(r) = c (1-\exp [-\beta r^2])$ and $g(r) = \bar{C} r^2/2$ respectively. The influence function is $J(r) = 1-r$. Equations 94, 95, and 97 of \cite{CMPer-Lipton4} relate parameters $c, \beta, \bar{C}$ to the Lam\`e parameters $\lambda, \mu$ and the critical energy release rate $G_c$. In \autoref{tab:mat props}, we list the value of constants for Poisson's ratio $0.22$ and $0.245$. These ratios are computed using the relation established in \cite{CMPer-Lipton4}. The critical bond strain between material point $y$ and $x$ is $S_c = \bar{r}/\sqrt{|y-x|}$ where $\bar{r} = 1/\sqrt{2\beta}$.
We consider the central difference time discretization described by \autoref{eq:finite diff eqn u} and \autoref{eq:finite diff eqn v} on a uniform square mesh of mesh size $h$.
We can write the peridynamic force $\mathcal{L}^\epsilon(\hat{\bolds{u}}^k)(\bolds{x}_i)$ as follows
\begin{align}
\mathcal{L}^\epsilon(\hat{\bolds{u}}^k)(\bolds{x}_i) = \int_{H_\epsilon(\bolds{x}_i)} (w_1(\bolds{y}, \bolds{x}_i) + w_2(\bolds{y}, \bolds{x}_i)) d\bolds{y},
\end{align}
where $w_1$ and $w_2$ can be determined from expression of $\mathcal{L}^\epsilon$ in \autoref{eq:total peri force}. In the simulation we approximate $\mathcal{L}^\epsilon(\hat{\bolds{u}}^k)(\bolds{x}_i)$ as below
\begin{align}
\mathcal{L}^\epsilon(\hat{\bolds{u}}^k)(\bolds{x}_i) \approx \sum_{\bolds{x}_j \in D_h\cap H_\epsilon(\bolds{x}_i)} (w_1(\bolds{x}_j, \bolds{x}_i) +w_w(\bolds{x}_j, \bolds{x}_i)) V_j \bar{V}_{ij},
\end{align}
where $V_j = h^2$ for uniform mesh in $2$-d and $\bar{V}_{ij}$ is the volume correction.
The numerical results are presented in the following section.
\begin{table}
\centering
\addtolength{\tabcolsep}{+2pt}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{|l||r|r|}
\hline
Parameters $\setminus$ Poisson's ratio & $\nu = 0.22$ & $\nu = 0.245$\\
\hline\hline
$c$ & $4712.4$ & $4712.4$\\
\hline
$\bar{C}$ & $-1.0623\times 10^{12}$ & $-1.7349\times 10^{11}$\\
\hline
$\beta$ & $1.7533\times 10^{8}$ & $1.5647\times 10^{8}$\\
\hline
$\bar{r}$ & $5.3402\times 10^{-5}$ & $5.6529\times 10^{-5}$\\
\hline
\end{tabular}
\caption{Peridynamic material parameters assuming bulk modulus $K = 25 \, GPa$ and critical energy release rate $G_c = 500\, J/m^{-2}$. Density is $\rho = 1200\, kg/m^3$.}
\label{tab:mat props}
\end{table}
\begin{figure}
\centering
\includegraphics[scale=0.15]{setup_crack_propagation.pdf}
\caption{Material domain $D = [0,0.1\, m]^2$ with crack of length $0.02\,m$. The x-component of displacement is fixed along a collar of thickness equal to the horizon on top. On the bottom the velocity $\bolds{v}_x = \pm 1\, m/s$ along x-direction is specified on either side of the crack to make the crack propagate upwards.}\label{fig:setup crack prop}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}{.47\linewidth}
\includegraphics[scale=0.22]{crack_tip_full_t_34mus.png}
\caption{}
\end{subfigure}
\begin{subfigure}{.47\linewidth}
\includegraphics[scale=0.14]{crack_tip_t_34mus.png}
\caption{}
\end{subfigure}
\caption{(a) Color plot of damage function $Z$ on deformed material domain at time $t=34\,\mu s$. Dark blue represents undamaged material $Z<1$, $Z\approx 1$ is yellow at crack tip, red is softening material. The plot is for a horizon $\epsilon = 2\, mm$ and $h = \epsilon/8$. Here, the displacements are scaled by $100$ and damage function is cut off at $5$ to highlight the crack zone. The maximum displacement is $4.4 \, mm$ and the maximum value of $Z(x)$ is $82$ at $t=34\,\mu s$. (b) View near the crack tip.}
\label{fig:damage plot}
\end{figure}
\subsection{Crack propagation: Fracture energy and numerical convergence study}\label{ss:crack prop}
The problem is intentionally similar to the problem given in the simulation presented in\cite{CMPer-Lipton2}. We consider a 2-d domain $D=[0,0.1\, m]^2$ (with unit thickness in third direction) with vertical crack of length $0.02\, m$. Boundary conditions are described in \autoref{fig:setup crack prop}.
The simulation time is $T = 34\, \mu s$ and the time step is $\Delta t = 0.004\,\mu s$.
We run simulations for four different horizons $\epsilon = 8\, mm, 4\, mm, 2\, mm, 1\, mm$. For each horizon, we obtain the results for mesh sizes $h = \epsilon/2, \epsilon/4, \epsilon/8$. We take uniform square mesh of size $h$.
Material properties correspond to the Poisson's ration $\nu = 0.245$, see \autoref{tab:mat props}.
For the coarsest horizon $\epsilon = 8\, mm$, number of mesh nodes are (approximately) $0.9\times 10^3, 3.5\times 10^3 , 13.7\times 10^3$ for $h=4, 2, 1\, mm$ respectively. The memory consumed are $10$ MB, $16$ MB, $95$ MB respectively. For $\epsilon = 1\, mm$, number of nodes are $42\times 10^3, 167\times 10^3, 668\times 10^3$ for $h=0.5, 0.25, 0.125\, mm$ respectively. The memory consumed are $44$ MB, $370$ MB, $4400$ MB respectively. All computations were performed on a single workstation in parallel using $20$ threads.
\subsubsection{Fracture energy of crack zone}
The extent of damage at material point $\bolds{x}$ is given by the function $Z(\bolds{x})$
\begin{align}\label{eq:damage}
Z(\bolds{x}) &= \max_{\bolds{y} \in H_\epsilon(\bolds{x}) \cap D} \frac{S(\bolds{y},\bolds{x};\bolds{u})}{S_c}.
\end{align}
We define the crack zone as set of material points which have $Z > 1$. We compute the peridynamic energy of crack zone and compare it with the Griffith's fracture energy. For a crack of length $l$, the Griffith's fracture energy (G.E.) will be $G.E. = G_c \times l$. The peridynamic fracture energy (P.E.) is given by
\begin{align*}
P.E. = \int_{\substack{\bolds{x} \in D,\\ Z(\bolds{x}) \geq 1}} \left[ \frac{1}{\epsilon^d \omega_d} \int_{H_\epsilon(\bolds{x})} |\bolds{y} - \bolds{x}| \mathcal{W}^\epsilon(S(\bolds{y},\bolds{x},\bolds{u}))\,d\bolds{y} \right] d\bolds{x},
\end{align*}
where $\mathcal{W}^\epsilon(S(\bolds{y},\bolds{x},\bolds{u}))$ is the bond-based potential, see \autoref{tensilepot}. For the choice of $f(r) $ and $g(r)$, only bond-based potential $f$ contributes to the fracture energy, therefore $P.E.$ is computed only from bond-based interaction.
\begin{figure}[h]
\centering
\begin{subfigure}{.47\linewidth}
\includegraphics[scale=0.35]{crack_length_vs_fracture_energy_horizon_2mm.pdf}
\caption{Horizon $\epsilon = 2\, mm$}
\end{subfigure}
\begin{subfigure}{.47\linewidth}
\includegraphics[scale=0.35]{crack_length_vs_fracture_energy_horizon_4mm.pdf}
\caption{Horizon $\epsilon = 4\, mm$}
\end{subfigure}
\caption{Crack length vs peridynamic fracture energy (P.E.) and Griffith's fracture energy (G.E.). G.E. is simply $G_c \times l$ where $G_c = 500$.}
\label{fig:crack zone energy}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.3]{error_in_fracture_energy.pdf}
\caption{Percentage error between peridynamic fracture energy and Griffith's fracture energy.}\label{fig:error crack zone energy}
\end{figure}
\autoref{fig:damage plot} shows the plot of $Z$ at time $t=34\,\mu s$ for horizon $\epsilon = 2\, mm$. The figure on the right shows the $Z$ field near a crack tip.
In \autoref{fig:crack zone energy} we plot the peridynamic and Griffith's fracture energy as a function of crack length. We see better agreement between the two energies up to larger length of crack for coarse horizon. In \autoref{fig:error crack zone energy} we plot the error in fracture energy at different times.
\begin{figure}[h]
\centering
\includegraphics[scale=0.35]{rate_t_34mus.pdf}
\caption{Convergence rate with respect to mesh size for different fixed size of horizons.}\label{fig:rate}
\end{figure}
\subsubsection{Convergence rate}
Consider a fixed horizon $\epsilon$ and three different mesh sizes $h = \epsilon/2, \epsilon/4, \epsilon/8$. We compute the convergence rate as follows. Let $\bolds{u}_1,\bolds{u}_2,\bolds{u}_3$ be approximate solutions corresponding to meshes of size $h_1,h_2,h_3$, and let $\bolds{u}$ be the exact solution. We write the error as $||\bolds{u}_h - \bolds{u}|| =C h^\alpha$ for some constant $C$ and $\alpha>0$, and fix the ratio of mesh size $h_1/h_2 = h_2/h_3 = r$, to get
\begin{align*}
\log( ||\bolds{u}_1 - \bolds{u}_2||) &= C + \alpha \log h_2, \\
\log( ||\bolds{u}_2 - \bolds{u}_3||) &= C + \alpha \log h_3.
\end{align*}
Recall that the norm $||\cdot||$ is $L^2$ norm. From above two equations, it is easy to see that the rate of convergence $\alpha$ is
\begin{align}\label{eq:rate formula}
\dfrac{\log( ||\bolds{u}_1 - \bolds{u}_2||) - \log( ||\bolds{u}_2 - \bolds{u}_3||)}{\log(r)}.
\end{align}
The convergence result for four different horizons is shown in \autoref{fig:rate}. From this figure we see that for $\epsilon=1\, mm$ the convergence rate is greater than $1$ for simulation times below $25\mu s$.
For all other horizons the rate is larger than $1$ for simulation times below $35\mu s$. Here the discrepancy is due to the error accumulation at each time step and can be reduced some what by taking smaller time steps.
The simulations show a rate of convergence that agrees with the a-priori estimates given in \autoref{thm:convergence}.
\subsection{Bending test with pre-crack}\label{ss:bending}
We consider a 2-d material domain (with unit thickness in third direction) $D = [0,0.25\, m]\times [0,0.05\,m]$ with single and double vertical cracks. We fix horizon to $\epsilon = 0.010\, m$ and mesh size $h = \epsilon/4 \, mm$. The boundary conditions are described in \autoref{fig:setup bending} for single crack. For the double crack problem, the two vertical cracks are symmetrically located at distance $0.02\, m$ along x-axis from the mid point $x=0.125 \, m, y=0$. With time step $\Delta t = 0.0014\,\mu s$ we run simulations upto time $T = 350 \, \mu s$. Material properties correspond to the Poisson's ration $\nu = 0.22$, see \autoref{tab:mat props}.
\begin{figure}
\centering
\includegraphics[scale=0.1]{setup_bending_with_crack.pdf}
\caption{Material domain $D = [0,0.25\, m]\times [0,0.05\,m]$ with single verticle crack of length $0.015\,m$ at mid point of bottom edge. We apply linear in time distributed load, along negative y-direction, on part of the top edge. At any time $t$, the load is zero at the end points of loading line (red line) and is $f_{max} \times t$ at the midpoint. We take constant $f_{max} = -1.0\times 10^{13}$. We fix a vertical displacement on two support regions shown in the figure.}\label{fig:setup bending}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}{.47\linewidth}
\scalebox{1.0}[1.25]{\includegraphics[scale=0.2]{bending_crack_single_t_130mus.png}}
\caption{$t = 130\, \mu s$}
\end{subfigure}
\begin{subfigure}{.47\linewidth}
\scalebox{1.0}[1.25]{\includegraphics[scale=0.2]{bending_crack_double_t_180mus.png}}
\caption{$t = 180\, \mu s$}
\end{subfigure}
\begin{subfigure}{.47\linewidth}
\scalebox{1.0}[1.25]{\includegraphics[scale=0.2]{bending_crack_single_t_180mus.png}}
\caption{$t = 180\, \mu s$}
\end{subfigure}
\begin{subfigure}{.47\linewidth}
\scalebox{1.0}[1.25]{\includegraphics[scale=0.2]{bending_crack_double_t_220mus.png}}
\caption{$t = 220\, \mu s$}
\end{subfigure}
\begin{subfigure}{.47\linewidth}
\scalebox{1.0}[1.25]{\includegraphics[scale=0.2]{bending_crack_single_t_190mus.png}}
\caption{$t = 190\, \mu s$}
\end{subfigure}
\begin{subfigure}{.47\linewidth}
\scalebox{1.0}[1.25]{\includegraphics[scale=0.2]{bending_crack_double_t_240mus.png}}
\caption{$t = 240\, \mu s$}
\end{subfigure}
\caption{Damage profile under bending load. Plots on left are for single crack and plots on right are for double crack.}
\label{fig:damage plot bending}
\end{figure}
In \autoref{fig:damage plot bending} damage profile at various times are shown for both single and double crack problem. In \autoref{fig:crack zone energy bending} we plot the fracture energy as a function of total crack length. The error in energy remain below $5\%$ till $185 \, \mu s$ for single crack problem and $232\, \mu s$ for double crack problem. As we can see from \autoref{fig:damage plot bending}, after time $185\, \mu s$ for single crack and $232\, \mu s$ for double crack, the spread of damage around crack is higher and therefore peridynamic fracture energy is higher.
\begin{figure}
\centering
\begin{subfigure}{.46\linewidth}
\includegraphics[scale=0.35]{crack_length_vs_fracture_energy_bending_single_crack.pdf}
\caption{Single crack}
\end{subfigure}
\begin{subfigure}{.46\linewidth}
\includegraphics[scale=0.35]{crack_length_vs_fracture_energy_bending_double_crack.pdf}
\caption{Double crack}
\end{subfigure}
\caption{Crack length vs peridynamic fracture energy (P.E.) and Griffith's fracture energy (G.E.).}
\label{fig:crack zone energy bending}
\end{figure}
\section{Proof of Lipschitz continuity for the non-local force}\label{s:proofs}
In this section, we prove \sautoref{Proposition}{prop:lipschitz} and \sautoref{Proposition}{prop:lipschitz L2}.
\subsection{Proof of Proposition 1}\label{ss:proof prop 2}
Recall that $I = [0,T]$ is the time domain, $X=\Cholderz{D;\bbR^{d}}\times\Cholderz{D;\bbR^{d}}$, and $F^\epsilon(y,t) = (F^\epsilon_1(y,t), F^\epsilon_2(y,t))$, where $F^\epsilon_1(y,t) = y^2$ and $F^\epsilon_2(y,t) = \mathcal{L}^\epsilon(y^1) + \bolds{b}(t)$. Given $t\in I$ and $y=(y^1,y^2), z=(z^1, z^2)\in X$, we have
\begin{align}\label{eq:norm of F diff in X}
&\normX{F^\epsilon(y,t) - F^\epsilon(z,t)}{X} \notag \\
&= \Choldernorm{y^2 - z^2}{D;\bbR^{d}} + \Choldernorm{\mathcal{L}^{\epsilon}(y^1) - \mathcal{L}^\epsilon(z^1)}{D;\bbR^{d}}
\end{align}
and
\begin{align}\label{eq:norm of F in X}
\normX{F^\epsilon(y,t)}{X} = \Choldernorm{y^2}{D;\bbR^{d}} + \Choldernorm{\mathcal{L}^{\epsilon}(y^1)}{D;\bbR^{d}} + b,
\end{align}
where $b = \sup_t ||\bolds{b}(t)||_{\Cholder{}}$.
Thus, to prove \autoref{eq:lipschitz property of F} and \autoref{eq:bound on F} of \sautoref{Proposition}{prop:lipschitz} we need to study the terms associated with $\mathcal{L}^\epsilon$ in the equations listed above. The peridynamic force $\mathcal{L}^\epsilon$ is sum of two forces, the tensile force $\mathcal{L}^\epsilon_T$ and the dilatational force $\mathcal{L}^\epsilon_D$. So for $\bolds{u},\bolds{v} \in \Cholderz{D;\bbR^{d}}$ we have
\begin{align}\label{eq:lipschitz norm of per force diff}
&\Choldernorm{\mathcal{L}^{\epsilon}(\bolds{u}) - \mathcal{L}^{\epsilon}(\bolds{v})}{D;\bbR^{d}} \notag \\
&\leq \Choldernorm{\mathcal{L}^{\epsilon}_T(\bolds{u}) - \mathcal{L}^{\epsilon}_T(\bolds{v})}{D;\bbR^{d}} + \Choldernorm{\mathcal{L}^{\epsilon}_D(\bolds{u}) - \mathcal{L}^{\epsilon}_D(\bolds{v})}{D;\bbR^{d}}
\end{align}
and
\begin{align}\label{eq:lipschitz norm of per force}
\Choldernorm{\mathcal{L}^{\epsilon}(\bolds{u})}{D;\bbR^{d}} \leq \Choldernorm{\mathcal{L}^{\epsilon}_T(\bolds{u})}{D;\bbR^{d}} + \Choldernorm{\mathcal{L}^{\epsilon}_D(\bolds{u})}{D;\bbR^{d}}.
\end{align}
We conclude listing estimates that will be used in the sequel.
For $\bolds{u} \in \Cholderz{D;\bbR^{d}}$ and $\omega\in \Cholderz{D;[0,1]}$ one easily deduces the estimates
\begin{align}
|\bolds{u}(\bolds{x} + \epsilon {\boldsymbol{\xi}}) - \bolds{u}(\bolds{x})| &\leq (\epsilon|{\boldsymbol{\xi}}|)^\gamma ||\bolds{u}||_{\Cholder{}}, \notag \\
|\bolds{u}(\bolds{x}+ \epsilon {\boldsymbol{\xi}}) - \bolds{u}(\bolds{y} + \epsilon{\boldsymbol{\xi}})| &\leq |\bolds{x} - \bolds{y}|^\gamma ||\bolds{u}||_{\Cholder{}},\notag \\
|\omega(\bolds{x}+ \epsilon {\boldsymbol{\xi}}) - \omega(\bolds{y} + \epsilon{\boldsymbol{\xi}})| &\leq |\bolds{x} - \bolds{y}|^\gamma ||\omega||_{\Cholder{}},
\end{align}
for $\bolds{x}, \bolds{y} \in D$ and ${\boldsymbol{\xi}} \in H_1(\mathbf{0})$. Since $\bolds{u}$ and $\omega$ are extended by zero outside $D$ these estimates also hold for all points outside $D$.
\subsubsection{Lipschitz continuity in H\"older space}
In this subsection, we provide upper bounds on \autoref{eq:lipschitz norm of per force diff}.
\paragraph{Non-local tensile force}\label{sss:bond-based proof}
For any $\bolds{u},\bolds{v} \in \Cholderz{D;\bbR^{d}}$, we provide upper bounds on
\begin{align}\label{eq:lipshitz norm bond force}
&\Choldernorm{\mathcal{L}^{\epsilon}_T(\bolds{u}) - \mathcal{L}^{\epsilon}_T(\bolds{v})}{D;\bbR^{d}} \notag \\
&= \sup_{\bolds{x}\in D} |\mathcal{L}^{\epsilon}_T(\bolds{u})(\bolds{x}) - \mathcal{L}^{\epsilon}_T(\bolds{v})(\bolds{x})| \notag \\
&+ \sup_{\bolds{x}, \bolds{y} \in D, \bolds{x} \neq \bolds{y}} \frac{|(\mathcal{L}^{\epsilon}_T(\bolds{u})(\bolds{x}) - \mathcal{L}^{\epsilon}_T(\bolds{v})(\bolds{x})) - (\mathcal{L}^{\epsilon}_T(\bolds{u})(\bolds{y}) - \mathcal{L}^{\epsilon}_T(\bolds{v})(\bolds{y}))|}{|\bolds{x} - \bolds{y}|^\gamma}.
\end{align}
Applying \autoref{eq:per bondbased simple force expression} and proceeding as in section
\autoref{sss:consistency} we see that
\begin{align}\label{eq:est 0}
& |\mathcal{L}^\epsilon_T(\bolds{u})(\bolds{x}) - \mathcal{L}^\epsilon_T(\bolds{v})(\bolds{x}) | \notag \\
&= \left\vert \frac{2}{\epsilon \omega_d} \int_{H_1(\mathbf{0})} \omega_{\boldsymbol{\xi}}(\bolds{x}) \frac{J(|{\boldsymbol{\xi}}|)}{\sqrt{s_{\boldsymbol{\xi}}}} \left[ f'(\bar{\bu}_{\boldsymbol{\xi}}(\bolds{x})\cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}}) - f'(\bar{\bv}_{\boldsymbol{\xi}}(\bolds{x})\cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}}) \right] \bolds{e}_{{\boldsymbol{\xi}}} d{\boldsymbol{\xi}} \right\vert \notag \\
&\leq \frac{2}{\epsilon \omega_d} \int_{H_1(\mathbf{0})} \omega_{\boldsymbol{\xi}}(\bolds{x}) \frac{J(|{\boldsymbol{\xi}}|)}{\sqrt{s_{\boldsymbol{\xi}}}} \left\vert f'(\bar{\bu}_{\boldsymbol{\xi}}(\bolds{x})\cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}}) - f'(\bar{\bv}_{\boldsymbol{\xi}}(\bolds{x})\cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}}) \right\vert d{\boldsymbol{\xi}} \notag \\
&\leq \frac{2C^f_2}{\epsilon \omega_d} \int_{H_1(\mathbf{0})} \frac{J(|{\boldsymbol{\xi}}|)}{s_{\boldsymbol{\xi}}} \left\vert \bar{\bu}_{\boldsymbol{\xi}}(\bolds{x}) - \bar{\bv}_{\boldsymbol{\xi}}(\bolds{x})\right\vert d{\boldsymbol{\xi}}.
\end{align}
A straightforward calculation gives the estimate
\begin{align*}
&|\bolds{u}_{\boldsymbol{\xi}}(\bolds{x}) - \bolds{v}_{\boldsymbol{\xi}}(\bolds{x})| = |\bolds{u}(\bolds{x} + \epsilon {\boldsymbol{\xi}}) - \bolds{u}(\bolds{x}) - (\bolds{v}(\bolds{x}+ \epsilon{\boldsymbol{\xi}}) - \bolds{v}(\bolds{x}))| \notag \\
&\leq |\bolds{u}(\bolds{x} + \epsilon {\boldsymbol{\xi}}) - \bolds{v}(\bolds{x}+ \epsilon{\boldsymbol{\xi}})| + |\bolds{u}(\bolds{x}) - \bolds{v}(\bolds{x})| \leq 2 ||\bolds{u} - \bolds{v}||_{\Cholder{}}
\end{align*}
and on applying this \autoref{eq:est 0} we get
\begin{align}\label{eq:lipshitz bound bondbased 1}
|\mathcal{L}^\epsilon_T(\bolds{u})(\bolds{x}) - \mathcal{L}^\epsilon_T(\bolds{v})(\bolds{x}) | &\leq \frac{4 C^f_2 \bar{J}_1}{\epsilon^2} ||\bolds{u} - \bolds{v}||_{\Cholder{}},
\end{align}
where $\bar{J}_1$ is given by \autoref{eq:def bar J}.
Next we derive a bound on
\begin{align*}
&\frac{|(\mathcal{L}^{\epsilon}_T(\bolds{u})(\bolds{x}) - \mathcal{L}^{\epsilon}_T(\bolds{v})(\bolds{x})) - (\mathcal{L}^{\epsilon}_T(\bolds{u})(\bolds{y}) - \mathcal{L}^{\epsilon}_T(\bolds{v})(\bolds{y}))|}{|\bolds{x} - \bolds{y}|^\gamma} \notag \\
&= \frac{1}{|\bolds{x} - \bolds{y}|^\gamma} \bigg\vert\frac{2}{\epsilon\omega_d} \int_{H_1(\mathbf{0})} \frac{J(|{\boldsymbol{\xi}}|)}{\sqrt{s_{\boldsymbol{\xi}}}} \left[\omega_{\boldsymbol{\xi}}(\bolds{x}) ( f'(\bar{\bu}_{\boldsymbol{\xi}}(\bolds{x})\cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}}) - f'(\bar{\bv}_{\boldsymbol{\xi}}(\bolds{x})\cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}}) ) \right. \notag \\
&\qquad \left. - \omega_{\boldsymbol{\xi}}(\bolds{y}) ( f'(\bar{\bu}_{\boldsymbol{\xi}}(\bolds{y})\cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}}) - f'(\bar{\bv}_{\boldsymbol{\xi}}(\bolds{y})\cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}}) )\right] \bolds{e}_{{\boldsymbol{\xi}}} d{\boldsymbol{\xi}} \bigg\vert.
\end{align*}
Let
\begin{align}\label{eq:def H}
H &:= \frac{1}{|\bolds{x} - \bolds{y}|^\gamma} \bigg\vert \omega_{\boldsymbol{\xi}}(\bolds{x}) ( f'(\bar{\bu}_{\boldsymbol{\xi}}(\bolds{x})\cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}}) - f'(\bar{\bv}_{\boldsymbol{\xi}}(\bolds{x})\cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}}) ) \notag \\
& \qquad - \omega_{\boldsymbol{\xi}}(\bolds{y}) ( f'(\bar{\bu}_{\boldsymbol{\xi}}(\bolds{y})\cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}}) - f'(\bar{\bv}_{\boldsymbol{\xi}}(\bolds{y})\cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}}) )\bigg\vert.
\end{align}
Then
\begin{align}\label{eq:bondbased ineq 1}
&\frac{|(\mathcal{L}^{\epsilon}_T(\bolds{u})(\bolds{x}) - \mathcal{L}^{\epsilon}_T(\bolds{v})(\bolds{x})) - (\mathcal{L}^{\epsilon}_T(\bolds{u})(\bolds{y}) - \mathcal{L}^{\epsilon}_T(\bolds{v})(\bolds{y}))|}{|\bolds{x} - \bolds{y}|^\gamma} \notag \\
&\leq \frac{2}{\epsilon\omega_d} \int_{H_1(\mathbf{0})} \frac{J(|{\boldsymbol{\xi}}|)}{\sqrt{s_{\boldsymbol{\xi}}}} H d{\boldsymbol{\xi}}.
\end{align}
To analyze $H$ we consider the function $\bolds{r}: [0,1] \times D \to \mathbb{R}^d$ given by
\begin{align}
\bolds{r}(l,\bolds{x}) &:= \bar{\bv}_{\boldsymbol{\xi}}(\bolds{x}) + l(\bar{\bu}_{\boldsymbol{\xi}}(\bolds{x}) - \bar{\bv}_{\boldsymbol{\xi}}(\bolds{x})),
\end{align}
and $\partial \bolds{r}(l,\bolds{x})/\partial l = \bar{\bu}_{\boldsymbol{\xi}}(\bolds{x}) - \bar{\bv}_{\boldsymbol{\xi}}(\bolds{x})$. We write
\begin{align}\label{eq:est 1}
& f'(\bar{\bu}_{\boldsymbol{\xi}}(\bolds{x})\cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}}) - f'(\bar{\bv}_{\boldsymbol{\xi}}(\bolds{x})\cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}}) \notag \\
&= \int_0^1 \frac{\partial f'(\bolds{r}(l,\bolds{x})\cdot\bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}})}{\partial l} dl \notag \\
&=\int_0^1 \frac{\partial f'(\bolds{r}\cdot\bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}})}{\partial \bolds{r}} \bigg\vert_{\bolds{r} = \bolds{r}(l,\bolds{x})} \cdot \frac{\partial \bolds{r}(l,\bolds{x})}{\partial l} dl \notag \\
&=\int_0^1 f''(\bolds{r}(l,\bolds{x})\cdot\bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}}) \frac{\bolds{e}_{\boldsymbol{\xi}}}{\sqrt{s_{\boldsymbol{\xi}}}} \cdot (\bar{\bu}_{\boldsymbol{\xi}}(\bolds{x}) - \bar{\bv}_{\boldsymbol{\xi}}(\bolds{x})) dl.
\end{align}
and similarly we have
\begin{align}\label{eq:est 2}
&f'(\bar{\bu}_{\boldsymbol{\xi}}(\bolds{y})\cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}}) - f'(\bar{\bv}_{\boldsymbol{\xi}}(\bolds{y})\cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}}) \notag \\
&= \int_0^1 f''(\bolds{r}(l,\bolds{y})\cdot\bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}}) \frac{\bolds{e}_{\boldsymbol{\xi}}}{\sqrt{s_{\boldsymbol{\xi}}}} \cdot (\bar{\bu}_{\boldsymbol{\xi}}(\bolds{y}) - \bar{\bv}_{\boldsymbol{\xi}}(\bolds{y})) dl.
\end{align}
Substituting \autoref{eq:est 1} and \autoref{eq:est 2} into \autoref{eq:def H} gives
\begin{align*}
H &= \frac{1}{|\bolds{x} - \bolds{y}|^\gamma} \bigg\vert \int_0^1 \left[ \omega_{\boldsymbol{\xi}}(\bolds{x}) f''(\bolds{r}(l,\bolds{x})\cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}})(\bar{\bu}_{\boldsymbol{\xi}}(\bolds{x}) - \bar{\bv}_{\boldsymbol{\xi}}(\bolds{x})) \right. \notag \\
& \qquad \left. - \omega_{\boldsymbol{\xi}}(\bolds{y}) f''(\bolds{r}(l,\bolds{y})\cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}})(\bar{\bu}_{\boldsymbol{\xi}}(\bolds{y}) - \bar{\bv}_{\boldsymbol{\xi}}(\bolds{y})) \right] \cdot \frac{\bolds{e}_{\boldsymbol{\xi}}}{\sqrt{s_{\boldsymbol{\xi}}}} d{\boldsymbol{\xi}} \bigg\vert. \notag \\
&\leq \frac{1}{|\bolds{x} - \bolds{y}|^\gamma} \frac{1}{\sqrt{s_{\boldsymbol{\xi}}}} \int_0^1 \bigg\vert \omega_{\boldsymbol{\xi}}(\bolds{x}) f''(\bolds{r}(l,\bolds{x})\cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}})(\bar{\bu}_{\boldsymbol{\xi}}(\bolds{x}) - \bar{\bv}_{\boldsymbol{\xi}}(\bolds{x})) \notag \\
& \qquad - \omega_{\boldsymbol{\xi}}(\bolds{y}) f''(\bolds{r}(l,\bolds{y})\cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}})(\bar{\bu}_{\boldsymbol{\xi}}(\bolds{y}) - \bar{\bv}_{\boldsymbol{\xi}}(\bolds{y})) \bigg\vert d{\boldsymbol{\xi}}.
\end{align*}
We now add and subtract $\omega_{\boldsymbol{\xi}}(\bolds{x}) f''(\bolds{r}(l,\bolds{x})\cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}})(\bar{\bu}_{\boldsymbol{\xi}}(\bolds{y}) - \bar{\bv}_{\boldsymbol{\xi}}(\bolds{y}))$, and note $0\leq \omega_{\boldsymbol{\xi}} \leq 1$, to get
\begin{align}\label{eq:est 3}
H &\leq \frac{1}{|\bolds{x} - \bolds{y}|^\gamma} \frac{1}{\sqrt{s_{\boldsymbol{\xi}}}} \int_0^1 |f''(\bolds{r}(l,\bolds{x}) \cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}})| |\bar{\bu}_{\boldsymbol{\xi}}(\bolds{x}) - \bar{\bv}_{\boldsymbol{\xi}}(\bolds{x}) - \bar{\bu}_{\boldsymbol{\xi}}(\bolds{y}) + \bar{\bv}_{\boldsymbol{\xi}}(\bolds{y})| dl \notag \\
&+ \frac{1}{|\bolds{x} - \bolds{y}|^\gamma} \frac{1}{\sqrt{s_{\boldsymbol{\xi}}}} \int_0^1 |\omega_{\boldsymbol{\xi}}(\bolds{x}) f''(\bolds{r}(l,\bolds{x})\cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}}) - \omega_{\boldsymbol{\xi}}(\bolds{y}) f''(\bolds{r}(l,\bolds{y})\cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}})| \notag \\
&\qquad \qquad|\bar{\bu}_{\boldsymbol{\xi}}(\bolds{y}) - \bar{\bv}_{\boldsymbol{\xi}}(\bolds{y})|dl \notag \\
&=: H_1 + H_2,
\end{align}
where we denoted first and second term on right hand side as $H_1$ and $H_2$.
Using the estimate
\begin{align*}
\frac{|\bar{\bu}_{\boldsymbol{\xi}}(\bolds{x}) - \bar{\bv}_{\boldsymbol{\xi}}(\bolds{x}) - \bar{\bu}_{\boldsymbol{\xi}}(\bolds{y}) + \bar{\bv}_{\boldsymbol{\xi}}(\bolds{y})|}{|\bolds{x} - \bolds{y}|^\gamma} &\leq 2 ||\bolds{u} - \bolds{v}||_{\Cholder{}}.
\end{align*}
and $|f''(r)| \leq C^f_2$ we see that
\begin{align}\label{eq:est 4}
H_1 &\leq \frac{C^f_2}{|\bolds{x} - \bolds{y}|^\gamma \sqrt{s_{\boldsymbol{\xi}}}} \int_0^1 |\bar{\bu}_{\boldsymbol{\xi}}(\bolds{x}) - \bar{\bv}_{\boldsymbol{\xi}}(\bolds{x}) - \bar{\bu}_{\boldsymbol{\xi}}(\bolds{y}) + \bar{\bv}_{\boldsymbol{\xi}}(\bolds{y})| dl \notag \\
&= \frac{C^f_2}{|\bolds{x} - \bolds{y}|^\gamma \sqrt{s_{\boldsymbol{\xi}}}} |\bar{\bu}_{\boldsymbol{\xi}}(\bolds{x}) - \bar{\bv}_{\boldsymbol{\xi}}(\bolds{x}) - \bar{\bu}_{\boldsymbol{\xi}}(\bolds{y}) + \bar{\bv}_{\boldsymbol{\xi}}(\bolds{y})| \notag \\
&\leq \frac{2C^f_2}{\sqrt{s_{\boldsymbol{\xi}}}} ||\bolds{u} - \bolds{v}||_{\Cholder{}}.
\end{align}
To bound $H_2$, we add and subtract $\omega_{\boldsymbol{\xi}}(\bolds{x}) f''(\bolds{r}(l,\bolds{y})\cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}})$ and further split the terms
\begin{align}\label{eq:est 5}
H_2 &\leq \int_0^1 \frac{| f''(\bolds{r}(l,\bolds{x})\cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}}) - f''(\bolds{r}(l,\bolds{y})\cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}})|}{|\bolds{x} - \bolds{y}|^\gamma \sqrt{s_{\boldsymbol{\xi}}}} |\bar{\bu}_{\boldsymbol{\xi}}(\bolds{y}) - \bar{\bv}_{\boldsymbol{\xi}}(\bolds{y})| dl \notag \\
&+ \int_0^1 \frac{|\omega_{\boldsymbol{\xi}}(\bolds{x}) - \omega_{\boldsymbol{\xi}}(\bolds{y})|}{|\bolds{x} - \bolds{y}|^\gamma \sqrt{s_{\boldsymbol{\xi}}}} | f''(\bolds{r}(l,\bolds{y})\cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}})| |\bar{\bu}_{\boldsymbol{\xi}}(\bolds{y}) - \bar{\bv}_{\boldsymbol{\xi}}(\bolds{y})| dl \notag \\
&=: H_3 + H_4,
\end{align}
where we used the fact that $0\leq\omega_{\boldsymbol{\xi}} \leq 1$ in first term.
We consider $H_3$ first. With $|f'''(r)| \leq C^f_3$ and $0\leq l,1-l \leq 1$ for $l\in [0,1]$, we have
\begin{align*}
&\frac{| f''(\bolds{r}(l,\bolds{x})\cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}}) - f''(\bolds{r}(l,\bolds{y})\cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}})|}{|\bolds{x} - \bolds{y}|^\gamma} \notag \\
&\leq \frac{C^f_3}{\sqrt{s_{\boldsymbol{\xi}}}} \frac{|\bolds{r}(l,\bolds{x}) - \bolds{r}(l,\bolds{y})|}{|\bolds{x} - \bolds{y}|^\gamma} \notag \\
&\leq \frac{C^f_3}{\sqrt{s_{\boldsymbol{\xi}}}} \frac{|1-l||\bar{\bv}_{\boldsymbol{\xi}}(\bolds{x}) - \bar{\bv}_{\boldsymbol{\xi}}(\bolds{y})| + |l||\bar{\bu}_{\boldsymbol{\xi}}(\bolds{x}) - \bar{\bu}_{\boldsymbol{\xi}}(\bolds{y})|}{|\bolds{x} - \bolds{y}|^\gamma} \notag \\
&\leq \frac{C^f_3}{\sqrt{s_{\boldsymbol{\xi}}}} \left( \frac{|\bar{\bv}_{\boldsymbol{\xi}}(\bolds{x}) - \bar{\bv}_{\boldsymbol{\xi}}(\bolds{y})|}{|\bolds{x} - \bolds{y}|^\gamma} + \frac{|\bar{\bu}_{\boldsymbol{\xi}}(\bolds{x}) - \bar{\bu}_{\boldsymbol{\xi}}(\bolds{y})|}{|\bolds{x} - \bolds{y}|^\gamma} \right)
\end{align*}
Following estimates
\begin{align*}
\frac{|\bar{\bv}_{\boldsymbol{\xi}}(\bolds{x}) - \bar{\bv}_{\boldsymbol{\xi}}(\bolds{y})|}{|\bolds{x} - \bolds{y}|^\gamma} &\leq 2 ||\bolds{v}||_{\Cholder{}},\quad \frac{|\bar{\bu}_{\boldsymbol{\xi}}(\bolds{x}) - \bar{\bu}_{\boldsymbol{\xi}}(\bolds{y})|}{|\bolds{x} - \bolds{y}|^\gamma} \leq 2 ||\bolds{u}||_{\Cholder{}}
\end{align*}
delivers
\begin{align}\label{eq:est 6}
\frac{| f''(\bolds{r}(l,\bolds{x})\cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}}) - f''(\bolds{r}(l,\bolds{y})\cdot \bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}})|}{|\bolds{x} - \bolds{y}|^\gamma} &\leq \frac{2C^f_3}{\sqrt{s_{\boldsymbol{\xi}}}} (||\bolds{u}||_{\Cholder{}} + ||\bolds{v}||_{\Cholder{}}).
\end{align}
We use the inequality above together with the estimate
\begin{align*}
|\bar{\bu}_{\boldsymbol{\xi}}(\bolds{y}) - \bar{\bv}_{\boldsymbol{\xi}}(\bolds{y})| &\leq 2 s_{\boldsymbol{\xi}}^\gamma ||\bolds{u} - \bolds{v}||_{\Cholder{}}
\end{align*}
to get
\begin{align}\label{eq:est 7}
H_3 &\leq \frac{4C^f_3}{s^{1-\gamma}_{\boldsymbol{\xi}}} (||\bolds{u}||_{\Cholder{}} + ||\bolds{v}||_{\Cholder{}}) ||\bolds{u} - \bolds{v}||_{\Cholder{}}.
\end{align}
We now consider $H_4$ in \autoref{eq:est 5}. Using $|f''(r)|\leq C^f_2$, $|\bar{\bu}_{\boldsymbol{\xi}}(\bolds{y}) - \bar{\bv}_{\boldsymbol{\xi}}(\bolds{y})|\leq 2 ||\bolds{u} - \bolds{v}||_{\Cholder{}}$, and the following estimate
\begin{align}
\frac{|\omega_{\boldsymbol{\xi}}(\bolds{x}) - \omega_{\boldsymbol{\xi}}(\bolds{y})|}{|\bolds{x} - \bolds{y}|^\gamma} &= \frac{|\omega(\bolds{x} + \epsilon {\boldsymbol{\xi}}) \omega(\bolds{x}) - \omega(\bolds{y} + \epsilon{\boldsymbol{\xi}}) \omega(\bolds{y})|}{|\bolds{x} - \bolds{y}|^\gamma} \notag \\
&\leq \frac{|\omega(\bolds{x}+\epsilon{\boldsymbol{\xi}})| |\omega(\bolds{x}) - \omega(\bolds{y})|}{|\bolds{x} - \bolds{y}|^\gamma} + \frac{|\omega(\bolds{y})| |\omega(\bolds{x} + \epsilon \bolds{y}) - \omega(\bolds{y} + \epsilon{\boldsymbol{\xi}})|}{|\bolds{x} - \bolds{y}|^\gamma} \notag \\
&\leq 2 ||\omega||_{\Cholder{}},
\end{align}
we have
\begin{align}\label{eq:est 8}
H_4 &\leq \frac{4 C^f_2 ||\omega||_{\Cholder{}}}{\sqrt{s_{\boldsymbol{\xi}}}}||\bolds{u} - \bolds{v}||_{\Cholder{}}.
\end{align}
Applying the inequalities \autoref{eq:est 7} and \autoref{eq:est 8} to \autoref{eq:est 5} gives
\begin{align}\label{eq:est 9}
H_2 &\leq \left[ \frac{4C^f_3}{s^{1-\gamma}_{\boldsymbol{\xi}}} (||\bolds{u}||_{\Cholder{}} + ||\bolds{v}||_{\Cholder{}}) + \frac{4 C^f_2 ||\omega||_{\Cholder{}}}{\sqrt{s_{\boldsymbol{\xi}}}} \right] ||\bolds{u} - \bolds{v}||_{\Cholder{}}.
\end{align}
Applying the upper bounds on $H_1$ and $H_2$ shows that
\begin{align}\label{eq:est 10}
H &\leq \left[ \frac{4C^f_3}{s^{1-\gamma}_{\boldsymbol{\xi}}} (||\bolds{u}||_{\Cholder{}} + ||\bolds{v}||_{\Cholder{}}) + \frac{4 C^f_2 (1+||\omega||_{\Cholder{}})}{\sqrt{s_{\boldsymbol{\xi}}}} \right] ||\bolds{u} - \bolds{v}||_{\Cholder{}}.
\end{align}
We substitute the upper bound on $H$ in \autoref{eq:bondbased ineq 1} to find that
\begin{align}\label{eq:lipshitz bound bondbased 2}
&\frac{|(\mathcal{L}^{\epsilon}_T(\bolds{u})(\bolds{x}) - \mathcal{L}^{\epsilon}_T(\bolds{v})(\bolds{x})) - (\mathcal{L}^{\epsilon}_T(\bolds{u})(\bolds{y}) - \mathcal{L}^{\epsilon}_T(\bolds{v})(\bolds{y}))|}{|\bolds{x} - \bolds{y}|^\gamma} \notag \\
&\leq \frac{2}{\epsilon \omega_d} \int_{H_1(\mathbf{0})} \frac{J(|{\boldsymbol{\xi}}|)}{\sqrt{s_{\boldsymbol{\xi}}}} \left[ \frac{4C^f_3}{s^{1-\gamma}_{\boldsymbol{\xi}}} (||\bolds{u}||_{\Cholder{}} + ||\bolds{v}||_{\Cholder{}}) \right. \notag \\
&\quad \left. + \frac{4 C^f_2 (1+||\omega||_{\Cholder{}})}{\sqrt{s_{\boldsymbol{\xi}}}} \right] ||\bolds{u} - \bolds{v}||_{\Cholder{}} d{\boldsymbol{\xi}} \notag \\
&= \left[ \frac{8C^f_3 \bar{J}_{3/2-\gamma}}{\epsilon^{5/2 - \gamma}} (||\bolds{u}||_{\Cholder{}} + ||\bolds{v}||_{\Cholder{}}) + \frac{8C^f_2(1+||\omega||_{\Cholder{}}) \bar{J}_{1}}{\epsilon^2} \right] ||\bolds{u} - \bolds{v}||_{\Cholder{}},
\end{align}
where $\bar{J}_\alpha$ is defined in \autoref{eq:def bar J}. Application of \autoref{eq:lipshitz bound bondbased 1} and \autoref{eq:lipshitz bound bondbased 2} deliver
\begin{align}\label{eq:lipshitz bound bondbased 3}
&\Choldernorm{\mathcal{L}^{\epsilon}_T(\bolds{u}) - \mathcal{L}^{\epsilon}_T(\bolds{v})}{} \notag \\
&\leq \left[ \frac{8C^f_3 \bar{J}_{3/2-\gamma}}{\epsilon^{5/2 - \gamma}} (||\bolds{u}||_{\Cholder{}} + ||\bolds{v}||_{\Cholder{}}) + \frac{8C^f_2(2+||\omega||_{\Cholder{}}) \bar{J}_{1}}{\epsilon^2} \right] ||\bolds{u} - \bolds{v}||_{\Cholder{}},
\end{align}
and we have established the Lipschitz continuity of the non-local force due to tensile strain.
Now we establish the Lipschitz continuity for the non-local dilatational force.
For any $\bolds{u},\bolds{v} \in \Cholderz{D;\bbR^{d}}$ we write
\begin{align}\label{eq:lipshitz norm state force}
&\Choldernorm{\mathcal{L}^{\epsilon}_D(\bolds{u}) - \mathcal{L}^{\epsilon}_D(\bolds{v})}{D;\bbR^{d}} \notag \\
&= \sup_{\bolds{x}\in D} |\mathcal{L}^{\epsilon}_D(\bolds{u})(\bolds{x}) - \mathcal{L}^{\epsilon}_D(\bolds{v})(\bolds{x})| \notag \\
&+ \sup_{\bolds{x}, \bolds{y} \in D, \bolds{x} \neq \bolds{y}} \frac{|(\mathcal{L}^{\epsilon}_D(\bolds{u})(\bolds{x}) - \mathcal{L}^{\epsilon}_D(\bolds{v})(\bolds{x})) - (\mathcal{L}^{\epsilon}_D(\bolds{u})(\bolds{y}) - \mathcal{L}^{\epsilon}_D(\bolds{v})(\bolds{y}))|}{|\bolds{x} - \bolds{y}|^\gamma}.
\end{align}
The potential function $g$ can either be a quadratic function, e.g., $g(r) =\beta r^2/2$ or it can be a convex-concave function, see \autoref{ConvexConcaveFunctionGa}. Here we present the derivation of Lipschitz continuity for the convex-concave type $g$. The proof for the quadratic potential functions $g$ is identical.
Let $g$ be a bounded convex-concave potential function with bounded derivatives expressed by \autoref{eq:def Cgi}.
As in previous sections we use the notation \autoref{eq:notations consist} and \autoref{eq:def bar J} and begin by estimating $|\theta(\bolds{x};\bolds{u}) - \theta(\bolds{x};\bolds{v})|$ where $\theta(\bolds{x};\bolds{u})$ is given by \autoref{eq:hydro strain}. Application of the inequality $|\bar{\bu}_{\boldsymbol{\xi}}(\bolds{x}) - \bar{\bv}_{\boldsymbol{\xi}}(\bolds{x})| \leq 2 ||\bolds{u} - \bolds{v}||_{\Cholder{}}$, and a straightforward calculation shows that
\begin{align}\label{eq:est 11}
|\theta(\bolds{x};\bolds{u}) - \theta(\bolds{x};\bolds{v})| &\leq 2\bar{J}_0 ||\bolds{u} - \bolds{v}||_{\Cholder{}}.
\end{align}
We now bound $|\theta(\bolds{x};\bolds{u}) - \theta(\bolds{y}; \bolds{u})|$ as follows
\begin{align}
|\theta(\bolds{x};\bolds{u}) - \theta(\bolds{y}; \bolds{u})| &= \bigg\vert\frac{1}{\omega_d}\int_{H_1(\mathbf{0})}J(|{\boldsymbol{\xi}}|) \left[\omega(\bolds{x}+\epsilon{\boldsymbol{\xi}}) \bar{\bu}_{\boldsymbol{\xi}}(\bolds{x}) - \omega(\bolds{y} + \epsilon{\boldsymbol{\xi}}) \bar{\bu}_{\boldsymbol{\xi}}(\bolds{y}) \right] \cdot \bolds{e}_{\boldsymbol{\xi}} d{\boldsymbol{\xi}} \bigg\vert \notag \\
&\leq \frac{1}{\omega_d}\int_{H_1(\mathbf{0})}J(|{\boldsymbol{\xi}}|) \vert \omega(\bolds{x}+\epsilon{\boldsymbol{\xi}}) \bar{\bu}_{\boldsymbol{\xi}}(\bolds{x}) - \omega(\bolds{y} + \epsilon{\boldsymbol{\xi}}) \bar{\bu}_{\boldsymbol{\xi}}(\bolds{y}) \vert d{\boldsymbol{\xi}} \notag \\
&\leq \frac{1}{\omega_d}\int_{H_1(\mathbf{0})}J(|{\boldsymbol{\xi}}|) |\omega(\bolds{x}+\epsilon{\boldsymbol{\xi}})| \vert \bar{\bu}_{\boldsymbol{\xi}}(\bolds{x}) - \bar{\bu}_{\boldsymbol{\xi}}(\bolds{y}) \vert d{\boldsymbol{\xi}} \notag \\
& \; + \frac{1}{\omega_d}\int_{H_1(\mathbf{0})}J(|{\boldsymbol{\xi}}|) |\omega(\bolds{x} + \epsilon {\boldsymbol{\xi}}) - \omega(\bolds{y}+ \epsilon{\boldsymbol{\xi}})| \vert \bar{\bu}_{\boldsymbol{\xi}}(\bolds{y}) \vert d{\boldsymbol{\xi}},
\end{align}
where we used $|\bolds{e}_{\boldsymbol{\xi}}|=1$ and Cauchy's inequality in the first equation, added and subtracted $\omega(\bolds{x}+\epsilon {\boldsymbol{\xi}})\bar{\bu}_{\boldsymbol{\xi}}(\bolds{y})$ in the second equation and used the triangle inequality. Applying $|\bar{\bu}_{\boldsymbol{\xi}}(\bolds{x}) - \bar{\bu}_{\boldsymbol{\xi}}(\bolds{y})| \leq 2|\bolds{x} - \bolds{y}|^\gamma ||\bolds{u}||_{\Cholder{}}$, $|\omega(\bolds{x}+\epsilon {\boldsymbol{\xi}}) - \omega(\bolds{y} + \epsilon{\boldsymbol{\xi}})| \leq |\bolds{x} - \bolds{y}|^\gamma ||\omega||_{\Cholder{}}$, and $|\bar{\bu}_{\boldsymbol{\xi}}(\bolds{y})|\leq 2||\bolds{u}||_{\Cholder{}}$ gives
\begin{align*}
|\theta(\bolds{x};\bolds{u}) - \theta(\bolds{y}; \bolds{u})| &\leq \frac{1}{\omega_d} \int_{H_1(\mathbf{0})} J(|{\boldsymbol{\xi}}|) 2 |\bolds{x} - \bolds{y}|^\gamma ||\bolds{u}||_{\Cholder{}} d{\boldsymbol{\xi}} \notag \\
&\; + \frac{1}{\omega_d} \int_{H_1(\mathbf{0})} J(|{\boldsymbol{\xi}}|) |\bolds{x} - \bolds{y}|^\gamma ||\omega||_{\Cholder{}} 2 ||\bolds{u}||_{\Cholder{}},
\end{align*}
i.e.,
\begin{align}\label{eq:est 12}
|\theta(\bolds{x};\bolds{u}) - \theta(\bolds{y}; \bolds{u})|
&\leq 2\bar{J}_0(1+||\omega||_{\Cholder{}}) ||\bolds{u}||_{\Cholder{}} |\bolds{x} - \bolds{y}|^\gamma.
\end{align}
We note that estimate \autoref{eq:est 11} and \autoref{eq:est 12} holds for all $\bolds{x},\bolds{y} \in D$ as well as for $\bolds{x}$ and $\bolds{y}$ in the layer of thickness $2\epsilon$ surrounding $D$.
Using \autoref{eq:per statebased simple force expression} we have
\begin{align}
|\mathcal{L}^\epsilon_D(\bolds{u})(\bolds{x}) - \mathcal{L}^\epsilon_D(\bolds{v})(\bolds{x})| &= \bigg\vert \frac{1}{\epsilon^2 \omega_d} \int_{H_1(\mathbf{0})} \omega_{\boldsymbol{\xi}}(\bolds{x}) J(|{\boldsymbol{\xi}}|) [g'(\theta(\bolds{x}+\epsilon{\boldsymbol{\xi}};\bolds{u})) + g'(\theta(\bolds{x};\bolds{u})) \notag \\
&\qquad \qquad -g'(\theta(\bolds{x}+\epsilon{\boldsymbol{\xi}};\bolds{v})) - g'(\theta(\bolds{x};\bolds{v})) ] \bolds{e}_{\boldsymbol{\xi}} d{\boldsymbol{\xi}} \bigg\vert \notag \\
&\leq \frac{1}{\epsilon^2 \omega_d} \int_{H_1(\mathbf{0})} J(|{\boldsymbol{\xi}}|) \bigg\vert g'(\theta(\bolds{x}+\epsilon{\boldsymbol{\xi}};\bolds{u})) + g'(\theta(\bolds{x};\bolds{u})) \notag \\
&\qquad \qquad -g'(\theta(\bolds{x}+\epsilon{\boldsymbol{\xi}};\bolds{v})) - g'(\theta(\bolds{x};\bolds{v})) \bigg\vert d{\boldsymbol{\xi}} \notag \\
&\leq \frac{1}{\epsilon^2 \omega_d} \int_{H_1(\mathbf{0})} J(|{\boldsymbol{\xi}}|) \left\{ \bigg\vert g'(\theta(\bolds{x}+\epsilon{\boldsymbol{\xi}};\bolds{u})) -g'(\theta(\bolds{x}+\epsilon{\boldsymbol{\xi}};\bolds{v})) \bigg\vert \right. \notag \\
&\qquad \qquad \left.+ \bigg\vert g'(\theta(\bolds{x};\bolds{u})) - g'(\theta(\bolds{x};\bolds{v})) \bigg\vert \right\} d{\boldsymbol{\xi}}.
\end{align}
Since $|g'(r_1) - g'(r_2)|\leq C^g_2 |r_1 - r_2|$, we have
\begin{align*}
|g'(\theta(\bolds{x};\bolds{u})) - g'(\theta(\bolds{x};\bolds{v}))| &\leq C^g_2 |\theta(\bolds{x};\bolds{u}) - \theta(\bolds{x};\bolds{v})| \notag \\
&\leq 2 C^g_2 \bar{J}_0 ||\bolds{u} - \bolds{v}||_{\Cholder{}},
\end{align*}
where we used \autoref{eq:est 11}. Similarly we have
\begin{align*}
|g'(\theta(\bolds{x}+\epsilon{\boldsymbol{\xi}};\bolds{u})) - g'(\theta(\bolds{x}+\epsilon {\boldsymbol{\xi}};\bolds{v}))| &\leq 2 C^g_2 \bar{J}_0 ||\bolds{u} - \bolds{v}||_{\Cholder{}}.
\end{align*}
and we arrive at the estimate
\begin{align}\label{eq:lipshitz bound statebased 1}
|\mathcal{L}^\epsilon_D(\bolds{u})(\bolds{x}) - \mathcal{L}^\epsilon_D(\bolds{v})(\bolds{x})| &\leq \frac{4C^g_2\bar{J}_0^2}{\epsilon^2} ||\bolds{u} - \bolds{v}||_{\Cholder{}}.
\end{align}
Now we estimate
\begin{align*}
& \frac{|(\mathcal{L}^{\epsilon}_D(\bolds{u})(\bolds{x}) - \mathcal{L}^{\epsilon}_D(\bolds{v})(\bolds{x})) - (\mathcal{L}^{\epsilon}_D(\bolds{u})(\bolds{y}) - \mathcal{L}^{\epsilon}_D(\bolds{v})(\bolds{y}))|}{|\bolds{x} - \bolds{y}|^\gamma}.
\end{align*}
We write
\begin{align*}
\mathcal{L}^\epsilon_D(\bolds{u})(\bolds{x}) - \mathcal{L}^\epsilon_D(\bolds{v})(\bolds{x}) &= \frac{1}{\epsilon^2 \omega_d} \int_{H_1(\mathbf{0})} \omega_{\boldsymbol{\xi}}(\bolds{x}) J(|{\boldsymbol{\xi}}|) [g'(\theta(\bolds{x}+\epsilon{\boldsymbol{\xi}};\bolds{u})) + g'(\theta(\bolds{x};\bolds{u})) \notag \\
&\qquad \qquad -g'(\theta(\bolds{x}+\epsilon{\boldsymbol{\xi}};\bolds{v})) - g'(\theta(\bolds{x};\bolds{v})) ] \bolds{e}_{\boldsymbol{\xi}} d{\boldsymbol{\xi}}
\end{align*}
and
\begin{align*}
\mathcal{L}^\epsilon_D(\bolds{u})(\bolds{y}) - \mathcal{L}^\epsilon_D(\bolds{v})(\bolds{y}) &= \frac{1}{\epsilon^2 \omega_d} \int_{H_1(\mathbf{0})} \omega_{\boldsymbol{\xi}}(\bolds{y}) J(|{\boldsymbol{\xi}}|) [g'(\theta(\bolds{y}+\epsilon{\boldsymbol{\xi}};\bolds{u})) + g'(\theta(\bolds{y};\bolds{u})) \notag \\
&\qquad \qquad -g'(\theta(\bolds{y}+\epsilon{\boldsymbol{\xi}};\bolds{v})) - g'(\theta(\bolds{y};\bolds{v})) ] \bolds{e}_{\boldsymbol{\xi}} d{\boldsymbol{\xi}}.
\end{align*}
to find
\begin{align}
&|(\mathcal{L}^\epsilon_D(\bolds{u})(\bolds{x}) - \mathcal{L}^\epsilon_D(\bolds{v})(\bolds{x})) - (\mathcal{L}^\epsilon_D(\bolds{u})(\bolds{y}) - \mathcal{L}^\epsilon_D(\bolds{v})(\bolds{y}))| \notag \\
&= \bigg\vert \frac{1}{\epsilon^2 \omega_d} \int_{H_1(\mathbf{0})} J(|{\boldsymbol{\xi}}|)\notag \\
&\; \bigg( \omega_{\boldsymbol{\xi}}(\bolds{x})[g'(\theta(\bolds{x}+\epsilon{\boldsymbol{\xi}};\bolds{u})) + g'(\theta(\bolds{x};\bolds{u})) -g'(\theta(\bolds{x}+\epsilon{\boldsymbol{\xi}};\bolds{v})) - g'(\theta(\bolds{x};\bolds{v})) ] \notag \\
&\; - \omega_{\boldsymbol{\xi}}(\bolds{y})[g'(\theta(\bolds{y}+\epsilon{\boldsymbol{\xi}};\bolds{u})) + g'(\theta(\bolds{y};\bolds{u})) -g'(\theta(\bolds{y}+\epsilon{\boldsymbol{\xi}};\bolds{v})) - g'(\theta(\bolds{y};\bolds{v})) ]\bigg) \bolds{e}_{\boldsymbol{\xi}} d{\boldsymbol{\xi}} \bigg\vert \notag \\
&\leq \frac{1}{\epsilon^2 \omega_d} \int_{H_1(\mathbf{0})} J(|{\boldsymbol{\xi}}|) \notag \\
&\; \bigg\vert \omega_{\boldsymbol{\xi}}(\bolds{x})[g'(\theta(\bolds{x}+\epsilon{\boldsymbol{\xi}};\bolds{u})) + g'(\theta(\bolds{x};\bolds{u})) -g'(\theta(\bolds{x}+\epsilon{\boldsymbol{\xi}};\bolds{v})) - g'(\theta(\bolds{x};\bolds{v})) ] \notag \\
&\; - \omega_{\boldsymbol{\xi}}(\bolds{y})[g'(\theta(\bolds{y}+\epsilon{\boldsymbol{\xi}};\bolds{u})) + g'(\theta(\bolds{y};\bolds{u})) -g'(\theta(\bolds{y}+\epsilon{\boldsymbol{\xi}};\bolds{v})) - g'(\theta(\bolds{y};\bolds{v})) ] \bigg\vert d{\boldsymbol{\xi}} \notag \\
&= \frac{1}{\epsilon^2 \omega_d} \int_{H_1(\mathbf{0})} J(|{\boldsymbol{\xi}}|) \bigg\vert \bigg( \omega_{\boldsymbol{\xi}}(\bolds{x})[g'(\theta(\bolds{x}+\epsilon{\boldsymbol{\xi}};\bolds{u})) -g'(\theta(\bolds{x}+\epsilon{\boldsymbol{\xi}};\bolds{v}))] \notag \\
&\qquad \qquad - \omega_{\boldsymbol{\xi}}(\bolds{y})[g'(\theta(\bolds{y}+\epsilon{\boldsymbol{\xi}};\bolds{u})) -g'(\theta(\bolds{y}+\epsilon{\boldsymbol{\xi}};\bolds{v}))] \bigg) \notag \\
&\qquad \qquad + \bigg( \omega_{\boldsymbol{\xi}}(\bolds{x})[g'(\theta(\bolds{x};\bolds{u})) -g'(\theta(\bolds{x};\bolds{v}))] \notag \\
&\qquad \qquad - \omega_{\boldsymbol{\xi}}(\bolds{y})[g'(\theta(\bolds{y};\bolds{u})) -g'(\theta(\bolds{y};\bolds{v}))] \bigg) \bigg\vert d{\boldsymbol{\xi}},
\end{align}
where we have rearranged the terms in last step. Application of the triangle inequality gives
\begin{align}
&|(\mathcal{L}^\epsilon_D(\bolds{u})(\bolds{x}) - \mathcal{L}^\epsilon_D(\bolds{v})(\bolds{x})) - (\mathcal{L}^\epsilon_D(\bolds{u})(\bolds{y}) - \mathcal{L}^\epsilon_D(\bolds{v})(\bolds{y}))| \notag \\
&\leq \frac{1}{\epsilon^2 \omega_d} \int_{H_1(\mathbf{0})} J(|{\boldsymbol{\xi}}|) \bigg( \bigg\vert \omega_{\boldsymbol{\xi}}(\bolds{x})[g'(\theta(\bolds{x}+\epsilon{\boldsymbol{\xi}};\bolds{u})) -g'(\theta(\bolds{x}+\epsilon{\boldsymbol{\xi}};\bolds{v}))] \notag \\
&\qquad \qquad - \omega_{\boldsymbol{\xi}}(\bolds{y})[g'(\theta(\bolds{y}+\epsilon{\boldsymbol{\xi}};\bolds{u})) -g'(\theta(\bolds{y}+\epsilon{\boldsymbol{\xi}};\bolds{v}))] \bigg\vert \notag \\
&\qquad \qquad + \bigg\vert \omega_{\boldsymbol{\xi}}(\bolds{x})[g'(\theta(\bolds{x};\bolds{u})) -g'(\theta(\bolds{x};\bolds{v}))] \notag \\
&\qquad \qquad - \omega_{\boldsymbol{\xi}}(\bolds{y})[g'(\theta(\bolds{y};\bolds{u})) -g'(\theta(\bolds{y};\bolds{v}))] \bigg\vert \bigg) d{\boldsymbol{\xi}}.
\end{align}
Now write $h_{\boldsymbol{\xi}}:\mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}^+$ given by
\begin{align}\label{eq:def h}
h_{\boldsymbol{\xi}}(\bolds{x},\bolds{y}) &:= \bigg\vert \omega_{\boldsymbol{\xi}}(\bolds{x})[g'(\theta(\bolds{x};\bolds{u})) -g'(\theta(\bolds{x};\bolds{v}))] - \omega_{\boldsymbol{\xi}}(\bolds{y})[g'(\theta(\bolds{y};\bolds{u})) -g'(\theta(\bolds{y};\bolds{v}))] \bigg\vert.
\end{align}
and
\begin{align}\label{eq:est 13}
&|(\mathcal{L}^\epsilon_D(\bolds{u})(\bolds{x}) - \mathcal{L}^\epsilon_D(\bolds{v})(\bolds{x})) - (\mathcal{L}^\epsilon_D(\bolds{u})(\bolds{y}) - \mathcal{L}^\epsilon_D(\bolds{v})(\bolds{y}))| \notag \\
&\leq \frac{1}{\epsilon^2 \omega_d} \int_{H_1(\mathbf{0})} J(|{\boldsymbol{\xi}}|) (h_{\boldsymbol{\xi}}(\bolds{x}+ \epsilon{\boldsymbol{\xi}}, \bolds{y}+ \epsilon{\boldsymbol{\xi}}) + h(\bolds{x},\bolds{y}) ) d{\boldsymbol{\xi}}.
\end{align}
We now estimate $h_{\boldsymbol{\xi}}(\bolds{x},\bolds{y})$ for any $\bolds{x},\bolds{y} $ in $D$ and in the layer of thickness $\epsilon$ surrounding $D$.
Proceeding as before we define $r:[0,1]\times D \to \mathbb{R}$ as follows
\begin{align}\label{eq:def r}
r(l,\bolds{x}) &:= \theta(\bolds{x};\bolds{v}) + l(\theta(\bolds{x};\bolds{u}) - \theta(\bolds{x};\bolds{v})),
\end{align}
so $\frac{\partial r(l,\bolds{x})}{\partial l} = \theta(\bolds{x};\bolds{u}) - \theta(\bolds{x};\bolds{v})$. We also have
\begin{align}\label{eq:est 14}
g'(\theta(\bolds{x};\bolds{u})) -g'(\theta(\bolds{x};\bolds{v})) &= g'(r(1,\bolds{x})) - g'(r(0,\bolds{x})) \notag \\
&= \int_0^1 \frac{\partial g'(r(l,\bolds{x}))}{\partial l} dl \notag \\
&= \int_0^1 g''(r(l,\bolds{x})) (\theta(\bolds{x};\bolds{u}) - \theta(\bolds{x};\bolds{v})) dl.
\end{align}
Similarly,
\begin{align}\label{eq:est 15}
g'(\theta(\bolds{y};\bolds{u})) -g'(\theta(\bolds{y};\bolds{v})) &= \int_0^1 g''(r(l,\bolds{y})) (\theta(\bolds{y};\bolds{u}) - \theta(\bolds{y};\bolds{v})) dl.
\end{align}
Substitution of \autoref{eq:est 14} and \autoref{eq:est 15} in $h_{\boldsymbol{\xi}}(\bolds{x},\bolds{y})$ gives
\begin{align*}
h_{\boldsymbol{\xi}}(\bolds{x},\bolds{y}) &= \bigg\vert \int_0^1 ( \omega_{\boldsymbol{\xi}}(\bolds{x}) g''(r(l,\bolds{x})) (\theta(\bolds{x};\bolds{u}) - \theta(\bolds{x};\bolds{v})) \notag \\
&\qquad \qquad - \omega_{\boldsymbol{\xi}}(\bolds{y}) g''(r(l,\bolds{y})) (\theta(\bolds{y};\bolds{u}) - \theta(\bolds{y};\bolds{v})) ) dl \bigg\vert \notag \\
&\leq \int_0^1 \bigg\vert \omega_{\boldsymbol{\xi}}(\bolds{x}) g''(r(l,\bolds{x})) (\theta(\bolds{x};\bolds{u}) - \theta(\bolds{x};\bolds{v})) \notag \\
&\qquad \qquad - \omega_{\boldsymbol{\xi}}(\bolds{y}) g''(r(l,\bolds{y})) (\theta(\bolds{y};\bolds{u}) - \theta(\bolds{y};\bolds{v})) \bigg\vert dl.
\end{align*}
Adding and subtracting $\omega_{\boldsymbol{\xi}}(\bolds{x})g''(r(l,\bolds{x})) (\theta(\bolds{y};\bolds{u}) - \theta(\bolds{y};\bolds{v}))$ gives
\begin{align}\label{eq:est 16}
h_{\boldsymbol{\xi}}(\bolds{x},\bolds{y}) &\leq \int_0^1 |\omega_{\boldsymbol{\xi}}(\bolds{x})|\, |g''(r(l,\bolds{x}))|\, |(\theta(\bolds{x};\bolds{u}) - \theta(\bolds{x};\bolds{v})) - (\theta(\bolds{y};\bolds{u}) - \theta(\bolds{y};\bolds{v}))| dl \notag \\
&+ \int_0^1 |\omega_{\boldsymbol{\xi}}(\bolds{x})g''(r(l,\bolds{x})) - \omega_{\boldsymbol{\xi}}(\bolds{y})g''(r(l,\bolds{y}))|\, |\theta(\bolds{y};\bolds{u}) - \theta(\bolds{y};\bolds{v})| dl \notag \\
&=: I_1 + I_2,
\end{align}
For $I_1$, we note that $0\leq \omega(\bolds{x}) \leq 1$ and $|g''(r)| \leq C^g_2$ and proceed further to find that
\begin{align}
I_1 &\leq C^g_2 |(\theta(\bolds{x};\bolds{u}) - \theta(\bolds{x};\bolds{v})) - (\theta(\bolds{y};\bolds{u}) - \theta(\bolds{y};\bolds{v}))| \notag \\
&= C^g_2 |\theta(\bolds{x};\bolds{u} - \bolds{v}) - \theta(\bolds{y};\bolds{u}-\bolds{v})|.
\end{align}
Using the estimate given in \autoref{eq:est 12} we see that
\begin{align}\label{eq:est 17}
I_1 &\leq 2 \bar{J}_0 C^g_2 (1+||\omega||_{\Cholder{}})||\bolds{u} - \bolds{v}||_{\Cholder{}} |\bolds{x} - \bolds{y}|^\gamma.
\end{align}
Now we apply the inequality given in \autoref{eq:est 11} to $I_2$ to find that
\begin{align*}
I_2 &\leq 2 \bar{J}_0 ||\bolds{u} - \bolds{v}||_{\Cholder{}} \int_0^1 |\omega_{\boldsymbol{\xi}}(\bolds{x})g''(r(l,\bolds{x})) - \omega_{\boldsymbol{\xi}}(\bolds{y})g''(r(l,\bolds{y}))| dl.
\end{align*}
Adding and subtracting $\omega_{\boldsymbol{\xi}}(\bolds{x})g''(r(l,\bolds{y}))$ gives
\begin{align*}
I_2 &\leq 2 \bar{J}_0 ||\bolds{u} - \bolds{v}||_{\Cholder{}} \int_0^1 |\omega_{\boldsymbol{\xi}}(\bolds{x})|\, |g''(r(l,\bolds{x})) - g''(r(l,\bolds{y}))| dl \notag \\
&\quad + 2 \bar{J}_0 ||\bolds{u} - \bolds{v}||_{\Cholder{}} \int_0^1 |\omega_{\boldsymbol{\xi}}(\bolds{x}) - \omega_{\boldsymbol{\xi}}(\bolds{y})|\, |g''(r(l,\bolds{y}))| dl \notag \\
&\leq 2 C^g_3 \bar{J}_0 ||\bolds{u} - \bolds{v}||_{\Cholder{}} \int_0^1 |r(l,\bolds{x}) - r(l,\bolds{y})| dl \notag \\
&\quad + 2 C^g_2 \bar{J}_0 ||\bolds{u} - \bolds{v}||_{\Cholder{}} \int_0^1 |\omega_{\boldsymbol{\xi}}(\bolds{x}) - \omega_{\boldsymbol{\xi}}(\bolds{y})| dl.
\end{align*}
The quantity $|r(l,\bolds{x}) - r(l,\bolds{y})| $ (see \autoref{eq:def r}) can be estimated as follows
\begin{align}
&|r(l,\bolds{x}) - r(l,\bolds{y})| \notag \\
&= | (1-l) \theta(\bolds{x};\bolds{v}) + l\theta(\bolds{x};\bolds{u}) - ((1-l) \theta(\bolds{y};\bolds{v}) + l\theta(\bolds{y};\bolds{u}))| \notag \\
&\leq |1-l|\, |\theta(\bolds{x};\bolds{v}) - \theta(\bolds{y};\bolds{v})| + |l|\, |\theta(\bolds{x};\bolds{u}) - \theta(\bolds{y};\bolds{u})| \notag \\
&\leq |\theta(\bolds{x};\bolds{v}) - \theta(\bolds{y};\bolds{v})| + |\theta(\bolds{x};\bolds{u}) - \theta(\bolds{y};\bolds{u})| \notag \\
&\leq 2\bar{J}_0 (1+||\omega||_{\Cholder{}}) ||\bolds{v}||_{\Cholder{}} |\bolds{x} - \bolds{y}|^\gamma + 2\bar{J}_0 (1+||\omega||_{\Cholder{}}) ||\bolds{u}||_{\Cholder{}} |\bolds{x} - \bolds{y}|^\gamma \notag \\
&= 2\bar{J}_0 (1+||\omega||_{\Cholder{}})(||\bolds{u}||_{\Cholder{}} + ||\bolds{v}||_{\Cholder{}}) |\bolds{x} - \bolds{y}|^\gamma,
\end{align}
where we used the fact that $l\in [0,1]$ and \autoref{eq:est 12}. Using the inequality above and $|\omega_{\boldsymbol{\xi}}(\bolds{x}) - \omega_{\boldsymbol{\xi}}(\bolds{y})| \leq 2|\bolds{x} - \bolds{y}|^\gamma ||\omega||_{\Cholder{}}$ we get
\begin{align}\label{eq:est 18}
I_2 &\leq 2 C^g_3 \bar{J}_0 ||\bolds{u} - \bolds{v}||_{\Cholder{}} 2\bar{J}_0 (1+||\omega||_{\Cholder{}})(||\bolds{u}||_{\Cholder{}} + ||\bolds{v}||_{\Cholder{}}) \, |\bolds{x} - \bolds{y}|^\gamma \notag \\
&\quad + 2 C^g_2 \bar{J}_0 ||\bolds{u} - \bolds{v}||_{\Cholder{}} 2|\bolds{x} - \bolds{y}|^\gamma |\,|\omega||_{\Cholder{}} \notag \\
&\leq 4\bar{J}_0 (1+||\omega||_{\Cholder{}})\, [ C^g_3 \bar{J}_0(||\bolds{u}||_{\Cholder{}} + ||\bolds{v}||_{\Cholder{}}) + C^g_2] \, ||\bolds{u} - \bolds{v}||_{\Cholder{}} \,|\bolds{x} - \bolds{y}|^\gamma.
\end{align}
Substituting \autoref{eq:est 17} and \autoref{eq:est 18} into \autoref{eq:est 16} gives
\begin{align}\label{eq:est 19}
&h_{\boldsymbol{\xi}}(\bolds{x}, \bolds{y}) \notag \\
&\leq 6\bar{J}_0 (1+||\omega||_{\Cholder{}})\, [ C^g_3 \bar{J}_0(||\bolds{u}||_{\Cholder{}} + ||\bolds{v}||_{\Cholder{}}) + C^g_2] \, ||\bolds{u} - \bolds{v}||_{\Cholder{}} \,|\bolds{x} - \bolds{y}|^\gamma.
\end{align}
We now apply \autoref{eq:est 19} to \autoref{eq:est 13} and divide both sides by $|\bolds{x} - \bolds{y}|^\gamma$ to see that
\begin{align}\label{eq:lipshitz bound statebased 2}
&\frac{|(\mathcal{L}^\epsilon_D(\bolds{u})(\bolds{x}) - \mathcal{L}^\epsilon_D(\bolds{v})(\bolds{x})) - (\mathcal{L}^\epsilon_D(\bolds{u})(\bolds{y}) - \mathcal{L}^\epsilon_D(\bolds{v})(\bolds{y}))|}{|\bolds{x} - \bolds{y}|^\gamma} \notag \\
&\leq \frac{1}{\epsilon^2 \omega_d} \int_{H_1(\mathbf{0})} J(|{\boldsymbol{\xi}}|) \notag \\
&\quad 2\times 6\bar{J}_0 (1+||\omega||_{\Cholder{}})\, [ C^g_3 \bar{J}_0(||\bolds{u}||_{\Cholder{}} + ||\bolds{v}||_{\Cholder{}}) + C^g_2] \, ||\bolds{u} - \bolds{v}||_{\Cholder{}} \, d{\boldsymbol{\xi}} \notag \\
&= \frac{12\bar{J}^2_0 (1+||\omega||_{\Cholder{}})\, [ C^g_3 \bar{J}_0(||\bolds{u}||_{\Cholder{}} + ||\bolds{v}||_{\Cholder{}}) + C^g_2]}{\epsilon^2} ||\bolds{u} - \bolds{v}||_{\Cholder{}}.
\end{align}
Collecting results inequalities \autoref{eq:lipshitz bound statebased 1} and \autoref{eq:lipshitz bound statebased 2} deliver the upper bound given by
\begin{align}\label{eq:lipshitz bound statebased 3}
&||\mathcal{L}^\epsilon_D(\bolds{u}) - \mathcal{L}^\epsilon_D(\bolds{v})||_{\Cholder{}} \notag \\
&\leq \frac{16\bar{J}^2_0 (1+||\omega||_{\Cholder{}})\, [ C^g_3 \bar{J}_0(||\bolds{u}||_{\Cholder{}} + ||\bolds{v}||_{\Cholder{}}) + C^g_2]}{\epsilon^2} ||\bolds{u} - \bolds{v}||_{\Cholder{}}.
\end{align}
\paragraph{Lipschitz continuity for $\mathcal{L}^\epsilon(\bolds{u})$}
Using \autoref{eq:lipshitz bound bondbased 3} and \autoref{eq:lipshitz bound statebased 3} we get
\begin{align}
&||\mathcal{L}^\epsilon(\bolds{u}) - \mathcal{L}^\epsilon(\bolds{v})||_{\Cholder{}} \notag \\
&\leq \bigg( \frac{8C^f_3 \bar{J}_{3/2-\gamma}}{\epsilon^{5/2 - \gamma}} (||\bolds{u}||_{\Cholder{}} + ||\bolds{v}||_{\Cholder{}}) + \frac{8C^f_2(2+||\omega||_{\Cholder{}}) \bar{J}_{1}}{\epsilon^2} \notag \\
&\: + \frac{16\bar{J}^2_0 (1+||\omega||_{\Cholder{}})\, [ C^g_3 \bar{J}_0(||\bolds{u}||_{\Cholder{}} + ||\bolds{v}||_{\Cholder{}}) + C^g_2]}{\epsilon^2} \bigg)||\bolds{u} - \bolds{v}||_{\Cholder{}}.
\end{align}
Let $\alpha(\gamma)$ defined as follows: $\alpha(\gamma) = 0$ if $\gamma \geq 1/2$ and $\alpha(\gamma) = 1/2 - \gamma $ if $\gamma \leq 1/2$. It is easy to verify that, for all $\gamma \in (0,1]$ and $0< \epsilon \leq 1$
\begin{align}
\max \bigg\{ \frac{1}{\epsilon^2}, \frac{1}{\epsilon^{5/2 - \gamma}} \bigg\} \leq \frac{1}{\epsilon^{2+\alpha(\gamma)}}.
\end{align}
Using $\alpha(\gamma)$ and renaming the constants we have
\begin{align}
&||\mathcal{L}^\epsilon(\bolds{u}) - \mathcal{L}^\epsilon(\bolds{v})||_{\Cholder{}} \notag \\
&\leq \frac{L_1 (1+||\omega||_{\Cholder{}}) (1+ ||\bolds{u}||_{\Cholder{}} + ||\bolds{v}||_{\Cholder{}})}{\epsilon^{2+\alpha(\gamma)}} ||\bolds{u} - \bolds{v}||_{\Cholder{}}.
\end{align}
To complete the proof of \autoref{eq:lipschitz property of F}, we substitute the inequality above into \autoref{eq:norm of F in X} to obtain
\begin{align}
&\normX{F^\epsilon(y,t) - F^\epsilon(z,t)}{X} \notag \\
&\leq ||y^2 - z^2||_{\Cholder{}} + \frac{L_1 (1+||\omega||_{\Cholder{}}) (1+ ||y^1||_{\Cholder{}} + ||z^1||_{\Cholder{}})}{\epsilon^{2+\alpha(\gamma)}} ||y^1 - z^1||_{\Cholder{}} \notag \\
&\leq \frac{L_1 (1+||\omega||_{\Cholder{}}) (1+ ||y||_X + ||z||_X)}{\epsilon^{2+\alpha(\gamma)}} ||y - z||_X,
\end{align}
and \autoref{eq:lipschitz property of F} is proved.
\subsubsection{Bound on the non-local force in the H\"older norm}
In this subsection, we bound $||\mathcal{L^\epsilon}(\bolds{u})||_{\Cholder{}}$ from above. It follows from \autoref{eq:per bondbased simple force expression} and a straightforward calculation similar to the previous sections that
\begin{align}
|\mathcal{L}^\epsilon_T(\bolds{u})(\bolds{x})| &\leq \frac{2C^f_1 \bar{J}_{1/2}}{\epsilon^{3/2}}, \notag \\
\frac{|\mathcal{L}^\epsilon_T(\bolds{u})(\bolds{x}) - \mathcal{L}^\epsilon_T(\bolds{u})(\bolds{y})|}{|\bolds{x}-\bolds{y}|^\gamma} &\leq \frac{4C^f_2 \bar{J}_1 ||\bolds{u}||_{\Cholder{}} + 4 C^f_1 \bar{J}_{1/2} ||\omega||_{\Cholder{}}}{\epsilon^2}.
\end{align}
Next we consider the non-local dilatational force $\mathcal{L}^\epsilon_D$. We show how to calculate the bounds for the case of a convex-concave potential function $g$. When $g$ is quadratic we can still proceed along identical lines. We use the formula for $\mathcal{L}^\epsilon_D(\bolds{u})(\bolds{x})$ given by \autoref{eq:per statebased simple force expression} and perform a straightforward calculation to obtain the upper bound given by
\begin{align}
|\mathcal{L}^\epsilon_D(\bolds{u})(\bolds{x})| &\leq \frac{2C^g_1 \bar{J}_{0}}{\epsilon^{2}}.
\end{align}
We have the estimate
\begin{align}
&|\mathcal{L}^\epsilon_D(\bolds{u})(\bolds{x}) - \mathcal{L}^\epsilon_D(\bolds{u})(\bolds{y})| \notag \\
&\leq \frac{1}{\epsilon^2\omega_d} \int_{H_1(\mathbf{0})} J(|{\boldsymbol{\xi}}|) \bigg\vert \omega_{\boldsymbol{\xi}}(\bolds{x}) (g'(\theta(\bolds{x}+\epsilon {\boldsymbol{\xi}};\bolds{u})) + g'(\theta(\bolds{x};\bolds{u})))) \notag \\
&\qquad \qquad - \omega_{\boldsymbol{\xi}}(\bolds{y}) (g'(\theta(\bolds{y}+\epsilon {\boldsymbol{\xi}};\bolds{u})) + g'(\theta(\bolds{y};\bolds{u})))) \bigg\vert d{\boldsymbol{\xi}} \notag \\
&\leq \frac{1}{\epsilon^2\omega_d} \int_{H_1(\mathbf{0})} J(|{\boldsymbol{\xi}}|) \bigg\vert \omega_{\boldsymbol{\xi}}(\bolds{x}) g'(\theta(\bolds{x}+\epsilon {\boldsymbol{\xi}};\bolds{u})) - \omega_{\boldsymbol{\xi}}(\bolds{y}) g'(\theta(\bolds{y}+\epsilon {\boldsymbol{\xi}};\bolds{u}))\bigg\vert d{\boldsymbol{\xi}} \notag \\
&+ \frac{1}{\epsilon^2\omega_d} \int_{H_1(\mathbf{0})} J(|{\boldsymbol{\xi}}|) \bigg\vert \omega_{\boldsymbol{\xi}}(\bolds{x}) g'(\theta(\bolds{x};\bolds{u})) - \omega_{\boldsymbol{\xi}}(\bolds{y}) g'(\theta(\bolds{y};\bolds{u}))\bigg\vert d{\boldsymbol{\xi}}.
\end{align}
Using $|\omega_{\boldsymbol{\xi}}(\bolds{x}) - \omega_{\boldsymbol{\xi}}(\bolds{y})| \leq 2|\bolds{x} - \bolds{y}|^\gamma ||\omega||_{\Cholder{}}$, $|g'(r_1) - g'(r_2)| \leq C^g_2 |r_1 - r_2|$, $|g'(r)| \leq C^g_1$, and the estimate on $|\theta(\bolds{x};\bolds{u}) - \theta(\bolds{y};\bolds{u})|$ given by \autoref{eq:est 12}, we obtain
\begin{align}
&|\mathcal{L}^\epsilon_D(\bolds{u})(\bolds{x}) - \mathcal{L}^\epsilon_D(\bolds{u})(\bolds{y})| \notag \\
&\leq\frac{ [2 \bar{J}_0 C^g_2 (1+||\omega||_{\Cholder{}})||\bolds{u}||_{\Cholder{}} + 2C^g_1||\omega||_{\Cholder{}} ||\bolds{u}||_{\Cholder{}} ]}{\epsilon^2}\, |\bolds{x} - \bolds{y}|^\gamma.
\end{align}
Last we combine results and rename the constants to get
\begin{align}
\label{rhs est}
||\mathcal{L}^\epsilon(\bolds{u})||_{\Cholder{}} &\leq \frac{L_2(1+||\omega||_{\Cholder{}}) (1+||\bolds{u}||_{\Cholder{}})}{\epsilon^2}.
\end{align}
This completes the proof of \autoref{eq:bound on F}.
\subsection{Proof of Proposition 2}\label{ss:proof prop 4}
Given $\bolds{u},\bolds{v} \in L^2_0(D;\mathbb{R}^d)$ we find upper bounds on the Lipschitz continuity of the nonlocal force with respect to the $L^2$ norm. Motivated by the inequality
\begin{align}
||\mathcal{L}^\epsilon(\bolds{u}) - \mathcal{L}^\epsilon(\bolds{v})||_{L^2} &\leq ||\mathcal{L}^\epsilon_T(\bolds{u}) - \mathcal{L}^\epsilon_T(\bolds{v})||_{L^2} + ||\mathcal{L}^\epsilon_D(\bolds{u}) - \mathcal{L}^\epsilon_D(\bolds{u})||_{L^2}.,
\end{align}
we bound the Lipschitz continuity of the nonlocal forces due to tensile strain and dilatational strain separately.
We study $\mathcal{L}^\epsilon_T$ first. It is evident from \autoref{eq:per bondbased simple force expression} and using the estimate $|f'(r_1) - f'(r_2)|\leq C^f_2 |r_1 - r_2|$, and arguments similar to previous sections that we have
\begin{align}\label{eq:est 19.1}
&|\mathcal{L}^\epsilon_T(\bolds{u})(\bolds{x}) - \mathcal{L}^\epsilon_T(\bolds{v})(\bolds{x})| \notag \\
&\leq \frac{2}{\epsilon\omega_d} \int_{H_1(\mathbf{0})} \frac{J(|{\boldsymbol{\xi}}|)}{\sqrt{s_{\boldsymbol{\xi}}}} |f'(\bar{\bu}_{\boldsymbol{\xi}}(\bolds{x}).\bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}}) - f'(\bar{\bv}_{\boldsymbol{\xi}}(\bolds{x}).\bolds{e}_{\boldsymbol{\xi}}/\sqrt{s_{\boldsymbol{\xi}}})| d{\boldsymbol{\xi}} \notag \\
&\leq \frac{2C^f_2}{\epsilon^2 \omega_d} \int_{H_1(\mathbf{0})} \frac{J(|{\boldsymbol{\xi}}|)}{|{\boldsymbol{\xi}}|} |\bar{\bu}_{\boldsymbol{\xi}}(\bolds{x}) - \bar{\bv}_{\boldsymbol{\xi}}(\bolds{x})| d{\boldsymbol{\xi}},
\end{align}
where we also substituted $s_{\boldsymbol{\xi}} = \epsilon |{\boldsymbol{\xi}}|$.
We apply \autoref{eq:ineq symm square} to \autoref{eq:est 19.1} with $C=\frac{2C^f_2}{\epsilon^2}$, $\alpha = 1$, and $p({\boldsymbol{\xi}}) = |\bar{\bu}_{\boldsymbol{\xi}}(\bolds{x}) - \bar{\bv}_{\boldsymbol{\xi}}(\bolds{x})|$ to get
\begin{align}
& ||\mathcal{L}^\epsilon_T(\bolds{u}) - \mathcal{L}^\epsilon_T(\bolds{v})||_{L^2}^2 \notag \\
&\leq \int_D |\mathcal{L}^\epsilon_T(\bolds{u})(\bolds{x}) - \mathcal{L}^\epsilon_T(\bolds{v})(\bolds{x})|^2 d\bolds{x} \notag \\
&\leq \int_D \left(\frac{2C^f_2}{\epsilon^2} \right)^2 \frac{\bar{J}_1}{\omega_d} \int_{H_1(\mathbf{0})} \frac{J(|{\boldsymbol{\xi}}|)}{|{\boldsymbol{\xi}}|} |\bar{\bu}_{\boldsymbol{\xi}}(\bolds{x}) - \bar{\bv}_{\boldsymbol{\xi}}(\bolds{x})|^2 d{\boldsymbol{\xi}} d\bolds{x} \notag \\
&= \left(\frac{2C^f_2}{\epsilon^2} \right)^2 \frac{\bar{J}_1}{\omega_d} \int_{H_1(\mathbf{0})} \frac{J(|{\boldsymbol{\xi}}|)}{|{\boldsymbol{\xi}}|} \left[ \int_D |\bar{\bu}_{\boldsymbol{\xi}}(\bolds{x}) - \bar{\bv}_{\boldsymbol{\xi}}(\bolds{x})|^2 d\bolds{x} \right] d{\boldsymbol{\xi}},
\end{align}
where we interchanged integration in last step. Using
\begin{align}
\int_D |\bar{\bu}_{\boldsymbol{\xi}}(\bolds{x}) - \bar{\bv}_{\boldsymbol{\xi}}(\bolds{x})|^2 d\bolds{x} &\leq 2 ||\bolds{u} - \bolds{v}||^2_{L^2}
\end{align}
we conclude that
\begin{align}\label{eq:est 20}
||\mathcal{L}^\epsilon_T(\bolds{u}) - \mathcal{L}^\epsilon_T(\bolds{v})||_{L^2} \leq \frac{4 C^f_2 \bar{J}_1}{\epsilon^2} ||\bolds{u} - \bolds{v}||_{L^2}.
\end{align}
In estimating $||\mathcal{L}^\epsilon_D(\bolds{u}) -\mathcal{L}^\epsilon_D(\bolds{v})||_{L^2}$ we will consider convex-concave $g$ noting that the case of quadratic $g$ is dealt in a similar fashion. From \autoref{eq:per statebased simple force expression} and using estimate $|g'(r_1) - g'(r_2)|\leq C^g_2 |r_1 - r_2|$, and proceeding as before we have
\begin{align}\label{lipd inequal}
&|\mathcal{L}^\epsilon_D(\bolds{u})(\bolds{x}) - \mathcal{L}^\epsilon_D(\bolds{v})(\bolds{x})| \notag \\
&\leq \frac{1}{\epsilon^2 \omega_d} \int_{H_1(\mathbf{0})} J(|{\boldsymbol{\xi}}|) [|g'(\theta(\bolds{x}+\epsilon{\boldsymbol{\xi}};\bolds{u})) - g'(\theta(\bolds{x}+\epsilon{\boldsymbol{\xi}};\bolds{v}))| \notag \\
&\qquad \quad + |g'(\theta(\bolds{x};\bolds{u})) - g'(\theta(\bolds{x};\bolds{v}))|] d{\boldsymbol{\xi}} \notag \\
&\leq \frac{C^g_2}{\epsilon^2 \omega_d} \int_{H_1(\mathbf{0})} J(|{\boldsymbol{\xi}}|) [|\theta(\bolds{x}+\epsilon{\boldsymbol{\xi}};\bolds{u}) - \theta(\bolds{x}+\epsilon{\boldsymbol{\xi}};\bolds{v})| + |\theta(\bolds{x};\bolds{u}) - \theta(\bolds{x};\bolds{v})|] d{\boldsymbol{\xi}} \notag \\
&= \frac{C^g_2}{\epsilon^2 \omega_d} \int_{H_1(\mathbf{0})} J(|{\boldsymbol{\xi}}|) [|\theta(\bolds{x}+\epsilon{\boldsymbol{\xi}};\bolds{u}-\bolds{v}) | + |\theta(\bolds{x};\bolds{u}-\bolds{v})|] d{\boldsymbol{\xi}}.
\end{align}
Squaring \autoref{lipd inequal} and applying inequality \autoref{eq:ineq symm square} with $C=\frac{C^g_2}{\epsilon^2}$, $\alpha = 0$, and $p({\boldsymbol{\xi}}) = |\theta(\bolds{x}+\epsilon{\boldsymbol{\xi}};\bolds{u}-\bolds{v}) | + |\theta(\bolds{x};\bolds{u}-\bolds{v})|$ gives
\begin{align}\label{l2bound}
&||\mathcal{L}^\epsilon_D(\bolds{u}) - \mathcal{L}^\epsilon_D(\bolds{v})||^2_{L^2} \notag \\
&\leq \int_D \left(\frac{C^g_2}{\epsilon^2} \right)^2 \frac{\bar{J}_0}{\omega_d}\int_{H_1(\mathbf{0})} J(|{\boldsymbol{\xi}}|) (|\theta(\bolds{x}+\epsilon{\boldsymbol{\xi}};\bolds{u}-\bolds{v}) | + |\theta(\bolds{x};\bolds{u}-\bolds{v})|)^2 d{\boldsymbol{\xi}} d\bolds{x} \notag \\
&\leq \left(\frac{C^g_2}{\epsilon^2} \right)^2 \frac{\bar{J}_0}{\omega_d}\int_{H_1(\mathbf{0})} J(|{\boldsymbol{\xi}}|) \left[ \int_D 2(|\theta(\bolds{x}+\epsilon{\boldsymbol{\xi}};\bolds{u}-\bolds{v}) |^2 + |\theta(\bolds{x};\bolds{u}-\bolds{v})|^2) d\bolds{x} \right] d{\boldsymbol{\xi}},
\end{align}
where we used Cauchy's inequality and exchanged integration in the last step. It is easy to verify that
\begin{align*}
\int_D |\theta(\bolds{x}+\epsilon {\boldsymbol{\xi}}; \bolds{u})|^2 d\bolds{x} &\leq 2 \bar{J}_0^2 ||\bolds{u}||_{L^2}^2
\end{align*}
holds for all ${\boldsymbol{\xi}} \in H_1(\mathbf{0})$. Combining this estimate and \autoref{l2bound} we see that
\begin{align}\label{eq:est 21}
||\mathcal{L}^\epsilon_D(\bolds{u}) - \mathcal{L}^\epsilon_D(\bolds{v})||_{L^2} &\leq \frac{4 C^g_2 \bar{J}_0^2}{\epsilon^2 }||\bolds{u} - \bolds{v}||_{L^2}.
\end{align}
Estimates \autoref{eq:est 20} and \autoref{eq:est 21} together delivers (after renaming the constants)
\begin{align}
||\mathcal{L}^\epsilon(\bolds{u}) - \mathcal{L}^\epsilon(\bolds{v})||_{L^2} &\leq \frac{L_3}{\epsilon^2} ||\bolds{u} - \bolds{v}||_{L^2},
\end{align}
where $L_3$ is given by \autoref{eq:def L3}.
This completes the proof of \sautoref{Proposition}{prop:lipschitz L2}.
\section{Energy stability of the semi-discrete scheme}\label{ss:stab proof}
In this section, we establish \autoref{thm:stab semi} for convex-concave potential functions $g$ as well as for quadratic potential functions.
We recall the semi-discrete problem introduced in \autoref{semidiscrete}.
We first introduce the semi-discrete boundary condition by setting $\hat{\bolds{u}}_i(t) = \mathbf{0}$ for all $t$ and for all $\bolds{x}_i \notin D$.
Let $\{\hat{\bolds{u}}_i(t)\}_{i,\bolds{x}_i\in D}$ denote the semi-discrete approximate solution which satisfies the following evolution, for all $t\in [0,T]$ and $i$ such that $\bolds{x}_i\in D$,
\begin{align}\label{eq:stab 1}
\ddot{\hat{\bolds{u}}}_i(t) = \mathcal{L}^\epsilon(\hat{\bolds{u}}(t))(\bolds{x}_i) + \bolds{b}(\bolds{x}_i,t),
\end{align}
where $\hat{\bolds{u}}(t)$ is the piecewise constant extension of $\{\hat{\bu}(t)\}_{i,\bolds{x}_i \in D}$, given by
\begin{align*}
\hat{\bolds{u}}(t,\bolds{x}) = \sum_{i, \bolds{x}_i \in D} \hat{\bolds{u}}_i(t) \chi_{U_i}(\bolds{x}).
\end{align*}
Let $\hat{\mathcal{L}^\epsilon}(\hat{\bu}(t))(\bolds{x})$ be defined as
\begin{align*}
\hat{\mathcal{L}^\epsilon}(\hat{\bu}(t))(\bolds{x}) = \sum_{i, \bolds{x}_i \in D} \mathcal{L}^\epsilon(\hat{\bu}(t))(\bolds{x}_i) \chi_{U_i}(\bolds{x})
\end{align*}
and define $\hat{\bolds{b}}(t)$ similarly. From \autoref{eq:stab 1} noting the definition of piecewise constant extension
\begin{align}\label{eq:stab 2}
\ddot{\hat{\bolds{u}}}(\bolds{x},t) &= \hat{\mathcal{L}^\epsilon}(\hat{\bolds{u}}(t))(\bolds{x}) + \hat{\bolds{b}}(\bolds{x},t) \notag \\
&= \mathcal{L}^\epsilon(\hat{\bolds{u}}(t))(\bolds{x}) + \hat{\bolds{b}}(\bolds{x},t) + \sigma(\bolds{x},t),
\end{align}
where the error term $\sigma(\bolds{x},t)$ is given by
\begin{align}
\sigma(\bolds{x},t) &:= \hat{\mathcal{L}^\epsilon}(\hat{\bolds{u}}(t))(\bolds{x}) - \mathcal{L}^\epsilon(\hat{\bolds{u}}(t))(\bolds{x}).
\end{align}
We split $\sigma$ into two parts
\begin{align}
\sigma(\bolds{x},t) &= \left[ \hat{\mathcal{L}^\epsilon_T}(\hat{\bolds{u}}(t))(\bolds{x}) - \mathcal{L}^\epsilon_T(\hat{\bolds{u}}(t))(\bolds{x}) \right] + \left[\hat{\mathcal{L}^\epsilon_D}(\hat{\bolds{u}}(t))(\bolds{x}) - \mathcal{L}^\epsilon_D(\hat{\bolds{u}}(t))(\bolds{x}) \right] \notag \\
&=: \sigma_T(\bolds{x},t) + \sigma_D(\bolds{x},t).
\end{align}
Multiplying both sides of \autoref{eq:stab 2} by $\dot{\hat{\bu}}(t)$ and integrating over $D$ gives
\begin{align}\label{eq:stab 2.1}
(\ddot{\hat{\bolds{u}}}(t), \dot{\hat{\bu}}(t)) &= (\mathcal{L}^\epsilon(\hat{\bu}(t)), \dot{\hat{\bu}}(t)) + (\hat{\bolds{b}}(t), \dot{\hat{\bu}}(t)) + (\sigma(t), \dot{\hat{\bu}}(t)),
\end{align}
where $(\cdot, \cdot)$ denotes the $L^2$-inner product.
\subsection{Estimating $\sigma$}
We proceed by estimating $L^2$-norm of $\sigma(t)$. It follows easily from \autoref{eq:per bondbased simple force expression} that
\begin{align}
|\sigma_T(\bolds{x},t)| &\leq \frac{4C^f_1 \bar{J}_{1/2}}{\epsilon^{3/2}} \quad \Rightarrow ||\sigma_T(t)||_{L^2} \leq \frac{4 C^f_1 \bar{J}_{1/2}\sqrt{|D|}}{\epsilon^{3/2}}.
\end{align}
We now deal with two cases of $g$ separately.
\textbf{1. Convex-concave type $g$: }In this case, we can easily show from \autoref{eq:per statebased simple force expression} that
\begin{align}
|\sigma_D(\bolds{x},t)| &\leq \frac{4C^g_1 \bar{J}_0}{\epsilon^2} \quad \Rightarrow ||\sigma_D(t)||_{L^2} \leq \frac{4 C^g_1 \bar{J}_0\sqrt{|D|}}{\epsilon^2}.
\end{align}
\textbf{2. Quadratic type $g$: }In this case we have $g'(r) = g''(0) r$. Let $\bolds{x} \in U_i$, i.e. in the unit cell of the $i^{th}$ mesh node. To simplify the calculations let $\bolds{u} = \hat{\bu}(t)$ (and later we will use the fact that $\hat{\bu}$ is piecewise constant function). From \autoref{eq:per statebased simple force expression}, we have
\begin{align}
|\sigma_D(\bolds{x},t)| &= |\mathcal{L}^\epsilon_D(\bolds{u})(\bolds{x}_i) - \mathcal{L}^\epsilon_D(\bolds{u})(\bolds{x})| \notag \\
&= \bigg\vert \frac{g''(0)}{\epsilon^2 \omega_d} \int_{H_1(\mathbf{0})} J(|{\boldsymbol{\xi}}|) \bigg[ \omega_{{\boldsymbol{\xi}}}(\bolds{x}_i) (\theta(\bolds{x}_i + \epsilon {\boldsymbol{\xi}}; \bolds{u}) + \theta(\bolds{x}_i;\bolds{u})) \notag \\
&\qquad - \omega_{{\boldsymbol{\xi}}}(\bolds{x}) (\theta(\bolds{x} + \epsilon {\boldsymbol{\xi}}; \bolds{u}) + \theta(\bolds{x};\bolds{u})) \bigg]\bolds{e}_{\boldsymbol{\xi}} d{\boldsymbol{\xi}} \bigg\vert.
\end{align}
Now consider the function $\bolds{a}(\bolds{x},{\boldsymbol{\xi}})$ defined as
\begin{align}\label{eq:def a}
\bolds{a}(\bolds{x}, {\boldsymbol{\xi}}) = \theta(\bolds{x}+\epsilon{\boldsymbol{\xi}}; \bolds{u}).
\end{align}
We then have
\begin{align}
&|\sigma_D(\bolds{x},t)| \notag \\
&= \bigg\vert \frac{g''(0)}{\epsilon^2 \omega_d} \int_{H_1(\mathbf{0})} J(|{\boldsymbol{\xi}}|) \bigg[ \omega_{{\boldsymbol{\xi}}}(\bolds{x}_i) (\bolds{a}(\bolds{x}_i,{\boldsymbol{\xi}}) + \bolds{a}(\bolds{x}_i, \mathbf{0})) \notag \\
&\qquad - \omega_{{\boldsymbol{\xi}}}(\bolds{x}) (\bolds{a}(\bolds{x},{\boldsymbol{\xi}}) + \bolds{a}(\bolds{x}, \mathbf{0})) \bigg]\bolds{e}_{\boldsymbol{\xi}} d{\boldsymbol{\xi}} \bigg\vert \notag \\
&\leq \frac{g''(0)}{\epsilon^2 \omega_d} \int_{H_1(\mathbf{0})} J(|{\boldsymbol{\xi}}|) (|\bolds{a}(\bolds{x}_i,{\boldsymbol{\xi}})| + |\bolds{a}(\bolds{x}_i,\mathbf{0})| + |\bolds{a}(\bolds{x},{\boldsymbol{\xi}})| + |\bolds{a}(\bolds{x},\mathbf{0})|) d{\boldsymbol{\xi}}.
\end{align}
Let
\begin{align}\label{eq:def b}
b_{\boldsymbol{\xi}} := |\bolds{a}(\bolds{x}_i,{\boldsymbol{\xi}})| + |\bolds{a}(\bolds{x}_i,\mathbf{0})| + |\bolds{a}(\bolds{x},{\boldsymbol{\xi}})| + |\bolds{a}(\bolds{x},\mathbf{0})|
\end{align}
then using the inequality \autoref{eq:ineq symm square} with $C=\frac{g''(0)}{\epsilon^2}$, $\alpha = 0$, and $p({\boldsymbol{\xi}}) = b_{\boldsymbol{\xi}}$, we get
\begin{align}\label{eq:stab 3}
|\sigma_D(\bolds{x},t)|^2 &\leq\left(\frac{g''(0)}{\epsilon^2} \right)^2 \frac{\bar{J}_0}{\omega_d} \int_{H_1(\mathbf{0})} J(|{\boldsymbol{\xi}}|) b_{\boldsymbol{\xi}}^2 d{\boldsymbol{\xi}}.
\end{align}
Thus on an interchange of integration we have
\begin{align}\label{eq:stab 4}
||\sigma_D(t)||^2_{L^2} &= \int_D |\sigma_D(\bolds{x},t)|^2 d\bolds{x} \notag \\
&\leq \sum_{i,\bolds{x}_i \in D} \int_{U_i} |\sigma_D(\bolds{x},t)|^2 d\bolds{x} \notag \\
&\leq \left(\frac{g''(0)}{\epsilon^2} \right)^2 \frac{\bar{J}_0}{\omega_d} \int_{H_1(\mathbf{0})} J(|{\boldsymbol{\xi}}|) \left[\sum_{i,\bolds{x}_i \in D} \int_{U_i} b^2_{\boldsymbol{\xi}} d\bolds{x} \right] d{\boldsymbol{\xi}}.
\end{align}
We denote the term inside square bracket as $I$ and estimate it next. Recalling the definition of $b_{\boldsymbol{\xi}}$ in \autoref{eq:def b} and using the identity $(\sum_{n=1}^4 c_n)^2 \leq 4 \sum_{n=1}^4 c^2_n$ we have
\begin{align}\label{eq:stab 5}
I &\leq 4 \sum_{i,\bolds{x}_i \in D} \int_{U_i} (|\bolds{a}(\bolds{x}_i,{\boldsymbol{\xi}})|^2 + |\bolds{a}(\bolds{x}_i,\mathbf{0})|^2 + |\bolds{a}(\bolds{x},{\boldsymbol{\xi}})|^2 + |\bolds{a}(\bolds{x},\mathbf{0})|^2) d\bolds{x}.
\end{align}
For $\bolds{x}$ either in $D$ or in layer of thickness $\epsilon$ surrounding $D$ take ${\boldsymbol{\xi}} \in H_1(\mathbf{0})$ and from the definition of $a(\bolds{x},{\boldsymbol{\xi}})$ we have
\begin{align}
|\bolds{a}(\bolds{x},{\boldsymbol{\xi}})|^2 &= |\theta(\bolds{x}+\epsilon {\boldsymbol{\xi}}, \bolds{u})|^2 \notag \\
&= \bigg\vert \frac{1}{\omega_d} \int_{H_1(\mathbf{0})} \omega(\bolds{x} + \epsilon {\boldsymbol{\xi}} + \epsilon \boldsymbol{\eta}) J(|\boldsymbol{\eta}|) \bar{\bu}_{\boldsymbol{\eta}}(\bolds{x} + \epsilon{\boldsymbol{\xi}}) \cdot \bolds{e}_{\boldsymbol{\eta}} d\boldsymbol{\eta} \bigg\vert^2 \notag \\
&\leq \bigg\vert \frac{1}{\omega_d} \int_{H_1(\mathbf{0})} J(|\boldsymbol{\eta}|) (|\bolds{u}(\bolds{x} + \epsilon{\boldsymbol{\xi}} + \epsilon \boldsymbol{\eta})| + |\bolds{u}(\bolds{x} + \epsilon {\boldsymbol{\xi}})|) d\boldsymbol{\eta} \bigg\vert^2,
\end{align}
where we used the fact that $0\leq \omega(x) \leq 1$ and definition of $\bar{\bu}_{\boldsymbol{\eta}}(\bolds{x}+\epsilon{\boldsymbol{\xi}})$. We now apply inequality \autoref{eq:ineq symm square} with $C = 1$, $\alpha = 0$ and $p(\boldsymbol{\eta}) = |\bolds{u}(\bolds{x} + \epsilon{\boldsymbol{\xi}} + \epsilon \boldsymbol{\eta})| + |\bolds{u}(\bolds{x} + \epsilon {\boldsymbol{\xi}})|$ to obtain
\begin{align}
|\bolds{a}(\bolds{x},{\boldsymbol{\xi}})|^2 &\leq \frac{\bar{J}_0}{\omega_d} \int_{H_1(\mathbf{0})} J(|\boldsymbol{\eta}|) (|\bolds{u}(\bolds{x} + \epsilon{\boldsymbol{\xi}} + \epsilon \boldsymbol{\eta})| + |\bolds{u}(\bolds{x} + \epsilon {\boldsymbol{\xi}})|)^2 d\boldsymbol{\eta} \notag \\
&\leq \frac{2\bar{J}_0}{\omega_d} \int_{H_1(\mathbf{0})} J(|\boldsymbol{\eta}|) (|\bolds{u}(\bolds{x} + \epsilon{\boldsymbol{\xi}} + \epsilon \boldsymbol{\eta})|^2 + |\bolds{u}(\bolds{x} + \epsilon {\boldsymbol{\xi}})|^2) d\boldsymbol{\eta},
\end{align}
where we have also used the inequality $(a+b)^2 \leq 2 a^2 + 2b^2$. This inequality holds for all $\bolds{x}$ and ${\boldsymbol{\xi}}$ which includes $\bolds{x} = \bolds{x}_i$ and ${\boldsymbol{\xi}} = \mathbf{0}$.
With estimate on $|\bolds{a}(\bolds{x},{\boldsymbol{\xi}})|^2$ and the fact that $\bolds{u}$ is a piecewise constant function defined over unit cells $U_i$, we immediately have
\begin{align}
\sum_{i,\bolds{x}_i \in D} \int_{U_i} |\bolds{a}(\bolds{x},{\boldsymbol{\xi}})|^2 d\bolds{x}
&\leq 4 \bar{J}_0^2 ||\bolds{u}||_{L^2}^2 = 4 \bar{J}_0^2 ||\hat{\bu}(t)||_{L^2}^2
\end{align}
where we substituted $\hat{\bu}(t)$ for $\bolds{u}$.
Combining above estimate with \autoref{eq:stab 5} we get
\begin{align}
I &\leq 64 \bar{J}^2_0||\hat{\bu}(t)||_{L^2}^2.
\end{align}
Finally, we use e the bound on $I$ and substitute it into \autoref{eq:stab 4} to show
\begin{align}
&||\sigma_D(t)||^2_{L^2} \leq \left(\frac{g''(0)}{\epsilon^2} \right)^2 \frac{\bar{J}_0}{\omega_d} \int_{H_1(\mathbf{0})} J(|{\boldsymbol{\xi}}|) 64 \bar{J}^2_0||\hat{\bu}(t)||_{L^2}^2 d{\boldsymbol{\xi}} \notag \\
\Rightarrow &||\sigma_D(t)||_{L^2} \leq \frac{8 g''(0) \bar{J}^2_0}{\epsilon^2} ||\hat{\bu}(t)||_{L^2}.
\end{align}
On renaming the constants the bound on $\sigma(t)$ can be summarized as
\begin{align}\label{eq:bound on sigma}
&||\sigma(t)||_{L^2} \notag \\
&\leq \begin{cases}
\frac{4 C^f_1 \bar{J}_{1/2}\sqrt{|D|}}{\epsilon^{3/2}} + \frac{4 C^g_1 \bar{J}_0\sqrt{|D|}}{\epsilon^2} \leq \frac{C}{\epsilon^2} \quad \text{for convex-concave }g,\\
\frac{4 C^f_1 \bar{J}_{1/2}\sqrt{|D|}}{\epsilon^{3/2}} + \frac{8 g''(0) \bar{J}^2_0}{\epsilon^2} ||\hat{\bu}(t)||_{L^2} \leq \frac{C_1 + C_2 ||\hat{\bu}(t)||_{L^2}}{\epsilon^2} \quad \text{for quadratic }g \,.
\end{cases}
\end{align}
\subsection{Energy inequality}
\label{energy inequality}
From \autoref{eq:stab 2.1} and noting the identity
\begin{align}
\frac{d}{dt}\mathcal{E}^\epsilon(\hat{\bu})(t) = (\ddot{\hat{\bu}}(t), \dot{\hat{\bu}}(t)) - (\mathcal{L}^\epsilon(\hat{\bu}(t)), \dot{\hat{\bu}}(t))
\end{align}
we have
\begin{align}\label{eq:stab 6}
\frac{d\mathcal{E}^\epsilon(\hat{\bu})(t)}{dt} &= (\hat{\bolds{b}}(t), \dot{\hat{\bu}}(t)) + (\sigma(t), \dot{\hat{\bu}}(t)) \notag \\
&\leq (||\hat{\bolds{b}}(t)||_{L^2} + ||\sigma(t)||_{L^2}) \,||\dot{\hat{\bu}}(t)||_{L^2}.
\end{align}
When $g$ is convex-concave we can apply identical steps as in the proof of Theorem 5 of \cite{CMPer-JhaLipton} together with the estimate \autoref{eq:bound on sigma} to obtain
\begin{align}
\sqrt{\mathcal{E}^\epsilon(\hat{\bu})(t)} &\leq \sqrt{\mathcal{E}^\epsilon(\hat{\bu})(0)} + \frac{t C}{\epsilon^2} + \int_0^t ||\hat{\bolds{b}}(s)||_{L^2} ds
\end{align}
for all $t\in [0,T]$. This completes the proof of energy stability for convex-concave potential functions $g$.
We now address the case of quadratic potential functions $g$. We introduce the energy $\bar{\mathcal{E}}^\epsilon(\hat{\bu})(t)$ given by
\begin{align*}
\bar{\mathcal{E}}^\epsilon(\bolds{u})(t) := \mathcal{E}^\epsilon(\bolds{u})(t) + \frac{1}{2}||\bolds{u}(t)||^2_{L^2}.
\end{align*}
Differentiation shows that
\begin{align*}
\frac{d\mathcal{E}^\epsilon(\hat{\bu})(t)}{dt} &= \frac{d\bar{\mathcal{E}}^\epsilon(\hat{\bu})(t)}{dt} - (\hat{\bu}(t), \dot{\hat{\bu}}(t)).
\end{align*}
Thus from \autoref{eq:stab 6} we get
\begin{align}\label{eq:stab 7}
\frac{d\bar{\mathcal{E}}^\epsilon(\hat{\bu})(t)}{dt} &\leq (||\hat{\bolds{b}}(t)||_{L^2} + ||\sigma(t)||_{L^2}) \,||\dot{\hat{\bu}}(t)||_{L^2} + (\hat{\bu}(t), \dot{\hat{\bu}}(t)) \notag \\
&\leq (||\hat{\bolds{b}}(t)||_{L^2} + C_1/\epsilon^2) \,||\dot{\hat{\bu}}(t)||_{L^2} + (C_2/\epsilon^2 + 1)||\hat{\bu}(t)||_{L^2} \, \,||\dot{\hat{\bu}}(t)||_{L^2}.
\end{align}
From the definition of energy $\bar{\mathcal{E}}^\epsilon$ we have
\begin{align}
||\hat{\bu}(t)||_{L^2} &\leq \sqrt{2 \bar{\mathcal{E}}^\epsilon(\hat{\bu})(t)} \hbox{ and},\notag \\
||\dot{\hat{\bu}}(t)||_{L^2} &\leq \sqrt{2 \bar{\mathcal{E}}^\epsilon(\hat{\bu})(t)}.
\end{align}
Using the above inequalities in \autoref{eq:stab 7} along with Cauchy's inequality gives
\begin{align}
\frac{d\bar{\mathcal{E}}^\epsilon(\hat{\bu})(t)}{dt} &\leq ||\hat{\bolds{b}}(t)||^2_{L^2} + \frac{C_1^2}{\epsilon^4} + 3(\frac{C_2}{\epsilon^2} + 1) \bar{\mathcal{E}}^\epsilon(\hat{\bu})(t).
\end{align}
Using the integrating factor $\exp[-3(C_2/\epsilon^2 + 1)t]$ we recover the inequality
\begin{align}
\bar{\mathcal{E}}^\epsilon(\hat{\bu})(t) &\leq \exp[3(C_2/\epsilon^2 + 1)t] \bigg( \bar{\mathcal{E}}^\epsilon(\hat{\bu})(0) \notag \\
&\quad \quad + \int_0^t (\frac{C_1^2}{\epsilon^4} + ||\hat{\bolds{b}}(s)||^2_{L^2}) \exp[-3(C_2/\epsilon^2 + 1)s] ds \bigg).
\end{align}
This completes the proof of \autoref{thm:stab semi}.
\section{Conclusions}
\label{s:conclusions}
In this article, we present an a-priori convergence analysis for a class of nonlinear nonlocal state based peridynamic models. We have shown that the convergence rate applies, even when the fields do not have well-defined spatial derivatives. The results are valid for two different classes of state-based peridynamic models depending on the potential functions associated with the dilatational energy. For both models the potential function characterizing the energy due to tensile strain is of convex-concave type while the potential function for the dilatational strain can be either convex-concave or quadratic.
The convergence rate of the discrete approximation to the true solution in the mean square norm is given by $C(\Delta t+h^\gamma/\epsilon^2)$. Here the constant depends on the H\"older and $L^2$ norm of the true solution and its time derivatives.
The Lipschitz property of the nonlocal, nonlinear force together with boundedness of the nonlocal kernel plays an important role. It ensures that the error in the nonlocal force remains bounded when replacing the exact solution with its approximation. This, in turn, implies that even in the presence of mechanical instabilities the global approximation error remains controlled by the local truncation error in space and time. This is supported by numerical results with crack propagation. The analysis shows that the method is stable and one can control the error by choosing the time step and spatial discretization sufficiently small.
\newcommand{\noopsort}[1]{}
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Péter Halmosi (; born 25 September 1979) is a Hungarian retired professional footballer who played as a midfielder. At international level, he won 35 caps for his country. His previous clubs include Szombathelyi Haladás, Grazer AK, Debrecen, Plymouth Argyle and Hull City.
Biography
Born in the ancient western city of Szombathely, Halmosi grew up in a family with significant sporting background. His father, Zoltán, won 11 caps playing for Hungary, and his mother also was involved in professional sport, representing Hungary as an athlete at several international events.
Career
Halmosi played his first game for his hometown club Szombathelyi Haladás at the age of 18 and stayed at the club until he was 23. In 2002, he joined Austrian side Grazer AK making 17 appearances and scoring 3 goals for the club. After playing several games for the reserves he returned to Hungary signing for Debrecen. His move to Debrecen turned out to be a successful stay and he spent four years with the club.
Halmosi next joined Plymouth Argyle, initially on loan until the end of the 2006–07 season, with a view to a permanent move at the end of the season. A £400,000 permanent deal with Argyle was completed on 16 May 2007, making him the record signing for the club at that time. He had been linked with moves to other clubs in the United Kingdom including Celtic and Blackburn Rovers. Halmosi made his Argyle debut in a 3–1 away win against Norwich City on 13 January 2007.
On 15 July 2008, Halmosi was spotted at Hull City's pre-season friendly against North Ferriby United. One day later, he signed a four-year contract with Hull for a fee in excess of £2 million, becoming, at the time, their record signing and their 6th of the summer. He made his Hull debut in the friendly match against Chesterfield on 22 July 2008 at Saltergate. His competitive debut came shortly after the hour mark in Hull City's first ever top flight game, against Fulham on 16 August 2008, as a substitute for Nick Barmby. He scored his first goal for the club in an FA Cup victory over against Sheffield United on 14 February 2009. On the final day of the January transfer window Halmosi joined Hungarian side Szombathelyi Haladas on loan until the end of the 2009–10 season.
On 4 January 2011, Halmosi left Hull City by mutual consent. Ten days later, he returned to Szombathelyi Haladás on a three and a half year contract.
Career statistics
Club
Other includes the UEFA Champions League and UEFA Cup.
International
Honours
Debrecen
Hungarian League: 2005, 2006
Hungarian Super Cup: 2005, 2006
Szombathelyi Haladás
Hungarian Cup runner-up: 2002
Individual
Named in the Hungarian National Championship I all-star team (nemzetisport.hu): 2010–11
Zilahi Prize: 2005
References
External links
Péter Halmosi at magyarfutball.hu
1979 births
Living people
Sportspeople from Szombathely
Hungarian footballers
Hungary international footballers
Association football midfielders
Szombathelyi Haladás footballers
Grazer AK players
Debreceni VSC players
Plymouth Argyle F.C. players
Hull City A.F.C. players
Nemzeti Bajnokság I players
Austrian Football Bundesliga players
English Football League players
Premier League players
Hungarian expatriate footballers
Expatriate footballers in Austria
Expatriate footballers in England
Hungarian expatriate sportspeople in Austria
Hungarian expatriate sportspeople in England
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Q: Do c++ guarantee header-initialized static const member to share a single instance across compile units and libraries? Let's consider a code
header:
class uid
{
public:
uid () {++i; }
static int i;
};
class foo
{
public:
const static uid id;
}
source:
static int uid::i = 0;
The header could be included into several source files, shared between compiler units and libraries.
Is it guaranteed that there would be only one instance off foo::id, that foo::id::id() would be called once at run-time and, the most important thing, would foo::id.i be the same everywhere in the program and it's libraries? On the other hand another shared header could have bar class with it's own static const uid id which is expected to differ from foo's one. Is it also guaranteed? If so, where actually foo::id symbol is stored, especially in case of shared (dynamic-linked) libraries.
On some reason c++ disables
class foo
{
public:
const static int id = create_uid(); // forbidden
}
allowing only compile-time const initialisation or initialization in source file. So there was some reason to disable this kind of approach.
A: Yes, that is guaranteed.
The symbol lives in the translation unit built from the source file where the object was defined. Indeed, that's why we have to define it in one!
The linker makes sure all the references from the copies of that header across your project all match up to the sole definition.
As for why C++ doesn't let you initialise static members inline: it does, but not in C++98. You need C++17 for that.
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De Symfonie nr. 3 in c-mineur, opus 78, ook bekend als de Orgelsymfonie, is een orkestwerk van de Franse componist Camille Saint-Saëns.
Hoewel de Nederlandse bijnaam suggereert dat dit werk een orgelsymfonie (voor orgelsolo) zou zijn, is dit niet het geval. Saint-Saëns zelf noemde het een Symphonie avec orgue ("met orgel"). Het orgel speelt alleen mee in het tweede en vierde deel van de symfonie:
Adagio - Allegro moderato
Poco adagio
Allegro moderato - Presto
Maestoso
Saint-Saëns schreef de symfonie in opdracht van de Royal Philharmonic Society in Londen ter gelegenheid van hun 73e seizoen. Hij droeg het werk op aan zijn net overleden goede vriend Franz Liszt. De symfonie heeft een duur van 35 minuten.
De muziek
1. Adagio - Allegro moderato
Het eerste deel begint langzaam en aarzelend, waarna de sfeer energieker wordt wanneer de pauken en strijkers de symfonie openen. De houtblazers spelen contrasterende dalende en stijgende melodieën. De openingssectie wordt herhaald en uitgewerkt, waarna de muziek wat ingetogener wordt ter voorbereiding op het vredigere Poco adagio
2. Poco adagio
In het Poco adagio valt voor het eerst het orgel te horen in een begeleidende stem. De expressieve openingsmelodie wordt gespeeld door de complete strijkerssectie, met uitzondering van de contrabassen. De violen spelen later een variatie op dit thema. Ter afsluiting wordt het thema gespeeld door het orgel, begeleid door pizzicato spelende strijkers.
3. Allegro moderato - Presto
De onrust van het eerste deel keert terug. Vooral de blazerssectie heeft een prominente rol. De piano komt hier voor het eerst aan bod met een aantal stijgende toonladders. Bij de herhaling van het presto komt een nieuw thema naar voren dat gespeeld wordt door de koperinstrumenten en de strijkers. Hier lijkt enige frictie in samenklank te ontstaan. De rust keert geleidelijk terug.
4. Maestoso
Een vol akkoord op het orgel kondigt het Maestoso aan. Strijkers en piano spelen een vloeiend thema, dat door het orgel wordt herhaald met begeleiding van het volle orkest. Een vrolijke stemming overheerst. Tegen het einde van het vierde deel neemt het tempo en de spanning toe. Het korte sluitstuk met uitgebreide toonladders van strijkers en houtblazers besluit het stuk in volle glorie.
Thema
Het thema dat door de gehele symfonie heen klinkt en vooral in het Maestoso (in majeur) breed naar voren komt (E-D-E-C-D-E-G-A-G), is afgeleid van het in kleine terts geschreven Dies Irae, Dies Illa (Es-D-Es-C-D-Bes-C-C) uit de rooms-katholieke dodenmis. De keuze van dit thema - in 1874 verwerkt in zijn symfonisch gedicht Danse Macabre - kan geïnspireerd zijn door het overlijden van Liszt, die zelf zijn Totentanz voor piano en orkest op het Dies Irae baseerde.
Bezetting
De symfonie is geschreven voor een orkestbezetting bestaande uit drie fluiten (waaronder één piccolo), twee hobo's, althobo, twee klarinetten, basklarinet, twee fagotten, contrafagot, vier hoorns, drie trompetten, drie trombones, tuba, pauken, triangel, bekkens, grote trom, strijkers (eerste viool, tweede viool, altviool, cello, contrabas), piano (twee- en vierhandig) en orgel.
Afgeleide muziek
Het in majeur optredende Dies Irae-thema uit het vierde deel is de basis van de popsong If I had words van Scott Fitzgerald en Yvonne Keeley uit 1977.
Compositie van Camille Saint-Saëns
3 Saint-Saëns
Compositie voltooid in 1886
Compositie voor orgel en orkest
|
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| 8,224
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On the occasion of graduations, we remind recent graduates and their families to adhere to specific rules of conduct that guarantee the dignity and decorum of the university institution and prevent damage to persons and property, which are contained in the Rules of Conduct for candidates and their families during graduation sessions.
In particular: it is absolutely forbidden to affix printed matter or other material both inside and outside the buildings; the classrooms in, in which the graduation sessions are held, must be vacated immediately after the candidate has been proclaimed; the University buildings are not places where graduation parties may be organised. This applies to both internal and external areas. It follows that it is forbidden to set up refreshment areas with the serving of food or beverages; noise, shouting and verbal outbursts that disrupt lessons are prohibited.
Finally, it is worth recalling the provisions of 'Article 41 - Code of Conduct' of the current University Regulations:
University staff and students must, also through their behaviour, promote the smooth running of university business and civilised coexistence.
Students who, by their behaviour, obstruct the regular and proper conduct of academic activities and cause damage to the University's assets, without prejudice to the application of the criminal and administrative sanctions provided for by the legislation in force, may be subject to disciplinary proceedings, in compliance with the principle of adversarial proceedings. The sanctionable conduct, the applicable procedure and sanctions, as well as their enforceability, are laid down in specific regulations to which reference is made.
For university staff, please refer to the provisions of the code of ethics and the legal and collective bargaining rules governing disciplinary matters.
|
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"redpajama_set_name": "RedPajamaCommonCrawl"
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Q: Why doesn't my break work? I am programming in C# in Microsoft Visual Studio 2010.
Somewhere in my code, I'm using break keyword in a while loop after checking if the line's value is null or not.
I am sure that line value is null but still the break command doesn't work.
How can I fix this?
while (!myfile.EndOfStream)
{
string line = myfile.ReadLine();
if (line == null) { break; }
else
{ ...}
}
A: Its not null, you are getting either an empty string back or string with White Space (Line breaks etc).
Its better if you modify your check as :
if (string.IsNullOrWhiteSpace(line)) { break; } //checks null and white space
// or empty string
String.IsNullOrWhiteSpace is provided with .Net framework 4.0 or higher.
A: The null check is not needed. It is already being enforced by the EndOfStream check. It can be removed.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,710
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Celia Worthington discusses the case of Pye (Oxford) v Graham which held the law of adverse possession contravenes a land owners right to peaceful enjoyment of his possessions.
Most of us are familiar with the phrase "Possession is nine tenths of the law". This came from the laws relating to land ownership where ownership evolved from land that was in possession of the occupier. Thereafter it was used as a form of currency with the King bestowing title to land on Lords in return for services. However notwithstanding the identity of the legal owner being known, it has always been possible to dispossess a land owner of his lands by occupying them. This law of "adverse possession" was eventually formalised in Northern Ireland in the Limitation (Northern Ireland) Order 1989.
Basically if the occupier of lands often known as 'the squatter' has gone into possession of the land for twelve years without any concealment, force or consent then the title owner could be dispossessed and prevented from bringing proceedings for return of the land.
The modern doctrine of adverse possession is relied upon and applied in numerous every day conveyancing transactions. A very common scenario is where boundaries on the ground have moved over the years so that when a survey is carried out it is discovered that there is extra land over and above that contained in the deeds. Statutory declarations would be sought from occupiers and placed with the deeds so that these could be used in any subsequent application to the land registry for registration or to assist defend an action for trespass instigated by the title holder.
Case law over time has established the criteria that the squatter has to meet providing practitioners with clear guidelines that they can use to advise clients involved in such cases. However, a recent case in the European Court of Human Rights entitled J A Pye (Oxford) Limited and another –v Graham UKHL has ruled that the law of adverse possession contravenes a land owners rights to peaceful enjoyment of his possessions.
The case went through the English Courts culminating in a House of Lords decision that the occupier (Graham) had dispossessed the title holder (Pye) of farm land occupied by Graham for more that twelve years. Pye took the case to Europe alleging that his human rights had been infringed. The land in question, development land in Berkshire, is purportedly worth millions.
Pye had allowed Graham to occupy the land for grazing purposes in 1977 and recognising that it could have development potential had entered into a licence whereby Graham was allowed to occupy the land until 31 December 1983. The licence allowed Graham to use the land for grazing or mowing. It expressly stated that after its expiry there would have to be a new contract. However no further agreement was ever entered into and even though Pye requested Graham to vacate the land Graham remained in occupation farming the land. Eventually Pye took possession proceedings and Graham defended them, eventually succeeding in the House of Lords. Even though there were strong facts in the case in favour of Graham, the Lords were reluctant to find in Graham's favour as in effect this allowed him to retain a piece of land which he had farmed for nothing and was now worth a small fortune. Mr Graham had died in a shooting accident before the trial and the action was continued by his widow.
The decision at the European Court has now ruled against Mrs Graham and thrown the whole framework of adverse possession into disarray. Practitioners await with keen anticipation the result of the UK's appeal of the European decision to the Grand Chamber of the European Court of Human Rights. If the decision is upheld and no amending legislation is implemented then the future of adverse possession laws in Northern Ireland is in the hands of local judges. It is expected and, for those suffering from the uncertainties Pye has created, hoped that a test case will be taken in the High Court in Belfast allowing for eventual appeal to the House of Lords to ascertain the compatibility or otherwise of our local legislation with the human rights convention. Some commentators believe that the European Courts were swayed to find in favour of Pye due to the value of the land in question and the particular facts of the case. Others believe that cases will now on the whole only be looked at more favourably for the squatter in circumstances where the squatter has no knowledge of the identity of the land owner.
In the meantime, until certainty is brought back into play, it seems that the cards are stacked against the squatter.
Celia Worthington is senior partner of the Commercial Department of Worthingtons Solicitors Belfast Office. Celia specialises in commercial property, banking, telecoms and corporate and commercial law. She advises a number of UK wide corporate clients and well known local charities and is currently Chairperson of Abbeyfield (NI) a local housing association.
|
{
"redpajama_set_name": "RedPajamaC4"
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| 1,180
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{"url":"https:\/\/codereview.stackexchange.com\/questions\/194756\/find-primes-using-sieve-of-eratosthenes-with-python","text":"# Find primes using Sieve of Eratosthenes with Python\n\nI was Googling for a 'correct' or 'efficient' implementation of the Sieve of Eratosthenes. I found them all more complicated than what I came up with. I am by no means a Python expert, so I was wondering if something is wrong with my code.\n\ndef get_primes(max_int):\nnumbers = range(2, max_int)\nwhile len(numbers) > 0:\nnumbers = [ num for num in numbers if num % numbers[0] != 0 or num == numbers[0] ] # only keep numbers that are NOT divisible by the prime\nyield numbers[0] # yield the prime\nnumbers = numbers[1:] # make a new numbers-array, the first num is a new prime\n\n\nCall it with\n\nprimes = get_primes(1000)\n\n\nAnd it gives you a generator of primes up until 1000.\n\nYour code looks correct to me, it does produce prime numbers in the given range.\n\nPEP8 Online reports some code style violations with respect to whitespace and line length, but it is not too bad.\n\nBut: This is not the Sieve of Eratosthenes. The Sieve of Eratosthenes keeps a (single) list of all candidate numbers, and computes multiples of each found prime to mark subsequent composite numbers in the list. Your algorithm computes the remainder of all remaining candidates instead, and creates new lists in each step.\n\nAs a simple benchmark I ran\n\nprint(sum(p for p in get_primes(100000)))\n\n\nwith your code, this takes approximately 5 seconds on my MacBook.\n\nIt can be improved slightly by filtering the list only once in each step and not twice:\n\ndef get_primes(max_int):\nnumbers = range(2, max_int)\nwhile len(numbers) > 0:\nyield numbers[0]\nnumbers = [num for num in numbers if num % numbers[0] != 0]\n\n\nThis reduces the time to 4.6 seconds.\n\nBut any \u201creal\u201d Eratosthenes sieve is much faster. As an example, with this one from Rosetta code\n\ndef primes_upto(limit):\nis_prime = [False] * 2 + [True] * (limit - 1)\nfor n in range(int(limit**0.5 + 1.5)): # stop at sqrt(limit)\nif is_prime[n]:\nfor i in range(n*n, limit+1, n):\nis_prime[i] = False\nreturn [i for i, prime in enumerate(is_prime) if prime]\n\n\nthe above benchmark runs in 0.08 seconds.\n\n\u2022 It seems I did not understand The Sieve enough, your answer helped me clarify. I've coded the implementation correctly now and it is indeed way faster. \u2013\u00a0Jelle van der Zwaag Jun 2 '18 at 17:01\n\nIf you want to speed it up further, you can use slice assignment instead of the inner \"for\" loop:\n\ndef primes_slice(limit):\nis_prime = [False] * 2 + [True] * (limit - 1)\nfor n in range(int(limit**0.5 + 1.5)): # stop at sqrt(limit)\nif is_prime[n]:\nis_prime[n*n::n] = [False] * ((limit - n*n)\/n + 1)\nreturn [i for i, prime in enumerate(is_prime) if prime]\n\n\nThat lets C code do the former inner loop for you under the covers, at the expense of some complicated stuff on the right side of the slice assignment. Even that can go away if you use the bitarray package for is_prime, which also reduces the memory footprint considerably. This tweak gives me about a 2x speed boost with n = 10,000,000\n\n$time .\/prime_rosetta_slice.py 1e7 664579 real 0m1.134s user 0m0.797s sys 0m0.344s$ time .\/prime_rosetta_upto.py 1e7\n664579\n\nreal 0m2.242s\nuser 0m1.875s\nsys 0m0.281s\n\n\nFor the next performance improvement, try keeping only odd numbers > 2 in is_prime and then manually returning 2 as a special case.","date":"2019-12-16 14:23:08","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5071727633476257, \"perplexity\": 2412.7900390262575}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-51\/segments\/1575540565544.86\/warc\/CC-MAIN-20191216121204-20191216145204-00231.warc.gz\"}"}
| null | null |
Bresilioidea è una superfamiglia di gamberetti. È probabilmente un gruppo polifiletico, contenente cinque famiglie, che possono o meno essere correlate fra loro.
Tassonomia
Agostocarididae
Alvinocarididae
Bresiliidae
Disciadidae
Pseudochelidae
Altri progetti
Collegamenti esterni
Caridea
|
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| 7,630
|
<?xml version="1.0" encoding="utf-8"?>
<LinearLayout ="vertical" ="@id/background" ="fill_parent" ="wrap_content"
xmlns:android="http://schemas.android.com/apk/res/android">
<RelativeLayout style="@style/location_list_item_layout">
<ImageView ="@id/quiz_logo_black" ="true" style="@style/location_list_item_flag" />
<com.quizup.lib.widgets.textViews.GothamTextView ="@color/dark_gray" ="@id/location" ="fill_parent" ="wrap_content" ="1" ="@id/button" ="@id/quiz_logo_black" ="true" ="true" style="@style/location_list_item_label" />
<com.quizup.lib.widgets.buttons.BorderButton ="@id/button" ="wrap_content" ="wrap_content" ="@string/pick" ="true" ="true" style="@style/location_list_item_button" />
</RelativeLayout>
<include ="@id/line" layout="@layout/layout_divider_line_light" />
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"redpajama_set_name": "RedPajamaGithub"
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| 1,446
|
GrimoireJS
========
GrimoireJS: Magic item generator for the Pathfinder tabletop RPG
written by Russell Champoux
This project contains a series of function under the 'Grimoire' module.
* The primary file to include is grimoire.rb
Grimoire currently only rolls for items based on the Pathfinder Core Rulebook
I hope to add roll tables for the Pathfinder Advanced Player's Guide soon
While the primary purpose of this project is to be used as a library accessed by other projects, you can manually generate items by following these steps
1. Open grimoire.html in a web browser
2. Hit F12 to open the developer console
3. All commands are within the following objects:
* Grimoire
* Armors
* Weapons
* Deities
* Intelligence
* Potions
* Rings
* Rods
* Scrolls
* Shields
* Staves
* Wands
* Weapons
* WondrousItems
Tip: All get*() function take a "type" argument that can be either 'minor', 'medium', or 'major'
Tip: Try running Grimoire.getItems(2,2,2) to generate several minor, medium, and major items
Send questions, feedback, or bug reports/fixes to RussellChamp [at] gmail [dot] com
|
{
"redpajama_set_name": "RedPajamaGithub"
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| 5,914
|
<?xml version="1.0" encoding="utf-8"?>
<Monster xmlns:xsd="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">
<Name>human zombie</Name>
<Attack1>Claw 1d8</Attack1>
<Difficulty>5</Difficulty>
<BaseLevel>4</BaseLevel>
<BaseExperience>41</BaseExperience>
<Speed>6</Speed>
<BaseAC>8</BaseAC>
<BaseMR>0</BaseMR>
<Alignment>-3 (chaotic)</Alignment>
<Frequency>(by normal means) 1 (Very rare)</Frequency>
<GenoCidable>Yes</GenoCidable>
<Weight>1450</Weight>
<NutritionalValue>200</NutritionalValue>
<Size>Medium</Size>
<Resistances>conveyed None</Resistances>
<ResistancesConveyed>None</ResistancesConveyed>
<Characteristics>
<string>does not breathe. </string>
<string>is mindless. </string>
<string>has a head, a couple of arms, and a torso. </string>
<string>does not eat. (*) </string>
<string>is undead. </string>
<string>is normally generated hostile. </string>
<string>can follow you to other levels. </string>
<string>has infravision. </string>
<string>appears in small groups normally. </string>
<string>can leave an old human corpse.</string>
</Characteristics>
</Monster>
|
{
"redpajama_set_name": "RedPajamaGithub"
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| 8,480
|
{"url":"https:\/\/astronomy.stackexchange.com\/questions\/40252\/how-to-manually-locate-a-certain-declination-in-the-sky-for-a-given-location-n","text":"# How to manually locate a certain declination in the sky for a given location? (no sextant available)\n\nConsider this example. I am standing in a flat surface, at latitude X, looking to the equator, in a random day of the year. (For simplicity, let's assume X > 23.5\u00ba)\n\nI want to find the position of the sun at noon on the winter solstice (that is, the highest point in the sky of the sun, the day which this reaches the lowest altitude), location which has a declination of 23.5\u00ba.\n\nWe know the sextant does exactly that. But say I don't have a sextant. I want to find an alternative method to do the same.\n\nIf my latitude is X, then such position of the sun is X-23.5\u00ba above the horizon (looking North). I could then divide this result by 90, and get a proportion of the sky between the horizon and my zenit where that point is. But this is too rough.\n\nPerhaps one way would be to have a long stick and a measuring tape. For a given value of X-23.5\u00ba there must be some relationship between my eye, the vertical height of the stick and my horizontal distance to the stick which allows me to find exactly the position in the sky.\n\nOr perhaps another method?\n\n\u2022 Are you able to manufacture a tool to help with this? Dec 6 '20 at 19:41\n\u2022 @JamesK Absolutely. As long as it is rudimentary Dec 6 '20 at 21:20\n\u2022 Seems like pretty much anything that works is effectively a sextant. If you can get something with a joint with enough friction to stay wherever you leave it, but still lets you move it easily, you can move it so it matches the declenation. Dec 7 '20 at 4:34\n\n## 1 Answer\n\nJacob's staff is probably all you need. It is a pole with a cross arm. By positioning the arm at a certain point on the pole and sighting down the pole one can get a position on the sky. A skilled user can achieve sub-degree accuracy with such a tool.\n\nThe position of the cross beam is easy enough to calculate by trigonometry, or by careful construction of triangles.\n\n$$\\tan(\\theta\/2)= c\/2l$$\n\nWhere $$\\theta$$ is the angle above the horizon, $$l$$ is the position of the crossbar from the eye and $$c$$ is the length of the crossbar.\n\n\u2022 Thanks. The tricky bit of the tool is to keep it horizontal at the eye level but perhaps a table could help. Dec 7 '20 at 12:43\n\u2022 Btw, woudln't something like a quadrant be much easier? Just select the angle which then you can project to the sky. Like the last photo here. No need to do calculations about horizontal and vertical length. Dec 7 '20 at 12:52","date":"2021-12-07 10:00:39","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 4, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7249361276626587, \"perplexity\": 563.3558557772637}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-49\/segments\/1637964363337.27\/warc\/CC-MAIN-20211207075308-20211207105308-00179.warc.gz\"}"}
| null | null |
\section{Introduction}
Galaxy clusters are the largest gravitationally bound objects in the Universe. Observations of galaxy clusters, in particular, estimations of the gravitation potential based on measurements of the velocity dispersions of member galaxies, suggested that Galaxy cluster masses were much larger than the values inferred by summing contributions from luminous matter, and lead to the proposal of non-luminous (i.e., dark) matter (DM)~\citep{Zwicky1937}. Since then, many observations, as well as N-body cosmological simulations, suggest that galaxy clusters contain a high amount of DM, making them prime targets to search for indirect DM signals \citep[see, e.g.][]{Colafrancesco2006,Clowe:2006aa,2009PhRvD..80b3005J,2009PhRvL.103r1302P,MASC2011,Klypin2011,Pinzke2011,Gao2012a,Hellwing:2015aa}.
Weakly interacting massive particles (WIMPs) constitute promising particle DM candidates. Among prominent WIMP candidates is the neutralino, which in many supersymmetric models is the lightest stable supersymmetric particle, allowing it to account for the observed relic DM density in the Universe. In many of these models, the neutralino can self-annihilate into particle-antiparticle pairs which subsequently may produce other particles, including $\gamma$ rays\xspace. The $\gamma$ rays\xspace will propagate undeflected by the interstellar\ magnetic fields and thus reveal the location of the DM annihilation~\citep[see, e.g.][for a review of searches for indirect DM searches using $\gamma$ rays\xspace]{Bertone:2005aa,Feng:2010aa,Bringmann:2012aa,Conrad:2015aa}. Several predictions of the expected DM annihilation rate in cosmological environments, such as galaxy clusters, and the associated signals in $\gamma$-ray\xspace\ data show that current space-borne $\gamma$-ray\xspace\ detectors like the Large Area Telescope (LAT) on board the \emph{Fermi}\ satellite \citep{LAT} will be unlikely to detect this signal in case the DM is smoothly distributed~\citep[see, e.g.][]{Pinzke2011}.
However, a smooth DM distribution is indeed not expected and recent cosmological N-body simulations predict instead that DM virialized regions, known as halos, contain a large number of smaller virialized and highly concentrated substructures called \emph{sub-halos} \citep{2005Natur.435..629S,2008Natur.454..735D,2008MNRAS.391.1685S}. Since the DM annihilation signal is proportional to the DM density squared, these highly concentrated sub-halos are expected to significantly boost the annihilation signal relative to the purely smooth DM scenarios \citep[e.g.,][]{2008ApJ...686..262K,2008MNRAS.391.1685S,2008A&A...479..427L,2008MNRAS.384.1627P,2009JCAP...06..014M}. The exact signal enhancement depends on the abundance, distribution and internal structural properties of the sub-halos. However, sub-halo properties are uncertain below the halo mass resolution of state-of-the-art N-body cosmological simulations, $\sim O(10^5\,\ensuremath{M_{\odot}})$ for Milky Way size halos \citep{2008Natur.454..735D,2008Natur.456...73S,Hellwing:2015aa} and $10^8\,\ensuremath{M_{\odot}}$ for simulations of galaxy clusters \citep[e.g.,][]{Gao2012a}. Thus, extrapolations of the relevant properties are required over several orders of magnitude in halo mass below the mass resolution limit in order to account for the whole halo mass range that is predicted to exist in the Universe, and more specifically in clusters. Here, the fractional enhancement of $\gamma$-ray\xspace flux due to sub-halos is called the sub-halo boost, or boost-factor in short. As recently discussed e.g. in \citet{MASC2013}, sub-halo boosts are very sensitive to the way these extrapolations are performed and boost estimates can vary drastically depending on the assumptions \citep[e.g.,][]{2010PhRvD..81d3532K,MASC2011,Pinzke2011,Gao2012a,2012PDU.....1...50K,2012MNRAS.425..477N,2014MNRAS.441.1329Z}.
A debate is ongoing as to whether the extrapolation with a power law to lower-mass halos \citep{Pinzke2011,Gao2012a} is justified or too optimistic~\citep[see, e.g.][]{2008Natur.454..735D,MASC2011}.\footnote{Note that this inherent theoretical uncertainty is alleviated if decaying DM is considered \citep{Dugger:2010aa,Huang2012}.}
A further challenge arises from the fact that cosmic-ray\ (CR) interactions in the intra-cluster medium (ICM) may also give rise to $\ensuremath{\mathrm{MeV}}\xspace--\ensuremath{\mathrm{GeV}}\xspace$ $\gamma$ rays\xspace, which if observed, would be difficult to distinguish from a DM-induced signal \citep[see, e.g.][]{Pinzke2011,2012JCAP...07..017A}. Despite intensive efforts, to date no $\gamma$ rays\xspace\ from clusters have been detected aside from those attributed to individual active galaxies \citep[see, e.g.][]{2010ApJ...717L..71A,Huang2012,Huber2013,Ackermann2013}.
The closest galaxy cluster is Virgo at a distance of about $\medit{15.4\pm0.5\,\rmn{Mpc}}$, subtending several degrees on the sky~\citep{virgo_fouque}. The cluster consists of several sub-clusters which are located around giant elliptical galaxies, most prominently M87 and M49 as well as around the two smaller clusters associated with M100 and M60 \citep{2002astro.ph..6272S}. These sub-clusters are in the process of merging with one another, while the system is dominated by the most massive sub-cluster centered on M87. For the remainder of this work we refer to the sub-cluster centered on M87 as Virgo-I and the sub-cluster centered on M49 as Virgo-II. Also relevant for an analysis of $\gamma$ rays\xspace\ is the fact that M87 harbors a known active galactic nucleus \citep[AGN,][]{2009ApJ...707...55A,2014ApJ...788..165H} which dominates the emission both in X-rays and $\gamma$ rays\xspace. Due to its proximity, Virgo is also an interesting target for the search of a DM-induced $\gamma$-ray\xspace\ signal with the \emph{Fermi}-LAT.
Earlier studies, concentrating on Virgo-I, tested for point-like or mildly extended ($\leq 1\fdg2$) emission towards the center of the cluster and yielded upper limits on the integrated $\gamma$-ray\xspace\ flux of $14.1\times10^{-9}\mathrm{ph\,cm^{-2}\,s^{-1}}$ and $17.1\times10^{-9}\mathrm{ph\,cm^{-2}\,s^{-1}}$ respectively~\citep{2010ApJ...717L..71A}, assuming a power-law spectrum of the $\gamma$-ray\xspace\ emission with photon index $\Gamma=2$ in an energy band from 200~\ensuremath{\mathrm{MeV}}\xspace to 100~\ensuremath{\mathrm{GeV}}\xspace.
Claims of $\gamma$-ray\xspace\ emission induced by DM annihilation were put forward by \cite{HanI}, using a \emph{very extended} DM-induced emission profile considering only Virgo-I.\footnote{We use the term very extended to distinguish mildly extended emission (up to $\sim2--3\deg$ in radius) from larger extensions up to $\gtrsim7\deg$ in radius that are considered here. The former was tested in a recent work, yielding null results on extended $\gamma$-ray\xspace\, emission \citep{Ackermann2013}.} Later studies attribute this putative signal to an incomplete point source model of the region~\citep{Macias,HanII}.
Here we present a comprehensive analysis of the Virgo region searching for very extended emission and discuss the various systematic effects relevant for this analysis. In Section~\ref{sec:ana} we discuss our data selection and analysis. Sections~\ref{sec:virgoCenter} and \ref{sec:extended} elaborate on the details of finding and characterizing extended excess emission, while Section~\ref{sec:IEM} is devoted to the discussion of the uncertainties associated with the interstellar emission model (IEM). The aforementioned possibility of additional point sources constituting a putative signal is discussed in Section~\ref{sec:ps}. As a result of these studies we devise an improved background model for the Virgo region. We use this model to derive new limits on the WIMP DM annihilation cross section (Section~\ref{sec:dmVirgo}) and in the case of CR-induced $\gamma$-ray\xspace production on the its flux. We compare our results to theoretical predictions in order to constrain relevant CR quantities (Section~\ref{sec:crVirgo}). We conclude and summarize our work in Section~\ref{sec:conclusion}.
\section{Observations \& Data Analysis}\label{sec:ana}
The main instrument on board the \emph{Fermi}\ satellite, the LAT, is a pair-conversion telescope sensitive to $\gamma$ rays\xspace in the energy range from 20~\ensuremath{\mathrm{MeV}}\xspace\ to $>300~\ensuremath{\mathrm{GeV}}\xspace$. For a more detailed description the reader is referred to \cite{LAT} and for the characterization of the on-board performance to \cite{fermi_inst2012}.
We analyzed archival \emph{Fermi}-LAT\ data between MJD 54682.7 (2008-08-04) and MJD 55789.5 (2011-08-16) corresponding to roughly three years of Pass~7 data. We chose Pass~7 data because the predicted $\gamma$-ray\xspace\ signal from possible DM annihilation stretches over several degrees in the case of the Virgo cluster. With the release of the reprocessed P7 data (P7REP), the LAT collaboration also released a new template to describe the Galactic foreground emission which is tuned to P7REP data. This model contains a component which is derived from re-injecting residual emission above a scale radius of about $2^{\circ}$\citep{Casandjian:2015aa}. Residuals larger than this scale radius are absorbed into the model and thus the model cannot be used to search for emission with larger extension. The release of P7REP data also implied a switch in the data processing pipeline limiting the available Pass~7 data to the three years which we used in this analysis. The reprocessing primarily impacts the energy reconstruction as it accounts for a time-dependent change of the calorimeter calibration constants \citep{ReprocessingPaper}, which in case of a signal may result in a moderate ($2--3\%$) shift towards higher photon energies with respect to Pass~7 data as well as an improvement in the high energy LAT point spread function (PSF), which however does not constitute any significant impact on our analysis. {\edit{We also would like to point out that the recently released Pass~8 event selection provides $\sim25$\% increased effective area above $1\,\ensuremath{\mathrm{GeV}}\xspace$~\citep{Ackermann-M.:2015aa}~ and increases the data live time by a factor of two compared to the data set used in our analysis of the Virgo region.}} We will show in subsequent sections that our findings and interpretation are limited by systematic uncertainties in the interstellar emission model (IEM). \edit{These uncertainties can not be overcome by statistics but require better understanding of the CR distribution and its interaction in our Galaxy which is used to derive the IEM. Moreover, the IEM released with Pass~8 contains a data-driven residual component at the scale of 2 degrees. Hence, using this model is not suitable in the case of the spatially extended Virgo cluster.}
The data were processed using the \emph{Fermi} {\tt{ScienceTools}} version v9r28p0. \footnote{The software packages required for LAT analysis along with the templates used to model the interstellar\ and extragalactic emission are made publicly available through the Fermi Science Support Center \url{http://fermi.gsfc.nasa.gov/ssc/data/}}
We selected events with high probability of being $\gamma$ rays\xspace by choosing the Source event class. In order to evade $\gamma$-ray\xspace\ contamination generated by CRs interacting with the atmosphere of the Earth, we removed events with a LAT zenith angle $>100\deg$. We excised time periods around bright solar flares and gamma-ray bursts and applied a rocking angle cut of 52\deg . Furthermore, we restricted our analysis to the 100~\ensuremath{\mathrm{MeV}}\xspace\ to 100~\ensuremath{\mathrm{GeV}}\xspace\ energy range and used the P7SOURCE\_V6 instrument response functions.
To model the Galactic foreground emission caused by CRs interacting with the gas and radiation fields in our Galaxy, we use the {\tt{gll\_iem\_v02.fit}} model. This IEM is the standard IEM\xspace provided by the \emph{Fermi}-LAT\ collaboration for point source analysis in the un-reprocessed flavor. The isotropic $\gamma$-ray\xspace\ emission is accounted for by the {\tt{isotropic\_iem\_v02.txt}} model. For simplicity we refer to this set of IEM\xspace and isotropic model as our \emph{standard} model. We chose a $20\deg\times20\deg$ region of interest (ROI) centered on the center of Virgo-I ($\alpha_{2000}=187\fdg71$ and $\delta_{2000}=12\fdg39$) and performed a binned likelihood analysis with $0\fdg1$ spatial bins and 30 logarithmic bins in energy.\footnote{The coordinates for Virgo-II are taken to be $\alpha_{2000}=187\fdg45$ and $\delta_{2000}=8\fdg00$.} A counts map of the ROI together with the position of all sources from the LAT 2-year catalog~\citep{2FGL} sources is shown in Fig.~\ref{fig:cmap}. In addition to the two aforementioned diffuse model components, our background model contains all sources within a $30^{\circ}$ radius around the Virgo-I center that are listed in the 2FGL catalog~\citep{2FGL}. The Virgo ROI contains mostly extragalactic sources which may be variable and thus the two-year source parameters in the 2FGL catalog might be bad approximations for the three-year data. Another challenge arises by performing an analysis down to 100 MeV. At this energy the \emph{Fermi}-LAT\ PSF with 68\% containment radius is about $7^{\circ}$~\citep{fermi_inst2012} and thus even far away but strong sources which are not modeled correctly might easily increase the significance of a very extended profile located at the cluster center. To account for this we free the normalization and spectral index of all sources within 5\deg from the center coordinates of either Virgo-I or II in addition to the bright sources in the ROI (see below). The sources left free in our fit are marked by crosses in Fig.~\ref{fig:cmap}.
\begin{figure}[tbp]
\centering
\includegraphics[width=.9\columnwidth]{f1}
\caption{ Counts map of the Virgo ROI between $100\,\ensuremath{\mathrm{MeV}}\xspace$ and $100\,\ensuremath{\mathrm{GeV}}\xspace$ smoothed by a $\sigma=0\fdg3$ gaussian kernel. 2FGL sources with free parameters in the likelihood fit are marked by crosses and those that have fixed parameters are marked by boxes. The prominent AGN M87 is almost in the cluster center. The cyan and yellow circles correspond to the angle subtending the virial radius, of Virgo-I and Virgo-II, respectively (see Section~\ref{sec:dmVirgo} for details). We show the excess identified in Section~\ref{sec:extended} as a magenta contour.}
\label{fig:cmap}
\end{figure}
Using a likelihood analysis we construct a test statistic ($TS$) following~\citet{ts} to evaluate the improvement of the likelihood fit to the ROI when adding a new source by defining $TS=-2(\log\mathcal{L}_{0} - \log\mathcal{L})$. $\mathcal{L}_{0}$ refers to the maximum likelihood value for the null hypothesis and $\mathcal{L}$ for the value for the alternative hypothesis (including an additional source such as the cluster itself). In the case of one additional degree of freedom the significance can be written as $\sigma=\sqrt{TS}$ (see Section~\ref{sec:extended} for details regarding the gauging of this quantity). We look for new point-like excesses and derive $TS$-maps by placing a test point source at the location of each pixel of the map and maximizing the likelihood using the {\tt{gttsmap}} tool.
To estimate the systematic uncertainties on source significance and flux properties caused by our limited knowledge of the IEM\xspace, we compare the standard IEM\xspace results to the results obtained with eight alternative models seeded with $\gamma$-ray\xspace emission maps generated by GALPROP\footnote{\url{http://galprop.stanford.edu}}~\citep{Vladimirov:2011aa} and additional templates (as described in detail in Section~\ref{sec:IEM}) \citep{2013arXiv1304.1395D}. We would like to stress that these models do not cover the entire systematic uncertainty associated with the IEM and the results are not expected to bracket the standard IEM results. Despite the potentially small coverage, using these models demonstrates the influence the IEM has on our result.
\section{Search towards the Virgo Cluster Center}\label{sec:virgoCenter}
Instead of starting our search for extended emission from the cluster center with a specific physical model, we consider a more generic model for any potential excess, namely a simple uniform disk with a given radius, $r_{\mathrm{disk}}$, centered on Virgo~I. The uniform disk profile is very successful in finding weak extended emission and is less prone to degeneracy with strong point sources such as M87. Moreover, a disk profile is usually sufficient to find sources of various shapes because, even for very strong sources, a discrimination between different emission profiles is usually not possible \citep[see, e.g.][]{fermi_extended}. For these reasons, we use the disk profile with a power-law spectral model for all our systematic studies of the extended excess (Sections~\ref{sec:virgoCenter} and \ref{sec:extended}). To further characterize the extended excess, we perform a $TS$ vs radius scan. The scan is performed in $0\fdg5$ steps of $r_{\mathrm{disk}}$ and shows a peak at $r_{\mathrm{disk}}=3\deg$ with an associated $TS$-value of 14.2 as shown in Fig.~\ref{fig:disk_r}. This finding is in agreement with \citet{HanI} who attributes most of their found emission to the innermost $3\deg$ of their profile.
\begin{figure}[tbp]
\centering
\includegraphics[width=.9\columnwidth]{f2}
\caption{$TS$ value for adding an extended source with a uniform disk profile fixed at the center of Virgo~I vs the radius of the disk ($r_{\textrm{disk}}$). There is a clear peak in the $TS$ distribution (at $r_{\textrm{disk}}=3\deg$) as it is expected from a finite-sized source of excess $\gamma$-ray\xspace . In case of an excess due to an overall residual photon distribution in the whole ROI a steady increase would be expected, which is incompatible with our findings.}
\label{fig:disk_r}
\end{figure}
Next, we considered the case of a DM-induced signal as proposed by~\citet{HanI} which would lead to a $\gamma$-ray\xspace\ contribution peaked at the cluster center, again considering Virgo-I as the center of the cluster. To this end we substitute the disk profile with the DM profile used in \citet{HanI} in our source model of the Virgo region. We reproduce their results, finding $TS\simeq23$, only if we fix the bright sources outside of $5\deg$ of Virgo-I to their catalog values as reported in 2FGL. However, when performing the analysis according to the description in Section~\ref{sec:ana}, the $TS$-value drops considerably ($TS\simeq17$).
\section{Origin of the putative extended emission}\label{sec:extended}
We scan the inner $10\deg\times10\deg$ of the Virgo region with a disk with a radius of $r_{\mathrm{disk}}=3\deg$ on a 0\fdg5 grid to find the best position for the origin of the putative extended excess. To obtain this $TS$ map we add a disk-like test source at each grid position and maximize the likelihood using {\tt{gtlike}}. We leave all sources within a radius of $5\deg$ from the position of the test source and all bright sources ($TS > 1000$) free to vary in normalization and spectral index. This approach is needed since in the presence of a real extended source, the degeneracy between the extended source and overlapping or very close-by point sources will decrease the significance of the extended sources. The resulting $TS$-map has peaks significantly offset from the centers of Virgo-I and Virgo-II. All of them are located in the lower right quadrant of our $TS$ map shown in Fig.~\ref{fig:tsmap_ext}.
To obtain a more detailed description we made a finer grid of 0\fdg2 spacing in the $4\deg\times4\deg$ region encompassing the highest $TS$ values. This finely binned $TS$-map is shown together with the coarse map in Fig.~\ref{fig:tsmap_ext}. From the fine map it is evident that a large region of the Virgo ROI yields $TS$ values above 25. In particular, there are two broad maxima seen which are spatially distinct from one another. Note that the typical $1\sigma$ localization contour for a source near threshold is about $0.1\deg$. Consequently, the peaks shown in Fig.~\ref{fig:tsmap_ext} appear to be larger than what would be expected from a point source.
{\edit{Repeating the study of $r_\rmn{disk}$ vs. TS at the two maxima positions we again find a clear peak at $r_{\mathrm{disk}}=3\deg$ and thus continue all subsequent analysis using a disk with $r_{\mathrm{disk}}=3\deg$. }}
\begin{figure}[tbp]
\centering
\includegraphics[width=.9\columnwidth]{f3}
\caption{ \emph{Top:} $TS$-map of a uniform disk with $r_{\mathrm{disk}}=3\deg$ using a 1\fdg0 grid for the $10\deg\times10\deg$ Virgo region. The open crosses mark the center position of Virgo-I and II. There is a concentration of high $TS$ values in the lower right quadrant indicating that the position of the centroid is not well defined. \emph{Bottom:} $4\deg\times4\deg$ region of the coarser $TS$-map shown in the top panel. The finer 0\fdg2 binning emphasizes two broad $TS$ maxima of equal height which are well separated by about 1\fdg5. The different contours indicate the 1, 2, and $3\sigma$ levels.}
\label{fig:tsmap_ext}
\end{figure}
In order to study the relationship between $TS$-values and significance, we performed an extended source search at 288 randomly selected sky positions (blank fields) to estimate the significance of finding a disk-shaped excess with $r_{\mathrm{disk}}=3\deg$. All test positions were selected so that their inner $5\deg$ do not overlap and hence each position can be treated as statistically independent. Furthermore, we only used positions with $|b|>20\degr$ and therefore exclude regions with bright Galactic $\gamma$-ray\xspace\ emission. The resulting TS distribution is shown in Fig.~\ref{fig:ts_hist} and is reasonably described by a $\chi^2$ distribution with one degree of freedom. We note that the number of tested blank fields is too small to sample any probability density function at $TS>8$. However, the absence of $TS>16$ along with the fact that the majority of blank fields yield $TS<8$ indicate that the significance of the excess we find here (up to $TS \sim 32$) is larger than $3\,\sigma$. We stress that this particular statement is only valid at high Galactic latitude from where we extracted our blank fields.
\begin{figure}[tbp]
\centering
\includegraphics[width=.9\columnwidth]{f4}
\caption{$TS$ distribution for adding an extended source with a uniform disk profile $r_{\textrm{disk}}=3\deg$ at randomly selected high Galactic latitude ($|b|>20\deg$) positions. The ROIs of the positions do not overlap within their inner $5\deg$. We do not find any $TS$ above 16 and most of the values lie below eight. The solid line is a $\chi^2$ distribution with one degree of freedom that reasonably well describes the data.}
\label{fig:ts_hist}
\end{figure}
\section{Interstellar Emission}\label{sec:IEM}
The interstellar $\gamma$-ray\xspace\ emission is the dominant $\gamma$-ray\xspace\ background in \emph{Fermi}-LAT\ analyses in the vicinity of the Galactic plane, while far from the Galactic plane the largest $\gamma$-ray\xspace\ contribution usually stems from discrete sources and isotropic $\gamma$-ray\xspace\ emission. However, in certain high-latitude regions the interstellar emission can contribute significantly to the $\gamma$-ray\xspace\ background and one of these regions is Virgo. Several features present in the IEM\xspace are contained within the Virgo ROI. The most striking features are the spatially uniform patch that is used to model the $\gamma$-ray\xspace\ emission from Loop~I~\citep{1962MNRAS.124..405L} and some filaments of HI around the center of the Virgo cluster as shown in Fig~\ref{fig:iem}.\footnote{See \url{http://fermi.gsfc.nasa.gov/ssc/data/access/lat/Model\_details/Pass7\_galactic.html} for the description of the model of Loop~I.}
\begin{figure}[tbp]
\centering
\includegraphics[width=.9\columnwidth]{f5}
\caption{Model counts map of the standard IEM\xspace model for the Virgo ROI above $1\,\ensuremath{\mathrm{GeV}}\xspace$. The cyan and yellow circles correspond to the angle subtending the virial radius, of Virgo-I and Virgo-II, respectively (see Section~\ref{sec:dmVirgo} for details). We show the excess identified in Section~\ref{sec:extended} as a magenta contour. Clearly visible is the patch that we associated with Loop~I as the bright light-blue band on the left side and several bright filaments, especially a donut-like shape close to the center of the ROI. The Virgo region does not show a uniform IEM\xspace as might be expected for high-latitude ROIs and must be treated accordingly when searching for extended emission.}
\label{fig:iem}
\end{figure}
Loop I is only approximately modeled in the standard IEM\xspace and could influence our analysis of the region. For most point sources in the Virgo region the IEM\xspace does not play a significant role since it contributes on average only about 110 counts per square degree above $100\,\ensuremath{\mathrm{MeV}}\xspace$. Yet, it must be considered carefully when analyzing weak and very extended emissions like our disk model in the previous section (accounting for about 800 predicted photons and about 28 counts per square degree). The isotropic $\gamma$-ray\xspace\ emission has the largest contribution to the diffuse $\gamma$-ray\xspace\ emission in the ROI and accounts for about 200 counts per square degree. We note that the spatial variation in the IEM\xspace $\gamma$-ray\xspace\ counts are up to a factor of two from the minimum value. These spatial variations in the IEM are about three times larger than the potential $\gamma$-ray\xspace\ counts from the disk profile. The weak disk emission could easily be caused by a missing feature in the IEM\xspace . Such a missing component would necessarily not be traceable by tracers of the interstellar gas (21-cm H\,{\sc I} line and 2.6-mm CO line), and makes the systematic evaluation of IEM\xspace\ influences on our analysis mandatory.
To assess the influence of the uncertainty in our knowledge of the IEM\xspace, besides the standard IEM\xspace, we use eight additional models. These models are seeded with simulated $\gamma$-ray\xspace\ intensities obtained from GALPROP, assuming CR halo heights of 4~kpc and 10~kpc, two different CR source populations, referred to as 'pulsars' and 'supernova remnants' (SNR) and two different HI spin temperatures (optically thin and 150K). Afterwards each model's spectral component is multiplied with a log-parabola function to provide a better model-data agreement and thus is not a direct output of GALPROP. For further details on the alternative models see~\cite{2013arXiv1304.1395D}. Most of the parameters varied in the alternative models are only expected to have very slight influence on the results for the Virgo region since they should not influence the local CR density very much and that is what is mostly sampled when looking in the direction of Virgo.
For each of these alternative models an individual modified isotropic diffuse contribution is used.
Each of these models is comprised of maps that are inferred from HI and CO tracers along with components modeling the large scale diffuse residual structures, such as Loop~I~\citep{fermi_loopI} and the \emph{Fermi}\ bubbles~\citep{fermi_bubbles2010,Bubbles_2014}. The HII emission that is inferred from GALPROP and based on the NE2001 model is added to the HI map. In addition each model also includes a model of emission from inverse Compton (IC) scattering of CR electrons on interstellar radiation also calculated by GALPROP. For these models the spectral line shifts of the HI and CO lines were used to derive maps for separate ranges of Galactocentric distance. Inside the Virgo ROI only HI is present and is located within the second and third ring HI templates (these rings correspond to Galactocentric distance ranges 4--8 kpc and 8--10 kpc, respectively). Beside these two HI components only the IC model and the Loop~I template are included in our fit. Each normalization of the alternative IEMs\xspace components is left free in the likelihood fit. The second HI ring only contributes to the Virgo ROI because of the HII emission that is added to the HI template. We note that there is considerable uncertainty in the estimation of the HII emission. The Loop~I template is a geometrical template adapted from~\cite{2007ApJ...664..349W} and based on a polarization survey at 1.4 GHz and it is modeled as two expanding shells centered on two local OB associations.
Our standard IEM on the other hand uses a uniform-patch Loop~I template whose shape was derived by visual inspection of the gamma-ray residuals when building the standard IEM\footnote{There is also another component in the Loop~I template of the standard IEM derived from from 408 MHz radio maps\
\citep{1982A&AS...47....1H} but this template is not visible in the Virgo ROI.}. There is currently no template of Loop~I available from observations at other wavelengths that adequately traces the gamma-ray emission observed by Fermi-LAT in the direction of Loop~I. By following two different approaches to define a Loop~I template into our models we can gain some insights into the influence of the Loop~I modeling on our results.
We randomly select one position from the bottom panel of Fig.~\ref{fig:tsmap_ext} where $TS\geq25$ and repeat our likelihood calculations using these alternative diffuse models.
The results do not have consistent $TS$ values between the individual IEMs\xspace as can be seen in Table~\ref{tab:iem}. While the standard, I, V and VII IEMs\xspace show high $TS$ values for an extended source, the other models yield considerably lower values.
Leaving the individual components of the IEM\xspace free to vary in the fit allows for a higher sensitivity to features in the IEM\xspace which might affect only one component map. Such a mis-modeling of a single component in the IEM\xspace could cause an extended excess roughly corresponding in shape with the mis-modeled or missing emission.
The large observed variation in $TS$-values of the disk emission for the alternative models provides an indication that the observed $\gamma$-ray\xspace\ emission may be due at least partially to an incomplete modeling of the IEM\xspace. In the direction of the Virgo cluster the interstellar gas is mostly local (within $\sim$1~kpc of the Sun) and thus only the local CRs should contribute to the IEM\xspace. The $\gamma$-ray\xspace\ emission caused by the local CR density is not expected to have large dependencies on the CR source distribution, CR halo height, or spin temperature and thus we would expect rather similar results for all models. However the alternative IEMs\xspace that have a relatively large $TS$ for the disk are associated with large predicted photon counts, overemphasising the contribution from HI ring 2 by increasing its amplitude by a factor O(10--30). Such an implausibly large increase in the contribution is explained by the shape of the model components of each IEM\xspace. HI ring 2 covers a region in the projected sky that is similar to the Loop I template in the standard model and by upscaling the normalization beyond the physically viable bounds of the HI component the overall fit is improved. Note that the extension of HI ring 2 in the ROI depends on the CR halo height and thus introduces some dependency of the $TS$ of the disk on this IEM\xspace parameter that is also not expected from the local CR density. The high normalization of the HI ring 2 demonstrates that some $\gamma$-ray\xspace\ emission is coming from its large region within the ROI that is not traced by the usual HI and CO tracers but can be partially compensated by overestimating their contribution in the ROI. The disk emission might just compensate a part of this large-area diffuse gamma-ray emission that is not overlapped by any other IEM\xspace component that could compensate it.
In general it is very difficult to obtain a conclusive picture as to the possible emission in the ROI. The IEM\xspace study suggests that while there may be some additional emission, the ROI contains several IEM\xspace components whose predicted emission overlap with the excess, making it difficult to disentangle their contribution in a log-likelihood\ fit to the $\gamma$-ray\xspace-data. A precise modeling of Loop I on the other hand, poses a considerable challenge since its $\gamma$-ray\xspace\ emission is not well traced by the radio emission. Hence, we used a geometric model for Loop I in the alternative IEMs\xspace compared to the $\gamma$-ray\xspace\ residual-inferred template in the standard IEM\xspace. The low significance of the emission found in this work makes it extremely difficult to identify its origin due to the aforementioned issues but is likely caused by inaccuracies in the IEM\xspace .
To describe the location of the extended emission we devised a double disk patch that accounts for the observed residual $\gamma$-ray\xspace\ emission. This double disk consists of two disks with $r_{\mathrm{disk}}=3^{\circ}$, one at each of the maxima in our $TS$-map shown in Fig.~\ref{fig:tsmap_ext} and a uniform single power law emission of the whole profile. In this way the model covers most of the extended emission and we can easily trace its position in sky maps and include it in all relevant plots discussing possible counter parts. In all further analysis when we are deriving upper limits on $\gamma$-ray\xspace emission from DM annihilation or CR interaction we leave the newly found extended emission unmodeled to be conservative as we can not make a definitive statement about the origin of the extended emission. This results in weaker upper limits compared to when including the aforementioned double-disk model for the emission when deriving upper limits.
\begin{deluxetable*}{lcccccccc}
\tablewidth{0pt}
\tablecaption{Parameters for the 9 interstellar\ emission models used}
\tabletypesize{\small}
\vspace*{-0.3cm}
\startdata
\hline\hline
Model & Sources & Halo height & $T_{S}$ & $\log(L_0)$ & $\log(L_{\mathrm{disk}})$ &$TS_{\mathrm{disk}}$ & $F(E>1\,\ensuremath{\mathrm{GeV}}\xspace)$ & $\Gamma$\\
& & kpc & K & & & & $\times10^{-9}\textrm{cm}^{-2}\textrm{s}^{-1}$&\\
\hline
I & Pulsars & 10 & $10^5$ & -342483.4 &-342472.1&21.7&$1.5\pm0.1$&$1.85\pm0.02$\\
II & Pulsars & 4 & $10^5$ & -342484.0 & -342476.3 & 15.4 &$1.3\pm0.3$&$1.8\pm0.1$\\
III & Pulsars & 10 & $150$ & -342480.3 & -342474.0 & 12.6&$1.3\pm0.3$&$1.8\pm0.1$ \\
IV & Pulsars & 4 & $150$ & -342481.3 & -342474.0 & 14.6 &$1.3\pm0.3$&$1.8\pm0.1$ \\
V & SNR & 10 & $10^5$ &-342481.9 &-342471.3 & 21.2&$1.5\pm0.4$&$1.9\pm0.1$\\
VI & SNR & 4 & $10^5$ & -342484.0 &-342476.3 & 15.4&$1.3\pm0.3$&$1.8\pm0.1$\\
VII & SNR & 10 & $150$& -342479.2 & -342469.4 & 19.7&$1.5\pm0.3$&$1.8\pm0.1$\\
VIII & SNR & 4 &$150$& -342481.1 &-342474.1 & 14.1&$1.2\pm0.4$&$1.8\pm0.1$\\
\hline
Standard & \nodata & \nodata & \nodata & -342494.3 & -342480.4 & 27.4&$1.7\pm0.3$&$1.9\pm0.1$ \\
\hline
\enddata
\tablecomments{The models I-VIII have the normalization of each individual component left free in the fit while the standard model has fixed ratios between individual components. The columns contain from left to right, the model name, the source population causing the $\gamma$-ray\xspace\ emission, the CR halo height, the spin temperature, the reference log-likelihood , the log-likelihood\ with the added disk, the $\gamma$-ray\xspace\ flux and spectral index of the disk profile.\label{tab:iem}}
\end{deluxetable*}
\section{Point source search}\label{sec:ps}
Previously undetected (variable) point sources that are not contained in the 2FGL catalog provide a possibility to account for the extended excess we find. Such sources can be significant background sources in the three-year data set and should be included in the background model. This possibility was first addressed by~\cite{Macias} and later by~\cite{HanII}.
However, neither searched for possible extended emission (not from DM annihilation) offset from the cluster center. Therefore we performed our own point source search in the Virgo ROI to be able to compare these results with the same analysis set up as our extended source search. We note that we search for new point source candidates without the double disk profile described in the previous section. This allows us to construct a new alternative background model including the newly found point source candidates but not considering any additional extended source profile.
We divide the Virgo ROI in $0\fdg1\times0\fdg1$ grid positions and fit all 2FGL sources in the region to obtain the log-likelihood reference value. We then add one point source at one grid point and calculate the $TS$ of this source candidate and repeat this procedure at each grid point. We identify source candidates corresponding to grid points with $TS>15$ and use them as seeds for the \texttt{gtfindsrc} localization algorithm. Finally we add all source candidates into our Virgo ROI model and fit the region again to obtain a better reference model for the extended source investigation.
Following this procedure, we find eight new source candidates, three of which appear clustered in close vicinity to one another. We note that when including either of the three, the remaining candidates are not statistically significant. Thus we only keep the brightest candidate in our background model and discard the other two. In summary, our new background model contains six new source candidates listed in Table~\ref{tab:ps}. Four of the six sources lie in the vicinity of the center of Virgo-I and three of them are in the region where we find the extended emission (substantially offset from the center of Virgo-I). While adding these six source candidates to our model yields a much better description of the ROI, the closest candidate sources to the center of Virgo-I or Virgo-II are below the conventional source detection threshold ($TS>25$). Note that five out of these six sources are contained in the latest 4-year source catalog \citep[3FGL, see][]{3fgl}. To identify possible multi-wavelength counterparts to the $\gamma$-ray\xspace\ sources we searched in the 95\% error circle around each source in the NASA/IPAC Extragalactic Database (NED) for potential $\gamma$-ray\xspace\ emitters. We do not find convincing counterparts for the new point source candidates; however, that does not perclude the possibility that they are real sources since a considerable fraction of \emph{Fermi}-LAT\ sources are not associated with multi-wavelength counterparts \citep{2FGL,3fgl}. The locations of the new source candidates are shown in the model map\footnote{A model map is the predicted counts map calculated for the ROI from the maximum-likelihood values of the model parameters.} in Fig.~\ref{fig:ps_model}. While we do get slightly different results compared to the work of~\cite{Macias} and~\cite{HanII} the discrepancies can be attributed to the larger data set used in~\cite{Macias} and~\cite{HanII} and the slightly different analysis procedures, such as the number of source parameters freed in the search or the different definitions of the ROIs. Considering the very weak emission of all source candidates the consistency of the findings among these works is reasonable. {\edit{We also note that when including the disk profile in addition to the new point source candidates in the ROI we obtain a $TS$ value of only $\sim7.5$ for the disk emission.}}
With this improved background model in hand, {\edit{which contains new point source candidates but not the disk profile}}, we devote the remainder of this paper to study the Virgo cluster as a $\gamma$-ray\xspace emitter, either via DM annihilation or via CR interactions (Sections~\ref{sec:dmVirgo} and~\ref{sec:crVirgo}).
\begin{deluxetable}{cccccc}
\tablewidth{0pt}
\tablecaption{Point source candidates not included in the 2FGL catalog}
\vspace*{-0.3cm}
\startdata
\hline\hline
Source candidate & RA & DEC & 95\% error & $TS$ &3FGL source \\
\hline
I & 185.493 & 12.03&0.1& 23.5& J1223.2+1215\\
II & 184.167& 9.460 &0.1& 23.4 & \\
III &185.692 & 8.148 &0.1& 21.6 & J1223.3+0818\\
IV & 190.89& 16.183&0.07& 28.1& J1244.1+1615\\
V & 193.419 & 3.574&0.05& 49.6& J1253.7+0327\\
VI & 180.292 & 20.141 &0.09& 27.8& J1200.9+2010\\
\hline
\enddata
\label{tab:ps}
\tablecomments{Note that 3FGL employs four years of data and typically contains sources with $TS>25$.}
\end{deluxetable}
\begin{figure}[tbp]
\centering
\includegraphics[width=.9\columnwidth]{f6}
\caption{Model counts map generated with the best-fit parameters after adding six new point-source candidates to the Virgo ROI background model, integrated over the entire energy range ($100\,\ensuremath{\mathrm{MeV}}\xspace \leq E \leq 100\,\ensuremath{\mathrm{GeV}}\xspace$). The cyan and yellow circles correspond to the angle subtending the virial radius, of Virgo-I and Virgo-II, respectively (see Section~\ref{sec:dmVirgo} for details). We show the excess identified in Section~\ref{sec:extended} as a magenta contour. The new sources are marked by crosses (white).}
\label{fig:ps_model}
\end{figure}
\section{Search for Dark Matter Annihilation in Virgo}\label{sec:dmVirgo}
The $\gamma$-ray flux from annihilating DM particles of mass $m_{\chi}$ can be written as
\begin{equation}
\Phi_{\gamma}(E,\psi)=\frac{1}{4\pi}\frac{\ensuremath{\langle\sigma v\rangle}\xspace}{2 m_{\chi}} \sum_{j}{B_{j}\frac{\rmn{d}N_{j}(E)}{\rmn{d}E_{j}}} \times J(\psi).
\end{equation}
In the above equation, we have \ensuremath{\langle\sigma v\rangle}\xspace as the thermally-averaged product of DM self-annihilation cross section times velocity, and the sum runs over the final states the DM particles annihilate into with their specific $\gamma$-ray\xspace annihilation yields $\rmn{d}N_{j}(E)/\rmn{d}E_{j}$ and branching fraction $B_{j}$ per final state $j$. We define the astrophysical J-factor as the line-of-sight integral of the squared DM density towards the observational direction, $\psi$, integrated over a solid angle $\Delta\Omega$:\begin{equation} J(\psi)=\int_{\Delta\Omega}\left\{\int_{l.o.s.}{\rho^{2}{[l(\psi)]}}\mathrm{d}l\right\}\mathrm{d}\Omega'. \label{eq:los}\end{equation}
Density profiles $\rho(r)$ for plausible DM distributions can be expressed in terms of a generalized Hernquist profile \citep{Hernquist1990}:
\begin{equation}
\rho(r)=\rho_0\left(\frac{r}{r_s}\right)^{-\gamma}\left[1+\left(\frac{r}{r_s}\right)^{\alpha}\right]^{\frac{\gamma-\beta}{\alpha}},
\label{eq:nfw}
\end{equation}
where $\alpha,\beta$ and $\gamma$ are shape parameters. High-resolution cosmological N-body simulations of cold DM halos indicate that their density profiles are well described by a Navarro-Frenk-White (NFW) profile where $\alpha=1,\beta=3,\gamma=1$ \citep{Navarro1996,Navarro1997}. The quantity $r_s$ is the characteristic scale radius of the profile with $c$ as the concentration parameter such that $r_s = r_{200}/ c$. $r_{200}$ is defined to be the virial radius for which within the cluster mass $M_{200}$ is:
{\edit{
$M_{200}= 4 \pi \rho_c/3 \times 200 \times r_{200}^3,$
where $\rho_c$ is the critical density of the universe and $\rho_0$ is the characteristic density of the profile, $\rho_0= \frac{200}{3} \frac{c^3 \rho_c}{\ln{(1+c)}-c/(1+c)}.$
As for the concentration-mass relation, we adopt here the one proposed by \citet{Prada2012} for the WMAP5 cosmology. This relation was derived from the $\sim10$ billion particle Bolshoi and MultiDark large scale structure N-body cosmological simulations \citep{Klypin2011}.\footnote{We assume a $\Lambda$CDM cosmology, characterized through $\Omega_{m}=0.32$, $\Omega_{\Lambda}=0.68$ and $h=0.67$ \citep{PlanckCollaboration2013}} By evaluating the integral $M_{500}=\int_{0}^{r_{500}}{ \rmn{d}^{3}r\,\rho(r)}$ with $\rho(r)$ as defined in Eq.~\eqref{eq:nfw}, we can numerically determine the value for $M_{200}$ for which this equation is satisfied, using the reported values for $M_{500}$ and $r_{500}$ from \citet{chen2007}.}}
We find $M_{200}=5.6\times10^{14}\,\ensuremath{M_{\odot}}$ as the mass of Virgo-I with $c=4.21$.
The integrated $J$-factor, for $r<r_{200}$ and assuming an NFW DM density profile, at a large angular diameter distance $D_a$ from Earth to the center of the cluster can be approximated by:
\begin{equation} J^{\rm NFW} = \frac{4\pi}3 \rho_0^2 r_s^3 \frac{1}{{D_a}^2}.\end{equation}
\subsection{Contribution from Substructure\label{sec:DMsubstructure}}
Some fraction of the DM should reside in sub-halos within the NFW-like primary halo. The presence of sub-halos implies both a flattening of the surface brightness profile \citep[see, e.g.][]{MASC2011,Gao2012a} and an enhancement (\emph{boost}, denoted $b$) of the J-factor which may increase the total annihilation signal by orders of magnitudes. Here $b=0$ corresponds to the case of the smooth NFW halo without the inclusion of additional substructure.
For the normalization of the DM substructure signal strength, we adopt a fiducial substructure model that follows the works by \citet{MASC2013}, assuming a moderate total boost factor of $b=33$ as given by their proposed parametrization of the boost for the Virgo mass (DM-I). We contrast this \emph{conservative} model with an \emph{optimistic} model that implicitly adopts a power-law extrapolation of the mass-concentration relation to the smallest (unresolved in simulations) halo masses, yielding $b=1200$ \citep[DM-II,][]{Gao2012b}.
For the spatial morphology of the DM-induced gamma-ray emission, including the predicted DM substructure signal, we adopt the form from a recent study of high resolution cosmological DM simulations of cluster-size halos \citep{Gao2012a}. The projected luminosity profile from the substructure is approximated by:
\begin{equation}\label{eq:sub}
I_{\mathrm{sub}}(\theta) = \frac{16 b\times L_{\gamma}^{\mathrm{NFW}}}{\pi \ln(17)} \frac{1}{\theta_{200}^2+16 \theta^2},\quad \mathrm{for}\, \theta\le \theta_{200},
\end{equation}
where $\theta$ is the distance from the cluster center in degrees and $\theta_{200}$ is the angle subtending the virial radius, given by $\theta_{200}=\arctan(r_{200}/D_{a})\times 180\deg/\pi$, and $L_{\gamma}^{\mathrm{NFW}}$ is the total $\gamma$-ray\xspace luminosity of the halo within the virial radius \citep{Gao2012a}, defined as $L_{\gamma}=4\pi \times \Phi_{\gamma}\times D_{a}^{2}$.
\begin{figure}[tbp]
\centering
\includegraphics[width=.9\columnwidth]{f7}
\caption{Shown is the annihilation flux profile as function of subtended angle for Virgo-I. We show this quantity for our two substructure benchmark scenarios (DM-I and DM-II) as solid and dashed lines, respectively. The dotted profile corresponds to the case where no substructure is included. For boosted profiles, the expected surface brightness profile has a broader (angular) distribution than for the smooth NFW profile. Outside the virial radius the DM halo is truncated, and accordingly we truncate our templates outside the virial radius. The dotted line indicates the angular virial radius $\theta_{200}$ (see text for details).\label{fig:profile_vs_r}}
\end{figure}
Note that the work by \citet{MASC2013} does not address the change in the spatial morphology of the annihilation signal due to the presence of sub-halos. Yet, in previous works, \citet{MASC2011} have shown that moderate values for {\it b} also lead to a significant flattening of the annihilation profile in clusters. We found this flattening to agree reasonably well with the one implied by Eq.~\ref{eq:sub} for moderate values of $b$. Thus, from here on, we assume this approximation to be a good representation of the spatial morphology of the DM substructure signal in both the DM-I and DM-II setups, with only the value of {\it b} differing from one to another substructure scenario. The resulting expected surface brightness profiles for DM annihilation are shown in Fig.~\ref{fig:profile_vs_r}.
Our choice of models for the substructure is motivated by assuming a common mass scale for both setups, $M_{\mathrm{cut}}=10^{-6}\ensuremath{M_{\odot}}$, at which the matter power-spectrum is truncated. Below this scale, no DM halos are formed and thus no halos will contribute to the expected DM signal. What the minimal DM halo size is and what properties these substructures have is, to a large extent uncertain and may depend on the specific DM particle model. The mass range for $M_{\mathrm{cut}}$ may vary from $10^7$ to $10^{-12}\ensuremath{M_{\odot}}$, where the upper limit comes the fact that we observe dwarf galaxies with that mass \citep{McConnachie:2012aa}. The lower limit is more uncertain and depends on the DM particle properties \citep{Profumo2006,Bringmann2009}.\footnote{Note that the actual values depend not only on the specific DM particle but also on the cosmological evolution of DM halos.}
Recalling the discussion on the morphology of the cluster from Section~\ref{sec:extended}, we remark that while the mass of Virgo-II is only about $\sim13\%$ of that of Virgo-I, its concentration parameter, $c=5.58$, is larger than that of Virgo-I (see Table~\ref{tab:jfactors}). {\edit{Since the DM annihilation flux is proportional to the third power of the concentration, the predicted DM-induced $\gamma$-ray\xspace flux of Virgo-II corresponds to about one third of the predicted DM-induced $\gamma$-ray\xspace flux from Virgo-I.}} In order to account for this in our DM modeling, we consider the cluster as a merging system where each sub-cluster is modeled individually according to the description given in this section. We then co-add the two resulting templates to form a composite spatial map which we use in the likelihood analysis. The projected annihilation flux maps are shown for the different models we choose in Fig.~\ref{fig:DMproj}. In this figure we added a contour that indicates the spatial position of the excess we report in this paper and we stress that a DM origin is unlikely because of the large offset between the predicted DM annihilation profile and the contours of the $\gamma$-ray\xspace excess (see Section~\ref{sec:extended} for details).
\begin{figure}[tbp]
\centering
\includegraphics[width=.9\columnwidth]{f8}
\caption{\label{fig:DMproj}Predicted integrated $\gamma$-ray\xspace-flux projection for the entire cluster above 100\,\ensuremath{\mathrm{MeV}}\xspace for DM annihilation of a 100\,\ensuremath{\mathrm{GeV}}\xspace WIMP into 100\% $b\overline{b}$ with an annihilation cross section of $10^{-26}\mathrm{cm^{3}s^{-1}}$ for the DM models discussed in this article (\emph{top}: DM-I, \emph{bottom}: DM-II). Note the different scales. The dashed contour indicates the location of the $\gamma$-ray\xspace excess reported in Section~\ref{sec:extended}.}
\end{figure}
We summarize the main characteristics of our chosen models in Tab.~\ref{tab:jfactors}. We take the sum of the $J$-factors for Virgo-I and Virgo-II to be the total $J$-factor of the Virgo system.
\begin{deluxetable*}{lccccccc}
\tablewidth{0pt}
\tablecaption{Virgo subclusters and derived DM density profiles}
\tablehead{
\colhead{Sub-cluster} & \colhead{$M_{200}$} & \colhead{$r_{200}$} & \colhead{$\theta_{200}$} & \colhead{$c$\tablenotemark{a}} & \colhead{$J_{\mathrm{NFW}}$} & \colhead{$J_{\mathrm{DM-I}}$\tablenotemark{a}} & \colhead{$J_{\mathrm{DM-II}}$\tablenotemark{b}} \\
\colhead{} & \colhead{($\times10^{14}\mathrm{M_{\odot}}$)} & \colhead{($\mathrm{Mpc}$)} & \colhead{($^\circ$)} & \colhead{} & \colhead{($\times10^{17}$)} & \colhead{($\times10^{18}$)} & \colhead{($\times10^{20}$)} }
\startdata
M87 (Virgo-I) & 5.60 & 1.70 & 6.3 & 4.21 & 2.56 & 6.50 & 3.33 \\
M49 (Virgo-II) & 0.72\tablenotemark{c} & 0.88 & 3.8 & 5.58 & 1.85 & 5.36 & 0.75 \\
\enddata
\tablecomments{\label{tab:jfactors}Shown are the characteristic quantities used to derive the resulting $J$-factors for the Virgo cluster modeled as a merging system between the sub-clusters associated with M87 and M49. Columns from left to right are name, mass, virial radius, angular radius $\theta_{200}$, concentration parameter $c$, as well as $J$-factors for NFW and the DM models used in this analysis for each of the sub-clusters. All $J$-factors are given in units of $\mathrm{GeV^{2}cm^{-5}}$ and have been computed over a solid angle subtending the virial radius of each sub-cluster.}
\tablenotetext{a}{\citet{MASC2013}}
\tablenotetext{b}{\citet{Gao2012a}}
\tablenotetext{c}{\citet{chen2007}}
\end{deluxetable*}
\subsection{DM Flux Enhancement due to Inverse Compton Scattering\label{sec:ICS}}
In calculating the $\gamma$-ray spectra from DM annihilation to leptons, we include the effects of Inverse Compton (IC) scattering of background radiation by electrons and positrons that result from the annihilation.\footnote{By leptons we refer to $e^{\pm}$ and $\mu^{\pm}$. We refrain from including IC calculations for annihilation into $\tau^{\pm}$ since its decay signature is closer to hadronic final states and thus any IC contribution would be sub-dominant.} We calculate the IC component of the spectrum by conservatively assuming scattering only of the cosmic microwave background (CMB); other radiation fields such as starlight could also contribute but are sub-dominant. We use the program DMFit \citep{Jeltema2008,Ackermann:2014aa} for spectrum calculations and include IC according to the procedure outlined in \citet{2010JCAP...05..025A}.
In cluster environments, electrons and positrons lose energy via radiation (e.g. IC scattering and synchrotron emission) on much shorter timescales than they diffuse. We therefore neglect the effects of diffusion \citep[e.g.][]{Colafrancesco2006}. We also neglect energy losses due to synchrotron radiation. Synchrotron losses would significantly suppress the IC signal if the average magnetic field of the cluster, $\langle B \rangle > B_{\mathrm{CMB}}\sim 3~\mu$G, where $B_{\mathrm{CMB}}$ is the magnetic field that has the same energy density of the CMB (the IC scattering background). Suppression would be on the scale of $(B_{\mathrm{CMB}}/ \langle B \rangle)^2 $. While data on the intracluster magnetic field of Virgo are limited, simulations suggest an averaged magnetic field of $\sim O(0.1\textrm{--}1~\mu\mathrm{G})$ \citep{Dolag2005a}, too small for synchrotron emission to be significant.
\subsection{Limits on \ensuremath{\langle\sigma v\rangle}\xspace}
We derive upper limits on \ensuremath{\langle\sigma v\rangle}\xspace using the profile likelihood method~\citep{Rolke2005} as implemented in the {\tt MINOS}-subroutine of the {\tt MINUIT} package \citep{James1975} which is available through the \emph{Fermi} Science Tools. We define the 95~\% upper limit on \ensuremath{\langle\sigma v\rangle}\xspace as the value of \ensuremath{\langle\sigma v\rangle}\xspace for a given mass $m_\chi$ where twice the difference in the log-likelihood, $2\times\Delta\mathcal{L}=2.71$ with respect to the value of the log-likelihood\, for the best fit value.\footnote{For the background modeling we employ the same considerations as discussed in Section~\ref{sec:ana}.}
\begin{figure*}[htbp]
\centering
\includegraphics[width=.9\columnwidth]{f9}
\includegraphics[width=.9\columnwidth]{f10}
\caption{\label{fig:channels}Obtained 95\% CL upper limit on \ensuremath{\langle\sigma v\rangle}\xspace for various annihilation channels assuming our fiducial substructure models (\emph{top}: DM-I, \emph{bottom}: DM-II). Both $e^{\pm}$ and $\mu^{\pm}$ channels include the contribution from IC scattering with the CMB as detailed in Section~\ref{sec:ICS} which starts to dominate the predicted emission above $50\,\ensuremath{\mathrm{GeV}}\xspace$ for $e^{\pm}$ and $100\,\ensuremath{\mathrm{GeV}}\xspace$ for $\mu^{\pm}$. The dashed line corresponds to the annihilation cross section for a thermal WIMP.}
\end{figure*}
Fig.~\ref{fig:channels} shows the dependency of the upper limits on the chosen DM annihilation channel for our fiducial models. The most constrained channels are $\chi \chi \rightarrow b\overline{b}$ and $\chi \chi \rightarrow \tau^+ \tau^-$. Accounting for IC emission in the leptonic channels $e^\pm$ and $\mu^\pm$ improves the constraints we obtain from the prompt emission by two to three orders of magnitude, above DM masses of $50\,\ensuremath{\mathrm{GeV}}\xspace$ and $100\,\ensuremath{\mathrm{GeV}}\xspace$, respectively. The limits for $e^\pm$ are the most constraining for DM masses above $\sim110\,\ensuremath{\mathrm{GeV}}\xspace$, due to the enhanced flux predictions from IC.
In Fig.~\ref{fig:dm_diffuse} we show our derived upper limits on \ensuremath{\langle\sigma v\rangle}\xspace and their associated $TS$ values for the $\chi \chi \rightarrow b\overline{b}$ channel and contrast our standard IEM\xspace with results obtained from using the alternative diffuse models as discussed in Section~\ref{sec:IEM}. Our optimistic limits exclude thermal WIMP cross sections below $40\,\ensuremath{\mathrm{GeV}}\xspace$. The limits derived from the more conservative assumptions are a factor 20 weaker across the entire probed mass range. Even with the inclusion of additional point sources as done in this work, there is a residual $TS\sim4$ if we consider the more extended and elongated profile as predicted by our optimistic model (DM-II). For the DM-I case, this value is reduced even further. Considering the alternative diffuse models, the resulting limits are generally weaker, associated with residual $TS<5$ except for WIMP masses $\lesssim20\,\ensuremath{\mathrm{GeV}}\xspace$. For lower masses, the alternative models give rise to residual $TS$ peaking at $TS\sim9.5$ or $\sim3.1\sigma$. Low-mass DM models associated to relatively high $TS$-values for one diffuse model show a large spread ($\delta TS\simeq5$) in $TS$ for the alternative models.
Recalling the description of the alternative diffuse models in Section~\ref{sec:IEM}, these differ from the standard IEM\xspace\, by having various large-scale components fit freely to the data (e.g. Loop~I, IC, etc.). The extent of these large-scale components is comparable to the spatial extension of our cluster template which causes a degeneracy between the fit parameters for the diffuse components and Virgo. As a consequence we find that soft photons ($E\lesssim10\,\ensuremath{\mathrm{GeV}}\xspace$), which would otherwise be attributed to the background IEM\xspace\, are now included in the number of predicted photon counts from Virgo for a light WIMP model.\footnote{For illustration purposes, the reader is reminded that the typical $\gamma$-ray\xspace spectrum (energy flux) of, e.g., a 20~\ensuremath{\mathrm{GeV}}\xspace WIMP annihilating into $b\overline{b}$ peaks at $\sim2\,\ensuremath{\mathrm{GeV}}\xspace$ and results mainly in soft photons in the \ensuremath{\mathrm{MeV}}\xspace-\ensuremath{\mathrm{GeV}}\xspace range, which can explain the large spread towards the lowest WIMP masses shown in Fig~\ref{fig:dm_diffuse}.} Note that this effect appears to be even more pronounced as the spatial template for the Virgo cluster is even more extended than the disk used in our previous study (refer to Section~\ref{sec:extended} for a detailed discussion). Finally, we also remark that this issue is by construction less apparent for the standard IEM\xspace, since here all components are fixed to their relative best-fit contributions obtained from a likelihood fit to the entire $\gamma$-ray\xspace-sky.
\begin{figure}[htbp]
\centering
\includegraphics[width=.9\columnwidth]{f11}\\
\includegraphics[width=.88\columnwidth]{f12}
\caption{ \emph{Top}: Obtained 95\% CL upper limit on \ensuremath{\langle\sigma v\rangle}\xspace for a DM WIMP annihilating into $b\overline{b}$ in the mass range from 10~\ensuremath{\mathrm{GeV}}\xspace up to 2~\ensuremath{\mathrm{TeV}}\xspace. The shaded areas represent the range of limits obtained when replacing the standard IEM\xspace with the alternative models described in Section~\ref{sec:IEM}. Solid and dashed lines represent the limits obtained using the standard IEM\xspace for our conservative (DM-I) and our optimistic (DM-II) boost model, respectively. The dashed line corresponds to the annihilation cross section of a thermal WIMP. \emph{Bottom}: Shown are the associated $TS$ values with this choice of models. See the text for a discussion regarding the $TS$-values obtained with the alternative diffuse models. Note that in both plots we omit data points in which Minuit/MINOS did not reach convergence ($<10\%$ of the tested mass-model scan points).}
\label{fig:dm_diffuse}
\end{figure}
\section{Cosmic-Ray-Induced Gamma Rays}\label{sec:crVirgo}
An alternative production mechanism of $\gamma$ rays\xspace originating from the Virgo region may be due to CR interactions. $\gamma$ rays\xspace are mainly produced in IC interactions of relativistic electrons or via hadronic $pp$-collisions producing pions and $\gamma$ rays\xspace through $\pi^{0}\rightarrow2\gamma$ \citep{Brunetti:2012aa}. The dominant production mechanism of $\gamma$-ray\xspace\ from CRs in the ICM is still debated: either cosmic-rays are accelerated directly in structure formation shocks (including the effect of AGNs and supernovae) through diffusive shock acceleration or an aged population of cosmic-ray are reaccelerated in the turbulent plasma generated by e.g. merging clusters \citep[see, e.g.][for a review]{2014IJMPD..2330007B}.
Since there is no giant radio halo associated with the Virgo cluster and the central part of the cluster has properties similar to a cool-core cluster \citep{2011MNRAS.414.2101U}, we expect the $\gamma$ rays\xspace from a population of reaccelerated cosmic-rays \citep[see, e.g.][]{2013ApJ...762...69Z} to be too faint to be detectable by the \emph{Fermi}-LAT\ throughout its lifetime. However, there is a strong dependence on the uncertain turbulent profile. Indeed, \citet{Pinzke:2015aa} showed that for a flatter turbulent profile than what was previously assumed, the $\gamma$-ray\xspace emission could be in reach with Ferm-LAT in the coming years. To keep the CR analysis simple, we neglect these aforementioned models as well as other leptonic models \citep{2009JCAP...08..002K}. Instead, we focus on constraining the $\gamma$ rays\xspace produced in a pure hadronic scenario in that region. Specifically, we adopt a simple but realistic model for the predicted universality of the CR-spectra built up from diffusive shock acceleration in large-scale structure formation shocks \citep{2008MNRAS.385.1242P,2010MNRAS.409..449P}. Based on these considerations, in this section we derive constraints on the CR-induced $\gamma$-ray\xspace\ flux and related CR quantities from Virgo.
\subsection{Modeling and Results}
Following earlier works in \citet{Ackermann2013}, we consider two different (hadronic) models for the CR distribution, the simulation-based approach by \citet{Pinzke2011}, which predicts a $\gamma$-ray\xspace surface brightness which closely follows the X-ray emitting gas in the ICM, and a model in which the CRs are confined within the cluster virial radius but evenly distributed with no dependence on the ICM gas (flat model). The latter can thus be seen as a simplified proxy for CR-streaming models which can lead to more extended $\gamma$-ray\xspace brightness profiles \citep{2011A&A...527A..99E, 2013MNRAS.434.2209W, 2014MNRAS.440..663Z}. While the expected $\gamma$-ray\xspace morphology varies, we assume the spectrum to be approximated by the universal model as detailed in \citet{Pinzke2011} (the interested reader is referred to Figure~1 of that paper). Analogously with the results presented in Section~\ref{sec:DMsubstructure}, we construct a model which takes into account the merging state of the cluster by overlaying the spatial template inferred from X-ray profiles from Virgo-I with that of Virgo-II. We show the predicted flux maps in Fig.~\ref{fig:CRmorph}. Outside $r_{200}$ we take the predicted flux to be negligible.
\begin{figure*}
\begin{center}
\includegraphics[width=.9\textwidth]{f13}
\end{center}
\caption{Projected predicted, integrated CR-induced $\gamma$-ray\xspace\ flux (above $E=100\,\ensuremath{\mathrm{MeV}}\xspace$) for the models considered in this analysis (\emph{left:} simulation-based model following \citet{Pinzke2011}; \emph{right:} model in which the CRs follow a flat distribution) in units of $\mathrm{ph\,cm^{-2}\,s^{-1}\,sr^{-1}}$. Each model is a superposition of the individual CR-models derived for M87 and M49. For reference, we show the location of the excess as blue dashed contour. Note the different scales in both plots. \label{fig:CRmorph}}
\end{figure*}
In analogy with the results in the previous section, we use the profile likelihood method to derive 95\% upper limits on the CR-induced $\gamma$-ray\xspace\ flux. Our results
are shown in Table~\ref{tab:CRresults}.
\begin{deluxetable*}{ccccc}
\tablecaption{CR-models and derived limits}
\tablehead{\colhead{CR model} & \colhead{$F_{\gamma,\mathrm{pred}}(E>100\,\ensuremath{\mathrm{MeV}}\xspace)$} & \colhead{$F_{\gamma,95}(E>100\,\ensuremath{\mathrm{MeV}}\xspace)$} & \colhead{$\langle X_{\mathrm{CR}} \rangle$} & \colhead{\ensuremath{\zeta_{p,\mathrm{max}}}\xspace}\\
\colhead{} & \colhead{$(\times 10^{-9}\mathrm{ph\,cm^{-2}\,s^{-1}})$} & \colhead{$(\times 10^{-8}\mathrm{ph\,cm^{-2}\,s^{-1}})$} & \colhead{} & \colhead{}
}
\startdata
Simulation-based & 15.0 & 1.2 & 6\%&40\%\\
Flat CR\tablenotemark{a} &0.4 & 1.8 &\dots & \dots \\
\enddata
\tablecomments{\label{tab:CRresults}Shown are both predicted and observed integrated fluxes above 100~\ensuremath{\mathrm{MeV}}\xspace for our models for CR-induced $\gamma$ rays\xspace\ as discussed in the text. For the simulation-based model, the remaining columns denote the volume-averaged CR-to-thermal pressure ratio and the maximum acceleration efficiency for CR protons, respectively.\tablenotemark{b}}
\tablenotetext{a}{In order to provide a consistent description, we normalize each profile to the total CR number within $r_{200}$ for the simulation-based model.}
\tablenotetext{b}{As the observed flux for the flat CR-model is a factor $\sim45$ above the predictions, these limits cannot be used to constrain \ensuremath{\langle X_{\mathrm{CR}}\rangle}\xspace.}
\end{deluxetable*}
We exclude $\gamma$-ray\xspace integral fluxes above $1.2\times10^{-8}\mathrm{ph\,cm^{-2}\,s^{-1}}$ for the simulation-based CR model over the energy range 100~\ensuremath{\mathrm{MeV}}\xspace -- 100~\ensuremath{\mathrm{GeV}}\xspace, which is about a factor $\sim1.4$ stronger than previously published \citep{2010ApJ...717L..71A}. Using the flat model yields an integral flux limit of $1.8\times10^{-8}\mathrm{ph\,cm^{-2}\,s^{-1}}$ which is above the value that was published previously. This can however be explained by the fact that flat CR models are generically less constrained by current $\gamma$-ray\xspace\ data \citep{Ackermann2013,2014MNRAS.440..663Z}.
\subsection{Constraints on \ensuremath{\zeta_{p,\mathrm{max}}}\xspace and \ensuremath{\langle X_{\mathrm{CR}}\rangle}\xspace}
Two important quantities associated with CRs are the maximum efficiency with which CRs are accelerated in shocks, \ensuremath{\zeta_{p,\mathrm{max}}}\xspace, along with the volume-averaged CR-to-thermal pressure ratio, \ensuremath{\langle X_{\mathrm{CR}}\rangle}\xspace. Current limits exclude efficiencies above 21\% and values for $\ensuremath{\langle X_{\mathrm{CR}}\rangle}\xspace>1\%$ for purely hadronic models \citep{Ackermann2013,2014MNRAS.440..663Z}. As shown in \citet{Ackermann2013}, for the simulation-based CR model, we expect a linear relationship between the $\gamma$-ray\xspace flux (or the limit on the flux) and \ensuremath{\langle X_{\mathrm{CR}}\rangle}\xspace as well as \ensuremath{\zeta_{p,\mathrm{max}}}\xspace, respectively, with little variation across cluster masses and evolutionary stages \citep{Pinzke2011}.
As the additional point sources are not fully sufficient to model the entirety of the reported $\gamma$-ray\xspace\ excess, the resulting limits from CR physics are less constrained. We find $\ensuremath{\zeta_{p,\mathrm{max}}}\xspace\leq40\%$ and $\ensuremath{\langle X_{\mathrm{CR}}\rangle}\xspace\leq6\%$ within $r_{200}$ of the combined system, both of which ranges have been excluded in previous multi-sample studies \citep{Ackermann2013} (all limits have been derived at a 95\% confidence level).
\subsection{Systematic Uncertainties due to IEM\xspace modeling}
In order to assess the robustness of these results, we repeat the calculations in the previous section for our set of alternative diffuse models. We find that our derived constraints can be up to $\sim40\%$ better than those obtained with the standard IEM\xspace.
\subsection{Degeneracy of Results with M87}
In general, CR-induced models are substantially more centrally peaked than any of our previously considered DM-motivated models \citep[see, e.g.][for a study of various CR scenarios, in the Coma cluster]{2014MNRAS.440..663Z}. This implies the potential for degeneracy with M87 itself (now referring to the AGN and not to the sub-cluster). Detected with the \emph{Fermi}-LAT\ with only six months exposure, M87 \citep{2009ApJ...707...55A} is best modeled as a power law with $\Gamma=2.1$ which is harder than the tested CR-models (above $\sim1\,\ensuremath{\mathrm{GeV}}\xspace$ the CR-model by \citet{Pinzke2011} can be approximated as a power-law with $\Gamma=2.3$). When comparing the fit results of the spectral parameters of M87 (both index and normalization are left to vary freely in the fit), we find that these vary within the quoted uncertainty given in 2FGL when performing the likelihood fit including either CR-model discussed here. We also note that since the cluster is modeled as merging system rather than as a spherically symmetric object, this helps in breaking the degeneracy between M87 and any cluster-induced emission.
\section{Conclusion}\label{sec:conclusion}
We find no strong evidence for extended emission associated with the Virgo cluster center. Yet, using the standard IEM\xspace we find a statistically significant extended excess from a disk profile with radius 3\deg clearly offset from the cluster center. Our TS map reveals two well-separated maxima, both clearly offset from the two main sub-clusters associated with the giant ellipticals M87 and M49. This signature makes a DM origin unlikely. Also, as there is no indication of accelerated CRs, evidenced by either radio or X-ray emission, an astrophysical origin due to e.g. accelerated CRs in the ICM is questionable. We thus report upper limits on CR-scenarios and DM-induced $\gamma$ rays\xspace.
Similar to previous studies, we carry out a search for new point sources in order to account for the increased data volume with respect to the employed source catalog. We find six new candidates in accordance with similar studies by~\citet{Macias} and \citet{HanII}. These new candidates, however, have no reported counterparts in other wavebands. Five of them are contained in the 3FGL-catalog. We carry out an alternative IEM\xspace study which is essential for estimating systematic uncertainties associated with the search for $\gamma$-ray\xspace\ emission from very extended sources. In our case the inconsistency between the IEMs\xspace demonstrates that the Virgo region is an especially difficult section of the sky. The proximity to poorly understood Galactic foregrounds emitting $\gamma$ rays\xspace, like Loop~I, makes the search for extended emission from this region very challenging. Our study also reveals the challenges of searches for such low photon density sources even in at high Galactic latitudes.
Accounting for the complex dynamics of the cluster, we model its emission by co-adding the contributions from the major sub-clusters centered on M87 and M49, respectively. In particular for very extended models, as predicted if considering large amounts of DM halo substructure, the spatial morphology departs from spherical symmetry. Resulting limits for either DM- or CR-induced $\gamma$ rays\xspace are generally weaker than that of other targets, e.g. dwarf spheroidal for the case of DM-annihilation \citep{dwarf_paper} and from collective cluster studies \citep{Ackermann2013}. The DM limits from the Virgo analysis here, for instance, are about an order of magnitude above the thermal WIMP cross-section when assuming a realistic model for the sub-halo boost.
Finally we would like to stress that the main findings in this paper are expected to remain unchanged even if more data were to be included, as the uncertainties in the results are dominated by systematics associated with the IEM\xspace modeling. We emphasize that the improved source model used with the analysis roughly corresponds to the model presented in the current deepest $\gamma$-ray\xspace\ catalog, 3FGL. Also, while the predicted constraints on DM annihilation and CR processes improve by up to a factor of $\simeq1.4$ if all available data are considered (6 years instead of 3 years), targets other than the Virgo cluster may be better suited for analysis, e.g. the Coma cluster for CR processes and Fornax for DM prospects \citep[see, e.g.][for a discussion]{Pinzke2011,2012JCAP...07..017A}.\footnote{\edit{Note that recent work by \citet{Feyereisen:2015aa} suggests that a detection of DM-induced $\gamma$ rays\xspace from clusters is unlikely, especially in the presence of large substructure boosts, since the same signature constitutes an irreducible background to the isotropic $\gamma$-ray\xspace background \citep{Ackermann:2015ab}.}} While farther away, the predicted $\gamma$-ray\xspace emission from both clusters is expected to be within the detection reach of the LAT and their apparent extensions on the sky is significantly less than Virgo which helps to reduce the uncertainties associated with the foreground IEM\xspace modeling, thus allowing for a more robust analysis.
\section*{Acknowledgments}
AP is grateful to the Swedish Research Council for financial support.
The \textit{Fermi} LAT Collaboration acknowledges generous ongoing support from a number of agencies and institutes that have supported both the development and the operation of the LAT as well as scientific data analysis. These include the National Aeronautics and Space Administration and the Department of Energy in the United States, the Commissariat \`a l'Energie Atomique and the Centre National de la Recherche Scientifique / Institut National de Physique Nucl\'eaire et de Physique des Particules in France, the Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in Italy, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), High Energy Accelerator Research Organization (KEK) and Japan Aerospace Exploration Agency (JAXA) in Japan, and the K.~A.~Wallenberg Foundation, the Swedish Research Council and the Swedish National Space Board in Sweden.
Additional support for science analysis during the operations phase is gratefully acknowledged from the Istituto Nazionale di Astrofisica in Italy and the Centre National d'\'Etudes Spatiales in France.
Facilities: \facility{\emph{Fermi}}
|
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Good Friday to you! I am taking a little give-away break (don't worry...there are two more!) to bring you a little fall-inspired style board. It's chilly out and the tights and boots and close-toed shoes are in full rotation but it may not quite be full-on coat season just yet (my very favorite kind of weather). This is the time of year that it's all about the accessories! It's their time to shine before really keeping Jack Frost out becomes the main goal and we get all focused on this "function over fashion" nonsense. Pshaw! It is this time of year that I head for my most happy, colorful and textured knits.
Some of you may know that I have a penchant for anything patterned, and also a little bit of an accessory problem addiction. So when the two meet, I start hearing circus music in my head. There are so many wonderful options out there that I think this might be a 4 part series on scarves alone...but I thought I would start with my true north: stripes. And having just spend a small fortune on fun striped socks to usher in my boots and clogs (oh yes, clogs.) for fall, I will include some of my favorite finds there too.What is better than a perky little peek of pattern under a pant leg or sneaking out from above your boot line? I have thought about it, and not much comes to mind at the moment!
I do have to give a shout out to SmartWool here for making some of the very best and most adorable socks around! They are warm, but not too warm and they last forever! Nice job guys! Also! If you are in the Boston/Cambridge area and love socks, head to Cambridge Clogs for the best selection I have seen. Hands down.
Hope this helps you get a little Cozy! Let me know your faves!
We also have a winner from the Sweetheart Sponsor Love-bomb give-away!! Thanks for showing them such love! I hope you found some clever, beautiful and interesting things on the way!
Comment #41 from Heather G. who found delight in Amy Peter's Studio's Starlight Rings!
Congrats Heather! Emily will be in touch to get you your CGD Holiday loot!!
Side note on Holiday Products: Many of you are wondering when/how you can get a hold of some of the CGD holiday product. I know! It's time! Our holiday cards and boxed sets will be on the website in short order. I will make an announcement when they go live. As for the rest of the product, we have SUCH limited quantities and have no access to more on some of the items, that I will be posting them here on the blog so you guys get first dibs. We will do them first-come first-serve style through our office. It is not an ideal situation, but I don't think anyone knew how dang cute all this stuff was gonna be and how fast it would sell through (except for me, of course!). I will be starting that process next week. Stay tuned and THANK YOU for loving it as much as I do!
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"redpajama_set_name": "RedPajamaC4"
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Nonprofit mapped political path for state Sen. Sam Blakeslee
By Evan Halper and Patrick McGreevy, Los Angeles Times
Aug. 9, 2012 12 AM PT
SACRAMENTO — Sam Blakeslee said he was putting personal ambition aside. The maverick GOP lawmaker from San Luis Obispo announced he would leave politics to run a nonprofit bankrolled by a big donor.
His only aim at the California Reform Institute would be to promote common-sense solutions to big policy problems vexing Sacramento.
An early "Strategic Plan" for the nonprofit reviewed by The Times, however, lays out a different goal: "Devise and execute a plan that makes Blakeslee a politically viable candidate for Republican statewide office in 2014."
The July 2011 document maps out how the institute's policy proposals would be a vehicle to "create messenger credibility" needed to make "a Blakeslee gubernatorial bid feasible." It suggests the state senator, whose standing as a moderate makes him a rare GOP politician with statewide appeal, spend his final months in the Legislature grabbing the media spotlight with bills intended to provoke Democrats.
The plan was never meant for public consumption, and now it could be a legal liability. The IRS has tended to consider boosting the political prospects of a candidate an unacceptable mission for a nonprofit, which taxpayers subsidize.
"Why did they ever write this down, and where was their lawyer when they did?" said Frances Hill, a professor at the University of Miami School of Law and a national authority on the use of certain types of nonprofits, such as Blakeslee's.
The senator's blueprint, she said, suggests the California Reform Institute was designed as "a political campaign with a long view."
Blakeslee, a former Assembly minority leader who has crusaded to ban gifts to politicians and has been critical of lawmaker junkets organized by nonprofits, says the plan reflects early "blue-sky thinking" and was scrapped. He repeated in an interview last month that he will be done with politics when his Senate term ends this year.
"I am not planning on running for either statewide office or any other office," he said.
He has collected $23,000 this year in a campaign account for the 2014 treasurer's race.
Representatives of the IRS would not discuss Blakeslee's institute. But in June, the agency released a ruling that stripped tax-exempt status from another think tank, created by a different politician, whose name was omitted from the document, along with other details. The IRS concluded that the organization's activities, while not explicitly campaign oriented, dovetailed too closely with its founder's political agenda.
The California Reform Institute's nonprofit category, 501(c)4, allows for limited political activity if the entity promotes "social welfare," according to IRS regulations. Many politicians have used such organizations to promote policy agendas, raise money for their travel and fund athletic tournaments and other events popular with voters.
Crossroads GPS, co-founded by former White House advisor Karl Rove, is one; it bankrolls television ads attacking Democrats on policy issues.
A 501(c)4 "must be operated for a broad community benefit and not a private purpose," said Marc Owens, aWashington, D.C., attorney and former director of the exempt organizations division of the IRS.
He said the California Reform Institute was wise to toss out its initial strategic plan — but that doesn't necessarily put it in the clear. "The IRS will look at the reality and not just statements on whether or not this organization is being used for private benefit," Owens said.
Blakeslee's nonprofit was jump-started with $750,000 from Charles Munger Jr., a wealthy California GOP activist who has been working to push the party in a more moderate direction. Blakeslee's chief of staff was named as the institute's chief executive, according to papers the organization filed with the state in January, and another staffer was identified as its secretary.
The designated chief executive, Christine Robertson, said recently that she has stepped down for now. But a letter from Munger's attorney to the institute's officials' attorney said it is "anticipated" that she will become the compensated executive director when Blakeslee's term ends.
Blakeslee said he would take no salary. Munger says his own goal is strictly about policy.
"I did not discuss with Sen. Blakeslee the possibility of this being a vehicle that would propel him to statewide office," Munger said. "Whatever other ideas were developed in Blakeslee's shop, that's not what I signed on for and it's not what the institute is set up for."
The reach for higher office was taken out of the institute's bylaws by the time Munger committed his money and before Blakeslee went public with his plans in May. On paper, the California Reform Institute now exists to serve the people of California and not one particular politician.
The early strategy that Blakeslee and his associates drafted included plans for him to write pieces of legislation "knowing they will ultimately be thwarted" by Democrats and their allies. Such a move "provides important historical narrative that authenticates passion for major reforms before becoming a candidate for governor," the document said.
The legislation would be submitted with "heavy emphasis on public relations and press strategy," said the plan.
It also contained a timeline of other actions, including writing a book, enlisting lobbyists to help raise money and tapping into Munger's network of well-to-do donors. It ended in June of next year with: "Announce Candidacy."
evan.halper@latimes.com
Evan Halper
Evan Halper is a former staff writer who wrote about a broad range of policy issues out of Washington, D.C., with particular emphasis on how Washington regulates, agitates and very often miscalculates in its dealings with California. Before heading east, he was the Los Angeles Times bureau chief in Sacramento, where he spent a decade untangling California's epic budget mess and political dysfunction.
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\section*{Abstract}
Market dynamic is quantified in terms of the entropy $S(\tau,n)$ of the clusters formed by the intersections between the series of the prices $p_t$ and the moving average $\widetilde{p}_{t,n}$. The entropy $S(\tau,n)$ is defined according to Shannon as $\sum P(\tau,n)\log P(\tau,n)$, with $P(\tau,n)$ the probability for the cluster to occur with duration $\tau$.
\par
The investigation is performed on high-frequency data of the Nasdaq Composite, Dow Jones Industrial Avg and Standard \& Poor 500 indexes downloaded from the Bloomberg terminal. The cluster entropy $S(\tau,n)$ is analysed in raw and sampled data over a broad range of temporal horizons $M$ varying from one to twelve months over the year 2018. The cluster entropy $S(\tau,n)$ is integrated over the cluster duration $\tau$ to yield the \emph{Market Dynamic Index} $I(M,n)$, a synthetic figure of price dynamics. A systematic dependence of the cluster entropy $S(\tau,n)$ and the \emph{Market Dynamic Index} $I(M,n)$ on the temporal horizon $M$ is evidenced.
\par
Finally, the \emph{Market Horizon Dependence}, defined as $H(M,n)=I(M,n)-I(1,n)$, is compared with the horizon dependence of the pricing kernel with different representative agents obtained via a Kullback-Leibler entropy approach.
The \emph{Market Horizon Dependence} $H(M,n)$ of the three assets is compared against the values obtained by implementing the cluster entropy $S(\tau,n)$ approach on artificially generated series (Fractional Brownian Motion).
\section{Introduction}
Entropy, as a tool to quantify heterogeneity and dynamics in complex systems, has found a number of applications in different contexts \cite{crutchfield2012between,bandt2002permutation,grassberger1983characterization,marcon2014generalization,karpiarz2014international,rubido2018entropy}.
In economics and finance, the entropy ability to quantify heterogeneity and disentangle ordered and disordered patterns in data relevant to complex systems, has been adopted for portfolio selection to outperform traditional methods based on Markowitz covariance and Sharpe single-index models \cite{philippatos1972entropy,fernholz2002stochastic,ou2005theory,xu2011portfolio,usta2011mean,zhou2013portfolio,zhang2012possibilistic, bera2008optimal,demiguel2009optimal,rodder2010entropy,
chandrinos2018construction,gospodinov2017general,chen2017study,ormos2014entropy,lahmiri2018informational,lahmiri2018randomness,lahmiri2018long,lahmiri2017disturbances,lahmiri2017nonlinear}.
Entropy ability to quantify dynamics other than heterogeneity has gained interest to the aim of implementing entropy-derived tools to shed light on fundamental aspects of asset pricing dynamics beyond portfolio optimization \cite{Hansen1991implications,Hansen2014Nobel,Hansen2019Macroeconommic,backus2014sources,ghosh2017what}.
\par
Macroeconomic shocks are becoming increasingly important due to the growing connectedness of the assets in a global economy. The propagation of these shocks, which intrinsically are not diversifiable, cannot be averaged out by diversifying investments and, thus, even the best selection of portfolio assets might fail in keeping investors safe.
Asset pricing models aim at providing estimates of endogenous risk by quantifying market evolution in terms of a stochastic function: the {\em pricing kernel} $m_{t}$. Equilibrium prices $p_t$ of traded securities can be represented as the conditional expectation of the discounted future payoff $z_t$:
\begin{equation}
p_t= E\left [\frac{m_{t+1}}{m_{t}}z_{t+1}\right ] \hspace{15pt},
\end{equation}
where $m_{t+1}/m_{t}$ is known as the {\em stochastic discount factor}. The pricing kernel $m_{t}$ is factorizable into a function of the consumption growth $\mu_{t+1}$ times $\psi_{t}$ (a model specific term):
\begin{equation}
m_t= \mu_{t+1}\psi_{t} \hspace{15pt}.
\end{equation}
The standard consumption-based asset pricing model identifies the pricing kernel as a simple parametric function of the consumption growth ${C_t}$. In this framework, with time-separable power utility representative agent models, the function $\mu_{t+1}$ is simply proportional to $\Delta C_t = \log ({C_t}/{C_{t-1}})$. More sophisticated agent behaviours have been suggested to explain puzzling phenomena such as amplitude and cross-sectional dispersion of returns among different categories of financial assets, equity premia and risk-free rates.
\par
Pricing kernel dispersion and dynamics with different representative agents are modelled by using the Kullback-Leibler entropy in \cite{backus2014sources} and extending the findings of \cite{Hansen1991implications}. The work \cite{Hansen1991implications} was addressed to quantify standard deviation and volatility to define the pricing kernels bounds. A lower bound was provided for the
volatility of the permanent component of asset pricing kernels, showing that stochastic discount factors need to be very volatile to be consistent with high Sharpe ratios \cite{Hansen1991implications}.
A relative entropy minimization approach, based on the Kullback-Leibler divergence, is put forward in \cite{ghosh2017what} to extract the model dependent term $\mu_{t+1}$ and quantify the minimum amount of extra information to be embedded in the standard pricing kernel models for reproducing asset return correctly.
The Kullback-Leibler divergence between the probability distribution functions of the components $\mu_{t+1}$ and $\psi_{t}$ has been used as criterion to estimate the deviation of $m_{t+1}$ with respect to the simple consumption flow growth model. It was argued that the Kullback-Leibler divergence criterion is equivalent to maximize the entropy of the fundamental pricing kernel component \cite{ghosh2017what}.
\par
An information theoretical tool has been recently developed which yields the weights of the efficient portfolio by using the \emph{cluster entropy} estimated via the detrending moving average algorithm proposed in \cite{carbone2013information,carbone2007scaling,carbone2004analysis}. Interestingly, the works \cite{ponta2017detrending,ponta2018information} show that the cluster entropy of the volatility takes values depending on each market, as opposed to the entropy of the prices, which was shown to be approximately invariant across the markets. The \emph{Market Heterogeneity Index}, defined as the integral of the cluster entropy, provides a cumulative figure allowing a straightforward comparison with the portfolio weights obtained by the Sharpe ratio approach. The main advantage of the cluster entropy approach is not to require a specific distribution of returns like a symmetric Gaussian distribution. Such a distribution is quite elusive in real-world financial assets and thus hindering, in principle, the application of Markowitz-based portfolio models.
\par
In this work, we implement the cluster entropy approach for quantifying the intrinsic dynamics of prices and capturing the endogenous sources of risk over different temporal horizons.
The present work builds upon and extends the study \cite{ponta2018information} that was limited to extract the portfolio weights from the cluster entropy of the prices and volatility of the financial series over a constant time horizon (about 6 years from 1998 to 2004). Under the condition of constant temporal horizon, the cluster entropy of the prices has been found to be almost invariant across the markets in \cite{ponta2018information}.
\par
Here the focus is on gaining insights in the intrinsic dynamic ruling price evolution. Hence, the cluster entropy analysis is performed over multiple horizons. The horizon dependence was not studied in \cite{ponta2018information} that reported the quantitative comparison of cluster entropy observed in several markets over the same horizon (i.e. same time interval of six years 1998-2004).
\par
The ability of the cluster entropy approach to quantify the intrinsic dynamic of the prices is proved by analysing several assets. For the sake of simplicity in this work we report the results obtained on the three markets described in Table \ref{tab:data}.
\par
Cluster entropy has been analysed for prices of market indices (tick-by-tick prices from Jan $1^{st}$ to Dec. $31^{st}$ 2018) NASDAQ, DJIA and S$\&$P500 with length $N=6982017$, $N=5749145$ and $N=6142443$ respectively. Data have been downloaded from the terminal www.bloomberg.com/professional. The three financial markets have been selected based on homogeneity and similarity criteria. The three markets are traded in the same country and with same currency. Furthermore the assets are characterised by a comparable number of transactions over time. The similarity criteria rule out differences in the dynamics that might be due to external causes. Another condition ensuring that the observed behavior is genuinely related to the intrinsic price dynamics rather than exogenous factors is to keep limited the maximum extension of the temporal horizon. In the current study the max temporal extension is one year (12 Months) and the analysis has been performed on multiples of monthly subsets from one to twelve months.
\par
It is worth noting that though in literature many study have been performed to understand the asset pricing dynamics using low-frequency data, for example, to estimate the low-frequency components of returns, in this paper the analysis is performed on high-frequency data to investigate and capture the endogenous sources of risk. The data ranges over one single year. The huge data sets allow one to apply the cluster entropy algorithm over monthly segmented series with average lengths of the order of $\sim 500000$.
\par
A systematic dependence of the cluster entropy of the asset prices over varying temporal horizons has been observed, that could be related to the macroeconomic fundamental properties and exogenous dynamics rather than to simple variations across different markets.
\par
The manuscript is organized as follows. The main relationships relevant to understanding and implementing the cluster entropy approach are shortly recalled in Section \ref{Method}. The analysed data sets (financial assets and artificially generated series) are described in Section \ref{Data}. In Section \ref{Results} the cluster entropy and the Market Dynamic Index of the prices series as a function of the temporal horizon $M$ are reported together with a comparison against the Kullback-Leibler entropy results obtained by simulating the pricing kernel with different representative agent models.
The cluster entropy and the market dynamic index estimated for Fractional Brownian Motion (FBM) sequences are reported and discussed. The artificially generated FBM data are taken as reference to validate the accuracy of the deviations observed in the real-world assets markets and validate the findings via a standard T-paired test.
\section{Methods}
\label{Method}
In this section, we briefly recall the main definitions and equations used which are the core computational ingredients of the algorithm. \par
The cluster entropy is obtained by taking the intersection of the asset prices $p_t$ and its moving average $\tilde{p}_{t,n}$ for different moving average window $n$ \cite{ponta2017detrending,ponta2018information,carbone2013information,carbone2007scaling,carbone2004analysis}. For each window $n$, the subsets $\{p_t: t=s,....,s-n \}$ between two consecutive intersections are considered. The subsets are named \emph{clusters}. The clusters are exactly defined as the portions of the series between death/golden crosses according to the technical trading rules. Therefore, the information content has a straightforward connection with the trader's perspective on the price and volatility series. Then, the clusters are ranked according to their characteristic size, the duration $\tau$. The probability distribution function $P(\tau,n)$ of the cluster duration is obtained.
The present approach directly yields either power-law or exponential distributed cluster distributions, thus enabling us to separate the sets of inherently correlated/uncorrelated blocks along the sequence.
The continuously compounded return is defined by:
\begin{equation}
\label{returnlin}
r_t = p_t - p_{t-h} \hspace{10pt} ,
\end{equation}
where $p_t$ is the price at the time $t$, with $ 0<h<t<N $ and $N$ the maximum length of the time series.
Alternatively, one can consider the log-return defined as:
\begin{equation}
\label{returnlog}
r_t = \log p_t - \log p_{t-h} \hspace{10pt}.
\end{equation}
\par
The approach adopted in this work builds upon the idea of Claude Shannon to quantify the 'expected' information contained in a message extracted from a sequence $\{x_t \}$ \cite{shannon1948mathematical} by using the entropy functional:
\begin{equation}
S[P(x_t)] = \int_X p(x_t) \log p(x_t) dx_t \hspace{5pt},
\label{Shannon_int}
\end{equation}
with $P$ a probability distribution function associated with the sequence $\{x_t \}$.
For discrete sets, Eq. (\ref{Shannon_int}) reduces to:
\begin{equation}
S[P(x_t)] = \sum_X p(x_t) \log p(x_t) \hspace{5pt}.
\label{Shannon}
\end{equation}
Consider the time series $\{x_t \}$ of length $N$ and the moving average $\{\widetilde{x}_{t,n}\}$ of length $N-n$ with $n$ the moving average window.
The function $\{\widetilde{x}_{t,n}\}$ generates, for each $n$, a partition $\{\cal{C}\}$ of non-overlapping clusters between two consecutive intersections of $\{x_t \}$ and
$\{\widetilde{x}_{t,n}\}$. Each cluster $j$ has duration:
\begin{equation}
\label{l} \tau_j\equiv \|t_{j}-t_{j-1}\|
\end{equation}
\noindent
where the instances $t_{j-1}$ and $t_j$ refer to two subsequent intersections.
The probability distribution function $P(\tau,n)$ can be obtained by ranking the number of clusters ${\mathcal N}(\tau_1,n),{\mathcal N}(\tau_2,n), ..., {\mathcal N}(\tau_j,n)$ according to their length $\tau_1, \tau_2,..., \tau_j$ for each $n$. A stationary sequence of clusters $\cal{C}$ is generated with probability distribution function varying as \cite{carbone2013information}:
\begin{equation}
\label{Pl} P(\tau,n)\sim\tau^{-\alpha} {\mathcal F}\left({\tau},{n}\right) \hspace{5pt},
\end{equation}
with the factor ${\mathcal F}\left({\tau},{n}\right)$ taking the form $ \exp({-\tau}/{n})$, to account for the finite size effects when $\tau\gg n$, resulting in the drop-off of the power-law and the onset of the exponential decay.
The cluster entropy writes (the details of the derivation can be found in \cite{carbone2013information,carbone2004analysis}):
\begin{equation}
S[P(\tau_j,n)] = \sum_j P(\tau_j,n)\log P(\tau_j,n) \hspace{5pt},
\label{Shannon}
\end{equation}
that by using Eq.~(\ref{Pl}) simplifies to:
\begin{equation}
\label{lentropy2}
S(\tau,n)=S_0+\log\tau^{\alpha}+{\tau\over n}\hspace{5pt},
\end{equation}
where $S_0$ is a constant, $\log\tau^{\alpha}$ and $\tau/ n$ are related respectively to the terms $\tau^{-{\alpha}}$ and ${\mathcal F}(\tau,n)$.
The minimum value of the entropy is obtained for the fully ordered (deterministic) set of clusters with duration $\tau=1$. Eq.~(\ref{lentropy2}) in the limit $n\sim\tau\rightarrow1$ and $S_0\rightarrow-1$ reduces to $S(\tau,n)\rightarrow0$. Conversely, the maximum value of the entropy $S(\tau,n)=\log N^{\alpha}$ is obtained when $n\sim\tau\rightarrow N$ (with $N$ the maximum length of the sequence). This condition corresponds to the maximum randomness (minimum information) carried by the sequence, when a single longest cluster is obtained coinciding with the whole series.
\par
For a fractional Brownian motion, the exponent $\alpha$ is equal to the fractal dimension $D=2-H$ with $H$ the Hurst exponent of the time series. The term $\log\tau^{\alpha}$ can be thus interpreted as a generalized form of the Boltzmann entropy $S=\log\Omega$, where $\Omega = \tau^D$ corresponds to the fractional volume occupied by the fractional random walker.
The term $\tau/n$ represents an excess entropy (excess noise) added to the intrinsic entropy term $\log\tau^D$ by the partition process. It depends on $n$ and is related to the finite size effect discussed above.
\par
We stress the difference between the time series partitions obtained either by using equal size boxes or moving average clusters.
For equal size boxes, the excess noise term ${\tau / n}$ vanishes (as it becomes a constant that can be included in the constant term) thus the entropy reduces to the logarithmic term as found in Ref.~\cite{grassberger1983characterization}, which corresponds to the intrinsic entropy of an ideal fractional random walk. When a moving average partition is used, an excess entropy term ${\tau / n}$ emerges accounting for the additional heterogeneity introduced by the random partitioning process operated by the moving average intersections.
\par
To univocally quantify market properties through the entropy Eq.~(\ref{lentropy2}), a cumulative information measure has been defined as follows:
\begin{equation}
I(n)=\int_0^{\tau_{max}} S (\tau,n)d\tau \hspace*{5 pt},
\label{Integral}
\end{equation}
which, for discrete sets, reduces to:
\begin{equation}
I(n) = \sum_{\tau} S (\tau,n) \hspace{5pt}.
\label{Integrald}
\end{equation}
\par
The function $I(n)$ has been used to quantify cross-market heterogeneity in \cite{ponta2018information}. The cluster entropy of the volatility $v_T$ was integrated over the cluster duration $\tau$ to the purpose of obtaining the weights of the optimal portfolio.
\par
In this work, the function $I(n)$ will be used to quantify the intrinsic market dynamic. The cluster entropy of the prices will be integrated over the cluster duration $\tau$ to the purpose of obtaining the horizon dependence.
\par
As a concluding remark to this section, it is worth mentioning the relation between the cluster entropy approach adopted in this work, the multiscale entropy (MSE) and its variants \cite{costa2002multiscale,niu2015quantifying,humeau2015multiscale}.
The multiscale entropy provides insights into the complexity of fluctuations over a range of time scales and thus extends the standard one-sample entropy.
The computational implementation of multiscale entropy implies a coarse graining of the time series at increasingly time resolutions. Coarse graining the data basically means averaging different numbers of consecutive points to create different scales or resolutions of the signal. In the cluster entropy approach proposed here, the coarse graining of the signal is performed through the moving average, i.e. a time dependent averaging.
The multiscale entropy analysis aims at quantifying the interdependence between entropy and scale, achieved by evaluating sample entropy of univariate time series coarse grained at multiple temporal scales. This facilitates the assessment of the dynamical complexity of the system whose behavior is reflected by the time series data.
\section{Data}
\label{Data}
Prices of market indices traded in the US, namely NASDAQ, DJIA and S$\&$P500 are investigated.
Data sets have been downloaded from the terminal www.bloomberg.com/professional.
For each index, the data set includes tick-by-tick prices $p_t$ from January to December 2018.
Details (Ticker; Extended name; Country; Currency; Members; Length) as provided by Bloomberg for the three assets are reported in Table \ref{tab:data}. The length of each index is referred to the year 2018 (last column).
Different temporal horizons have been considered as monthly integer multiples of one-month period $M$ ranging from $M=1$ up to $M=12$. The individual lengths of the subsequences referred to the twelve time periods are reported for each index in Table \ref{tab:sampleddata}.
\par
To the purpose of performing cluster entropy analysis over sequences with constant lengths, the raw data are sampled, thus yielding data series with equal length.
The sampling frequency is defined for each series by dividing the length of the series corresponding to the longest horizon by the minimum, rounding the ratio to the nearest whole, that is used to sample the raw data.
\par
Consider for example the S$\&$P500 market ($3^{rd}$ column in Table \ref{tab:sampleddata}). The minimum value of the length is that at $M=1$ (January with $N=516635$) and the maximum value is longest horizon of interest (for example $N=5180006$ for horizon $M=10$ equal to ten months from January to October). As the sampling frequency is different for each series we consider the minimum value to perform the analysis with the same length.
\par
Furthermore for the sake of validating the obtained results, a set of computational tests have been performed on artificially generated series of different lengths $N$. The artificial series have been generated by means of the FRACLAB tool available at: https://project.inria.fr/fraclab/. The artificial series have been generated with lengths $N$ corresponding to those of the financial markets under investigation (Table \ref{tab:sampleddata}). Further details and results are reported in the following Sections.
\section{Results}
\label{Results}
\par
Probability distribution $P(\tau,n)$ and entropy $S(\tau,n)$ have been calculated for a large set of prices series by means of the procedure summarized in Section \ref{Method}. The series of the NASDAQ, DJIA and S$\&$P500 indexes described in Section \ref{Data} have been used for the investigation.
\par
Fig.~\ref{Fig:entropyrawpriceM1} shows the cluster entropy $S(\tau,n)$ calculated by using raw data prices. In particular, the plots refer to one month of data ($M=1$). The series lengths are $N=586866$, $N=516644$ and $N=516635$ respectively for NASDAQ, DJIA and S$\&$P500 as given in Table \ref{tab:sampleddata}.
\par
Fig.~\ref{Fig:entropyrawpriceM12} shows the cluster entropy $S(\tau,n)$ calculated by using raw data prices, as in Fig.~\ref{Fig:entropyrawpriceM1}, but here the series refer to a horizon of twelve months ($M=12$). The series lengths are $N= 6982017$, $N=5749145$ and $N=6142443$ respectively for NASDAQ, DJIA and S$\&$P500 as one can find in the last row of Table \ref{tab:sampleddata}.
\par
Fig.~\ref{Fig:entropysampledpriceM1} shows the cluster entropy $S(\tau,n)$ calculated by using the prices series of the sampled data. The plots refer to the first month of data ($M=1$). All the series have same length $N=492035$.
\par
Fig.~\ref{Fig:entropysampledpriceM12} shows the cluster entropy $S(\tau,n)$ calculated by using the prices series of the sampled data. The plots refer to twelve months ($M=12$). All the series have same length $N=492035$.
\par
Different curves in each figure correspond to moving average values varying from $n=30\hspace{2pt}\mathrm{s}$, $n=50\hspace{2pt}\mathrm{s}$, $n=100\hspace{2pt}\mathrm{s}$, $n=150\hspace{2pt}\mathrm{s}$, $n=200\hspace{2pt}\mathrm{s}$ $\ldots$ up to $n=1500\hspace{2pt}\mathrm{s}$ (with step $100\mathrm{s}$).
\par
One can note that the entropy curves exhibit a behaviour consistent with Eq.~(\ref{lentropy2}). At small values of the cluster duration $\tau \leq n$, entropy behaves as a logarithmic function. At large values of the cluster duration $\tau \geq n$ the curves increase linearly with the term ${\tau /n}$ dominating.
${S}(\tau,n)$ is $n$-invariant for small values of $\tau$, while its slope decreases as $1/n$ at larger $\tau$, as expected according to Eq.~(\ref{lentropy2}), meaning that clusters with duration $\tau > n $
are not power-law correlated, due to the finite-size
effects introduced by the partition with window $n$. Hence, they are characterized
by a value of the entropy exceeding the curve $\log \tau^D$, which corresponds to power-law correlated clusters. It is worthy to remark that clusters with same duration $\tau$
can be generated by different values of the moving average window $n$.
At a constant value of $\tau$, larger entropy values are obtained as $n$ increases.
\par
The entropy ${S}(\tau,n)$ of the NASDAQ, DJIA and S$\&$P500 prices (shown in Fig.~\ref{Fig:entropyrawpriceM1}, Fig.~\ref{Fig:entropyrawpriceM12}, Fig.~\ref{Fig:entropysampledpriceM1} and Fig.~\ref{Fig:entropysampledpriceM12}) is representative of a quite general behaviour observed in several markets analysed by using the proposed cluster entropy approach.
\bigskip
\par
In the following, we will discuss how to quantify the horizon dependence of the asset prices by using the cluster entropy function $S (\tau,n)$ estimated over different periods $M$. To this purpose, we use the \emph{cumulative information measure} function defined in Eq.~(\ref{Integral}).
\par
The quantity $I(M,n)$ is calculated by using the values of the entropy $S (\tau,n)$ of the asset prices $p_t$ estimated over several periods $M$, ranging from one to twelve months, by using raw and sampled data.
The first period ($M=1$) of the price sequences is taken in correspondence of January 2018 for all the assets. Multiple period sequences have been built by considering $M=2$ (January and February 2018) and, so on, up to $M=12$ (one year from January to December 2018). Details concerning lengths of the series corresponding to the temporal horizons $M$ are reported in Table \ref{tab:sampleddata}.
\par
The \emph{cumulative information measure} $I(M,n)$ has been plotted in Fig.~\ref{Fig:integral} for the prices of the NASDAQ, DJIA and S$\&$P500. One can observe a dependence of the function $I(M,n)$ at different $M$ horizons.
\par
$I(M,n)$ is the same for all $M$ implying that the horizon dependence $H(M,n)$ is negligible at small scales (small $n$/ small $\tau$ values). Conversely, at large $n$ values, i.e. with a broad range of cluster lengths $\tau$ spanning more than one decades of values in the power law distribution, a horizon dependence $H(M,n)$ varying with $M$ is found.
\par
For identically distributed sequences of clusters, $I(M,n)$ does not change with $M$ regardless of the value of $n$. This, has been shown in Fig. \ref{Fig:entropysampledpriceM1612Artif} where the cluster entropy $S (\tau,n)$ of artificially generated series (fractional random walks) are shown. One can note that the curves are practically unchanged at varying horizons $M$ and cluster duration $\tau$.
The departure from the {\em iid} case can be taken as a measure of price dynamics.
\par
Furthermore, by comparing the figures corresponding to the different assets a dependence of the function $I(n)$ is observed. In the case of the NASDAQ the variation seems larger than for the S$\&$P500, and even larger than for the DJIA.
\section{Discussion and Conclusions}
\label{Discussion}
Next, the main results of the analysis of the \emph{cluster entropy} $S (\tau,n)$ and the \emph{cumulative information measure} $I(M,n)$ are compared with the results obtained by using information theoretical approaches by other authors.
\par
To build a cluster entropy index of horizon dependence, i.e. a synthetic numerical parameter with the ability to provide an estimate of the horizon dependence, we consider the entropy integral $I(n)$ defined by Eq.~(\ref{Integrald}) at one-period ($M=1$) and at multiples of one period $M$ respectively defined as $I(1,n)$ and $I(M,n)$. The quantity $I(M,n)$, defined above on the basis of Eq.~(\ref{Integrald}) is called \emph{Market Dynamic Index}.
\par
To the purpose of comparing our results with those of paper \cite{backus2014sources}, the horizon dependence $H(M,n)$ is calculated as:
\begin{equation}
\label{horizon}
H(M,n) = I(M,n)-I(1,n) \hspace{5pt}.
\end{equation}
Values of \emph{Market Dynamic Index} $I(M,n)$ and horizon dependence $H(M,n)$ calculated by using the NASDAQ, DJIA and S$\&$P500 data are reported in Table \ref{tab:horizonsNASDAQ}. The quantity $I(1,n)=I(1)$ is a reference value of the one-period entropy (lower bound). It is taken as $I(1)=0.0049$, $I(1)=0.0214$ and $I(1)=0.0197$ respectively for power utility, recursive utility and difference habit agent models of the consumption growth following \cite{backus2014sources}. The value $I(12,n)$ has been obtained from the curves in Fig.~\ref{Fig:integral} for the prices of the NASDAQ, DJIA and S$\&$P500. $H(12,n)$ is the difference between $I(12,n)$ and $I(1,n)$ on account of Eq.~(\ref{horizon}).
\par
Next, the values of the horizon dependence obtained by using the cluster entropy will be checked against those obtained by using different representative agent models for the definition of the pricing kernel in \cite{backus2014sources}.
The pricing kernel dynamics has been quantified by a measure of entropy dependence on the investment horizon for popular asset pricing models. The pricing kernel accounts for the stochastic dynamic evolution of asset returns, which in their turn contain information about the pricing kernel. The analysis is based on the Kulback-Leibner divergence (also known as relative entropy) of the true probability distribution of the prices with respect to the risk-adjusted probability. On account of those results, it was argued that a realistic asset pricing model should have substantial one-period entropy and modest horizon dependence to justify equity mean excess returns and bond yields at once.
\par
The Kullback-Leibler (KL) divergence of the continuous probability measure $p(x)$ with respect
to some probability measure $p^*(x_t)$, writes:
\begin{equation}
\label{KL1}
{KL}(P||P^*) = \int_X p(x_t) \log \left( \frac{ p(x_t)}{p^*(x_t)}\right) dx_t
\end{equation}
Eq.~(\ref{KL1}) can be interpreted as the expectation of the function $\log { p(x_t)}/{p^*(x_t)}$ with respect to the probability $p(x_t)$:
\begin{equation}
\label{KL2}
{KL}(P||P^*)= E \left[\log \left( \frac{ p(x_t)}{p^*(x_t)}\right)\right]
\end{equation}
It can be easily shown that the relative entropy Eq.~(\ref{KL1}) reduces to Eq.~(\ref{Shannon_int}) for constant probability $p^*(x_t)$.
\par
Investigation of asset price dispersion and dynamics has been put forward by using a variant of the Kullback-Leibler (KL) divergence of the pricing kernels $m_{t,t+n}$ expressed in terms of the ratio between the true and risk-adjusted distribution \cite{backus2014sources}. In this work, different representative agent models have been considered to quantify the {\em Market Horizon Dependence} $H(M)$ :
\begin{equation}
H(M)= I(M) -I(1)
\end{equation}
with the quantity $I(M)$ defined as:
\begin{equation}
I(M)= \frac{EL_t(m_{t,t+M})}{M} \hspace{10pt}.
\end{equation}
where ${EL_t(m_{t,t+M})}$ is defined as the average of the relative entropy of the pricing kernel, and $I(1)$ is calculated at the month $M=1$. A summary of the horizon dependence obtained by estimating the Kullback-Leibler (KL) entropy with pricing kernels generated by different representative agent models according to the approach of \cite{backus2014sources} is reported in Table \ref{tab:hdconstant}.
\par
To further validate the behaviour observed in real-world financial markets, simulations have been performed on artificial data generated by means of the FRACLAB tool available at: https://project.inria.fr/fraclab/. The FRACLAB tool generates Fractional Brownian Motion series with assigned Hurst exponent $H$. The Hurst exponent corresponding to financial prices is generally assumed to be $H\sim 0.5$. In Fig.~\ref{Fig:entropysampledpriceM1612Artif}, the cluster entropy curve is shown for a FBM series with $H=0.5$ and different lengths $N$. For the curves shown in Fig.~\ref{Fig:entropysampledpriceM1612Artif} the artificial series has been generated with a total length equal to the one of the NASDAQ index ($N=6982017$). Then the artificial series has been divided in twelve consecutive segments with the same lengths of the NASDAQ sub-sequences (values of first column of Table \ref{tab:sampleddata}). Figures refer respectively to the first segment ($M=1$), the first sixth segments ($M=6$) and twelve segments ($M=12$).
\par
To the purpose of fully appreciating the different behaviour of real world market series compared to those exhibited by the artificially generated sequences, the market dynamic index has been calculated for the artificial series (Fig. \ref{Fig:integralMDIArtif}). The Market Dynamic Index has quite a constant value at the different horizons $M$ and moving average clusters $n$, thus exhibiting a behaviour different than real market dynamic indexes shown in Fig.~\ref{Fig:integral}.
\par
Last but not least, results of statistical significance tests are reported in Table \ref{tab:significance}. The test has been performed by using the paired t-test to check the null hypothesis $h=0$ that the cluster entropy values obtained on the real-world financial markets and those obtained on the artificial series (FBMs with $H=0.5$ assumed as benchmark) come from distributions with equal mean and same variance with a probability $p$.
\par
One can note in Table \ref{tab:significance} that the probability $p$ that the null hypothesis holds true ranges from $0.5154$ to $0.7584$. This confirms that the NASDAQ market behaves quite differently from the traditional interpretation of independent elementary stochastic process of price variations, as the FBM with $H=0.5$ implies.
The S\&P 500 exhibits an intermediate tendency to behave as ideal market being the probability $0.7399\leq p \leq 0.9248$. The probability for Dow Jones ranges within the interval $0.8892\leq p \leq 0.9434$. Thus it seems that the DJIA index reproduces more closely the behaviour of the fully independent stochastic process involved in the FBM with $H=0.5$.
\par
From an economic perspective the results have shown how financial market apparently very similar in terms of regional features, size and volumes may exhibit different horizon dependence. The obtained results are very robust from a statistical point of view. Thus they can represent a valid basis for developing investment tools to quantify risks and extremely useful for investors in classifying markets and choosing their strategy.
\clearpage
\newpage
\clearpage
\newpage
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Q: lossing memory after running series of queries over and over I have sql script that contains CREATE TEMPORARY TABLE, INSERT INTO...SELECT, DROP TEMPORARY TABLE. As I run this script over and over, my used memory (RAM) increases until an error says MySQL server has gone away. What could be the reason? I was thinking the memory used gets back to its previous size soon as the temporary table has been dropped.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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<li class="navelem"><b>sf</b></li><li class="navelem"><a class="el" href="classsf_1_1InputStream.html">InputStream</a></li> </ul>
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<div class="title">sf::InputStream Class Reference<span class="mlabels"><span class="mlabel">abstract</span></span><div class="ingroups"><a class="el" href="group__system.html">System module</a></div></div> </div>
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<p>Abstract class for custom file input streams.
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<p><code>#include <<a class="el" href="InputStream_8hpp_source.html">InputStream.hpp</a>></code></p>
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<tr class="memitem:a4b2eb0f92323e630bd0542bc6191682e"><td class="memItemLeft" align="right" valign="top">virtual </td><td class="memItemRight" valign="bottom"><a class="el" href="classsf_1_1InputStream.html#a4b2eb0f92323e630bd0542bc6191682e">~InputStream</a> ()</td></tr>
<tr class="memdesc:a4b2eb0f92323e630bd0542bc6191682e"><td class="mdescLeft"> </td><td class="mdescRight">Virtual destructor. <a href="#a4b2eb0f92323e630bd0542bc6191682e">More...</a><br /></td></tr>
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<tr class="memitem:a8dd89c74c1acb693203f50e750c6ae53"><td class="memItemLeft" align="right" valign="top">virtual Int64 </td><td class="memItemRight" valign="bottom"><a class="el" href="classsf_1_1InputStream.html#a8dd89c74c1acb693203f50e750c6ae53">read</a> (void *data, Int64 size)=0</td></tr>
<tr class="memdesc:a8dd89c74c1acb693203f50e750c6ae53"><td class="mdescLeft"> </td><td class="mdescRight">Read data from the stream. <a href="#a8dd89c74c1acb693203f50e750c6ae53">More...</a><br /></td></tr>
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<tr class="memdesc:a76aba8e5d5cf9b1c5902d5e04f7864fc"><td class="mdescLeft"> </td><td class="mdescRight">Change the current reading position. <a href="#a76aba8e5d5cf9b1c5902d5e04f7864fc">More...</a><br /></td></tr>
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<tr class="memdesc:a599515b9ccdbddb6fef5a98424fd559c"><td class="mdescLeft"> </td><td class="mdescRight">Get the current reading position in the stream. <a href="#a599515b9ccdbddb6fef5a98424fd559c">More...</a><br /></td></tr>
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<div class="textblock"><p>Abstract class for custom file input streams. </p>
<p>This class allows users to define their own file input sources from which SFML can load resources.</p>
<p>SFML resource classes like <a class="el" href="classsf_1_1Texture.html" title="Image living on the graphics card that can be used for drawing. ">sf::Texture</a> and <a class="el" href="classsf_1_1SoundBuffer.html" title="Storage for audio samples defining a sound. ">sf::SoundBuffer</a> provide loadFromFile and loadFromMemory functions, which read data from conventional sources. However, if you have data coming from a different source (over a network, embedded, encrypted, compressed, etc) you can derive your own class from <a class="el" href="classsf_1_1InputStream.html" title="Abstract class for custom file input streams. ">sf::InputStream</a> and load SFML resources with their loadFromStream function.</p>
<p>Usage example: </p><div class="fragment"><div class="line"><span class="comment">// custom stream class that reads from inside a zip file</span></div>
<div class="line"><span class="keyword">class </span>ZipStream : <span class="keyword">public</span> <a class="code" href="classsf_1_1InputStream.html">sf::InputStream</a></div>
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<div class="line"> Int64 <a class="code" href="classsf_1_1InputStream.html#a8dd89c74c1acb693203f50e750c6ae53">read</a>(<span class="keywordtype">void</span>* data, Int64 size);</div>
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<div class="line"> Int64 <a class="code" href="classsf_1_1InputStream.html#a76aba8e5d5cf9b1c5902d5e04f7864fc">seek</a>(Int64 position);</div>
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<div class="line"> Int64 <a class="code" href="classsf_1_1InputStream.html#a599515b9ccdbddb6fef5a98424fd559c">tell</a>();</div>
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<div class="line"> Int64 <a class="code" href="classsf_1_1InputStream.html#a311eaaaa65d636728e5153b574b72d5d">getSize</a>();</div>
<div class="line"></div>
<div class="line"><span class="keyword">private</span>:</div>
<div class="line"></div>
<div class="line"> ...</div>
<div class="line">};</div>
<div class="line"></div>
<div class="line"><span class="comment">// now you can load textures...</span></div>
<div class="line"><a class="code" href="classsf_1_1Texture.html">sf::Texture</a> texture;</div>
<div class="line">ZipStream stream(<span class="stringliteral">"resources.zip"</span>);</div>
<div class="line">stream.open(<span class="stringliteral">"images/img.png"</span>);</div>
<div class="line">texture.<a class="code" href="classsf_1_1Texture.html#a786b486a46b1c6d1c16ff4af61ecc601">loadFromStream</a>(stream);</div>
<div class="line"></div>
<div class="line"><span class="comment">// musics...</span></div>
<div class="line"><a class="code" href="classsf_1_1Music.html">sf::Music</a> music;</div>
<div class="line">ZipStream stream(<span class="stringliteral">"resources.zip"</span>);</div>
<div class="line">stream.open(<span class="stringliteral">"musics/msc.ogg"</span>);</div>
<div class="line">music.<a class="code" href="classsf_1_1Music.html#a4e55d1910a26858b44778c26b237d673">openFromStream</a>(stream);</div>
<div class="line"></div>
<div class="line"><span class="comment">// etc.</span></div>
</div><!-- fragment -->
<p>Definition at line <a class="el" href="InputStream_8hpp_source.html#l00041">41</a> of file <a class="el" href="InputStream_8hpp_source.html">InputStream.hpp</a>.</p>
</div><h2 class="groupheader">Constructor & Destructor Documentation</h2>
<a class="anchor" id="a4b2eb0f92323e630bd0542bc6191682e"></a>
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<td class="memname">virtual sf::InputStream::~InputStream </td>
<td>(</td>
<td class="paramname"></td><td>)</td>
<td></td>
</tr>
</table>
</td>
<td class="mlabels-right">
<span class="mlabels"><span class="mlabel">inline</span><span class="mlabel">virtual</span></span> </td>
</tr>
</table>
</div><div class="memdoc">
<p>Virtual destructor. </p>
<p>Definition at line <a class="el" href="InputStream_8hpp_source.html#l00049">49</a> of file <a class="el" href="InputStream_8hpp_source.html">InputStream.hpp</a>.</p>
</div>
</div>
<h2 class="groupheader">Member Function Documentation</h2>
<a class="anchor" id="a311eaaaa65d636728e5153b574b72d5d"></a>
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<td class="memname">virtual Int64 sf::InputStream::getSize </td>
<td>(</td>
<td class="paramname"></td><td>)</td>
<td></td>
</tr>
</table>
</td>
<td class="mlabels-right">
<span class="mlabels"><span class="mlabel">pure virtual</span></span> </td>
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<p>Return the size of the stream. </p>
<dl class="section return"><dt>Returns</dt><dd>The total number of bytes available in the stream, or -1 on error </dd></dl>
<p>Implemented in <a class="el" href="classsf_1_1FileInputStream.html#aabdcaa315e088e008eeb9711ecc796e8">sf::FileInputStream</a>, and <a class="el" href="classsf_1_1MemoryInputStream.html#a6ade3ca45de361ffa0a718595f0b6763">sf::MemoryInputStream</a>.</p>
</div>
</div>
<a class="anchor" id="a8dd89c74c1acb693203f50e750c6ae53"></a>
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<td class="memname">virtual Int64 sf::InputStream::read </td>
<td>(</td>
<td class="paramtype">void * </td>
<td class="paramname"><em>data</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">Int64 </td>
<td class="paramname"><em>size</em> </td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
</table>
</td>
<td class="mlabels-right">
<span class="mlabels"><span class="mlabel">pure virtual</span></span> </td>
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</div><div class="memdoc">
<p>Read data from the stream. </p>
<p>After reading, the stream's reading position must be advanced by the amount of bytes read.</p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">data</td><td>Buffer where to copy the read data </td></tr>
<tr><td class="paramname">size</td><td>Desired number of bytes to read</td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>The number of bytes actually read, or -1 on error </dd></dl>
<p>Implemented in <a class="el" href="classsf_1_1FileInputStream.html#ad1e94c4152429f485db224c44ee1eb50">sf::FileInputStream</a>, and <a class="el" href="classsf_1_1MemoryInputStream.html#adff5270c521819639154d42d76fd4c34">sf::MemoryInputStream</a>.</p>
</div>
</div>
<a class="anchor" id="a76aba8e5d5cf9b1c5902d5e04f7864fc"></a>
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<td class="memname">virtual Int64 sf::InputStream::seek </td>
<td>(</td>
<td class="paramtype">Int64 </td>
<td class="paramname"><em>position</em></td><td>)</td>
<td></td>
</tr>
</table>
</td>
<td class="mlabels-right">
<span class="mlabels"><span class="mlabel">pure virtual</span></span> </td>
</tr>
</table>
</div><div class="memdoc">
<p>Change the current reading position. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">position</td><td>The position to seek to, from the beginning</td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>The position actually sought to, or -1 on error </dd></dl>
<p>Implemented in <a class="el" href="classsf_1_1FileInputStream.html#abdaf5700d4e1de07568e7829106b4eb9">sf::FileInputStream</a>, and <a class="el" href="classsf_1_1MemoryInputStream.html#aa2ac8fda2bdb4c95248ae90c71633034">sf::MemoryInputStream</a>.</p>
</div>
</div>
<a class="anchor" id="a599515b9ccdbddb6fef5a98424fd559c"></a>
<div class="memitem">
<div class="memproto">
<table class="mlabels">
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<td class="memname">virtual Int64 sf::InputStream::tell </td>
<td>(</td>
<td class="paramname"></td><td>)</td>
<td></td>
</tr>
</table>
</td>
<td class="mlabels-right">
<span class="mlabels"><span class="mlabel">pure virtual</span></span> </td>
</tr>
</table>
</div><div class="memdoc">
<p>Get the current reading position in the stream. </p>
<dl class="section return"><dt>Returns</dt><dd>The current position, or -1 on error. </dd></dl>
<p>Implemented in <a class="el" href="classsf_1_1FileInputStream.html#a768c5fdb3be79e2d71d1bce911f8741c">sf::FileInputStream</a>, and <a class="el" href="classsf_1_1MemoryInputStream.html#a7ad4bdf721f29de8f66421ff29e23ee4">sf::MemoryInputStream</a>.</p>
</div>
</div>
<hr/>The documentation for this class was generated from the following file:<ul>
<li><a class="el" href="InputStream_8hpp_source.html">InputStream.hpp</a></li>
</ul>
</div><!-- contents -->
</div>
<div id="footer-container">
<div id="footer">
SFML is licensed under the terms and conditions of the <a href="http://www.sfml-dev.org/license.php">zlib/png license</a>.<br>
Copyright © Laurent Gomila ::
Documentation generated by <a href="http://www.doxygen.org/" title="doxygen website">doxygen</a> ::
</div>
</div>
</body>
</html>
|
{
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| 7,658
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Day two at the USCCB meeting
This article appears in the USCCB Fall 2014 feature series. View the full series.
In commenting on the synod on the family at the USCCB meeting yesterday, Cardinal Timothy Dolan said there were two synods: the raucous, politicized, controversial synod portrayed by some, and the actual synod, which was a study in consensus, even a little dull. It is true that notwithstanding the comments of a few prelates, some who attended the synod and some who did not, they achieved a remarkable consensus at the synod. But the voices of the other prelates were loud and insistent, and we in the press did not make that up.
If I may borrow Cardinal Dolan's metaphor, there are two Catholic Churches in the U.S. today. One Church is thrilled by Pope Francis, glad not to feel that everything is their fault, happy that they no longer feel the lash of judgment because they cannot measure up to the moral standards articulated by certain conservative commentators, delighted to know that it is OK not to be obsessed exclusively by certain issues, even -- what was unimaginable for most just a short time ago -- proud to be Catholic again.
The other Church is meeting in the ballroom in Baltimore this week. There is no excitement. The agenda is very pre-"VatiLeaks". The obsession with abortion, contraception and same-sex marriage rolls on in dreary predictability. Everyone is "in a state of agreement, or silent in a false and quietist peace," the very thing Pope Francis said would have worried him if it had characterized the recent synod. It characterizes the meeting of the USCCB so far. It is bizarre to me that the encomiums to Pope Francis are formulaic at best or absent entirely. So far as the public discussions go, you would not know that this is an interesting, let alone exciting, time to be a Catholic. The whole world knows. The cat is out of the bag. And the bishops seem to be asking, "What is a cat?"
Am I being too harsh? I don't think so. The poor did not get much of a mention from the podium, and even when they did, it did not ring true, as when Archbishop Salvatore Cordileone spoke about the relationship between poverty and the breakdown in marriage. There was no acknowledgement that people living in poverty have a harder time making marriages work. The acknowledgement was reversed, with the implication that maybe those people would not be so poor if they lived in the moral manner the bishops intend. +Cordileone always speaks with that air of moral superiority one encounters at the Ethics & Public Policy Center and over at The Catholic Thing.
And when did it become a sign of respect to refer to people in ways they do not, and would not, refer to themselves, e.g., "people who experience same-sex attraction"? In 2014, it is OK to say "gay people." In all my years working at Dupont Circle, I never heard someone say, "I am a person who experiences same-sex attraction." Who talks like this? It makes it seem as if the bishops say the word "gay," they will be infected somehow. And you do not have to be particularly morally sensitive to know that for people who were accused for centuries of the sin that "dare not speak its name," with all the ugly experiences of closeted life that flowed therefrom, refusing to say the word "gay" is exceedingly callous.
I was saddened that Archbishop Joseph Kurtz did not even mention the word "immigration" in his first presidential address. These texts are worked on by the staff, group-driven, and are rarely known for their eloquence, but they are usually comprehensive. The omission is especially strange seeing as concern for immigrants is not only an issue close to the Holy Father's heart, but because much of the summer, many of these bishops worked closely with their Catholic Charities to provide for the thousands of unaccompanied children coming across the border.
The omission is also bizarre because the bishops received some of their most powerful and positive media coverage in years on account of their care for the immigrant children. And earlier in the year, when several bishops traveled to the border near Tucson, Ariz., and said Mass at the fence, the images of Cardinal Sean O'Malley and Bishop Gerald Kicanas giving Holy Communion through the slats in the border fence were some of the most powerful images communicated to the Latino community, and to the rest of us, about the truth that in Christ there is no East nor West, no North nor South. The omission is glaring when you consider that a bipartisan immigration reform bill has been sitting on the desk the Speaker of the House for more than a year, and everyone knows the votes are there to pass it, but this Catholic politician need not fear any pressure from the bishops. How is one to interpret this omission? Was it a mere oversight? Or does it accurately reflect the priorities of the leadership of the conference?
I was not, at first, impressed by the nuncio's talk, but having reread it, it was better than I thought. His message was a very Pope Francis message, one of accompaniment. It was encouraging the bishops to be like Francis. I would have liked it better if he had been explicit -- as Catholics, still more as Catholic bishops, there is no unity apart from Peter. Still, he said, "Our young people today clearly are looking for a challenge, a goal, a purpose. They need to find meaning in their lives. They need to be attracted to Christ in positive ways by the example of so many declared and undeclared saints living in the Church today in the United States." You do not have to spend very long on a college campus to know that one way to make the attraction to Christ unlikely is to evidence an obsessive dismissal of the human dignity of gays and lesbians, yes? The nuncio speaks in diplomatic speak, but more than one bishop said they found his talk very inspiring and encouraging for a conference that is still struggling to understand what Pope Francis is about.
One notable exception to the un-Francis tone of the discussions came from Cardinal Donald Wuerl. In speaking about the synod, both in the meeting and at the press conference, he must have used the word "consensus" 25 times. His point was obvious -- the approach articulated by the synod, of accompaniment, of nonjudgmentalism: This was not merely a private peccadillo of the Supreme Pontiff. The approach commanded the assent of a decisive majority of the synod fathers. We all know which fathers were not in the decisive majority, those who resist the consensus. There is no need to marginalize them. They have marginalized themselves.
So we will see what today brings. I intend to take a nap when they are voting on some of the liturgical translations. I prefer the self-surrender of sleep to a sleep-inducing self-referential discussion. The elections will be interesting, especially the ballots for secretary of the conference and for communications chair. Already, Bishop William Murphy has withdrawn his name for nomination to the communications post, and it is not clear who might be nominated in his stead, but the smart money is that Bishop Chris Coyne, the first bishop to start a blog and to go on Twitter, will get the nod.
For the secretary's position, any bishop who has read Elisabetta Pique's wonderful new biography of Pope Francis will know that they probably do not want Archbishop Timothy Broglio representing the conference in meetings at the Vatican during this pontificate. Besides, Archbishop Gregory Aymond is a true consensus builder and is almost universally respected by the body of bishops.
Blog posts will continue through the day.
Peacocks in the headlights
Bishops' meeting lacks passion, leadership
Recapping the USCCB meeting
Bishops see new perspective on Middle East after Holy Land pilgrimage
USCCB president aims to meet with Obama, congressional leaders
What's Worse Than Partisanship?
WaPo & Benedict on Immigration
Links for 10/05/15
Pacem in terris at 50: The Video
|
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| 7,447
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The Ecology of Freedom: The Emergence and Dissolution of Hierarchy
Philosophy Politics
Publisher: AK Press
"The very notion of the domination of nature by man stems from the very real domination of human by human." With this succinct formulation, Murray Bookchin launches his most ambitious work, The Ecology of Freedom. An engaging and extremely readable book of breathtaking scope, its inspired synthesis of ecology, anthropology and political theory traces our conflicting legacies of hierarchy and freedom from the first emergence of human culture to today's globalized capitalism, constantly pointing the way to a sane, sustainable ecological future.
Murray Bookchin, cofounder of the Institute for Social Ecology, has been an active voice in the ecology and anarchist movements for more than 40 years. The author of numerous books and articles, he lives in Burlington, Vermont.
In Oakland, California on March 24, 2015 a fire destroyed the AK Press warehouse along with several other businesses. Please consider visiting the AK Press website to learn more about the fundraiser to help them and their neighbors.
|
{
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| 8,275
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ALTER TABLE task ADD COLUMN IF NOT EXISTS spark_extension JSONB;
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{
"redpajama_set_name": "RedPajamaGithub"
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| 9,813
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Konge i Danmark: Erik 7. 1396-1439
Se også 1433 (tal)
Begivenheder
Kinesiske skibe under admiral Zheng He når Kap Det Gode Håb på Afrikas sydspids. Det er kun et pludseligt skift i kejserens holdning til opdagelsesrejser, der hindrer, at kineserne finder søvejen til Europa, før europæerne finder søvejen til Kina!
Født
Dødsfald
Eksterne henvisninger
33
Artikler om enkelte år
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| 7,701
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{"url":"http:\/\/kjk.office.uec.ac.jp\/Profiles\/70\/0006981\/theses_e1.html","text":"Published papers\nNumber of published data : 13\n No. Classification Refereed paper Title Authorship Author Journal Volume\/issue\/page Publication date ISSN DOI URL 1 Paper Yes Statistical Estimation for CAPM with Long-Memory Dependence. Advances in Decision Sciences Volume 2012, Article ID 571034- 2012 2 Paper Yes Optimal portfolios with end-of-period target. Advances in Decision Sciences Volume 2012, Article ID 703465- 2012 3 Paper Yes Analysis of CL and Estimating Function Estimators for Financial Time Series Models. Advances in science, technology and environmentology B8, 57-65 2012 4 Paper No Portmanteau\u691c\u5b9a\u306e\u89e3\u6790 \u7d4c\u6e08\u7406\u8ad6 367, 19-32 2012 5 Paper Yes Asymptotic Optimality of Estimating Function Estimator for CHARN Model. Advances in Decision Sciences Volume 2012, Article ID 515494- 2012 6 Paper Yes Control variate method for stationary processes. Journal of Econometrics 165\/ No 1, 20-29 2011 7 Paper Yes Local Whittle LIkelihood Estimators and Tests for non-Gaussian Linear Processes. Statistical Inference for Stochastic Processes 13\/ No 3, 163-174 2010 8 Paper Yes Spectral analysis for intrinsic time processes. Statistics and Probability Letters 79\/ No 23, 2389-2396 2009 9 Paper Yes Systematic Approach for Portmanteau Tests in View of Whittle Likelihood Ratio. Journal of the Japan Statistical Society 39\/ No 2, 177-192 2009 10 Paper Yes Asymptotic efficiency of estimating function estimators for nonlinear time series models. Journal of The Japan Statistical Society 39\/ No 2, 209-231 2009 11 Paper Yes Asymptotic efficiency of conditional least squares estimators for ARCH models. Statistics and Probability Letters 78\/ No 2, 179-185 2008 12 Paper Yes Comparison of Whittle type portmanteau tests. Scientiae Mathematicae Japonicae 68\/ No 2, 247-254 2008 13 Paper Yes Note on Asymptotics of Whittle estimators for square transformed ARCH($\\infty$) models. Scientiae Mathematicae Japonicae 65\/ No 1, 43-51 2007","date":"2021-12-09 04:54:34","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3002617061138153, \"perplexity\": 5797.7481581966795}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-49\/segments\/1637964363659.21\/warc\/CC-MAIN-20211209030858-20211209060858-00297.warc.gz\"}"}
| null | null |
Guston är en civil parish i Storbritannien. Den ligger i grevskapet Kent och riksdelen England, i den sydöstra delen av landet, km öster om huvudstaden London.
Källor
Externa länkar
Civil parishes i Kent
|
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| 9,006
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Ormsby Brunson Thomas (August 21, 1832October 24, 1904) was an American lawyer, Republican politician, and Wisconsin pioneer. He was a member of the United States House of Representatives for three terms (1885–1891), representing Wisconsin's 7th congressional district. He also served five years in the Wisconsin Legislature and 13 years as district attorney of Crawford County, Wisconsin.
Early life and career
Ormsby Thomas was born in Sandgate, Vermont, but moved to the Wisconsin Territory with his parents when he was a young child. His parents settled in Prairie du Chien, in Crawford County. After receiving his early education, he was sent to the Burr Seminary in Manchester, Vermont, for an academic education. He then graduated from the State and National Law School in Poughkeepsie, New York, in 1856. He was admitted to the bar in Albany, New York, that year. He returned to Prairie du Chien to establish his own law practice.
He became active in local politics and was elected district attorney of Crawford County in 1858, running on the Democratic Party ticket. He was subsequently re-elected in 1860.
Civil War service
When the American Civil War started, in 1861, Thomas was the incumbent district attorney in Crawford County and was running for Wisconsin State Assembly. He was elected to represent Crawford County in the 1862 session of the Wisconsin Legislature.
After the end of the 1862 legislative session, Thomas went to work raising a company of volunteers for the Union Army from Crawford County. His company was enrolled as Company D in the 31st Wisconsin Infantry Regiment, and Thomas was formally commissioned captain in September 1862. For the first several months of their service, the regiment was solely tasked with training for battle, supervising conscription of draftees, and guarding Confederate prisoners of war at camps in Wisconsin. The regiment was reorganized at Racine, Wisconsin, in the Winter of 1862–1863, and mustered into federal service.
They were sent to the western theater of the war and assigned to protecting supplies and logistics in Middle Tennessee through all of 1863. Thomas resigned his commission in January 1864 and returned to Wisconsin.
Postbellum years
On returning to Wisconsin, Thomas was elected to another term in the Assembly, but was now running on the National Union Party ticket. This was a short-term alliance of Republicans and Pro-Union Democrats. Thomas served in the Union caucus in the 1865 session of the Legislature and remained associated with the Republican Party for the rest of his life. He was not a candidate for re-election in 1865, but was elected to another term in the Assembly in 1866.
After the legislative session in 1867, Governor Lucius Fairchild appointed Thomas to fill the vacant district attorney post in Crawford County, following the resignation of Joseph M. Wilcox. He served the remainder of Wilcox's term and was then elected to another term in 1868.
In 1869, he ran for Wisconsin State Senate in the 30th State Senate district, but was defeated by Democrat George Krouskop. He subsequently was elected to another term as district attorney in 1870. In 1872, he was a presidential elector for Ulysses S. Grant.
During the 1870s, Thomas was focused on his legal practice. He formed a partnership with Charles S. Fuller in 1876, which became a prominent and leading practice throughout the state. Fuller was later appointed a county judge in the 1880s.
Thomas resumed his political career in 1879, running for Wisconsin State Senate in the 4th State Senate district, which comprised Crawford County following the 1876 redistricting law. Thomas was easily elected running on the Republican ticket; he received 56% of the vote in a three-person race.
Congressional career
Thomas did not run for re-election in 1881, and in 1882, he waged a campaign for nomination to Congress, from Wisconsin's 7th congressional district. His principal opponent was believed to be Gideon Hixon, who had the support of delegates from La Crosse County, but after 29 ballots at the Republican 7th district convention, the delegates broke in favor of Cyrus M. Butt, of Viroqua. Butt went on to lose the general election to Democrat Gilbert M. Woodward.
Two years later, Thomas made another run for congress. This time, he won the nomination on the first ballot at the convention. He went on to defeat Woodward in the general election with 52% of the vote. He was subsequently re-elected in 1886 and 1888, serving in the Forty-ninth, Fiftieth, and Fifty-first congresses. During the 51st Congress, Thomas was chairman of the House Committee on War Claims. He was defeated seeking a fourth term in 1890, in the wave election that saw Democrats claim 8 of Wisconsin's 9 congressional seats.
After leaving office, he resumed his legal career with his old partner, Judge C. S. Fuller. Fuller, however, committed suicide in 1897. In 1898, Thomas accepted the Republican nomination for another term as district attorney of Crawford County, and was elected in the Fall.
During his final term as district attorney, Thomas suffered from Bright's disease. His health continued to deteriorate after he left office in January 1903. He died at his home in Prairie du Chien on October 24, 1904. At the time of his death, he was described as the oldest member of the State Bar of Wisconsin.
Personal life and family
Ormsby B. Thomas married Sarah E. Rosencrans at Prairie du Chien in June 1875. They had three children together before her death in 1884. Their first child, however, died in infancy.
Electoral history
Wisconsin Senate (1869)
| colspan="6" style="text-align:center;background-color: #e9e9e9;"| General Election, November 2, 1869
Wisconsin Senate (1879)
| colspan="6" style="text-align:center;background-color: #e9e9e9;"| General Election, November 4, 1879
U.S. House of Representatives (1884–1890)
References
External links
Thomas, Ormsby Brunson 1832 - 1904 at Wisconsin Historical Society
1832 births
1904 deaths
People from Sandgate, Vermont
People from Prairie du Chien, Wisconsin
State and National Law School alumni
Union Army officers
People of Wisconsin in the American Civil War
Wisconsin lawyers
District attorneys in Wisconsin
Republican Party members of the Wisconsin State Assembly
Republican Party Wisconsin state senators
Republican Party members of the United States House of Representatives from Wisconsin
19th-century American politicians
|
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| 3,666
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How Sands Anderson Vaccine Injury Lawyers Help You
Vaccines Covered by the National Vaccine Injury Compensation Program
Vaccine Injury Information
Brachial Neuritis
SIRVA
Immune Thrombocytopenia Purpura (ITP)
Vasovagal Syncope
Other Vaccine-Related Injuries
Vaccine Injury Claims Process
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Your Vaccine Injury Lawyers
Contact the Sands Anderson Vaccine Injury Legal Team
Vaccine Injury Legal Team
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Shoulder Injury Related to Vaccine Administration (SIRVA)
Covered Vaccines
Elizabeth M. Muldowney
Ramon Rodriguez, III, MD
Blogs and Information
About the VICP
Transverse Myelitis Caused by Vaccines
What is transverse myelitis?
Transverse myelitis is a rare neurological condition in which the entire width of the spinal cord becomes inflamed. This can cause nerve damage and pain. As a result, people with this condition can have weakness or numbness in their limbs and problems with movement.
Transverse myelitis affects males and females of all ages and races. In rare cases, transverse myelitis can be caused by vaccines, but often the cause is unknown, in which case it's called idiopathic transverse myelitis. It can also happen from other conditions, such as viral infections or multiple sclerosis.
What are the symptoms of transverse myelitis?
Symptoms of transverse myelitis can include:
Back pain, or shooting pain in the legs or arms or around the torso
Trouble with coordination
Feelings of numbness, tingling, coldness or burning
Bowel and bladder problems
In addition to these more common symptoms, people with transverse myelitis may have muscle spasms, headaches, fevers or loss of appetite. In rare cases this condition can cause breathing problems.
How is transverse myelitis diagnosed?
Your doctor will ask about your medical history and do a thorough neurologic examination. An important step is to rule out other conditions, such as a herniated or slipped disc, spinal stenosis, abscesses, vitamin deficiencies and others.
The following tests are used to diagnose transverse myelitis:
Blood tests. These tests can rule out other disorders, including infections, and can also show certain antibodies found in people with transverse myelitis.
Magnetic resonance imaging (MRI). An MRI of the spine or brain can help identify an underlying cause for transverse myelitis.
Lumbar puncture (spinal tap). This test uses a needle to remove a small sample of spinal fluid to look for increases in proteins or other signs of infection.
How is transverse myelitis treated?
Treatments for transverse myelitis are aimed at reducing the inflammation in the spinal cord and managing symptoms, as well as treating any infection that may have caused this condition. They may include:
Medications. Drugs can help reduce pain, treat infections or relieve complications, such as bladder problems, muscle spasms or stiffness, depression and others.
Intravenous immunoglobulin (IVIG). This treatment uses antibodies from healthy donors to remove unhealthy antibodies from circulation.
Intravenous (IV) corticosteroids. These drugs help decrease swelling and inflammation in the spine and also reduce immune system activity, which can contribute to inflammation.
Plasma exchange (plasmapheresis). This procedure uses a machine to assist immune system activity by exchanging the blood plasma for certain fluids. These fluids help remove antibodies and other substances that may be causing the inflammation.
Treatments for transverse myelitis are usually given in a hospital or rehabilitation facility with a specialized medical team. People whose breathing is affected by this condition may need to be placed on a respirator.
People with this condition usually recover with only minor side effects, or none at all. A few people will have a recurrence of transverse myelitis, and in rare cases people may have permanent damage that affects their ability to do ordinary, daily tasks.
Can vaccines cause transverse myelitis?
Transverse myelitis from a vaccine is rare. However, there are several vaccines that have been linked to this condition. They include vaccines for seasonal influenza (flu shot), hepatitis B, measles mumps rubella (MMR) and diphtheria tetanus pertussis (DTP). These vaccines are included in the National Vaccine Injury Compensation Program, which helps people with vaccine injuries recover their losses.
When transverse myelitis is caused by a vaccine, the symptoms can begin from a few days up to three months after the vaccination.
Have you been injured?
If you think you may have developed this condition from a vaccine, contact us right away. Our experienced vaccine injury attorneys can file a claim for you with VICP, at no cost to you.
Vaccine Injury Blogs and Information
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The information contained in our web site describes legal matters handled in the past by our attorneys. Of course, the results we have achieved depend upon a variety of factors unique to each matter. Because each matter is different, our past results cannot predict or guarantee a similar result in the future.
Copyright © 2021 Sands Anderson
Sands Anderson
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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Plopu este un sat în comuna Podu Turcului din județul Bacău, Moldova, România.
Note
Plopu, Podu Turcului
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"redpajama_set_name": "RedPajamaWikipedia"
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**_They're playing with fire in this reissue of Jill Shalvis's red-hot Blaze..._**
Firefighter Aidan Donnelly has always battled the flames with trademark icy calm. That is, until a blazing old flame returns—in the shape of sizzling soap star Mackenzie Stafford! Aidan wants to pour water over the unquenchable heat between them. But that just creates more steam....
Kenzie is not the delicate, fragile female she looks like. She has one clear objective, and nothing will stand in the way of her goal—well, nothing but the red-hot touch of a certain dangerously sexy fireman, that is!
Originally published in 2008
"Riveting suspense laced with humor and heart is her hallmark and Jill Shalvis always delivers."— _USA TODAY_ bestselling author Donna Kauffman
## Flashback
#### _Jill Shalvis_
# _CONTENTS_
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
Excerpt from _Playing Dirty_ by Taryn Leigh Taylor
Excerpt from _Lost and Found Sisters_ by Jill Shalvis
# _CHAPTER 1_
THE FIRE BELL RANG for the fourth time since midnight, interrupting Aidan Donnelly in the middle of a great dream in which he was having some fairly creative, acrobatic sex with a gorgeous blonde. The last thing he wanted was to be shaken awake, but apparently sex, imaginary or otherwise, wasn't on his card for the evening.
He was on the last few hours of a double shift from hell. The loudspeaker mounted in one corner of the bunk room was going off, telling him and his crew that they would not be going home in one short hour after all, but back into the field on yet another emergency call.
Putting the blonde back where she belonged, in the file in his brain labeled Hot Erotic Fantasy, Aidan got up to the tune of a bunch of moans and groans from his crew.
So close. He'd been so close to three desperately needed days off...
Across the room Eddie kicked aside the latest issue of _Time,_ which had an entire company of firefighters on the cover. "A lot of good being the sexiest occupation does us," the firefighter grumbled, "when we're too exhausted to take advantage of it."
"Some of us don't need beauty sleep." This from Sam, Eddie's partner. "Like, say, Mr. 2008 here." He slid a look Aidan's way, but Aidan found himself too tired to rise to the bait.
Through no fault of his own, he'd been named Santa Rey's hottest firefighter for 2008. This dubious honor came along with another—being put on the cover of Santa Rey's annual firefighter's calendar. "I told you, I didn't submit my name."
Eddie grinned in the middle of dressing. "No, we did, Mr. 2008."
Aidan gave him a shove, and Eddie fell back to the mattress, snorting out a laugh as he staggered upright again and grabbed his boots. "Yeah, like being that pretty is a hindrance."
"I am not pretty."
No one answered him in words as they pulled on their gear, but several made kissy noises as they headed toward their rigs. Still groggy, and definitely out of sorts, Aidan took the shotgun position next to Ty, his temporary partner, on loan from a neighboring firehouse, since his usual partner Zach was still off on medical leave.
Eddie and Sam grabbed their seats, as well as Cristina and Aaron, another on-loan firefighter, and they were all off into the dark night—or more accurately, the dark predawn morning—following the ambulance, which had pulled out first. The air was thick with dew, and salty from the ocean only one block over. For now the temperature was cool enough, but by midday the California August heat would be in full bloom, and they'd all be dying. Aidan got on the radio to talk to dispatch. "It's an explosion," he told the others grimly.
"Where?" Ty asked.
"The docks." Which could be anywhere from the shipping area, to the houseboats filled with year-round residents. "Only one boat's on fire, but several others are threatened by the flames, with no word on what caused the explosion."
Behind him, Eddie swore softly, and Aidan's thoughts echoed the sentiment. Explosions were trickier than a regular fire, and far more unpredictable.
"Are they calling for backup?" Sam asked.
They needed it. Firehouse Thirty-Four was sorely overworked and dangerously exhausted going into the high fire season. They'd had a rough month. Aidan's partner and best friend Zach had been injured after digging into the mysterious arsons that had plagued Santa Rey. Mysterious arsons that were now linked to one of their own.
Blake Stafford.
Just the thought brought a stab of fresh pain to Aidan's chest. Now Zach was off duty and Blake was dead, leaving them all devastated.
Cristina was especially devastated, and with good reason. She'd been Blake's partner, and the closest to him. She'd suffered like hell over his loss, and also over the arsons he'd been accused of committing.
She blamed herself, Aidan knew, which was ridiculous. She couldn't have stopped Blake.
As it turned out, none of them could have stopped him.
Aidan considered himself pretty damn tough and just about one-hundred-percent impenetrable, but losing Blake had been heart-wrenching. He missed him, and hated what he'd been accused of. He didn't want to believe Blake was dead, and he sure as hell didn't want to believe Blake guilty of arson, and the resulting death of a small boy—none of them did, but the evidence was there. He could hardly even stand thinking about it—classic denial, Aidan knew, but it was working for him. "Dispatch's sending rigs from Stations Thirty-Three and Thirty-Five."
No one said anything to this, but they were all thinking the same thing—it'd take those stations at least ten extra minutes to get on scene from their locations—and the sense of dread only increased as they pulled up to the docks.
Turned out that the fire wasn't at the shipping docks, but where the smaller, privately owned boats were moored at four long docks, each with ten bays. Possibly forty boats in total, many of them occupied.
Chaos reined in the predawn. Their senior officer was usually first on scene, setting up a command center, but he was coming from another fire and was five minutes behind them. The sky was still dark, with no moon, and the visibility wasn't helped by the thick plumes of black smoke choking the air out of their lungs. Flames leaped fifty feet into the air, coming from a boat halfway down the second of the four docks. Aidan took a quick count, and his stomach tightened with fear. There were boats on either side of the flaming vessel, and more on the opposite side of the dock.
Not good.
As they accessed their equipment and laid out lines, three police squad cars tore into the lot, followed by the command squad, all of whom leaped to work evacuating the surrounding docks. Aidan and company needed to contain the flames, but the explosion burned outrageously hot. He could feel that mind-numbing heat from a hundred feet back. With the chief now on scene, barking orders through their radios, Aidan and the others moved with their hoses, their objective to keep the flames from spreading to any of the other boats. They were halfway there when it came.
A sharp, terrified scream.
The sound raised the hair on the back of Aidan's neck, and he dropped everything to run toward the burning boat, Ty right behind him.
The scream came again, clearly female, and Aidan sped up. No one knew better than a firefighter what it was like to be surrounded by flames, to have them lick at you, toy with you. It was sheer, horrifying terror.
They had to get to her first.
Behind them came Sam, Eddie, Cristina and Aaron, directing water on the flames to clear Aidan and Ty's path down the dock toward the boat. Twenty feet,then ten,and that's when he saw her. A woman standing on the deck of the burning boat, wobbling, the flames at her back.
"Jump!" he yelled, wondering why she didn't just make the short leap to the dock—she could have made a run for safety. " _Jump_ —"
Another explosion rocked them all. Aidan skidded to a halt, spinning away and crouching down as debris flew up into the air to match the intensifying flames. The chief was shouting into the radio, demanding a head count. Aidan lifted his head and checked in as he took in the sights. The boat was still there. With his heart in his throat, he searched for a visual on the woman—
_There._ In the same spot she'd been before, still on the deck but on the floor now, holding her head. _Goddammit._ He got to his feet, took a few running steps, and dove onto the boat.
She nearly jumped out of her skin when he landed next to her. "It's okay." He dropped to his knees at her side to try to get a good look and see how badly she was injured, but the smoke had choked out any light from the docks and she was nothing but a slight shadow. A slight shadow who was hunched over and coughing uncontrollably.
"The boat," she managed. "It k-keeps b-blowing up—"
"Can you stand?"
"Yes. I—" She let out a sound that tugged at his memory, but he pushed that aside when she nodded. She got up with his help, twisting away from him to stare up at the flames shooting up the mast and sails. "Ohmigod..."
He pulled her closer to his side, intending to jump with her to the dock and the hell off this inferno, but several things hit him at once.
The name of the boat painted across the out side of the cabin, flickering in and out of view between the flames.
_Blake's Girl._
No. It couldn't be. Then came something of far more immediate concern—the rumbling and shuddering of the deck beneath their feet. "We have to move."
"No. No, please," she gasped. "You have to save the boat."
"Us first." He couldn't have put together a more coherent sentence because of all that was going through his head. _Blake's Girl..._
Blake's boat. God, he'd all but forgotten that Blake had owned a boat.
Then there was the woman in his arms, facing away from him, but invoking that niggling sense of familiarity. There was something about her wild blond curls, about the sound of her voice—
The warning signals in his brain peaked at once. In just the past thirty seconds, the flames had doubled in strength and heat. The deck beneath their feet trembled and quivered with latent simmering violence.
They were going to blow sky high. Whipping toward the dock he got another nasty surprise—the flames had covered their safe exit.
On the other side of those monstrous flames stood Ty, Eddie and Sam, hoses in hand, battling the fire from their angle, which wasn't going to help Aidan and his victim in time. Cristina was there, too, with Aaron, and even in the dark he sensed their urgency, their utter determination to keep him safe.
They'd so recently lost one of their own; there was no way they were going to let it happen again.
"Ohmigod," the woman at his side gasped, staring, as if mesmerized, at the sight of the flames closing in on them.
She wasn't the only one suddenly mesmerized, and for one startling heartbeat, Aidan went utterly still, as for the first time he caught a full glimpse of her.
He knew that profile.
He knew her. _"Kenzie?"_
At the sound of her name on his lips, uttered in a low, hoarse, surprised voice, her head whipped toward his, eyes wide. Her wavy blond hair framed a pale face streaked with dirt and some blood, but was still beautiful, hauntingly so.
She was Mackenzie Stafford, Blake's sister. Kenzie to those who knew and loved her, Sissy Hope to the millions of viewers who watched her on the soap opera _Hope's Passion_.
She was not a stranger to Aidan, but not because of her television stardom. He knew her personally.
_Very_ personally. "Kenzie."
"I can't—I can't hear you."
People never expected fire to be noisy, but it was. The flames crackled and roared at near ear-splitting decibels as they devoured everything in their path.
Including them if they didn't move, a knowledge that was enough to pull his head out of his ass and get with the program. Old lover or not, he still had to get her out of there alive. But she was looking at him through Blake's eyes, and his heart and gut wrenched hard. There was maybe twenty feet of water between _Blake's Girl_ and the next boat, which was starting to smoke as well, and would undoubtedly catch on fire any second. It didn't matter. They had no choice. "Kenzie, when I say so, I want you to hold your breath."
"D—do I know you?"
He wore a helmet and all his equipment, and in the dark, not to mention the complete and utter chaos around them, there was no way she could see him clearly. Still, he had to admit it stung. "It's me, Aidan. Hold your breath now, on my count."
"Aidan, my God."
"Ready?"
"The boat's going to go, every inch of it, isn't it?"
Yep, including the few square inches they were standing on. In fact, it was going to go much more quickly than he'd have liked. Since they couldn't get to the dock, it was into the ocean for them, where they'd wait for rescue.
"No," she said, shaking her head. "There's got to be another way."
Unfortunately there wasn't, and he quickly stripped out of his jacket and gear because the protection they offered wouldn't be worth the seventy-five pounds of extra weight while treading water and holding up Kenzie to boot. At least she was conscious. She didn't appear to have on any shoes, or anything particularly heavy on her person, all of which were points in her favor."On three, okay? Remember to hold your breath."
"I don't think—"
"Perfect. Go with that. One—" He nudged her in front of him, pushing her to the railing.
"Aidan—"
"Two—"
_"Are you crazy?"_
_"Three."_
"Hell, no. I'm not going into the—"
He dropped her into the water, and she screamed all the way down.
# _CHAPTER 2_
KENZIE HIT THE ICY OCEAN, and as she took in a huge mouthful of water, she realized she'd forgotten to hold her breath, a thought that was completely eradicated when _Blake's Girl_ exploded into the early dawn.
In the brilliant kaleidoscope, she barely registered the splash next to her, or the two strong arms that came around her, supporting her as flying pieces of burning debris hit the water all around them.
Aidan. My God, Aidan...That it was him boggled her mind. She tried to remind him that she could swim on her own, but the shock of the cold water sapped both her voice and the air in her lungs, and also hampered the working of her brain.
She'd never experienced anything like it. Never in her life had she been so hot and so frozen at the same time. The heat came from the flames, so high above them now that she was in the water, but no less terrifying. And yet, an icy cold had taken over her limbs, making movement all but impossible, weighing her down, sitting on her chest, sucking the last of the precious air from her overtaxed lungs.
Someone was screaming, and Kenzie envied their ability to draw air into their lungs because her own felt as constricted as if she had a boa slowly squeezing the life out of her.
The scream came again.
_Huh?_
It sounded sort of like her.
And then she realized, as if from a great distance, that it _was_ her screaming, which meant that somehow she was breathing. Okay, that was good. So was the man holding her in the water, tucking her head against him, shielding her from the pieces falling out of the sky at his own risk. Without him, she'd have gone down like a heavy stone and she knew it.
"Shh," he was murmuring. "I've got you. It's okay, Kenzie, it's going to be okay..."
She was hurt, but not so hurt as to stop the memories bombarding her at the sound of his voice. How could she not have _instantly_ recognized him?
He was the first man who'd ever broken her heart.
He'd ditched his helmet and she could see his face now. He didn't look happy to see her, and honestly, on that point, if he hadn't been saving her sorry ass, they'd have been perfectly in sync. "Aidan." She could see the fire reflected in his eyes. _Blake's Girl_ was really blazing now. "My God, we almost—"
"I know." His short, dark hair was plastered to his head. Water ran in rivulets down his face, which was starkly pale. His long, inky-black eyelashes were spiky, and he had a cut above one eyebrow that was oozing blood. In spite of all of that, she had the most ridiculous thought: _wow,_ he looked good all fierce and intense and wet.
Aidan Donnelly, first real boyfriend. First...everything... She could hardly believe it, certainly couldn't process it,so she craned her neck, staring at the boat that looked like one big firecracker. "It just blew, and I—"
"Kenzie—"
"—I mean one minute I'm sitting there missing my brother, and the next..."
He looked into her eyes, his cool and composed. "It's going to be okay, but I need you to—"
"And it blew. I was just sitting there, surrounded by his things, missing him, and then _boom_. My Choos are probably halfway to China by now. I really liked those Choos."
"Kenzie," he said in a tone of authoritative calm. "I need you to listen to me now. Can you do that?"
She could take a gulp of air. But listening? The jury was still out on that one. Her ears were ringing. And the water was so damn cold. In fact, she was shaking and hadn't even realized it, shudders that wracked her entire body and rattled her teeth.
"Hold onto me, Kenzie. That's all you have to do, okay? Just hold onto me."
_Right._ Hold onto him. She'd grown up here in Santa Rey, and once upon a time she'd held onto him plenty. She'd held onto him, laughed with him, slept with him...
Actually, there'd never been much sleeping involved between them, a thought which brought an avalanche of others. Him fresh out of the firefighters' academy and possessing a body that had made her drool, not to mention the knowledge of how to use that body to make hers go wild...
But that had been what, six years ago? Hell, she could barely think, much else handle any math at the moment, so she couldn't be sure.
He was towing her out, away from the boat and any danger of falling debris, while shouting something to two firefighters on the other side of the burning vessel, both of whom had hoses on the fire.
She'd been in a fire before. On the set of her soap opera, _Hope's Passion,_ before it'd been cancelled. But that was under carefully controlled circumstances. This wasn't a TV show with lines for her to follow. This was the real thing, with no makeup department standing by to color in pretend injuries, dammit.
She'd have loved a script right about now, with a happy ending, please.
At least she was still breathing.
Hard to beat that.
_Blake's Girl_ hadn't gotten so lucky.
Neither had Blake. Oh, yeah, _there_ was the familiar rush of pain, slicing right through the numbness from the cold water, lancing her heart—the pain that had been with her since she'd learned Blake was dead. Making it worse, adding confusion and anger to her grief was the fact that he'd been accused of being an arsonist and murderer.
_God, Blake..._
Another chunk of burning debris fell from the still flaming boat, and she imagined it was something of Blake's, something she'd never see again. Or maybe it was her own suitcase, or her laptop, which wasn't a big loss in the scheme of things, but it held the scripts she'd been writing...
At least if she died, she would no longer be a freshly unemployed soap star.
It was so damn ironic—she'd never been able to come home when Blake had been alive because she'd been too busy working. Then days after he'd died, her soap had been cancelled. Now she could drive up all she wanted, and he was gone... Her first trip home in forever and it had been to see after his things, things that were now smoldering in the water around her.
"Don't give up on me," Aidan said. His eyes focused ahead on where he was swimming to, some point invisible to her. It was too dark to see their color clearly but she knew them to be a light brown with flecks of green that danced when he laughed.
He wasn't laughing now.
_Nope._
He glanced at her, then resumed swimming straight and sure, moving them away from the flames, which also meant away from any warmth, while she did as he'd asked and just held on. She could do nothing but. Like old times...
Why did it have to be _him,_ the guy who'd crushed her heart, stomped on her pride and then walked away from her without a backward glance?
Did _he_ hurt over the loss of Blake?
Did _he_ believe the lies?
Because that thought, and all the others that came with it, came close to defrosting her, she shoved them aside. The blessed numbness was working for her. She hadn't come to Santa Rey in the past six years, but Blake had visited her in L.A. on the set, whenever he could, and on top of his visits, they'd been in frequent contact by e-mail, texting and phone calls, and had remained close despite their physical distance. He was the only family she'd had.
And now he was gone.
Forever gone.
"Kenzie? You still with me?" Aidan's lean jaw was tight with tension and was scruffy, as if he hadn't had time to shave in a day or two. Or four.
"Unfortunately." She'd like to be anywhere but "with" him. She could feel his longer, stronger legs moving, bumping into hers, and it made her irrationally mad. She didn't want help, not from him, but when she wriggled free to prove herself fine, she went down like a stone. Straight beneath the surface of the icy water, where she promptly did the stupid thing of opening her mouth to breathe and got a lungful of extremely cold salt water for her efforts.
Thankfully, she was immediately hauled back up again and pulled against a hard chest, one hand fisted in the back of her shirt, the other arm across the backs of her thighs in a grip that could have rivaled Superman's.
Firefighter to victim.
Not ex-boyfriend to ex-girlfriend.
And wasn't that just the problem? Once upon a time he _really_ had had her, only he'd been the one to let go. He'd done it, he'd said, because of their respective careers and because he didn't like hiding their relationship from his friend Blake, but she knew the truth. It was because he'd decided she'd been falling in love with him and he hadn't been ready for love, so he'd shooed her away and had moved on.
She'd hated him for that for a good long time, for not giving himself a chance to feel what she'd felt, and, yeah, he'd been right—she _had_ been more than halfway in love with him. It'd taken a while,but eventually her anger had drained, and she'd acknowledged that he'd been right to break it off with her before she'd gotten even more hurt... But that hadn't eased her pain at the time.
Maybe she should consider herself lucky they were doing this reintroduction in an official capacity—him on the job, and her being just one in a blur of people he rescued. Less personal.
"Stop fighting me." His voice cut through the shocking noise of the night: the sirens, the shouting of the other firefighters and personnel, the ever-present, horrifying crackling of the flames, the small waves smacking into each other, waves that would be cresting over her head if it wasn't for Aidan's holding her with what appeared to be little to no effort. "I've got you."
"I don't want you to have me."
"Okay, roger that. But at the moment you don't have a choice."
"Of all the firefighters in this damn town..."
She thought she caught a flash of a grim smile. So he was no more thrilled than she was. He wasn't even looking directly at her, his attention instead focused on the boat behind her, and the dock behind that, reminding her that not only was he saving her hide, he was simultaneously looking for other people who needed help.
"I was alone on the boat," she told him.
"What were you doing?"
"Saying good-bye to Blake."
Sorrow, regret, and anguish all briefly flashed in his eyes. "Kenzie—"
"He didn't do those things you're all accusing him of, Aidan."
She had his attention now, all of it, and she'd forgotten the potency of having Aidan Donnelly giving her one-hundred-percent of his focus. _"He didn't."_
"Did he say something, anything to you at all, before he died?"
Died...Hearing the words from his mouth made Blake's death all the more real, as did being back here in her hometown, and it hit her hard. Throat so tight that she couldn't speak, she shook her head. No, Blake hadn't said anything at all, which made her feel even worse. "It wasn't him who set those fires. I know it."
"Kenzie," he said very gently, but she didn't want to hear it, didn't want to hear anything he said, so she shook her head again and closed her eyes, which brought an unexpected and horrifying sense of vertigo, making her clutch at him. "I want out."
"I know. They're coming for us right now."
That was good. Because something was definitely wrong. Her vision was getting fuzzy. Her brain was getting fuzzier. Scared and a little overwhelmed, she pressed her face into the crook of his neck, her nose to his throat, the position hauntingly familiar and at once flooding her with memories.
She'd been here before.
Okay, not here, not in the water, freezing, scared, but she'd been held by him, had pressed her face against his warm flesh and inhaled him in, absorbing the way he held her close, as if he'd never let anything happen to her.
He smelled the same, a scent she'd never quite managed to forget, and it was messing with her brain in spite of the fact that she'd just survived an explosion, a nighttime swim in the freezing ocean, and an uncomfortable reunion with the one and only guy she'd ever let break her heart.
_Dammit._ She blamed Blake. _Blake..._
"Kenzie." Aidan gave her a little shake. "Stay with me now."
_No, thanks..._
"Open your eyes," he demanded. "Come on, Kenzie. Stay awake, stay with me."
As opposed to giving in to the delicious lethargy slowly taking over? _Nah..._ "Too tired."
"I know, but you can do this. You can do anything, remember?"
She nearly smiled at the reminder of her own personal motto, but then remembered who was talking. Yeah, she'd once believed that she could do anything, with him at her side.
He'd proved her wrong.
_Oh, boy._ Her eyes _were_ closing. It'd be so easy to let them, to just drift off and not feel the cold anymore, but even in her fuzziness, she knew that was bad, so with great effort, she pried her eyes open.
And her gaze landed on him. The last time she'd seen him, she'd been so young. _They'd_ been so young. She'd just turned twenty-two, been signed by a Los Angeles agent, and had landed her first small walk-on role. He'd been two years older, fit and gorgeous, and on top of his world as a young firefighter.
Plastered against him, her hands clenched on his biceps, her legs entwined with his, her chest up against him the way it was, she could feel that he was still fit.
Very fit.
And thanks to the flames and also the spotlights from the guys on the dock keeping track of them, she also knew that he was still gorgeous. If he hadn't cut her loose without a backward glance, she'd be happy to see him.
_Very_ happy.
A group of firefighters had made their way through the flames to the end of the neighboring dock, and had secured it with criss-crossing lines of water. One of them leaped into the ocean, and with long, sure strokes swam toward them.
"Here," he called out to Aidan, holding out an arm for Kenzie.
"I've got her," Aidan said.
But Kenzie had had enough, of Aidan and his capable, strong arms, of his scent and especially of the memories. So she reached out for the second fire-fighter, going into his arms without looking back, arms that had never held her before, arms that didn't know her, arms that didn't evoke the past.
Even though she wanted to, she wouldn't look back.
# _CHAPTER 3_
BY THE TIME AIDAN HAULED himself out of the water, Ty had handed Kenzie off to the EMTs. Dustin and Brooke took her away from the flames and straight to their ambulance.
_Good._
Chilled, drenched to the skin, Aidan made his way through the organized mayhem to his rig, where he stripped down and pulled on dry gear, the questions coming hard and fast in his head.
What the hell had Kenzie been doing there? Odd timing, given that in all these years, she'd not shown up in Santa Rey, not once. At least that he was aware of. Blake had never mentioned any visits, but then again, why would he? He'd had no idea that Aidan had dated his baby sister, and then walked away rather than engage his heart. They'd never told him, knowing he wouldn't have liked it.
Nope, Kenzie hadn't been back, not even for Blake's memorial service, and yet suddenly here she was, on Blake's boat, a boat that just happened to blow sky high once she'd set foot on it.
Odd coincidence.
During the time the two of them had been in the water together, the sky had lightened. Dawn had arrived. The chief had put an explosives team in place, and had a plan to contain the fire. Aidan needed to get back into the thick of it, but first he had to see Kenzie and make sure for himself that she was okay. She'd had a head laceration and multiple cuts and wounds, and that had been before he'd tossed her into the water.
He looked through the horde of people working the flames—Eddie and Sam, Aaron, Ty and Cristina, plus the guys from Thirty-Three, all on hoses and past the explosives experts surveying the still burning shell of _Blake's Girl_ to where the ambulance was parked.
Kenzie was seated at the back of the opened rig between Dustin and Brooke. She was dripping everywhere, her clothes revealing what he already knew, that she was petite and in possession of a set of mouth-watering curves that had gotten only more mouth-watering in the past few years. She wore layered tees, the top one pink, ribbed and long-sleeved, unbuttoned to her waist, the one beneath white with pink polka-dots, opened to just between her breasts, both soaked through and suctioned to her body enough to expose her bra, which was also pink, lace and quite sheer.
He'd been a firefighter for years and he'd rescued countless victims, many female, some of whom had been as wet as Kenzie, and never, not one single goddamn time, had he ever stopped in the middle of a job to notice their breasts.
It was his first clue that he was in trouble, deep trouble—but when it came to Kenzie, that was nothing new. He chose to ignore his observation for now, for as long as he possibly could. His gaze dropped past her shirt with shocking difficulty, to a pair of button-fly jeans low on her hips, also dangerous territory because he'd always loved her legs, especially how bendy they could get...
_Don't go there._
She shoved her hair out of her face, which still looked far too pale, even a little green, although that didn't take away from her beauty. Once upon a time she'd been a gorgeous study of sexy, frou-frou feminine mystery to him.
Some things never changed.
As if she felt his gaze, she looked up, and from fifty feet, between which were other firefighters, equipment and general chaos, she found him.
Between them the air seemed to snap, crackle, pop.
Six years ago, the thought of a long-distance relationship had been as alien to him as a close-distance relationship, and he'd told himself he had no choice but to break things off, even though that had really just been an excuse.
He'd broken things off because she'd scared him, she'd scared him deep. And apparently, given the hard kick his heart gave his ribs, she still did.
She'd been able to get inside him, make him feel things that hadn't been welcome, and, yeah, he'd run like a little girl.
He felt like running now.
But this time it was Kenzie who turned away. Dustin unfolded a blanket and wrapped it around her shoulders, while Brooke checked her pupils, then dabbed at the various cuts on her face.
Kenzie sat still, eyes closed now, looking starkly pale but alive.
Alive was good.
She huddled beneath the blanket, cradling a wrist, nodding to something Brooke asked her. Aidan knew that Brooke and Dustin, both close friends, would take good care of her. They took good care of everyone, which meant that Kenzie was in the very best hands.
Still in the thick of the organized chaos around him, Aidan took a second to let his gaze sweep over her. She really did seem as okay as he could hope for, and he told himself to turn away.
He was good at that. After all, he'd learned to do so at a young age from his own family, who'd shuffled him around more than a deck of cards on poker night. Yeah, he was good at walking a way. Or at least good at pretending he didn't care when others walked away from him.
And after all, he'd done the same to her.
God, he'd been cruel to her all those years ago. Not that he'd meant to be. Going through the academy had been a life lesson for him. He _could_ belong to a "family." He _could_ make long-lasting friends. He _could_ love someone with all his heart.
But loving his fellow firefighters like the brothers they'd become was one thing.
Loving Kenzie had been another entirely.
Since she'd left, he'd seen her only on TV. As a rule, he didn't watch soaps. He didn't watch much TV at all, actually. If he wasn't working, he was renovating the fixer-upper house he'd bought last year, emphasis on _fixer-upper._ If he wasn't doing that, he was playing basketball, or something else that didn't cost any money because the fixer-upper had eaten his savings.
But there'd been the occasional night where he'd sat himself in front of a game and caught a promo for Kenzie's soap. There'd also been the few times at the station where one of the guys had flipped on the TV during her show.
Three times exactly—and yeah, he remembered each and every one. The first had been five years ago, and she'd been wearing the teeniest, tiniest, blackest, stringiest bikini in the history of teeny-tiny black string bikinis, her hair piled haphazardly on top of her head with a few wild curls escaping, looking outrageously sexy as she'd seduced her on-screen lover. It'd taken him a few attempts to get the channel changed, and even then it hadn't mattered. That bikini had stuck with him for a good long while.
The second time had been a few Christmases back. She'd been wearing a siren-red, slinky evening dress designed to drive men absolutely wild. She'd been standing beneath some mistletoe, looking up at some "stud of the month." Aidan hadn't been any quicker with the remote that time, and had watched the entire, agonizing kiss.
The third time had been for the daytime Emmys. She'd accepted her award, thanking Blake for always believing in her, and then had thanked some guy named Chad.
Chad.
What kind of a name was Chad?
And where was Chad now, huh? Certainly not hauling her off a burning boat and saving her cute little ass. Guys named Chad probably only swam when playing water polo.
In the ambulance, Dustin said something to Kenzie, and she opened her eyes, flashing a very brief smile, but it was enough.
She was okay.
Aidan forced himself to move, to get back to the job at hand, and it was a big one. The explosions had caught the boats on either side of _Blake's Girl,_ escalating the danger and damages. They had the dock evacuated, and as the sun streaked the sky, they were working past containment, working to get the flames one-hundred-percent out.
With one last look at Kenzie, Aidan entered the fray.
* * *
IT TOOK HOURS.
Aidan and his crew piled into their rigs just as the lunch crowd began to clutter the streets of Santa Rey. If he closed his eyes, he could still feel the imprint of Kenzie in his arms. He'd held onto her for what, three minutes tops? And yet she'd filled his head and his senses, and for those one-hundred-and-eighty seconds, time had slipped away, making him feel like that twenty-four-year-old punk he'd once been.
He'd been with Kenzie for one glorious summer, and she'd wanted to stay with him, which should have been flattering. She'd wanted to wear his ring and have a house and a white picket fence.
And his children.
But it hadn't been flattering at all.It'd been terrifying.
So he'd acted like a stupid, shortsighted guy. There was no prettying that up, or changing the memory. Fact was fact. He'd gotten a great job, and he'd had the world at his feet, including, he'd discovered, lots of women who found his chosen profession incredibly sexy.
He'd not been mature enough to realize what he already had; he'd been a first-class asshole. He'd sent Kenzie away, pretended not to look back and had filled his life with firefighting, women, basketball, wood-working, more women...
A hand clasped his shoulder. "Hey, Mr. 2008. Home sweet home."
"Shutup."They'd pulled into the station. He hopped out of the rig and went straight to Dustin, who was cleaning out the ambulance." The victim? How is she?"
Cristina poked her head out from the station kitchen. "Hey, guys, there's food—" At the sight of Dustin, who she'd gone out with several times before unceremoniously discarding him without explanation, she broke off. "Oh. _You're_ here."
Dustin looked at her drily. "What, is the food only for the staff that you _haven't_ slept with and dumped?"
Aidan winced at the awkward silence, and if he wasn't in such a desperate hurry to hear about Kenzie, he might have refereed for the two of them, because if anyone needed refereeing, it was these two. "The vic," he said again to Dustin.
"Sorry," Dustin said, turning back to him. "She's not bad, thanks to your quick thinking. A few second-degree burns, possible broken wrist, some lacerations."
"Her head trauma—"
"No concussion."
"Stitches?" he demanded, causing Dustin to take a quick glance at Cristina, who raised an eyebrow.
Aidan knew he was bad off when the two of them could share a worried look over him.
"No stitches," Dustin said. "You okay?"
"Yeah." Aidan took his first deep breath in hours, which prompted another long look between Dustin and Cristina.
"You sure?" Cristina asked.
_Jesus. "Yes."_ Leaving them alone to work through their issues, he headed inside the station. After he'd showered, cleaned up and clocked out, he got into his truck and debated with himself.
Home and oblivion were attractive choices.
Or he could go to the hospital, see Kenzie and get a question or two answered.
Not quite as attractive, because nothing about sitting with Kenzie and looking into her soulful eyes was going to be simple. Nope, that was a guaranteed trip to Heartbreak City.
Home, then, where he wouldn't have to do anything but fall facedown into his bed. Yeah, sounded good. He put his truck in gear.
And drove to the hospital.
* * *
KENZIE OPENED HER EYES and stared at a white ceiling. She was on a cot in the emergency room, her cuts and burns all cleaned and bandaged, her wrist wrapped, her head stitched back on—okay, so it'd only needed butterfly bandages. Now she was being "observed," although for what, she had no idea.
At least she was warm again, or getting there. She had three blankets piled on top of her, which helped, and a hospital gown, which didn't.
She'd just seen the fire investigator, Mr. Tommy Ramirez.Tommywasshort,dark,andquitetothepoint. The point being that he'd found it extremely odd that she'd been on Blake's boat at the time of its explosion.
She did, too, considering she'd only gotten to town that night. Closing her eyes, she frowned. She also found it odd that he was wasting his time questioning her instead of investigating the real perpetrator of the arsons, because her brother was _innocent_. No way had Blake set all those awful fires they were trying to pin on him. Blake, sweet, quiet, loving Blake, the brother who'd been there for her when their parents had died fifteen years ago, when they'd gone through foster care, when she'd wanted to go off to Hollywood. He'd never have hurt a fly much less purposely hurt another human being. And endanger a child?
Never.
God, she hated hospitals. They smelled like fear and pain and helplessness, and all of them combined reminded her of her own uncertain childhood. She wished she was back on the L.A. set of _Hope's Passion,_ acting the part of the victim instead of really being one. Comfort food would help. Maybe a box of donuts—
From the other side of her cubicle curtain came a rustling, and then the hair at the back of her neck suddenly stood up, as if she was being watched. Opening her eyes, she blinked the room into focus. Everything was white and... _blurry_. But not so much so that she missed the back of a guy's head as he ran off and out of sight. "Hey!"
He hadn't been wearing scrubs but a red T-shirt, so he couldn't have been hospital staff. Who'd come to see her and then leave without a word? She struggled to think but she was so tired, and a little woozy still, and when she let her eyes drift shut, she ended up dozing off...
* * *
"NOT THE SAME TYPEof point of origin as the other fires."
Kenzie opened her eyes and turned her head, taking in the curtain, now pulled all the way closed around her cot. She was a woman who liked change, who in fact thrived on it, but she had to say, she didn't like this change. Not at all.
How much time had passed?
"So you're saying what, Tommy, that the chief has you on a gag order?"
_Oh, boy._ She didn't need to peek around the curtain to know _that_ voice. That voice had once been the stuff of her daydreams, of her greatest fantasies. That voice had used to melt her bones away and rev her engines.
_Aidan._
"I'm not saying anything," Tommy said. "Except what I told Zach weeks ago. I'm on this. It's a kid glove case. So you need to back off."
"I want to see Kenzie when she wakes up."
_He'd_ been the one who'd looked in on her? She didn't know how she felt about that. Had he seen her sleeping? Had she been snoring?
Why hadn't he come back when she called out?
"Tell me this much at least," Aidan said, presumably still to Tommy. "Did either you or the chief even know Blake had a boat?"
"No, but I was waiting on a full investigative report from the county, and it would have shown up on there."
"And then you would've what, seized the property as evidence?"
"Yes, of course. To search it, just like we've done with his house. All the current evidence regarding the case points to Blake being in on the arson."
_In on the arson._ Kenzie absorbed the odd choice of words. Did he mean that he thought there could be more than one arsonist?
"So who beat you to the boat, Tommy? Who wanted to make sure there was no chance of extracting any evidence from it?"
The answer actually gave Kenzie hope—because it meant that someone _else_ could possibly be proven to be responsible for the arsons, maybe even someone who'd framed Blake.
"There's been at least seven highly destructive fires," Tommy said. "Adding up to millions of dollars in damages. The chief's ass is on the line, and so is mine. If Blake was still alive, he'd be behind bars. That he's not doesn't change anything. The investigation is ongoing."
"But it's possible he was working with someone," came Aidan's voice. "Is that what you're saying?"
"No comment."
"Do you know who?"
_"No comment."_
"You know something's off, Tommy, or you wouldn't be here."
"Yes," the investigator agreed tightly. "Something is off, and..."
Their voices lowered to a whisper. She leaned toward the curtain, but they were talking so quietly now she couldn't hear anything but...her name. Definitely, she'd heard her name.
Why were they talking about her?
She scooted even closer to the edge of the cot and cocked an ear, but still couldn't hear anything. _Dammit!_ Blake couldn't have done any of those things they'd accused him of. She knew it, and she was going to prove it herself if necessary, starting with eavesdropping on this conversation. Tommy said something Kenzie couldn't quite catch, so she leaned even further, and—
Fell off the cot to the floor. _"Ouch."_
At the commotion, the curtain whipped open. She tried to push herself upright but with one wrist useless and the other pinned beneath her, she was pretty much a beached fish. A nearly naked beached fish, with her butt facing a crowd of three: Tommy, the nurse and, oh, perfect—Aidan. She could see the tabloids now: Ex-Soap Star Mackenzie Caught Panty-less. "Ouch," she said again and rolled to her back, gasping when the cold linoleum hit her bare backside. She sighed just as someone dropped to his knees at her side, and then Aidan's face swam into her vision.
"Are you okay?" he demanded.
_Sure._ Sure, she was okay. If she didn't think about the fact that she'd just mooned him.
"Here." After helping him get her back on the cot, the nurse fussed a moment, checking all of Kenzie's various injuries. Luckily, Tommy had backed out of the room, vanishing, for now at least.
"What the hell were you doing?" Aidan demanded when the nurse left them alone, too.
"Oh, a little of this, a little of that—" Realizing her gown was twisted very high up on her thighs—which, of course, was nothing to what he'd just seen—she grabbed her blanket and tried to cover herself up. A little like closing the barn door after the horse had escaped, she knew, but she was mortified. Except the movement made her want to throw up, and she reached up, holding her head tightly.
"Here." He took over the task of covering her, quickly extricating his hands when he was done, not quite meeting her gaze as he sat at her side.
Awkward moment..."So," she said. "What are you doing here?"
"Looking in on you."
Yep. And he'd gotten to look in on far more than he'd probably intended.
"Are you all right?" he asked.
"Depends on your definition of _all right_."
At that, his eyes cut to hers and he sighed, scrubbing a hand over his face, his fingers rasping over the growth there. He looked and sounded exhausted. "I'm sorry, Kenzie."
"For what? That I just mooned you, or that I'm here at all?"
Aidan got to his feet, pulling the curtain shut again to give them privacy, privacy that she wasn't sure she wanted.
He'd changed his clothes. He wore a pair of jeans now, loose on his long legs, low on his hips, with a long-sleeved shirt unbuttoned over a gray T-shirt that seemed to emphasize his broad shoulders and tough, athletic build. "Your shirt isn't red," she said slowly.
"What?"
"Before, somebody in a red shirt was looking at me."
"When?"
"I don't know." She rubbed her temples. "I'm out of it."
"It was a tough night."
"Yeah." But _he_ didn't look like he'd just worked his ass off and managed to save her life to boot; he looked casual, relaxed.
Cool as a cucumber.
And so hauntingly familiar, not to mention gorgeous, that she couldn't keep her eyes on him. How unfair was it that he'd gotten even better-looking with age? "Thanks for stopping by, Aidan, but you can see I'm fine. You can go."
He looked doubtful.
"Seriously. I'm really okay."
She almost had him, she could tell, but then she ruined it by shivering.
Without a word, he grabbed another blanket and settled it over her. She appreciated his sense of duty, but what she would appreciate even more would be his vanishing.
Or her.
Yeah, that might be better. If she could just vanish on the spot. _Poof._ "Okay, now I'm good, thanks. Really."
"Really?"
"Yes. I mean you can't even look at me, so—"
Lifting his head, he met her eyes, his hot enough to singe her skin.
"Oh," she breathed, feeling her heart kick, hard.
"I can't look at you?" he repeated in low disbelief. "Are you kidding me? Kenzie, I can't do anything _but_ look at you."
# _CHAPTER 4_
AT AIDAN'S WORDS, Kenzie's breath caught and held. She didn't know how to take him, especially the way he was looking at her, as if maybe he could see all the way through her, to her heart and soul, right to the very center of her being, where all the hurt was so carefully bottled up.
She'd gotten over him. Years ago. She really had. She'd gotten over how he'd once made her laugh, made her think, made her happy...
Made her come...
No way could he possibly reach her now. Not with that hard body, not with the look in his eyes and definitely not with the memories.
Okay, maybe the memories got to her, just a little bit. For one glorious summer, he'd been the best part of her life—before he'd walked away without so much as a glance back, that is.
_Good. There_ was her anger, which would hopefully negate the fact that he was standing right here in the flesh looking good enough to...well...That thought made her want to sweat. But apparently she could be both over him and turned on by him at the same time, which confused her to say the least. She had no idea what that was about. No idea at all.
_None._
She'd moved on years ago from that young, sweet, innocent girl. Now she was a woman with a backbone of sheer steel that had gotten her through some tough times.
She knew people tended to look at her carefully cultivated outer package—thank you, stylist to the stars—an outer package that was petite and willowy, even fragile-looking, and completely underestimate her.
But on the inside she was one-hundred-percent survivor, thank you very much. She'd lived through losing her parents early, through a happy-as-it-could-be teenage-hood with just Blake. She'd lived through being in the public eye, through the ups and downs of TV fame and most recently, through the death of her brother. All of that would have cracked most women, but she wasn't easily cracked.
She would get to the bottom of this mess, no matter what she had to do in order to get there. _No matter what._ Even if she had to use her beauty, her checking account, her damn body.
She would do it.
Whatever it took.
_For Blake._
"I heard you talking to the investigator," she said softly.
Aidan's eyes met hers, and she wished like hell she could read his mind. But she couldn't, and he didn't say another word to help.
"I think he's wondering if I'm guilty of something."
He just looked at her some more.
"The only thing I'm guilty of is knowing that he hasn't done his job if he thinks Blake did those things."
At that, his face softened, and regret filled his eyes, along with a grimness that had her shaking her head before he even spoke.
"Don't say it," she warned, not willing to hear it, not from him. Not from anyone. Not when she was this close to a breakdown. A grief breakdown. "Don't." She _knew_ Blake, goddammit. She did. She didn't remember much about her parents before they'd died in a car crash, but she remembered Blake. Every bit of him. He was the boy who'd held her hand every time they'd had to move to a new foster home. He was the teenager who'd punched a boy in the face when he'd hurt her, he was the man who'd believed in her enough to work double shifts to pay for her publicity shots so she could pursue her acting dream.
He could _never_ have committed arson. She'd have sworn Aidan would have known that as well, but apparently she was wrong.
"There's evidence—" he began, but she shook her head.
"Circumstantial." She swallowed hard but a lump of emotion, the one that had been there since Blake's death, remained. "I see that you're no better a friend than you were a boyfriend."
He opened his mouth, but before he could respond, the nurse pulled aside the curtain and entered the cubicle, followed by a doctor. "Everyone out," the nurse ordered.
"I'm the only one here," Aidan said.
"So get out," the nurse responded sweetly.
Kenzie closed her eyes and lay back. She didn't look at Aidan again; in fact, she didn't open her eyes until she heard the rustling of the curtain, signaling he'd left.
Which was fine. Perfect, really. Because she'd sure as hell rather be alone than look into his eyes and see things she didn't want to see.
* * *
AIDAN EXITED the emergency room, feeling like a class-A jerk. Though how that was possible, what with his saving her life and all, he had no idea...
Okay, he knew.
She'd seen the look in his eyes; she'd understood something she hadn't wanted to understand—that he knew Blake was involved with those arson fires.
Aidan felt torn up about it, sick over it, but facts were facts. Blake had been placed at the scene of each arson by various witnesses. He had been depressed since losing Lynn, his partner before Cristina, in a fire the year before. His home had been seized and searched, and in his garage they'd found a stack of wire mesh trash cans, similar to the ones identified as the point of origin in each of the arsons.
Most damning, Aidan's partner, Zach, had also seen him holding a blowtorch just moments after Zach's house had been set on fire, with Zach and Brooke inside. Zach had almost died there.
And Blake _had_ died there, perhaps deliberately. He'd died, leaving all of them, Zach, Aidan and the other firefighters, even Tracy, the woman he'd had such a crush on, everyone, destroyed.
Kenzie was in denial. He got that. She was angry. He got that, too. She needed someone to vent that anger at, to place it on, and he'd been handy enough.
_I see that you're no better a friend than you were a boyfriend._
Yeah, that had been a direct hit. Having her look at him as if _he_ was the bad guy had really gotten to him, especially considering he still had the scrapes and bruises from saving her.
The late afternoon sun was sinking fast, cooling off the day. Having been up for two straight days now, he desperately needed sleep. He could close his eyes standing up right there in the hospital lot, and not wake up if a cyclone hit. He was so tired that he'd probably sleep completely dreamless. Well, except for maybe dreaming about Kenzie's bare ass. Yeah, now that he'd seen that again, he'd most likely dream about it for a good many hours.
Days.
Years.
"Aidan."
_Hell._ Tommy was leaning up against Aidan's truck, a file in his hands, mouth pinched tight, looking as if he had plenty of things to say, and all fantasies abou Kenzie's ass vanished. "What now?"
"I wasn't aware that you knew her personally."
"Who?"
"Come on, Aidan. Don't play with me. Mackenzie Stafford. You didn't say that you knew her."
He sighed. "So?"
"So it felt to me like maybe you knew her... _well_."
"Yeah. Once upon a time."
"Okay, and so once upon a time, did you know she was Blake's sister?"
Getting into tricky territory here. No one had known he and Kenzie had dated in the past. It'd been a quick, hot thing, _very_ hot, and he certainly hadn't been in any hurry to tell Blake he'd gotten his sister in bed. Kenzie hadn't told Blake, either, for her own reasons, and then when Kenzie had gone off to Los Angeles, it hadn't mattered anymore.
Did it matter now, with Blake dead? He couldn't see how it did. "Yeah, I knew she was Blake's sister."
"Did you know that boat was Blake's?"
"Where are we going with this, Tommy?"
"Did you?"
Aidan let out a breath. "Not until we were in the water and she told me."
Tommy nodded. "Because you always sit around with someone you're rescuing and chat about property ownership."
"I asked her why she was there, on that boat. I was under the impression that she was in Los Angeles."
"Yeah?" Tommy's eyes studied him, considering. "So just how well do you know her?"
"Irrelevant."
"I wonder if Blake would have thought so."
Aidan fished his keys out of his pocket. "I'm going home to sleep. For many, many hours. When I'm back on duty you can drill me all you want. Maybe I'll be able to think more clearly."
"Maybe I don't want you thinking more clearly."
"And what the hell does that mean?"
"It means I need answers now. Did you know she was staying on the boat? Did you maybe visit with her there before the fire?"
"I told you. No. And no."
"Ms. Stafford thinks Blake is innocent. That he was not only framed but possibly murdered, and she intends to prove it."
_Soundedright._ Kenzie might look like a pretty ball of fluff, but she had sharp wits and was loyal to a fault. She also had the tenacity of a bulldog. Once she got her brain wrapped around an idea, there was nothing anyone could do to change her mind. Not about falling in love with him, not about being an actress and most definitely not about believing that Blake couldn't be guilty of arson.
"So the question stands," Tommy said quietly. "How well do you know her?"
"Did." Well enough that when he'd looked into her eyes, he'd felt an odd stirring, a sensation almost like coming home. Yeah, once upon a time he'd known her well. As well as he'd known anyone. "Past tense."
"Good enough."
"For what?"
"To get you to tell her to stay the hell out of this investigation and not interfere."
"People don't tell Kenzie what to do."
"You're going to. Because the chief has put out the word. If anyone hinders this investigation, we'll have them arrested, Blake's sister or not."
_Great. Perfect._ If Aidan told her that, she'd jump in with both feet, because one thing he remembered and remembered well—nothing scared her. Nothing. "Seriously. It's not a good idea for me to tell her anything."
"Well, then, I hope she has bail money."
_Shit._ Aidan watched Tommy walk away, then he turned to his truck. Needing sustenance before he passed out cold for at least the next twelve hours straight, he stopped at Sunrise, the café that was the perpetual hangout for everyone at the station. The two-story building was right on the beach. Downstairs was food central, while the second floor was the living quarters for Sheila, the owner. The rooftop was the place to go to view the mountains, the ocean, the entire world it seemed, and to think.
Stepping inside, his sense of smell immediately filled with all the aromas he associated with comfort: coffee, burgers, pies...Sheila smiled at him, and as the sixty-two-year-old always did, fawned over him as he imagined a mother would.
His own mother wasn't too into fawning, at least not over him. She'd divorced his father when Aidan had been two, and he'd spent most of his childhood years being shuffled from family member to family member while she'd relived her wild youth. Granted, he'd been more than a handful of trouble, purposely going after it in a pathetic bid for attention, so in hindsight he didn't blame anyone for not keeping him around for long.
Eventually, he'd ended back up at his dad's, where the two of them had spent a few years doing their best to tolerate each other until, when Aidan had been fifteen, his dad had remarried and promptly given his new wife three babies in a row.
Aidan had landed at his mom's once again, a little bit rebellious and a lot angry, but by then his mother had settled down some, remarrying as well.
Now Aidan had five half brothers and sisters, and didn't quite belong on either side of the family.
Not that he'd had it as rough as Blakeand Kenziehad. He knew exactly why the brother and sister had been as close as they had, and exactly why Kenzie would fight tooth and nail to prove her brother's innocence.
What he didn't know was how to convince her to let the law handle things, or if he even had a right to ask such a thing of her.
Between a rock and a hard place.
He ate his fill, and by the time he set down his fork, he felt halfway human. He still needed his bed, badly, but with Tommy's words echoing in his head, he knew he had to try to talk to Kenzie again first. He needed to warn her to let Tommy do his job. For old times' sake.
Or so he told himself.
He pulled out his cell phone and called the hospital, but was told she'd been released.
Where would she go? Back to Los Angeles? No, she wouldn't leave Santa Rey, not until she did what she'd come to do, which was prove Blake's innocence, so he asked Sheila for the local phone book and a slice of key lime pie, both of which he took up to the roof. Sitting facing the ocean, he began calling. But as it turned out, Kenzie wasn't registered at any of the three hotels in the area, probably because there were two conventions in town and everything was fully booked. He looked at the remaining list of several dozen motels and B and Bs, and sighed. He'd made his way through the most likely candidates when Sheila came out on the roof with a fresh mug of coffee.
"What's up for you tonight?" Even with her bouffant hair, she barely came up to his shoulder. "You planning on saving any more damsels in distress?"
He didn't bother asking her how she knew about last night's fire—the gossip train in Santa Rey was infamous. "No damsels, distressed or otherwise. I have a bed in my immediate future."
"You sleeping alone these days?"
_Unfortunately, yeah._ The last woman he'd gone out with had found someone else, someone with more money and more time, and he'd gotten over her fairly quickly but hadn't yet moved on. He couldn't tell that to Sheila, though, or she'd set him up with her niece, as she'd been trying to do all year...
"My niece would be perfect for you, Mr. 2008."
He winced. "You saw the calendar."
"Honey, I saw, I bought, we all drooled. Now about my niece..."
Her niece was divorced with four kids, and while she was a very lovely woman, a waitress at Sunrise, in fact, he wasn't anxious to help create yet another fractured family. "I'm sorry, Sheila. But at the moment, I'm—"
"Enjoying being alone," Sheila finished for him with a sigh. "Yeah, yeah, I've heard it before."
Standing, he handed her back the phone book, then gave her a hug." How about you? You could marry me."
She cackled good and long over that one, and walked to the roof door. "If I was thirty years younger, you'd be sorry you said that..."
He laughed, but his smile faded fast enough. With no idea how to track down Kenzie, he left and drove home, thinking he'd just go horizontal for a little while and then figure it out, but as he drove up to his house, he saw a red convertible Mercedes Cabriolet in his driveway.
And the outline of a woman sitting on his porch, lit from behind by the setting sun.
She was wearing two hospital gowns layered over each other and a pair of hospital booties, reminding him that her clothes had gotten sliced and diced pretty good and probably any luggage she'd had on the boat was long gone.
Her hair, wild on the best of days, had completely rioted around her face in an explosion of soft waves, the long side bangs poking her in one eye and resting against her cheek and jaw, where she had a darkening bruise that matched the one above her other eye, accompanied by a two-inch-long butterfly-bandaged cut. She was cradling her splinted left wrist in her lap. Her good hand was cut up as well, and so were both her arms—nothing that appeared too deep or serious, but enough to make him wince for her. Her legs were more of the same.
She was alone and beat up, and hell if that didn't grab him by the throat and squeeze. Then there were those melt-me eyes that lifted to his and filled.
_Jesus._ He thought he was so damn tough but one soft sigh from those naked lips and he was a bowl of freaking jelly.
She had a plastic bag beside her, and one peek at it tugged at him harder than he could have imagined given what he did for a living and how often he'd seen this very thing.
Her clothes from the fire.
Probably all that she had left here in Santa Rey. In her unsplinted hand she clutched a small prescription bottle, most likely pain meds. _Hell._ He was such a goner.
"I haven't taken any yet," she whispered, shaking the bottle. "Couldn't, because I took a cab from the hospital to the docks where I had my car, which I drove here."
"Kenzie—"
"You had a package. It was torn, so I looked in." She lifted one of a stack of firefighter calendars, with his own mug and half-naked body on the cover.
"Nice," she said, a ghost of a smile crossing her lips. "Mr. 2008."
He bit back a sigh. "It's for charity."
"And you definitely contributed." She waggled her eyebrows, then winced. " _Ouch._ I'm not allowed in Blake's house—evidence. And the hotels are all booked up, just my luck. Did you know you have a convention of dog trainers in town? Why are there five hundred dog trainers in Santa Rey?"
"Because we let dogs on our beaches."
"Oh." She sighed. "So we let dogs on our beaches, but not me into a hotel. Kinda makes sense when you think about it."
How that made sense, he had no idea.
"Because my karma sucks."
"Okay, come on." Gently, he pulled her up, taking the bag. Letting her hold onto the medication, he led her inside, telling himself he was going to give her Tommy's warning and that was it.
Other than that, he was going to stay out of it entirely.
But holding onto her, he realized she was trembling, and as he took her into his living room, she went directly for his couch, which she sank onto with a grateful little sigh. "I think she went on vacation."
"Who?"
"My karma." She gave him an exasperated look, like he wasn't listening to her, and then very carefully leaned her head back and closed her eyes.
"Hey." Squatting down before her, he put his hands on her thighs, looking into her eyes when she opened them. "You okay?"
She let out a sound that might have been a laugh, or a sob.
He hoped to God it was the first. "Rough twenty-four hours," he murmured.
Another nod, carefully slow and precise, giving her away. She definitely wasn't laughing. In fact, she was in pain, lots of it; rising, he went into the kitchen for a glass of water. Bringing it back to her, he pried the prescription drugs from her fingers, read the label—yep, painkillers—and shook one out.
"I'm okay."
"You don't look it. You look like hell."
"You say the nicest things."
With another sigh, he once again hunkered down at her side. "Look, you've been through a lot. I know you're alone and..."
"If you say helpless, I'll slug you with my good fist."
Once upon a time she'd been the most amazing thing in his life.
_The. Most. Amazing. Thing._
On the outside she'd been so mind-blowingly, adorably, effortlessly sexy. Inside, she'd been pure warmth and sweetness, loyal to a fault, always believing the best in everyone, willing to defend what she believed in to the death if necessary.
From their very first moment together, she'd wreaked havoc with his common sense. Before her, nothing in his world had been warm or sweet or particularly loyal. She'd brought lightness into the dark.
Until he'd sent her away. "Not helpless," he said a little thickly. "Never helpless."
"Okay, then." She hugged herself and shivered.
With a frown, he moved to the fireplace. For late summer, the evening did have a chill to it, and she probably was still in some shock. He set up kindling and held a lit match to it until it flamed with a low _whoosh_.
With a startled cry, Kenzie shrank back from the small flames, covering her face.
_Yeah, still in shock._ He should have thought about how she'd feel about a flame of any kind, and cursing himself, he rose and went to her.
"I'm okay," she whispered, peeking out from between her fingers, very carefully not looking at the flickering fire. "It's the crackling." She grimaced. "And, okay, the sight. I don't know what's wrong with me."
"It's normal."
"I don't feel normal."
He didn't feed the small fire, letting it burn out. "I'm sorry. Let's go with the heater instead, okay?"
Once again she leaned her head back, carefully not moving a single inch more than she absolutely had to. "Thanks."
She was killing him. "Kenzie—"
"Could we not talk? It's threatening my head's precarious perch on my shoulders."
"Take the pill."
"I guess I could use a little oblivion. Okay, I could use a lot of oblivion..." Turning her head, she eyed the fireplace as if it were a spitting cobra. "You know, they don't call me Kenzie in Los Angeles."
"Or in the gossip rags."
Without moving another muscle, she arched an eyebrow, appearing to be genuinely surprised. He'd given himself away.
"You read them?"
"Hard to miss when you're going through the grocery store," he said defensively. "They're right next to the candy bars."
The smallest smile crossed her lips.
"You dated that underwear model. The one who danced naked on all the commercials. Chad."
"Chase. And he wasn't naked. He was wearing the underwear he was marketing. Which isn't that much less than what you're wearing in that calendar, Mr. 2008." She gave him a long look.
"Last year you went out with a European prince."
"Now that was just publicity."
He didn't know if he believed her, or cared.
Strike that. He cared. "Take the pill." He watched her chase it with the glass of water he offered.
Yeah, he cared.
_Dammit._
"Problem," she said, and licked a drop of water off her bottom lip.
He dragged his gaze up to hers. "What?"
"Even if there were no dogs. I still couldn't get a room. I have no money—my purse either burned up or is below several yards of water, probably both." Kenzie winced. "The hospital had to give me an emergency taxi voucher to get to my car. I'd be really screwed right now if my keys hadn't been in my pocket. Luckily, I also left my cell in the car, so I called my financial manager and he's overnighting emergency funds. But your address was the only one I could think to give him, and I have no place to go until it arrives. And now I can't drive." She shook the bottle of pills. "It's not recommended."
Their eyes met as the implications of her little speech sank in.
"Apparently, I still trust you," she whispered. "At least a little."
_Damn_ if that didn't cut right through everything to the heart of the matter. For better or worse, she trusted him, and he had to admit, that meant something to him. Plus, there was the other truth—there was no other place she could go. Like it or not, he was her only contact in town. Which meant...
She was staying here.
With him.
# _CHAPTER 5_
KENZIE SAT ON AIDAN'S COUCH absorbing the awkward silence. Her eyes were closed but she could feel him close. Thinking. Probably panicking. "Or if you loan me a few bucks, I'll call a cab."
"And go where?"
_Right._ Well, dammit, if he'd just give her some room, she could just sit and try to ignore him— _try_ being the key word.
It wasn't his good looks that held her interest. She'd had her fill of good-looking guys on a daily basis at work and she would have said Aidan wasn't that pretty, at least not soap-star pretty. Until she'd seen the calendar. Because holy cow, he'd looked pretty damn fine in eight-and-a-half-by-eleven color glossy, there was no doubt. But he was also tough, and far more rugged than that. There was just something about his eyes and mouth, and the laugh lines lining both that suggested he could be dangerous or outrageous, sweet or maybe not so much so, sheer trouble or the boy next door...
She knew all to be true.
What she didn't know was why she'd come _here,_ to his house.
Okay, she knew. He was the only familiar thing in her entire world. She'd gotten his address easily enough by calling his station, where some friendly firefighter had recognized her and cheerfully offered up direction. She'd driven here on auto-pilot, having no trouble remembering her way around Santa Rey, getting spooked only when she'd thought she was being followed by a gray sedan.
Which was ridiculous and paranoid. God, she needed a nap.
Aidan's house was tiny, and definitely old, but cozy. From the looks of things, he'd been remodeling it. The living room had lovely hardwood floors and gorgeous wood trim on all the windows, which looked out to the ocean and the rolling hills surrounding it.
He'd always been handy—with tools, with his mind, his words.
His body...
Yeah, he'd been really good in that department. In fact, it was fair to say he'd been her willing tutor, and she a most apt pupil.
But that thought led to others, including the fact that she'd once been young and stupid enough to believe in fairy tales. Aidan had been her prince, her happily-ever-after.
Until he hadn't been.
Luckily she was no longer young or stupid. She no longer dated men while dreaming of that white picket fence and two point four kids. Nope, she dated simply to have fun, and once in a while, to have good sex.
Easy come, easy go.
Too bad she and Aidan weren't having a go at things now, because she was finally with the program, she finally got the rules. They'd probably have a hell of a time.
An evening breeze came through an open window and she drew in a fresh breath. Her pain pill had begun to kick in, and she sank a little deeper into the very comfortable couch. The last time she'd been in Aidan's place, which back then had been an apartment, he'd owned a bed, a TV, a stereo and a box of condoms.
That'd been all they'd needed.
She hadn't been the only one to change. His needs had apparently upgraded. His couch was extra large, and double extra comfortable. There was a TV, triple extra large, and the perennial stereo. But he also had a desk with a computer on it, and some beautiful prints on the walls, which were painted in muted beachy colors.
No condoms in sight. That was undoubtedly for the best. But she liked the house. Low maintenance, calm, even warm and clean. Her place wasn't so different, which meant she felt far more at home here than she would have ever admitted out loud.
How ironic that she'd come back into town to handle Blake's affairs, and to raise hell on the arson charges, intending to stay as far out of Aidan's path as possible, only to end up here in his house, with nowhere else to go.
High on meds...
From the windows she could hear the waves slapping against the shore. Next to her, he was still, just sitting there breathing, soft and even, but she didn't look at him. Wasn't ready to look at him. Yet apparently her nose didn't get that memo because her nostrils quivered, trying to catch a quick whiff of the man—except all she could smell was herself and the smoke and soot stuck to her skin. "I stink."
"It's stress."
"No, not like that." She rolled her eyes, which hurt like a son-of-a-bitch. "Like smoke."
"You could take a shower." His voice was low, a little gritty, and a whole lot suggestive, although she knew that last was all her own imagination.
She couldn't help it, the guy had a voice that brought to mind slow, hot sex. Seriously, if he could bottle the sound, he'd have been rich.
"Kenzie? Do you want to take a shower?"
_Yes, please._ In her own place with her own things and her own thick, cozy, warm bathroom and fuzzy bunny slippers. And then she'd like a good DVD and a bag of popcorn, something to give her mind a mini-vacation from its current hell. "That would be nice, thanks."
He offered her a hand. She stared at it, and then into his face, which was solemnly watching her. "Just a hand," he murmured.
Knowing she was a bit wobbly, she put her hand in his bigger, warmer one and let him pull her up. She staggered into him, and for a moment he held her, and caving in to her own yearning, she pressed her face to his throat and was immediately overcome with memories.
But she didn't do memories, at least not anymore, so she forced herself to step free of him.
He led her down the hall and into what must have been his bedroom. The walls were a soft cream, which went beautifully with the cedar ceilings. But what caught her eye was the biggest bed she'd ever seen, piled high with a thick navy-blue comforter and a mountain of pillows. It was made, sort of. It was _boy_ -made, which meant the covers had been tugged up. His hamper appeared to be a pile of clothes in the corner, but other than that, the room was as warm and clean and welcoming as the rest of the house.
She shouldn't have been surprised. The Aidan she'd known had been rough-and-tumble tough, always cool and calm and impenetrable no matter the circumstances, which she imagined served him well in his field. She'd seen that in action on the boat and in the water.
But much like his house, he hada warm, soft, welcoming center. It was what had made him so damn likeable.
Now, with the dubious honor of a few years and some maturing, that likeability had turned into an undeniable sex appeal she discovered while standing there staring at his bed, feeling a rather inexplicable stirring deep in her belly.
"Here." With a hand to the small of her back, he gently nudged her all the way into the room, then passed by her, his arm brushing hers as he moved into the bathroom, which was all cool, white tile and more wood trim. He flipped on the shower, which was nearly as big as his entire kitchen.
"Wow," she said, staring at it.
He shrugged. "I like showers."
"I remember." The words slipped out of her mouth before she could stop them. _Damn,_ she really needed a script writer for this real-life thing.
His gaze slid to hers. Very slowly, he arched an eyebrow.
She turned away to blush in peace, but he turned her back toward him with a careful hand on her arm. "Kenzie?"
She stared at his chest, her vision a little compromised by the nice little pill she'd taken, but not so much so that she couldn't appreciate the view. "Yeah?"
"Do we need to talk?"
_Absolutely not._ "No."
She didn't want to discuss her carnal knowledge of his love of showering. Not when she remembered, in vivid Technicolor, taking more than a few with him. She remembered, for instance, the time he'd backed her up to the shower wall in his apartment, lifting her legs around his waist, thrusting into her until she couldn't have told him her own name. She remembered the feel of him, hot and thick inside her, remembered how it felt to be pressed between the hard wall and his harder body, the water pounding down over the top of them until she'd cried out so loudly his roommate had pounded on the bathroom door to make sure she was okay... They'd laughed so hard they'd barely been able to finish, but they'd managed.
They'd always managed.
The humbling truth was, once upon a time, he'd been able to make her come in less than three minutes, using nothing more than his mouth and his portable showerhead.
_God._
Just the reminder had her beginning to sweat and her knees wobbling. And if she was being honest, there were some other even more base reactions going on. She firmly ignored them all and lifted her chin. "No. We don't need to talk."
He nodded very solemnly, but she would have sworn his eyes had heated, and along with that heat was a sort of wry humor.
Oh, perfect. Now _he_ was remembering, too.
But what really cooked her goose was while she was squirming, nipples hard, thighs trembling, he was amused.
She ought to slug him. She thought about it, but just then, from the plastic hospital bag came the muffled sound of her cell ringing. Since it could only be someone she didn't want to talk to, like her agent wanting her to get in line for auditions before everyone else from her show snatched up all the jobs, she ignored it.
He gestured toward the steaming shower. "It was the first thing I redid in the house."
Thinking about his shower was infinitely more appealing than thinking about being unemployed. Thinking about him _in_ the shower? Priceless. But he was still looking just amused enough at her interest that she shrugged lightly. _Look at me not caring..._
But on the inside she was caring big-time, wondering how the hell to get him _un_ -amused and hot, because dammit she wanted him hot.
Why the hell she wanted it made no sense to her, none whatsoever, but she couldn't stop thinking about it. _She_ was hot, so _he_ needed to be the same. Call it petty revenge on the guy who'd once walked away from her. Call it desperation for a diversion from her real reason for being here. But she wanted him to want her. _Needed_ him to want her. She wanted that more than her next breath, and she wanted him to suffer for it.
Around them the steam started to rise, but instead of declaring his undying lust for her, he turned and walked back into his bedroom, vanishing from view.
Kenzie let out a breath. Weary, tired of her own smoky stench, she removed her splint and reached for the tie on her hospital gowns, then went still in surprise when Aidan reappeared.
His broad shoulders filling the doorway, his dark eyes met hers as he held out two folded towels. "You still like to use two?"
She blinked as he set them on the counter by the sink. "Yeah." She cleared her throat. "Thanks."
Jaw a little tight, he nodded, and very carefully didn't come any closer.
_Huh._ He didn't look that amused now. He looked, dared she think it, a little...hot.
Interesting.
He was going to give her some privacy. Privacy that, shock of all shocks, she didn't actually want. But there he went, turning away again.
"I'll be in the other room if you need anything," he said. "Just call for me."
_Wow._ He was being considerate, sweet and sensitive, none of the traits she would have associated with him. "You know, this would probably be a lot easier on me if you could continue to be the asshole that you once were."
"Yeah, there's a problem with that."
"Which is?"
"I'm not the same guy I was then."
She opened her mouth, not sure what she planned on saying, but it didn't matter because he walked away, shutting the door quietly behind him.
Kenzie stared at the closed door before stripping and then getting into the shower. Once there, she hissed when the water hit her various cuts but she stood beneath the spray anyway, for a very long time, before finally soaping up. It took five shampoos to get out the smoke smell and even then she wasn't sure she managed completely. By the time the hot water was gone, her skin was wrinkled like a prune and she smelled like Aidan. It was ridiculous but she kept lifting her arm to her nose so she could inhale the scent of him.
When she'd wrapped herself up in the towels, one on her head, one around her body, she opened the bathroom door and found Aidan sitting on the bed, his legs spread, his hands clasped between them, his face pensive. "Better?" he asked, looking for himself.
"Almost human."
A brief smile curved his lips as he held out bandages and antiseptic cream for her injuries. "I was wondering if you planned on drowning yourself in there."
"I'm angry and frustrated and devastated, but not stupid."
He let out a slow nod, his gaze dropping from her face to her body, studying the towel covering her from just beneath her armpits to mid-thigh. She was gratified to see an absolute lack of humor now.
Slowly he stood up, and something surged within her. Lust, which she beat back. Triumph, which she let take over. _Want me..._ Yeah, that worked for her, him wanting her. Because when he admitted that out loud, she was going to lift her chin, flat-out reject him and maybe feel just the tiniest bit better.
She hoped. God, she hoped. Because _something_ had to ease this knot in her chest. _Knot, hell._ It was a ball, a huge ball, and it was suffocating her. If she gave too much thought to it, it swelled even bigger and threatened to overcome her.
Then he walked toward her, and she shivered in anticipation because here it came, the him wanting her portion of the evening.
But he simply held out her cell phone. "It went off again when you were in the shower. Local cell number."
"Oh." She flipped it open and looked at it, having no idea who would be calling her locally. Blake had been her last tie to Santa Rey. In any case, whoever it was hadn't left a message so she set the phone down.
Aidan strode right past her, going to his dresser.
Okay, she could work with this. Maybe he was going for a condom. Which of course he wasn't going to need—
He held up a shirt. "You still like to sleep in just a T-shirt?"
She stared at the shirt in his hand, at the hand that had once been able to make her purr. She lifted her head, met his gaze, and smiled.
He gave her a little smile in return, and it was all the more sexy because it was a little baffled, a little bowled over, as if he was surprised, pleasantly so, to find her finally smiling at him.
But she wanted more than that. Needed more than that, and she thought maybe she knew what to do.
If she dared...
But she'd always been bold, especially in front of a camera. And if she closed her eyes, she could be bold here as well.
Doing just that, she then reached up, pulling out the end of the towel from between her breasts, and let the thing drop.
It hit the floor with a soft thud.
Naked as a jay bird, she opened her eyes.
Aidan, unflappable, cool, calm as the eye-of-a-storm Aidan, had gone still as stone, his only movement his Adam's apple when he swallowed hard.
She held out a hand for the proffered T-shirt.
He didn't let go of it, seemingly frozen into place, as he looked her over from head to toes and back again.
She'd never thought of herself as particularly vengeful, and especially didn't wish him harm after he'd saved her life, but he'd once been able to walk away from her without a back ward glance, and that had not only broken her heart, but destroyed her confidence.
The look on his face took a good part of that remembered pain away. "Thank you," she said, tugging on the T-shirt, practically having to pry it out of his fingers.
He didn't say a word, he didn't have to. The bulge behind the button fly of his jeans said it all, and with a little shimmying movement, she pulled the shirt over her head, letting it cover her body, before turning and walking out of the room, a real smile on her face for the first time since she'd heard about Blake's death.
# _CHAPTER 6_
THE MOMENT HE WAS ALONE in his bedroom, Aidan let out a long, slow breath. He needed to go after Kenzie to tell her she could have his bed to sleep in, but after the past sixty seconds, he needed a moment.
Or ten.
Or maybe a cold shower.
Bending for the towel she'd dropped, he winced. Still hard as a rock, but who wouldn't be? She had the body that most red-blooded males fantasized about—all soft, warm curves, and then there'd been her tan lines, outlining what looked like a string bikini.
God bless tan lines.
Yeah, he was going to need another moment. He calculated a few multiplication problems in his head, and then went after her. She stood in his living room with her back to him, facing the large picture window that looked out on a darkening sky. She wore the T-shirt he'd lent her, which thanks to the show she'd given him a moment ago, he now knew she had nothing on beneath it. Her shoulders were ramrod straight, her hands at her sides.
And he had no idea what she was thinking.
"I wanted to spread Blake's ashes into the ocean," she said softly to the window. "Off the bluffs. He would have liked that."
He let out a low breath, knowing what was coming next, hating what was coming next.
"Only there are no ashes."
The pain reverberated in her voice, and somehow bounced off his own chest, rolling over his heart. _Dammit._ He headed toward her.
"All I can do is put a marker next to our parents' graves." Her voice wobbled at this, but she didn't lose it, just stared out at the night. "He's innocent, Aidan."
The Kenzie he'd known had always believed the best of everyone, to a fault. Seemed that hadn't changed, only this time it was going to bite her on the ass.
"And I would have thought you'd think so, too," she said with more than a little accusation in her voice. She sighed, the sound soft and heart-breaking as it shuddered out of her.
"Look," he said. "Why don't you go to bed and get some sleep. You'll feel better if you do."
"I doubt that." But she finally turned from the window. The last of the day's light slanted in through the glass behind her, casting her in its soft glow, rendering the T-shirt just sheer enough to stop his heart.
Not sure how much more of her glorious body he could take without dropping to his knees and begging for mercy, he stayed right where he was instead of getting any closer to her.
Closer would be a mistake, especially with those hugely expressive eyes on his, and that look of grief all over her face.
"Sleep won't change anything that I'm feeling," she whispered. "He'll still be innocent."
"Kenzie, they found a scrapbook of all the fires in Blake's house. He was keeping track of them."
"That doesn't mean he's guilty."
"What _does_ it mean?"
"Something else." She hugged herself, looking miserable and alone, and hurting. "I wish we were friends," she said very quietly. "I wish that you hadn't hurt me, and that I didn't have the urge to hurt you back."
Feeling bad, feeling a whole host of things he shouldn't be feeling at all, he took her hand. "I'm sorry I hurt you back then. I'm sorry I let you go. But I was young and stupid, Kenz. I was a complete ass."
She lifted a shoulder, tacitly agreeing with him.
"I'd like to think that if we were seeing each other now," he said softly, "and one of us wanted out, that we'd do better. That we'd make the friendship work."
Another lift of her shoulder, with slightly less temper in it this time.
Okay, that was something, a step at least. Pulling her toward him, he turned to lead her back to his bed, where he was going to tuck her in and then walk away.
Be the good guy.
Only she tugged him back, and suddenly he was holding onto her and she was pressing her face into his throat and breathing in deep, and...and _hell_. He was in trouble, sinking fast. "I showered at the station," he murmured into her hair. "But I need another. I still smell like smoke, Kenz, and—"
"Right." Pulling free, she turned away. "Sorry."
And now she thought he didn't want to hold her, when that was _all_ he wanted. "Kenzie—"
"No, you're right. Absolutely right. Let's not go there." She smiled, and anyone who'd ever seen her smile for real would have recognized it as a first-class fake, but he didn't dare say a word about it because he had the feeling she was barely hanging on.
As was he.
She turned away. "You're right. Sleep might be best. But I'll take the couch—"
"No, don't be ridiculous. I—"
"Make no mistake, Aidan. I still want to hurt you. It's immature and extremely juvenile of me, but it's fact. So, no. I'm not sleeping in your bed." She walked back to the couch.
"Kenzie—"
"Please," she said, sinking down to the cushions and closing her eyes. "Could I have a blanket?"
"Of course." He went and got several, came back and spread them over her.
She didn't speak, or for that matter, move.
"Call me if you need anything," he finally said.
She gave no response to that, either, and he nodded even though she wasn't looking at him. "Okay then...night." He paused, but she still didn't say anything to release him from the strange torment he felt. In the end, he did as she seemed to want, and left her alone.
* * *
A FEW MINUTES LATER, Kenzie heard the shower go on, and in spite of herself, pictured Aidan stripping off his clothes and climbing in.
Soaping himself up...
Standing there beneath the steamy hot water all naked.
And unintentionally sexy.
Behind her, from somewhere else in the house, a phone rang. A machine clicked on and she heard Aidan's voice saying, "You know what to do at the beep."
Then came a "Hey, you" in a low, Marilyn Monroe-like purr. "It's Lori. You didn't call me back. I've been lonely for you, baby. Come over sometime soon, okay? I'll be waiting..."
Kenzie listened to the click as the machine went off and silence filled the house.
Seemed Aidan was still the guy who left women feeling lonely for him. She should return the favor. She should go...somewhere.
But as she listened to the shower running, she let out a long breath and admitted to herself—as silly as it seemed—there was something undeniably consoling about being here with him. She'd told him she trusted him a little, and that was as truthful as it was unsettling. Yes, she had nowhere else to go, but it was far more than that. At the moment, he was the only familiar, comforting presence in her life. At the moment, she wanted to be there, she really did, even knowing that the longer they spent together, the more they would grow closer, whether she liked it or not.
Only, she was afraid she would like it. A lot more than was wise.
* * *
AIDAN SURFACED from a deep, deep sleep, aware that something had woken him, but not sure what. He opened his eyes and saw his dark bedroom lit up in black and white by the faint glow of the moon slanting in through his horizontal blinds.
There, by his bed, stood an angel.
An angel in his T-shirt, in the same white swaths of moonlight as his room.
She was hurting, sad, scared...and why the hell hadn't he given her a suit of armor instead of just a T-shirt? Had he been looking for punishment? Because there it was, in flesh and blood and glorious curves and wild hair, and a face so hauntingly beautiful she took his breath. He was in trouble, deep trouble, because although he'd managed to resist opening his heart to her that first time, he wasn't quite sure he would be able to manage it this time.
Without a single word, she lifted his covers and scooted into the bed.
With him.
He was exhausted, beyond exhausted, and was afraid he didn't have the self-control to deal with this. _"Jesus,"_ he gasped as she pressed her icy feet to his.
"Sorry."
But she didn't pull them back. Nope, she tucked them beneath his, sucking the warmth out of him.
"Don't look at me like that," she whispered.
He had no idea what she was talking about. There was no way she could clearly see his expression, she couldn't see any more than he could in the strips of moonlight. He could see her eyes, not her nose. He could see her mouth, not her chin...
"I'm not sleepwalking, or pain-pill walking." She pressed a little closer, so that her legs entangled in his.
Now would probably be as good a time as any to remind her that he slept naked, but as he opened his mouth, she spoke first.
"And I'm not here for another broken heart like I got the last time." She poked a finger into his chest. "In fact, if anyone's going to have a broken heart this time, it's going to be you. So you can just wipe that look of pity off your face."
"Pity is the last thing I've got going on," he assured her. He lay there achingly close, freezing his ass off thanks to her feet. "So you're going to break my heart?"
"Going to do my damnedest."
"I never meant to break yours."
"At least let me think I'm getting my revenge, okay?"
Her toes were killing him. So were her legs, the ones all caught up in his. And somehow he had a thigh between hers...
She propped her head up with her good hand, staring at him in the oddly lit room. Now he could see her forehead and her nose, but not her eyes or her mouth.
"It really is going to be you nursing the heart this time," she whispered.
That could very well be. But honestly, he wasn't sure his bruised heart functioned enough to break. Hell, it was probably dried up from misuse. And yet...and yet lying there with her in his arms seemed to jump-start the organ. It ached, and not just because of their past, it ached for the here and now, for the woman she'd become.
"You," she repeated softly, even a little smugly, and for some reason, some sick reason, it was a turn-on.
And because he was weak and maybe just a little bit stupid, he put his hand on her hip and leaned in to see her better, which he couldn't. She was still in slatted black and white. "I meant what I said, Kenz. I'm sorry you got hurt."
"Good. I _want_ you sorry. Very, very sorry."
Yes, but did she want him aroused? Because he was. Her T-shirt had risen up enough to remind him she wasn't wearing panties.
Yeah, colossally stupid.
By now it had to be crystal clear to her that he was butt-ass naked. In the name of fair warning, he pulled her in a little closer.
"What are you doing?"
What was he doing? No idea. Bending his head, he rubbed his jaw to hers, bumped the tip of his nose to her earlobe.
With a shiver, she clutched at him and arched her neck, giving him better access.
Which he took.
"I can't remember what I was saying," she murmured.
He let out a breath in her ear and she shivered again, which he liked. He liked that a lot. "You were telling me how you're going to break my heart."
"That's right." Her fingers dug into the small of his back as she moved, the black and white shadows shifting over her. "I am. Aidan?"
"Yeah?"
"You're naked."
He'd been wondering when that would come up. Seeing as he was already quite "up"...
She gulped, and then did something he didn't expect. She rolled to her back and pulled him on top of her, allowing him to settle between her thighs, which were not cold like her feet, but warm and cushy and very, very welcoming.
"You should know," she whispered in his ear, making sure her lips brushed his flesh, causing a series of shivers of his own." I plan to make you beg for mercy this time."
_God._ "I'm close to begging right now," he admitted.
"Really?"
She sounded breathless as hell, which was another big turn-on. So many..."Really."
He was hard. She was soft, so soft, and pressing all that softness up against him. "If you're not sleepwalking, or having a bad dream," he wondered, "why are you in here?"
"No hotels, remember?"
"Why are you in bed with me?" he clarified.
Her hands glided up and down his back, going lower on each pass. "My feet were cold."
He pressed his feet to hers, and then his mouth to her throat. "Is that all?"
"Absolutely. That and the begging."
He let out a huff of low laughter against her skin, and then because his mouth was right there against her neck, and because she was touching his butt, and because she smelled good, he took a little nibble.
Her fingers dug into him, telling him how much she liked it but she shook her head. "No more touching until you beg."
"I wasn't touching, I was kissing."
"No kissing until you beg. No anything until you beg."
"I've never begged for this before."
"No? Well, it's good for your character to try new things."
He laughed again. Laughed while trying to get laid. That was new. "Okay." Lifting his head, he cupped her face between his hands and looked into her eyes. She was smiling, too, and it was good to see her doing so. It was good to see her period; his smile slowly faded. "Can I kiss you, Kenzie?"
"Is that the best you got?"
"Can I pretty-please kiss you?"
"Well, I _suppose..._ "
That was all he let her get out before he lowered his mouth to hers and kissed her. She let out a little murmur of surprise and what he sincerely hoped was pleasure, because _holy shit,_ it was like taking a time machine back in time, back to that sweet, hot, most amazing summer he'd once spent in her arms.
She made the sound again, the one that drove him crazy with wanting, and then she entwined her arms up around his neck, gliding her fingers into his short hair and tightening them, as if she didn't want him going anywhere.
_Fat chance._
When he slid his tongue to hers, it was another homecoming, and this time her shuddery sigh was pure, hungry delight with a sprinkle of unadulterated lust on top.
_Oh, yeah._ Pulling back just enough to look into her eyes, he found the same sense of bewildered wonderment across her face that he imagined was across his. Because, yes, they were attracted to each other because of their past, but suddenly it was much, much more than that. Then the next thing he knew, they'd lunged for each other again, trying to climb into each other's body, just like old times.
Only it was new, all so damn new, and all the more heart-wrenching and gripping for it. They were no longer young and stupid. They were old enough to know better, old enough to know exactly what they were doing, old enough that he knew that this time, there would be no escaping unscathed.
It didn't stop him.
# _CHAPTER 7_
OH. MY. GOD.
Kenzie struggled to think, but Aidan had taken her breath away And, as he surged up to his knees between her spread thighs, his hands fisted in the hem of his own shirt, his intention perfectly clear, he nearly stole her sanity—but she held on by a thread. "Wait," she gasped, putting a hand to his chest. "Hold it."
Still kneeling between her sprawled legs, his hands on the big T-shirt, about to strip her as naked as he was, he looked into her eyes. "Wait?"
She could have drowned in his gaze. Happily drowned. "You stopped begging."
He arched an eyebrow, which was highlighted by the slants of moonlight across his face. Stripes of light and dark, and in them, he was beautiful. "I mean it," she managed. "Absolutely nothing else happens here without some serious begging."
He stared at her, then lowered his head for a moment. When he lifted it again, she expected him to tell her he never begged for anything. That this—she—wasn't worth it. After all, she hadn't been once.
But he surprised her. "When we were together," he said quietly, "I dreamed about your body on the nights we didn't sleep in the same bed. Did you know that?"
"No." She shook her head. "You never said." He'd never said a lot of things. He'd held back so much.
_And to be honest, so did I..._
"I'd get off on it," he said, not holding back this time. Which did exactly what she hadn't wanted—it opened her heart to him.
"On you," he murmured. "For years afterward, I'd get off thinking about you."
She stared up at him. "You mean you..."
"Uh-huh. I jerked off." Leaning over her, he was nothing but a shadow until he bent even closer. Through the shutters, rectangles of light slashed over him as he let her look into his eyes, which were dark and scorching. "So much I'm lucky I'm not blind."
She laughed but also swallowed hard, surprisingly aroused at the thought of his touching himself while picturing her. "Oh."
"Yeah, oh." His eyes glittered with heat and memories and suddenly both the heat and memories were making her feel awfully warm from the inside out.
Actually, they were making her hot.
Very hot.
"Tonight, just looking at you..." He let out a long breath and shook his head. "It brings it all back, but it's even stronger."
His mouth was in the shadows. She couldn't see his lips moving but his voice washed over her, as did the images he evoked. He was bringing it all back for her, too.
"You were beautiful then," he said. "But you're even more beautiful now. I want to take this shirt off of you, Kenz. Please let me."
At his words, she nearly turned the tables and begged _him_. She could feel the T-shirt caught high on her thighs. His hips were holding her legs open to him, and with just a little nudge of the shirt, he'd be able to see all her god-given goodies, along with the fact that she was already wet.
"Please," he murmured. "Please let me."
_Oh, God._ "Yes."
He shifted, and then she could see his mouth, which rewarded her with a smile as he made his move, his fingers closing around the hem of the shirt, slowly tugging it up, revealing her body.
She'd wanted this, sought it out under the guise of getting her long-needed revenge, but that was really just a lie, and her first flicker of doubt hit.
_Just who was going to get hurt here...?_
The night air brushed over her breasts as he pulled the shirt all the way off and over her head. Her nipples hardened. Goose bumps spread over her flesh, and it wasn't because she was cold. There were five stripes of moonlight across her body, one across her eyes, her throat, another highlighting her breasts, her belly and her crotch. He couldn't have lined her up more perfectly for his perusal, and he definitely perused.
"Aidan—"
His hand stroked over her hip, and her breath backed up into her throat. She opened her mouth to say maybe she'd been hasty about this whole breaking his heart thing, but before she could, he'd put a hand on her inner thigh and pushed, further opening her to him.
The slants of shadows hampered his view, but he didn't seem bothered, not with his front row seat.
The only sound in the room came from him as he let out a groan. "God, Kenzie. You're so pretty." He lowered his head, then paused, his mouth a hairsbreadth away from her trembling belly. "I want to kiss. I want to taste. I want that more than I want my next breath. Please let me..."
As far as begging went, it was pretty good. "O-okay," she managed, and almost before the word was out, he'd nudged her legs open even wider, wedging them there with his broad shoulders. He slowly lowered his head. "Pretty please," he whispered across her flesh.
Her wet flesh.
_"Yes."_ Her heels dug into the mattress as he "pretty pleased" his tongue over her, and then his teeth, and then his warm lips, over and over again leaving her a panting, gasping, quivery mass of sensitized nerve endings, and when she exploded for him, he surged up, produced a condom and slid into her with one sure, powerful thrust.
"Oh," she gasped, reaching up to hold onto him because her world had just spun on its axis. The feel of him deep inside her—and he was deep, as deep as he could get—had her spiraling. Gone were all thoughts of hurting him, or revenge. She could think of nothing but this, but him. Not that she would admit such a thing. "You...you didn't beg for that."
Cupping her face, he tilted it up to his. "Pretty-please may I drive you out of your living mind?"
_Oh, God._
"Kenzie? May I?" His voice was thick with the same hunger and need that was driving her.
"Yes."
"Good. May I also pretty-please make you scream my name?"
In answer, she arched up, her breasts pressing into his hard, warm chest, her legs wrapping around his waist.
He groaned, a low, rough sound that scraped at all her good spots but he didn't move. "Can I?"
"I don't usually do much screaming."
He just smiled, and then took her mouth as he took her body, indeed driving her out of her mind with all too disturbing ease, and when she exploded again, she cried out his name.
Loudly.
She might have even screamed it.
As the blood finally slowed in her veins, as the roar of it lowered to a trickle in her head, she became aware of the fact that she was gripping him tight, holding him close with her arms and her legs, not letting him escape.
He didn't say a word, just nuzzled lazily at her neck as his breathing slowed.
Hers wasn't slowing. Embarrassed at how tightly she was holding him, she forced herself to let him go, certain he'd roll away.
But in perhaps the loveliest thing he'd done all night, he didn't. Instead, he remained right where he was, turning just his head to press his lips to her jaw, murmuring her name on a sigh.
It was one of those defining moments, where she suddenly knew the truth—she'd not exacted a single ounce of revenge. In fact, she'd made things worse.
She'd risked her own heart.
But for that one moment at least, she didn't care, because maybe he'd changed. Maybe things could be different this time, and—
"You screamed my name." He lifted his head, revealing a strong smile. "You begged." He out-and-out grinned then, not broken, not even a little bit. "We still work hard."
"There's no _we_." She pushed him off her, suddenly and irrationally irritated. "No we at all."
Completely oblivious to the picture he made sprawled out on the bed, buck naked, he put his hands behind his head and continued to smile like an idiot. "Are you telling me you have no desire to do that again?"
"None."
"Ah, Kenzie. You're such a pretty liar."
_Yeah._ Yeah, she was. A pretty liar, and a good liar. But she had no idea how else to hide the fact that she still had feelings for him in spite of their past—or maybe because of it. _God._ She needed to get out for a while, needed to clear her head. Get some answers. _Alone._
"Stay," he murmured.
"Okay." She looked at him. "I'll stay if you tell me this. Why did you really dump me?"
At that, his amusement faded. "I told you I was an idiot back then."
"Granted. Why else?"
He looked at her and she nearly backed down; she certainly held her breath, but he touched her face. "Because I didn't know what I had."
* * *
AIDAN SLEPT like the dead. Or like a man who'd been far too close to serious exhaustion. When he opened his eyes, he felt the various aches and pains from the fire, and from the mattress gymnastics he and Kenzie had executed, and was grateful to know he had two days off, because more sleep was on his To Do list. Much more.
So was more mattress gymnastics.
Considering that Kenzie was wrapped around him like a pretzel, that shouldn't be too difficult to manage. As he looked into her face, taking in each of the cuts and the bruises there in the light of day, he felt a tug in his belly.
He wished like hell he could say he was just hungry, but he knew the truth.
He was a goner.
She was as cut up and bruised as he was, more so, and if _he_ hurt like hell, he could only imagine how she felt. He was used to such injuries. She wasn't.
"I realize I've spent my days on a television set, where my worst injury was a paper cut from that day's script," she whispered, eyes still closed. "But I'm not feeling as bad as I probably look."
Her face was relaxed now; and he realized it hadn't been before—not on Blake's boat, not when she'd crawled in bed with him, not even when he'd stripped her out of his shirt and proceeded to make her scream.
That he'd undone her so easily didn't stroke his ego. She'd undone him just the same. It'd always been like that for them, a virtual explosion of need and lust and hunger.
But he'd attributed much of that to being young and horny. He hadn't anticipated a resurgence of those feelings, And he doubted she had either. But that's exactly what they'd gotten.
With a sigh, she slid out of his arms and off the bed. He enjoyed the view as she walked to the bathroom, but when she shut the door, his smile faded. She needed sustenance, and a bandage change. Getting up, he pulled on his jeans and went into the kitchen, where he grabbed a pan and eggs and went to work getting them both some protein so that they could go back to bed and burn it all off again.
His doorbell rang and Aidan stopped dicing peppers long enough to sign the clipboard of a pudgy guy in brown shorts, who handed him a slim package.
When he heard the shower go off, he finished the eggs and then grabbed his first-aid bag and knocked on the door. "Bandages, aspirin and breakfast. And your package from L.A. is here."
"Perfect timing—I've got to run."
"You mean back to Los Angeles?"
The door opened and steam came out. As did Kenzie wrapped in another of his towels. "Not back. Not yet."
The towel was tucked between her breasts, which pushed them up and nearly out, a fact he'd have taken the time to thoroughly enjoy except for the nasty bruise arcing along her left collar bone. "You need rest."
"I need clothes." She moved past him and into his bedroom. "Can I borrow a pair of sweats?"
"Sure." He opened his dresser and handed the clothes over.
"Thanks. I've really got to go."
She was going to go snoop. Get in Tommy's way. Get herself arrested. "Kenzie, listen to me. You need to stay out of the investigation. The chief doesn't want you digging—"
"I don't work for him. He can't tell me what to do."
"If you stay—"
"No. Thank you, but, no."
Usually in the light of day, with a woman in his bedroom, _he_ was the one who had to go. Usually.
Okay, always.
It felt odd to have the shoe on the other foot. Especially given the magnitude of what they'd shared last night, and he wasn't alone in feeling it, dammit. He knew he wasn't.
But Kenzie moved carefully away from him, slowly, as if still in pain, but with conviction. She was set on going, leaving him with a disconcerted feeling in his gut.
Was this how he'd made women feel? Like they'd already been forgotten? "Let's change your bandages—"
"I can do it on my own."
Seemed she was used to doing stuff on her own. That was new.
So was his unsettledness over the way this was going down.
"Yeah," she said at his quiet surprise. "I'm not the same helpless little thing I used to be."
"I never thought you were helpless."
"Well, I was. But I've grown up. I've changed. In many ways. And I don't need anyone's help. For anything."
He arched an eyebrow. "You needed me when we—"
"No. Well, yes, _yes,_ I needed you to save me from the fire, but—"
"That's not what I was talking about." He pointed to his bed.
"Oh, no. That was just me, breaking your heart. I warned you, remember."
Bullshit. That hadn't been just revenge. "Kenzie."
"Sorry. Got to go. Have to go." Once again she dropped her towel, which had the same magical effect on him as it had last night. While he stood there taking in the glorious sight of her naked body, she pulled on the sweats, kissed him on the cheek, then walked out of the room.
And, given the sound of the front door opening and then closing, out of his house.
And, most likely, out of his life.
Fitting justice really, as he'd once done the same to her. Moving to the living room, he looked out the window in time to catch her taillights as they vanished down his driveway.
_I've changed,_ she'd said, and she had.
But as the blood once again began a northward flow from behind the zipper of his pants back up to his brain, another thought managed to get his attention.
He'd changed as well. And he was going to prove it.
# _CHAPTER 8_
SOMEONE WAS KNOCKING on Aidan's door when he turned off the shower. _She'd come back._ With his pulse kicking, he grabbed a towel and wrapped it around his waist, heading for the door at a speed far faster than his usual get-there-when-I-get-there saunter.
Only it wasn't Kenzie at all. "Dammit."
His best friend and partner Zach just looked at him. "Nice to see you, too." Without waiting for an invitation, he pushed past Aidan and walked in.
Fair enough. Aidan had let himself into Zach's house plenty of times. Aidan shut the door behind Zach and shoved his fingers through his wet hair. "Sorry. Thought you were someone else."
Zach took in Aidan standing there dripping wet, wearing only a towel. "Clearly. Who is she?"
"How do you know it's a she?"
"Because if you're meeting a guy dressed like that, we have a whole different issue to talk about."
Aidan rolled his eyes and left Zach to go get some clothes. In his bedroom, he looked at his bed as he pulled on a clean shirt. The covers were tossed half on the floor, and on his nightstand were two empty condom wrappers.
And though it was crazy given that Kenzie had used his shampoo, his clothes and his soap, he'd have sworn he could smell her scent, some complicated mix of soft, determined, sexy woman. He stared at the bed, remembering how he'd felt when she'd crawled in with him, remembering how natural it'd been to kiss and touch her, to sink into her body and go to a place he hadn't been in a long time.
Then they'd slept together, and that had felt good, too, being all tangled up in each other again. Familiar, but new. Even better, if that was possible. Things hadn't been complicated in the dark.
Things had been amazing.
But she'd left.
When he walked back into the kitchen, he found Zach staring at the breakfast he'd made for Kenzie.
"You made breakfast," Zach said. "As in got out a pan and cooked something."
"Yeah. So?"
"You put out napkins."
"Let me repeat myself. So?"
"So you never put out napkins. Not when it's me or the other guys."
"Do you want to split the food with me or not?"
"You didn't cook this for me."
"You're right."
Zach raised an eyebrow.
"You're going to question a plate of food?" Aidan said. "Really?"
Zach didn't have to be asked twice. He grabbed a plate and pulled up a chair.
"I thought you and Brooke were going away for a few days since you haven't been cleared to go back to work yet."
"We are. We're leaving tomorrow morning. Wanted to see you first."
"Ah, that's so sweet. You're going to miss me."
"Actually, I'm not." Zach shoveled in some food, and looked at him. "I heard about the explosion. I should have been there."
Aidan looked at the cast on Zach's left wrist, remembered how close he'd come to losing him along with Blake, and felt the food get caught in his throat. "You're not healed yet."
"It's coming along though." He squeezed his fingers into a fist, then stretched them straight out. "I could be back at work, dammit. I have no idea why the chief's being so hard-assed about this. I'm willing and able."
"Enjoy your few days off. You and Brooke deserve it."
"Yeah." Zach sighed. "So is the boat a complete loss?"
"Unfortunately."
"Kenzie all right?"
"Heard about that, too, huh?"
"Yeah." Zach paused. "Was it awkward, considering your past with her?"
"To be the one rescuing her?"
"What else?"
_Yeah, genius, what else._ Maybe sleeping with her...But that hadn't been awkward. Not one little bit.
Zach was looking at him. "What am I missing?"
Aidan shook his head. "Nothing."
"Come on."
"Okay, nothing I want to talk about."
"That I buy," Zach said, and like the good friend he was, changed the subject. "I heard that Blake must have kept his accelerants on the boat, which is why it blew like it did."
That was one theory, Aidan was sure.
But he had another. "Well..."
"What?" Zach asked.
"You're going to tell me I'm crazy."
Zach stood up and went to the refrigerator for the milk. "All those times I thought those fires were arson, you were the only one who believed me. I'll be the last one to tell you that you're crazy."
"Yeah, but now we know that Tommy was behind you the entire time, he was just in the middle of his investigation. Still is, with the chief riding his ass to put an end to this."
"Yeah." Zach pushed away his plate. "So I wonder what they'd say now."
"About...?"
"About your not buying that boat fire was any more accidental than the other fires. Or me not buying it, either."
Aidan looked into his best friend's eyes and let out a breath. "That boat was blown up for a reason and I think that reason was to hide something. Something that someone didn't want found."
"What?"
"I don't know. And I'm betting Tommy and the Chief don't know either but they want to."
"It doesn't make sense," Zach said. "Blake's dead."
Aidan pushed away his plate. "Yeah." Goddamn, but he wasn't going to get used to that any time soon, the fact that Blake, a friend, _one of them_ for Christ's sake, was not only gone, but accused of arson. "Which means that he wasn't working alone and whoever the other person is, they're running scared of something."
"Or someone," Zach said. "Kenzie shows up out of the blue after what, six years? Seems kind of odd, doesn't it?"
Aidan's gut tightened. "Her brother's dead, Zach."
"Yes. Her arsonist brother. They were close, right?"
"What are you saying, that she's his cofelon?"
"Look, I don't want to think about Blake doing the things they've accused him of, either. And I really don't want to think about the fact that if he was still alive, he'd be in jail. But those are the facts."
Aidan scrubbed his hands over his face. "She _just_ got into town."
"You know that for sure?"
Actually, no, he didn't.
"Why was she on his boat?"
"Going through his things." Listen to him defend her. "Missing him."
Zach closed his eyes and rubbed them hard. "If that were true, wouldn't she have come sooner?"
"I don't know. I don't know anything except that Blake was all she had." Aidan got to his feet because he had to move, had to pace the length of the kitchen. "She's...devastated. Horrified. And pissed off that we all believe that Blake's guilty. I think she's going to go digging on her own and find out what she can."
"Which should make Tommy oh-so-happy."
"He's going to have her arrested if she hinders the investigation," Aidan admitted. "And she's going to hinder. It's in her nature. She intends to prove Blake innocent."
Zach raised a brow. "You got all that from pulling her out of the water?"
_Well, shit._ Aidan picked up his fork and shoveled some food in.
"You saw her after the fire. At the hospital."
"Yeah."
Zach paused. "And after that as well, I'm thinking."
"Yeah."
Zach peered around Aidan and into the living room, pointedly looking down the hallway.
"She's not still here."
"But she _was_ here? Jesus, Aidan. What would Tommy say?"
"Since when does that matter?"
"Since we both now know that he was on our side about the arsons all along. He'll be on this, too, you can guarantee it."
Yeah. In hindsight, sleeping with Kenzie been a pretty stupid thing to do. And yet, what else could he have done but given her a place to stay?
Except for that using up two condoms part. He probably could have not done that.
"We've got to let Tommy do his thing here," Zach said quietly.
"I can't believe you're suggesting I stay out of it, when you did the very opposite."
"And paid for it," Zach reminded him, lifting his casted wrist.
"She was hurting, Zach. And alone. Her purse had burned in the fire and she had nowhere else to go so I let her stay here. End of story."
"You could have lent her money. She's a famous soap diva—I think she'd have been good for it."
"The hotels were all booked up."
When Zach just looked at him, Aidan lifted a shoulder. "It was just bad luck on her part."
"Just bad luck, huh?Funny, you don't look so put out."
"Don't you have a fiancée to go home to?"
Zach grinned dopily. "Yeah."
"So go already."
Zach got up, then paused. "Look, Aidan, I know she meant something to you once, but—"
"She's Blake's sister."
"And _your_ ex. I'd think that'd be reason enough to stay away from her."
_Yeah._ One would think...
* * *
OPENING THE SLIM ENVELOPE she'd scooped from Aidan's kitchen table on her way out the door, Kenzie practically kissed the credit card she found inside. She needed some personal items, like clothes of her own, not to mention underwear. Not that she didn't love Aidan's sweats, because she did. They smelled like him. They felt like him.
Which was exactly why she had to get _out_ of them.
She did her best not to pout over the loss of her Choos, which she wasn't going to find at Wal-Mart, but the store was still one of God's greatest creations. When she'd bought and put on a peasant skirt, two layered tank tops and a pair of sandals, she got back into her car. She'd missed two calls on her cell, both from that same local number as before, but no messages, so she put it out of her head and drove to the docks. Then she sat in the parking lot nursing a hot chocolate and a blessed box of donuts, staring at the charred remains of Blake's boat.
She was alone except for the occasional car. One was a light-gray sedan that slowed as it passed her, the windows so dark that she couldn't see in. Probably another looky-loo like herself, except...except she'd seen a car like it before, somewhere...
She ate a donut.
Until a couple of weeks ago, before Blake's death, she hadn't had chocolate or donuts in months. Maybe years. She'd been on a strict eighteen-hundred-calorie diet, combined with a workout every single day, without fail. All to look good.
That's what TV stars did. They looked good. She was paid to.
Except she no longer had a TV show to look good for. Back in L.A., she knew the job-finding frenzy had already begun. All her co-stars were busy auditioning, and what was she doing? Eating donuts instead of facing the fact that she was unemployed.
Her cushy, easy, comfortable, fun job had come to an end.
Life over.
She looked at _Blake's Girl_ and felt the last donut congeal in her throat. No. Her job was over, not her life.
_Blake's_ life was over.
_God._ Brushing the sugar from her fingers, she got out of the car. She wasn't looking her best, but then again, there were no paparazzi in Santa Rey. And thanks to no one in the press making the connection between her and _Blake's Girl,_ there were no reporters to take pics of her pale, makeup-free face, or all of the bruises and cuts she'd sustained in the fire. Her wrist wasn't bothering her, but the splint was a pain in the butt. She hadn't been able to corral her hair into a ponytail, which meant it was flying wild around her face and in her eyes.
She could have asked Aidan for help but she'd rather have the wild hair than have his hands on her again.
Okay, that wasn't true, wasn't anywhere close to true, but she could pretend it was.
_Dammit._
For those few hours last night in his arms, she'd not been alone and lost and hurting. She'd been transported, taken out of herself.
And along the way, she'd forgotten to make him regret dumping her. _Nicely done._ Rolling her eyes at herself, she moved closer to the docks. The charred remains of _Blake's Girl_ were taped off with yellow crime scene tape.
She didn't know what that was about.
They thought Blake was a criminal? Fine. But they couldn't pin this one on him, he was already gone.
Gone...
Chest tight, she walked along the yellow tape, getting as close as she could, which wasn't close enough. No one was around, on the dock or otherwise, and she couldn't stop the thought—what if she ducked under the tape? Surely, as Blake's only living relative, she deserved to have a look.
The two boats on either side of _Blake's Girl_ were still there. Barely. One was nearly burned black, and in fact looked as if it might still be steaming. The other was half gone, and half untouched.
And between them? A shell of a boat, blackened and charred beyond recognition.
Blake's boat was completely destroyed.
Looking at it, she could see it as it'd been two nights ago, when she could stand on it and still feel her brother's presence, when his things had still been okay. She wished she'd gotten something of his, something, anything...
Maybe she could crawl beneath the tape and get onboard to comb through the torched remains, and thinking it, she bent down, but at the sound of an engine, stopped and turned.
It was the gray sedan again, making another pass of the parking lot.
Goose bumps rose on her arms as she got that same sensation of being watched she'd had at the hospital.
Who was following her?
It wasn't Aidan. No way. He'd make himself known, that was for damn sure. He had a way of making himself known...
Someone else then.
Tommy?
No. Tommy didn't have the resources to have her followed. She doubted anyone in Santa Rey did.
Then she remembered her earlier missed calls, and pulled out her phone, hitting the number.
No one answered.
She ran her hand along the yellow police tape, but the truth was, she didn't quite have the nerve to boldly defy the law.
At least not during the daylight hours.
But tonight...
Yeah, tonight.
Under the cover of darkness.
Turning away, she squeaked as she accidentally bumped into a hard wall.
A hard wall that was really a warm, hard chest she recognized all too well, along with the big, warm hands that settled on her arms.
# _CHAPTER 9_
THE COLLISION SET KENZIE back a step, but Aidan held her upright.
She tilted her head up, up, up...and looked into his face, which was unfortunately indecipherable.
"You okay?" he asked, his voice low and calm, and concerned.
Okay, concern was good. Concern implied that he hadn't noticed what she'd been about to do. But was she okay? _Hell, no._
Not even close.
"Are you?" His gaze swept down her body, then up again, as if categorizing her injuries, which reminded her of last night, when he'd also been categorized her body.
With his tongue.
"Yes," she managed. "I'm fine."
"Good. What the hell are you doing here?"
"Funny, I was going to ask you the same thing. Are you following me?"
"No."
"You're not driving a gray sedan and going everywhere I go?"
"I drive a truck, a blue one and I didn't follow you here. I got lucky on the first try. I figured you'd come here and try to do something stupid."
"I did nothing of the kind."
"You don't consider ducking beneath that yellow tape stupid?"
"Only if I'd gotten caught."
"Hello," he said, still holding on to her. His fingers tightened. _"Caught."_
"Yes, but you don't count."
He looked both boggled _and_ irritated. "And why is that?"
"Because what are you going to do, arrest me? Last night you were kissing me, touching me, fu—"
"Okay," he said with a low laugh. "Now just hold on a second—"
"I'm just saying." She narrowed her eyes and went for bravado, even though she could hardly breathe while looking at the big blackened sailboat that less than two days ago had been _Blake's Girl_.
Aidan had saved her.
He'd saved her and she was poking at him because she was all twisted up inside. So she let out a breath and looked into his face, where she found a surprising blend of sympathy and old affection mixed in with the frustration and fear.
"I came here to talk," he said. "Not arrest you. Jesus. Now what the hell is this about a gray sedan?"
"Nothing."
He just looked at her for a long moment. "What aren't you telling me?"
"Nothing."
"More like everything." He let out a breath. "Tommy expects you to let him do his job."
"I'm not going to get in his way. I'm going to help him."
"Now see, I don't think he likes help."
"Too bad for him."
"It's going to be too bad for you if you piss him off. He can and will have you arrested if you don't stay out of his way."
"Believe me, I plan to stay out of his way."
"Okay." He nodded. "New subject then."
_Uh-oh._
"Last night..."
Kenzie didn't know how she felt about last night. And because she didn't, she absolutely didn't want to talk about it. "Yeah. Now's not a good time for me."
"You don't think so?"
She shook her head.
His eyes lit with something that might have been wry humor. He'd been just as beat up as her yesterday, but unlike her, today he did not look like something the cat dragged in. No, he looked tall and fit, and in his loose cargoes and T-shirt, he seemed very in charge of himself and his world.
She, on the other hand, was in charge of exactly nothing at the moment. "Maybe later." And maybe not.
He hadn't taken his hands off of her arms, and if asked she'd have said she wasn't sure how she felt about that, but that would be a lie. At the moment, his support felt like a lifeline.
Her only lifeline. "Tell me something," she said very quietly, her eyes on his so she didn't miss any little nuance, because this was very, very important to her. "Arson. It's a well studied crime, right? The people who do it, most of them belong to a particular character type. Aggressive. Violent. Repeat offenders."
"Yes," he agreed. "How do you know this?"
"We did a whole plotline about an arsonist last year. Would you characterize Blake as aggressive or violent?"
"Not even close."
"Exactly," she said.
"Which doesn't prove anything. There's physical evidence—"
"Okay," she agreed. She knew about the evidence. "But most arsonists _want_ their work admired. Isn't that correct?"
"Yes, but—"
" _But_ Blake maintained his innocence. Tommy told me that much."
"Yes," Aidan agreed, his expression reflecting his worry for her, whether he wanted it to or not.
Which she didn't want to face. She meant to do two things when it came to Aidan, especially after last night. First: keep her distance. And second: leave _him_ pining for _her_.
It was going to be nearly impossible to handle the second while doing the first but she would give it her best shot. "So can't you concede that it's possible that you're wrong about Blake?"
"I'm not the one accusing him of anything."
She looked at him, really looked at him, and understood something she'd missed before. He didn't want to believe the worst of Blake any more than she did, and that was so much more than she expected from him, from anyone, that it was like a balm to all her fear and grief.
He wasn't against her or Blake. She wasn't completely alone, at least not in that moment, and she found herself closing the gap between them to wrap her arms around his broad shoulders, hugging him hard, so damn relieved to have him there with her.
With a rough sound, his arms came around her, too, and he pulled her in, letting her lean on him. "Kenzie," he whispered, bowing his head over hers. "It's okay. It's going to be okay."
_Yeah._ Keeping her distance from him was going to be damned tough.
So would be breaking his heart, but she was still going to do it. It was that, or see hers crushed again, and that was simply not going to happen.
* * *
Aidan had never been a hugging sort of guy. He loved physical contact, especially the naked kind, with the fairer sex, but touching just out of sheer affection and nothing else? That hadn't really been a part of his life. Having been the sort of child who'd made it difficult for others to like him, much less love him, he hadn't inspired a lot of affection growing up. And working with mostly guys all the time...well, they tended to shove and wrestle rather than hug.
So this, with Kenzie, should have felt awkward. Alien. At the very least it was an intrusion of his personal space that he would have thought would make him squirm to be free.
But it didn't. Even though a piece of her hair was poking him in the eye and she was stepping on his toe, and her nose—pressed against his throat—was icy enough to make him wince, he didn't move.
In fact, he tightened his arms on her, pressing his face into her hair, inhaling her as if he didn't want to let go.
Because he really didn't.
She was warm and soft and sweet, and when her fingers slid into his hair he nearly purred. His hand skimmed down her spine, pressing low on her back, urging her even closer as he just continued to breathe her in.
Just down the dock, two seagulls argued over some found treasure. Water slapped at the wood pylons. Beyond that, the devastation of the fire sat right before their eyes. Aidan didn't want her looking at it. "You need to get out of here."
"Yeah." She stepped back. "I know. I'm going."
He caught her hand, and when she looked at him questioningly, he saw the truth in her eyes. Wherever she was headed, it was to make trouble.
"I'm a big girl now."
_Yes._ She was a woman who could more than take care of herself. Which in no way eradicated the need within him to protect her. "Have you eaten?"
She stared at him, then let out a low breath. "I tell you I can take care of myself and you want to feed me? Even after I also told you that I only wanted to be with you in order to break your heart?"
"Yeah, see, about that..." He stroked a loose strand of hair off her face, letting his finger trace the rim of her ear, absorbing her little shiver. "I don't really believe you."
"Oh, it's true," she said with utter conviction. "I'm going to break your heart."
"That wasn't the only reason you stayed with me last night. Slept with me."
"Okay, true. You saved my life. I owed you."
He shook his head. "That wasn't it, either."
"What was it then, smart guy?"
"You like being with me."
A helpless laugh escaped her at that.
"I like being with you, too, Kenz."
She shook her head. "You're off your rocker."
"Already established. So. Food?"
She stared at him, then caved. "I guess I could eat."
She followed him in her car to Sunrise Café. Aidan had no idea why he took her there, other than that taking her back to his place, where they'd be alone, seemed like a really bad idea.
Sheila was thrilled to see him and gave him a huge hug, smiling with some speculation at Kenzie. Even though it was afternoon by then, Aidan ordered a large breakfast. When Kenzie tried to get just coffee, he merely doubled his order, and then took her up to the roof.
There was a long bench against the far wall, where they sat to watch the surf. It was rough, which didn't stop the surfers from enjoying it.
Kenzie stared out at the waves. "It's nice up here. A good place to think. You come here a lot?"
"I do."
"Sheila's fond of you."
"Very," he agreed.
She smiled at him, and just like that, melted his heart. "You've made some good ties," she said softly.
He got a little lost in her eyes, and leaned in with some half-baked idea of kissing her, and—
_"Come and get it!"_ Sheila yelled up from the bottom of the stairwell.
Sighing—what else could he do—Aidan led the way down to the crowded dining room. Sheila seated them, then brought them their plates, winking at Aidan before leaving.
Kenzie looked down at her loaded plate. "I'm not that hungry."
"Uh-huh." He nudged her fork closer to her fingers. "That's what you always used to say. You'd tell me you weren't hungry and then you'd eat everything off my plate, remember?"
Humor lit her eyes. "What I remember is that you were my boyfriend. You were supposed to share."
"So, what are you saying? That you wouldn't, say, eat off Chad's plate?"
"Chase. And he's vegan and doesn't eat anything that isn't completely raw, so, no, I wouldn't."
Aidan leaned over and stroked another stray strand of hair off her cheek. He had no idea why he kept finding excuses to touch her, other than she looked sad and just a little lost. She wore no makeup, and all those gorgeous blond waves had rioted around her face, a few long strands curling around her jaw. It was just Kenzie. No smoke and mirrors, no pomp or celebrity. Just the woman who'd once touched his heart.
And, apparently, still did.
So he did what he'd wanted to do on the roof—he leaned over their food and kissed her, just once, softly on the lips. When he pulled back, she gave a baffled little smile and touched her fingers to her mouth. "What was that for?"
Before he could answer, Zach walked up to their table. "Hey."
"Hey," Aidan said in surprise. "Kenzie, this is Zach. Zach, Kenzie is—"
"Blake's sister." Zach's eyes softened as he looked at her. "I miss your brother."
"Thank you," she murmured. "Me, too."
Zach turned to Aidan and handed him a file.
"What's this?"
"I wanted you to have it while I was gone. In case you need it for anything."
Aidan opened the file and instantly knew what he held. All the evidence Zach had gathered over the past few months on the mysterious arsons. Zach had been the first one to suspect something was going on and the first to go to Tommy for answers. Closing the file he met Zach's steady gaze. "Thanks. Want to join us?"
"Can't. Brooke's waiting for me. I just talked to Eddie and Sam. Did you know there was another explosion last night? The hardware store on Sixth."
"Injuries?"
"Several, and one death. Tracy Gibson."
Aidan's stomach dropped. The woman Blake had had a crush on for months before his death.
Kenzie divided her gaze between them. "Who's Tracy?"
"She was an employee at the hardware store," Zach told her. "Same setup as _Blake's Girl,_ " he said to Aidan, tapping the file with meaning. "So keep this."
Aidan understood. Zach thought he might need the info in the file when he was gone.
"Nice meeting you," Zach said to Kenzie. With a squeeze to Aidan's shoulder, he left.
"So what does that mean?" Kenzie asked. "If there was a similar explosion, maybe Blake's boat wasn't an accident."
"Maybe."
"A new serial arsonist?" she scoffed. "What are the chances of that in a small town like this?"
"I don't know."
"I know," she said. "Next to nil."
She was watching him with sadness still in her eyes, along with a sense of sharp intelligence that said she wasn't going to let this go. The brash tilt of her chin alluded to a strength of will, of passion, he knew first-hand, and suddenly he was afraid for her.
For her, _of her,_ and of the feelings she invoked inside him. Damn, not again...Not falling for her again, he told himself. But it didn't matter that he was seated across from her in a crowded café, surrounded by people.
She was all he saw.
He watched her push her food around the plate for a few minutes, then wrapped his fingers around her wrist, guiding her fork to a large bite of eggs and bringing it to her mouth.
She took it into her mouth, chewed and swallowed, all with her gaze never leaving his. "You keep looking at me like you care."
"I do."
"You shouldn't."
"Why not?"
"Because I'm not going to care about you back." At that, she broke eye contract and stared down at the food. "At least not like I did before."
"So you've mentioned."
"I mean it."
"I believe you." He also believed that she just might get her big wish, because looking at her sitting there, knowing _she'd_ be walking away from _him_ this time, caused a strange sensation deep inside him. He'd have sworn it was his heart rolling over and exposing its underbelly.
Kenzie took another bite of food as his cell phone buzzed. It was Dispatch. "Sorry," he said, standing. "I have to take this."
"No problem." She was suddenly engrossed in her food, not even looking up when he went outside to get good enough reception to hear that two firefighters had come down with the flu. They needed replacements for the next shift. So much for a day off—he was going back on duty, starting now.
He turned to go back inside the café and nearly bumped into Kenzie. "Sorry," she said, flashing a smile that didn't quite meet her eyes. "I've got to go."
_Huh._ That had been _his_ line.
"I paid the bill—"
He reached for his wallet. "Let me—"
But she put her hand over his and shook her head. "It's on me. Consider it a very small down payment."
"For what?"
"For what I owe you for saving my life."
"Kenzie—"
"Thank you," she said softly, looking into his eyes, making his head spin. "I'm not sure I said that enough. I am extremely grateful."
Wait. That sounded like a good-bye. "Okay, hold on a second. Are you—"
Going up on tiptoes, she put a hand to his chest, leaned in and kissed him on the jaw. She added a smile to the mix, one that went all the way to her eyes this time as she touched her fingers to her lips and then blew him another kiss.
Then she turned and walked away.
As he'd once done to her. "Kenzie."
But she'd already gotten into her car. Where the hell was she going? She revved the engine and was gone, out of the lot, perhaps out of his world. He stood there a moment, absorbing a barrage of emotions, starting with regret and ending with a surprising hurt, and then he shrugged it off and walked inside to say good-bye to Sheila. That's when his head stopped spinning and it hit him.
Kenzie had stolen his file.
# _CHAPTER 10_
UNFORTUNATELY FOR KENZIE, the doggie convention was still in town. She tried a couple of B and Bs and got excited when a cute front desk clerk recognized her and said he'd stir up a room. But then he picked up his phone and yelled, "Ma! Get out of the room, I've got a girl!"
Kenzie shouldn't have been surprised, since her karma was clearly still on vacation. She made the clerk leave his mother in the room and escaped. Back in her car, she sighed, feeling very alone.
She missed Blake.
And dammit, she already missed Aidan, too. Missed his voice, his smile, his touch.
How was that even possible? She'd just left him. She'd stolen his file for God's sake. No doubt he was cursing her right this minute.
And definitely _not_ missing her.
She pulled into the library and made herself comfortable on a large chair in a far corner, then opened the file. Almost immediately she felt an odd prickle of awareness, and then the hair on the back of her neck stood up.
She was being watched again.
She craned her neck left and then right, but no one in her immediate area was so much as looking at her. Behind her was a set of shelves, and she shifted, trying to see through a gap to the aisle on the other side.
Nothing.
Clearly she was still in the process of losing her mind. Determined, she went back to the file. Zach and Aidan had been thorough. There was a list of fire calls from Firehouse Thirty-Four over the past six months, five of them highlighted. The questionable fires, she realized.
The arsons Blake had ultimately been accused of starting.
Attached were details of those five properties: architectural plans, permits, a history of ownership, purchases and sales. Each had been plotted out on a map, and scrutinized up one side and down the other, including everything that had been found on site after the fire.
Zach had noted finding a metal mesh trash can at each site, and even had a picture of one, from the fire just before the one at Zach's own house. As she was looking at it, her cell phone vibrated. She nearly ignored it until she saw it was the same local cell phone number as before, and she grabbed it. "Hello?" she said breathlessly.
When several people in chairs nearby glared at her, especially one older woman going through a stack of history books, Kenzie hunched her shoulders, mouthed a "sorry" and whispered "hello" much more softly.
An equally soft voice spoke in return. "Forget about it, forget about _all_ of it, and go back to Los Angeles."
Kenzie clutched the phone. She couldn't tell if she recognized the speaker because the voice was purposely being disguised. "Is that a threat?"
"You're going to be stubborn. Goddammit."
_"Who is this?"_ she demanded.
"It doesn't matter. Just get the hell out of Santa Rey."
"So you _are_ threatening me."
"If I said yes, would you go?"
"No."
_"Shit."_ There was a beat of silence. "Okay, listen to me. There's only one way out of this."
_"What?"_ she said, forgetting to whisper, receiving more glares for that. With effort, she lowered her voice. "What do you mean?"
"Your laptop was destroyed in the boat fire?"
"How do you know that?"
"You have backup."
"What does that have to do with—" She went still as it hit her. She and Blake had shared files. Music files, movie files...they'd e-mailed and IM'd each other regularly. And once a week he'd send her a large backup file from his laptop so that if it ever crashed, she could just send him back what he needed. She'd done the same. She'd saved all her stuff, _and_ Blake's, in her Yahoo account. All she had to do was get to another computer. _"Who are you?"_
"Check the demos. That's the key."
"What?" Kenzie clutched the phone. "What does that mean? Who are—"
But she knew before she even finished her sentence that he was gone. But who was he? A friend of Blake's? _"Dammit."_
"Shh!" everyone around her hissed.
_Yeah, yeah, fine._ But the prickle in the back of her neck hadn't gone away. She got to her feet and moved to the end of the aisle, peeking around the corner just in time to catch sight of the back of a guy running away. No red shirt this time but she knew it was the same guy she'd seen at the hospital. She hightailed it after him, but when she got to the other end of the aisle, she plowed directly into the librarian.
"No running in the library!"
"Sorry." Kenzie stepped around her, but it was too late. Her helpful mysterious caller was gone. She turned back to the librarian. "Can I use an online computer?"
"You have to sign up."
"Okay, where?"
"We're closing in half an hour, and the computers are in use until then. How about the morning?"
"Fine." She'd spend tonight going through the boat and Blake's place for anything that could help her. Then she'd borrow Aidan's computer—if he let her—or come back here to prove that Blake had been set up. Because that was the only answer she was willing to accept.
Someone had framed him, was _still_ framing him.
And she was going to find out who.
* * *
AT THE STATION, Aidan was run ragged by one call after another. Near the end of the shift, his unit was called out to a secondary fire at the hardware store, where the explosion from two days ago had killed Tracy. Looking at the scene woke Aidan right up. The new fire wasn't from any smoldering spark left over from the explosion. No way. This fire had been set.
Purposely.
In a wire mesh trash can.
Tommy was already there, and at the look on Aidan's face, shook his head. "Don't start."
"Arson."
"I said don't start."
"Let me guess. We're not going to have this conversation."
"Bingo." Tommy sounded extremely tense. "And this time I'll tell you why." He got up in Aidan's face. "Because I'm close, okay? I'm very, _very_ close to finishing this. So you need to let me do just that. Got it?"
Aidan didn't see that he had a choice. Later, back at the station, he stretched out on the station couch, closing his bleary eyes, needing to think.
Somehow it was all connected, he just knew it...He fell asleep trying to piece it all together, and then dreamed of a certain hot, curvy, sweet woman. A hot, curvy, sexy woman who happened to also be a _thief._
He woke up when someone sat on him.
And then bounced on him.
Opening his eyes, he met Cristina's frowning ones. "Trying to sleep here."
"No, you're not. Your eyes are open."
"Watch this." He closed them again.
She bounced again, a maneuver that threatened to break his legs. "How's Blake's sister?"
"Why are you asking me?"
"Because you're sleeping with her. Is she okay?"
He shook his head. "How? How do you know what I barely know?"
"Rumor mill." Her derisive humor hid her misery. Cristina was hurting. Hurting over losing Blake, her partner. Hurting over somehow blowing it with Dustin. She was so hard on the outside that they all forgot how soft and sensitive she was deep inside. She'd loved Blake like a brother, and cared about Kenzie by default.
"How is she, Aidan?"
"I don't know," he answered honestly.
"What do you mean you don't know?"
"She hasn't returned my phone calls."
"So you're losing your touch, too." She broke off, momentarily distracted when Dustin walked into the room.
The tall, tough-bodied, soft-hearted EMT pushed up his glasses, glanced at Cristina and a muscle jumped in his jaw.
Cristina didn't appear to breathe. Five agonizing seconds passed, and finally, she looked away first.
Dustin merely sighed.
The two of them had been doing some kind of emotional tap dance for weeks now. Dustin said he wanted more. Cristina said she didn't.
Now the tension in the room was so thick Aidan could hardly even see them anymore. "Hey, here's an idea. You two could lust after each other in secret and then ignore each other in person. Because it's not awkward at all."
"Shut up, Aidan." Cristina sent a glare in Dustin's direction, one that said _you're an idiot._
Without a word, Dustin walked away, into the kitchen.
Cristina expelled a low breath.
"Looks like I'm not the only one losing my touch," Aidan noted. "What did you do?"
"How do you know I did something?"
_"Please."_
Cristina sighed. "He's got his panties all unraveled because I went out with an ex."
"Ouch."
"No. No ouch. It was just dinner for God's sake. No biggee."
"Yeah. But it was dinner with a guy you've gotten naked with."
She shrugged, but dejection had settled over her pretty features. "Whatever."
"Cristina."
"I told you, it was just dinner." She got off of his legs, making sure to get an elbow in his gut. "And if he can't see that then screw him."
"Why don't you just talk to him? Tell him the truth?"
"Talking isn't what I want." She headed outside, slamming the door as she went.
Aidan's cell rang and he leaped for it, hoping for Kenzie, but he got Tommy instead.
"Might want to get down to county," the inspector said in an undecipherable tone.
"Why?"
"Because I had your girlfriend arrested."
"You arrested Kenzie?"
"You have another girlfriend I don't know about?"
"She's not my—" He pinched the bridge of his nose. "What the hell happened?"
"She's in for trespassing and interfering with a crime scene, so you figure it out. You don't control your women very well."
"She's not my woman!"
"Either way, I'd hurry. Oh, and get your checkbook. This date's going to cost you big."
# _CHAPTER 11_
JAIL WASN'T NEARLY as adventurous as it'd been that time Kenzie had been arrested on her soap. Then she'd had a costume director and a makeup artist. Oh, and nice, soft, flattering lights. Plus she'd been able to walk off the set when the director had yelled "cut", and had sipped her iced tea and laughed it all off.
No such luxuries today.
Real life sucked.
She was given her phone call—which went to her attorney, who promised to work on getting her out. With Kenzie's own checkbook, of course.
After several hours in a holding cell, during which she contemplated the odd and unwelcome turn her life had taken, and also chewed on a few nails, she was handed her see-through baggie of personal belongings—that was twice in two days—and shown the door.
Standing in front of it wasn't her attorney, but her own gorgeous, personal savior.
Aidan was dressed in his firefighter uniform, which told her he'd come right from the job. He still wore his firefighter badass expression, too, and was looking more than a little bit temperamental as well.
_Yeah._ Not exactly thrilled to see her.
Nor was she thrilled to see him.
Okay, so a little part of her was. The bad girl part of her, which reared its horny head and begged _Oh, please can we have him just one more time?_
She ignored that and her quivery belly, and tried to brush past him.
"What, no thank you?" He shifted so that she was forced to bump into him.
Backing up, she put her hands on her hips and sent him a glare as mean as she could conjure up after a few hours spent in jail. "I didn't call you."
"Yeah. I noticed."
There were several people milling around, all from a different part of society than she was used to. The guy closest to her might have been fifty, or a hundred and fifty, it was hard to tell with the multitude of hats and coats he was wearing, despite it being summer. He pulled out a cigarette and a match, and even though she saw it coming, when he struck the match to the matchbox and the little _whoosh_ hit her ears, she cringed.
Aidan was there in a second, holding her steady, which only further embarrassed her. "Easy."
"Damn." She let out a shaky breath. "What _is_ that?"
"Post traumatic—"
She waggled a finger in his face. "Don't say it."
"—stress. Why didn't you call me, Kenzie?"
"Who did?"
"Tommy."
"Rat-fink bastard." It was coming back to her, her childhood here—the small town mentality, the utter lack of secrets, the way everyone stuck their nose in everyone else's business. She'd had enough of that from her early years to last her a lifetime.
She and Blake had been kept together as they'd gone into the child care protective services, where they'd landed in a total of three foster homes, each as kind and as warm as they could possibly be, and for that she was more than grateful, she was also lucky—but she'd never really settled into any of them. She didn't tend to settle, didn't tend to get comfortable; it was what had made her so certain Aidan was the one.
Look how that had blown up in her face.
When she'd gone off to Los Angeles and begun acting, she'd found heaven. Pretending to live someone else's life, already all scripted out? Perfect. She'd loved it. _Still_ loved it.
But a small part of her knew that she couldn't always rely on a script. That at some point she would have to wing it. She'd eventually need a life, a _real_ one, and she'd always figured that life would somehow be entwined with her brother's, maybe even right here in Santa Rey...
But now there was nothing for her here, nothing except proving Blake's innocence.
Aidan caught her arm as she stepped outside. She yanked free and he put up his hands, letting her step away from him as they walked outside. He leaned a hip against a tree, looking big and tall and attitude-ridden as he eyed her like she was a lit fuse.
His hair had been finger-combed at best. She could smell soap and man, and the potent mix of testosterone and pheromones boggled her mind. If she lived to be two hundred years old, she'd never understand her attraction to him. Back in her L.A. world, she had access to dozens of gorgeous men. Hundreds.
But while some had been nice dalliances, none of them had ever really gotten any where. Probably because a good number of the men she met were like her.
Pretend.
Not Aidan. He lived life with his eyes wide open, no script needed. His job demanded a lot of him, and he was tough because of it, but he hadn't ever shied away from something just because it was hard. Except for her.
"Thanks for bailing me out," she conceded.
"Need a ride to your car? Or are you going to manage that on your own, too?"
The sun was warm and bright, and she stood still in it for a moment, tilting her head up to it, inhaling deeply. Then she turned to the man who had once been her everything. Whether she liked it or not—and for the record, she didn't—he could still stop her heart, make her pulse race, and worst of all, make her hormones stand up and shimmy. "Yeah. A ride would be great, if you don't mind."
He let out a sound that told her what he thought of that, and took her to his truck.
"About that ride..." She slowed, dragging her feet. "Everything's still booked. Maybe there's something—"
"You know where there's something." He turned on the engine and pulled out of the lot. "At my place."
"Yeah." She shook her head. "No."
"Yeah no?"
She sighed. "It's just that staying with you seems like a whole lot of trouble I don't want to face."
"Why?"
"Because I don't want to lead you on."
"I thought you enjoyed exacting your revenge on my body."
With more than a slight twinge of regret and, _dammit,_ guilt, she avoided his gaze.
"Come on, Kenz, be honest. You're not afraid of hurting me. You're afraid _you'll_ get hurt."
Wasn't that the plain ugly truth.
"You made sure I understood that you'd changed," he said softly, looking over at her for a beat before returning his attention to the road. "Now you have to understand something. I've changed as well."
Yes. Yes, he had.
"Look, you wanted to know what happened all those years ago?" he asked. "I got scared, that's what the hell happened. I'd always lived my life without letting people inside my heart, where they could hurt me. But you got in, and, yeah, that terrified me. You're doing it again, by the way, getting in, and I'm not any more thrilled about it now than I was then."
Something warm slid through her at his words, and the low, rough tone in which they were spoken. Warm, and dangerously seductive.
He pulled into his driveway and shut off the engine, turning in his seat to face her. "You'll have to make do without the five-star rating." He paused a beat. "Although there are certain five-star services I _do_ offer."
When she met his gaze she saw the sparkle of pure wicked trouble in his eyes. _Oh, boy._ "Aidan—"
"I'm talking about my breakfasts, which you happened to miss out on. And then there's my massage specialty." He didn't add any obvious eyebrow waggle or other suggestive gesture, but his eyes crinkled and she knew he was _thinking_ suggestively.
_Yup._ Dangerously seductive. She already knew how erotic his touch could be, just how earthy, how naughty, and she wasn't ready to go back there. Not if she intended to be the one to walk away this time.
And there would be walking away when this was over...
Even while she was thinking it, he took her hand and led her to his door. Her instinct was to make a smartass comment to piss him off, chase him away, and yet she didn't do anything but allow him to open the door for her. Once she started to step inside, he stopped her. When she met his gaze, he asked, "You planning anything else I should know about?"
"Like?"
"Shit. Anything. It could be anything."
The sun was bright. The surf behind them loud and choppy. She loved the scent of the ocean. She'd missed that, working long, long days on set in the middle of Los Angeles. Now that she'd been cancelled, she could see taking a laptop out on the beach and just writing to her heart's content if she wanted. "My immediate plans involve a shower."
"That's all?" he asked so warily that she smiled.
"Yeah. That's all."
He touched the corner of her smiling mouth. "That's a good look for you."
"What are you talking about, I smile all the time."
"On TV, maybe. But I haven't seen much of it here."
"Well, maybe that's because I was in a fire, then facing the fact that my brother's dead, and then..." And then she'd been in his bed, naked, panting, sobbing his name, holding onto his head as his mouth and then his body had taken her to heaven—
" _That_ look," he said, pointing at her. "I want to know what you were thinking just then to put _that_ look on your face."
She crossed her arms over her suddenly aching breasts. "Nothing."
"You are such a liar," he chided softly.
He gestured her inside his place, and she took a better look around than she had when she'd been fresh out of the hospital, and then fresh out of his bed. She saw the pretty windows, the wood floors he'd done himself, and felt another ache, this one in her chest.
She knew that growing up, Aidan hadn't had much of a stable home life, either. He'd been shuffled around as much as she had. Going into the fire academy had changed his life, given him a team, but more than that, his first _real_ friendships. The kind of friendships that would last, the kind of friend that had his back no matter what. He still hadn't had any real understanding of what that meant when she'd gone off to Los Angeles, but she could tell it had come to him in the years since. There was an easy confidence about him, an air that said he'd been well liked, well taken care of...
Well loved.
Her heart did a little flop at that because she hadn't given herself the same. Oh, sure, she was liked. She'd been taken care of. But loved by someone other than Blake?
No.
And if she took away the fame, leaving just smalltown girl Kenzie Stafford, what would actually be left?
The answer was as unsettling as the thought, especially given that now she really was without that fancy job. "Aidan?"
He'd headed for the kitchen, but stopped and turned to her. "Yeah?"
"Thanks."
"For?"
"For bailing me out. For waiting to make sure I was okay."
He leaned back against the wall and studied her. "So why did you do it, Kenz? Why did you go back after I'd warned you not to—" He broke off and shook his head. "Never mind. I just heard my own words and realized _exactly_ why you did it. _Because_ I warned you not to."
"Am I that stubborn?"
"Hell, yeah, you're that stubborn."
She rolled her eyes, then caught the flash of humor in his. He was laughing at her, and not with her, which should have made her defensive and possibly bitchy, but in spite of herself, she let out a laugh, too. "Okay, so it wasn't the smartest thing I've done. But it was the right thing."
"How about stealing my file, was that the right thing, too?"
She let out a low breath. "I was wondering when we were going to get to that."
He just looked at her, big and bad and...patient. So damn patient. She pulled the file from her bag and handed it over. "Thanks."
"I'd say you're welcome, if I'd given it to you."
"You'd have done the same thing in my position."
"You think so?"
She looked into his compelling eyes and felt her breath catch. "Okay, no. You would have asked. But maybe you're a better person than I am."
His eyes expressed his surprise at that statement. They both knew she hadn't always considered him such a great guy. "People change," she whispered, mirroring his words back to her. "Right?"
"That's right." The smile hit his eyes before his lips slowly curved, and there was an answering quiver that began in her belly. _Oh, boy._ Not good. He was standing too close, and not being annoying or antagonistic, and suddenly it all seemed too intimate.
She started to turn away but that was cowardice, and if she was going to learn anything while being back here in Santa Rey, it was not going to be that, so she faced him again. "I really am sorry for dragging you into this. For getting arrested and you having to bail me out. For driving you crazy. Pick any of the above."
"You didn't drag me into anything."
"Maybe not, but I'm about to." She let out a breath. "I need to tell you something."
"Okay." When she didn't go on, he raised an eyebrow. "Is it something that's going to get you arrested again?"
"No. I'm kind of hoping to avoid repeating that experience."
"Good."
"But there are things you should know. Things you're not going to like."
"Try me."
"Okay. I've been getting calls from someone I think is trying to help me."
He stared at her. "Your local cell caller?"
"Yes. He told me the key, whatever that means, is in Blake's computer files."
"He?"
"I think so. But I can't place the voice, he's disguised it."
"How the hell does he know the key's in Blake's computer files?" Aidan asked her.
"I don't know."
"Blake's laptop was never found. I'm betting it went up in _Blake's Girl_."
"As did mine. But with a computer, I could access my backup files, which would include Blake's backup files."
"I have a computer." He was close enough that she could see the green swirling in his light brown eyes. The scar bisecting his left eyebrow, the lines on his face, only added character, and a sexiness she couldn't have explained to save her life.
His mouth was slightly curved and she knew if she leaned in and touched hers to it, his lips would be warm and skilled, and most of all, giving.
"I didn't think I'd be happy to see you," she murmured, stepping closer. "But I've been proven wrong on two accounts now. When you saved me from the fire, and when I came out of jail and saw you standing there."
"Just the two?"
"Well, _maybe_ one other time..."
Leaning close, he let his mouth brush her ear. "Try a couple."
At the reminder of how he'd made her come _several_ times, easily she might add, as if he knew her body better than she did, a little shiver of awareness went down her spine, chased by another one, this one pure anticipation.
He could do it again. He could take her there again, to heaven, to oblivion...Only this time it wouldn't be adrenaline. This time she'd go in with her eyes wide open. His needed to be as well.
"I was worried about you," he murmured. "You've got to stay out of this one, Kenzie. Stay out of Tommy's way."
Somehow her face was nuzzling his throat, and she was trying to breathe him in. "I'm going to prove Blake's innocence in all this," she told him, liking the feel of her lips against his skin. "No matter the cost."
"Even if the price is my friendship?"
Her throat actually tightened at the thought and she pulled back to look into his eyes. "Is it going to cost me that?"
"Depends." He took her hand, put it on his chest and offered her a smile. "You still intending on stomping all over my tender heart?"
At that, and the crinkles at the corners of his eyes, the ones telling her he was teasing her, she out and out laughed, feeling much of tension drain away. "Yes."
His hands went to her hips, pulling her closer, and she stared into his face, feeling so at home in his house that she found herself hesitating, not for the first time that day, and wishing she had a script for what came next.
"You're thinking again," he murmured.
"Yeah."
He leaned back against the front door, unexpectedly giving her space. Space she thought she'd wanted, but found she didn't want at all. "I really did intend to stomp all over your heart, you know. When I first saw you again, I wanted to hurt you the way you'd hurt me. But then we kissed."
"We did a lot more than kiss."
She flashed back to that night, when she'd climbed into bed with him, pressing her icy feet to his, then her body. She remembered realizing he was naked and warm and strong and hard... _God_. He'd been so utterly irresistible, she'd lost her head. And, yeah, they'd done a lot more than kiss. "Fine. We kissed, and then I decided I should sleep with you and then walk away. Perfect, neat revenge."
"Neat, maybe. But not perfect." His eyes were glittering with knowledge, hard won. "Because it wasn't as easy as you thought, was it?"
No, it hadn't been. Because it'd been amazing between them. So damned amazing. "Maybe I've been looking at this wrong."
He didn't move from the door, just kept looking at her, his eyes warm, his mouth curved, his body big and bad and so gorgeous she could hardly stand it.
She wanted him.
Again.
Still.
"Maybe it's not about sleeping with you once and walking away," she heard herself say. "Maybe it's about letting this thing take its own lead for as long as I'm here."
"'This thing'? You mean the way we apparently can't stay out of each other's pants?"
At the huskiness in his voice, her nipples hardened. "Yes."
# _CHAPTER 12_
AIDAN PUSHED AWAY from the door and came toward Kenzie, all easy, loose-limbed confidence, yet radiating an intensity that made her breath catch. He didn't stop until they were toe-to-toe, and she slowly tipped up her head to look into his inscrutable eyes.
"You want to have sex," he said silkily. "Here. Tonight. Now."
Her breath caught at his bluntness. "And then maybe again later."
"Later," he repeated, as if trying to process this.
"Maybe even until I leave Santa Rey. At which time we both walk away, eyes wide open."
He just stared at her for the longest moment. "What happened to trouncing on my heart?"
"It seems you were right. I don't really want to hurt you."
When he shot her a not-buying-it look, she caved. "Okay, so I want to hurt you less than I want to sleep with you again."
"You know, you'd think I'd be tough enough to walk away from such an overwhelmingly romantic offer," he said drily, sounding both intrigued and baffled. "But apparently..." He put his hands on her hips. "I'm not."
She offered a smile that was sheer nerve. "So...yes?"
His eyes never wavered, holding hers, leveling her as he pulled her in. "I don't know, Kenzie. I'm a little afraid..."
"Be serious."
His smile was crooked and impossibly endearing. "I am. This time you could really do it, whether you're trying to or not. This time, you just might take out my heart."
"Come on," she quipped, even as a part of her was afraid he was right, for both of them. "If we're just having a physical relationship and nothing else, how can we get hurt?"
With a soft laugh, he slid his hands up her spine, and then back again, low enough now to cup her butt and squeeze.
He was hard.
Bending his head, he put his mouth to her ear and let out a breath that made her shiver in longing. "Just a physical relationship, Kenzie? Is that all this is? Really?" He sank his teeth into her lobe and she shivered again.
"It—it's all it _should_ be," she managed.
Another soft, deprecating laugh rumbled through his chest, this one aimed at the both of them. "Okay, well as long as we're being honest, you should know..." His hands glided up her spine again, this time beneath her shirt to touch bare skin. "Even though you _are_ going to hurt me, it's not enough to make me say no. Truth is, nothing could..."
She opened her mouth to say something, but then he kissed the spot he'd just nipped at, soothing the ache as his fingers stroked over her skin. Her eyes drifted shut, and she slid her arms around his neck, pressing close. "No pain, no gain," she whispered, and he let out another low laugh as he lifted her up and carried her to his bedroom.
To his bed.
He settled over her, looking down into her face for a beat before lowering his head and taking her mouth with his demanding one.
If simply walking into his house had felt like a homecoming, then this, here, now, felt even more so. He felt like home, he smelled like home, and he tasted even better; she hesitated, thinking, _uh-oh_.
His hands came up to hold her face. "What?"
She stared up into his eyes and saw herself reflected there, as if they were one, and although it was deeply unsettling to realize that this time she could fall even harder for him—if she let herself—she also couldn't imagine walking away, without being in his arms again.
"Kenz?"
"Nothing, it's nothing." And she pulled him down for another kiss as the heat of him seeped into her bones, warming her with a sensual promise of what was to come. Those big, warm hands slid along her arms, lifting them up over her head, entwining their fingers as his mouth continued to plunder hers, delivering on that promise.
It was familiar, and it was comforting, and yet it was so, so much more as well. Not since being with Aidan six years ago had she given any thought to what it would be like to be with a guy long enough that he felt...like home. She was a woman who liked change, who liked the new and exciting, who lived off the lines someone else wrote for her each day.
But with Aidan, she knew what he felt like, what he tasted like, exactly how crazy he could drive her with a touch of a single finger, and yet being with him felt almost unbearably _right,_ and far more arousing than she could have ever imagined.
Still kissing her, he pulled off her top, then her skirt. Her new bra was a front hook, which didn't slow him down at all, and when he had her naked except her panties, he hooked his fingers in the thin strip of cotton on her hips and let his gaze meet hers. Then he tugged, slipping the underwear down her legs and off, sailing them over one shoulder. Towering over her, fully dressed while she was as naked as she could get, he let out a low breath. "You're so beautiful."
"And you're overdressed." Still in his fire gear, in fact...
"In a minute." He was kneeling between her legs. He spread his, which in turn spread hers, and his gaze took her in, in one fell swoop, heating her skin everywhere he looked. He traced his fingers over her breasts, her belly, her thighs.
Between.
When he bent his head with fierce intent, she sucked in a breath, a breath that clogged her throat when he replaced his fingers with his mouth.
"Aidan," she managed, hardly recognizing her own voice. "I—"
His tongue encircled her tender, sensitized flesh, making her quiver from the inside out, and she promptly forgot what she'd meant to say. While his tongue and fingers circled and teased and stroked, she gripped the sheets and stared down at him. His hair stood up, from her fingers, she realized. His eyes were closed, his expression dreamy as he brought her such bliss she could hardly even see, much less think.
But she didn't close her eyes. She watched him concentrate on her pleasure as if it were his own, took in his moves, the moves that were driving her right out of her ever-loving mind.
It was as if he knew what made her tick, inside and out. That was a terrifying thought, really. Because the girl he'd once known no longer existed, and since then...well, she hadn't really let anyone know her.
An ever-changing script.
That was her life.
A life she was no longer sure about. But having him take her apart the way he was, _that_ she was sure about.
He opened his eyes, so molten hot that they were nearly black, and looked up at her. He was sure, too, which should have stopped her cold, and she stirred. "Aidan—"
"Shh."
Then he swirled his tongue in a precise rhythm over ground zero, and she lost it.
Completely.
Lost.
It.
Panting for breath, arching up off the mattress and into his mouth, she dug her fingers into the sheets, throwing her head back at the peak, sobbing out his name.
Slowly he brought her back to planet earth. She closed her eyes, savoring the pleasure, still quivering and pulsing as he kissed his way back up her body, his tongue stroking a rib, a nipple, her throat...and then he cupped her face and smiled at her.
"You shushed me," she said, her voice sounding weak and raspy.
"It was for a good cause." He rocked his hips into hers.
"I'm going to get you back for that."
He smiled wickedly. "Should I be scared?"
"Terrified." Rolling him over, she sat on him and tugged his uniform shirt off. She could have spent a year lapping him up with nothing but her tongue. He had a tight body, toned from years of physical labor. His chest was broad, hard, his belly rippled with sinew and rising and falling in a way that assured her she was in no way alone in this almost chemical-like attraction they shared, which transcended both time and logic.
His hands went to the button on his pants to help speed up the process, and she ran her fingers up the taut, corded muscles of his abs. He unzipped, she tugged, and then nearly drooled at the sight of the part of him so happy to see her.
She licked her lips.
He groaned.
She kissed him, on the very tip.
"Kenz—" he choked out, tunneling his fingers through her hair.
Since her mouth was now full, she couldn't answer, and he said something completely unintelligible anyway, which, she had to admit, only egged her on. God, she loved rendering this big, bad, tough man completely incapable of speech. Loved the power that surged through her at the way he was breathing, saying her name.
Loved so much about it that it scared her. Scared her into being even more bold and brazen so that she didn't have to think about how much being with him meant to her.
How much he meant to her.
Using her hands and mouth, she drew him to the edge. "Two-minute warning," he groaned out, his hands fisted in the sheets at his sides as she ran her tongue up his length. "Okay, thirty seconds. _Maybe._ "
She kept going until he swore and grabbed a condom, nudging her to her back, his hands running up the undersides of her arms until they were over her head. His knee spread her legs, his thigh rubbing against the core of her.
"In," she gasped, arching into him. "In me now."
Lowering his body to hers, he nipped at her lower lip, then kissed her, hard and deep, his tongue slipping into her mouth at the very moment he slipped into her body. "Like that?"
She couldn't answer. Hell, she could hardly breathe.
"Kenzie?"
"Yes," she managed, then shuddered as he withdrew, only to thrust into her again. And again. _"Like that."_
The feel of him, thick and hot and filling her to the brink, had her gasping his name, wrapping her legs around his hips, leaving her unable to remember exactly what she was supposedly paying him back for. Her toes were curling, her skin feeling too tight for her body, which seemed to swell from the inside out. "Aidan—"
"Come," he demanded, grinding his teeth in what looked like agony. "I want to feel you come before I—"
She burst in mindless, blind sensation, and barely heard his strangled answering groan as he exploded.
For long moments afterward, they lay there entwined, panting and damp, and powerless to move, their breathing echoing loudly through the bedroom.
"Is it just me," she finally managed, "or does that get better and better?"
"Oh, yeah."
She fell quiet a moment, but then couldn't resist. "You think it'll keep happening? You know, until I leave?"
"If it does, it's likely to kill me."
"Yes." She sighed dreamily. "But what a way to go."
His soft huff of laughter was the last thing she remembered before she drifted off to sleep.
* * *
AIDAN WOKE UP SOMETIME LATER with a smile, his body ready for another round. In the pitch-dark, he rolled over for Kenzie.
And got nothing.
With a very bad feeling in his gut, he sat up. "You're gone, aren't you?" he said into the night.
When he got no answer, he tossed back the covers and got out of bed, but it was too late. She had left. He told himself he wasn't her keeper, and she could go wherever she wanted, but he'd been lulled into the impression that she hadn't been done with him yet.
She _wasn't_ done with him, not yet. Which meant she was probably out there looking to poke her nose into the arsons. Aidan hurriedly got dressed. He had no idea where she was but he needed to find out, because with whatever information she'd get, she'd go snooping into things that were guaranteed to piss off Tommy.
_Hell._ They'd just spent hours in his bed. And in his shower. And then his bed again. Hadn't he tired her out?
His stomach was grumbling and his head starting to pound when he picked up his cell phone and called hers; he was shocked when she answered.
"Hi," she said in that soft, breathless voice that had only a few hours before made him come.
Just hearing it stirred him halfway to life. He was little better than Pavlov's dogs. "Where are you?"
"Oh, out and about." She still sounded breathless.
"Kenzie, what are you doing?"
"Um...exercising?"
"That's a bad word to you."
"Not anymore. Do you have any idea how much work it takes to stay in TV shape?"
And then he heard it, the unmistakable sound of a sliding door either opening or closing. "Where are you?"
"Whoops, bad connection," she said.
He gnashed his teeth together. "We have a great connection. What are you up to?"
"Wow, I can hardly hear you..."
"Kenz—"
"Gotta go."
He didn't have to hear the click to know she'd shut her phone. Nor did he bother with swearing. Instead, he grabbed his keys and went after her, figuring her options were severely limited. She wouldn't have gone back to the docks because there were no sliding door there. So she was probably at Blake's house. He supposed she could also be at any one of the arsons Blake had been accused of, but most of them had been demo'd, and plus it seemed likely that if she was butting her nose in, she'd start at the top.
So would he.
He hit the jackpot on his first try. Pulling into the small house Blake had claimed as his own, he parked right next to Kenzie's flashy Mercedes. He got out of his truck and felt the hood of her car.
Still warm.
So she hadn't been there long. She was just damn lucky she hadn't gotten herself arrested again, considering the yellow tape surrounding the house. Just thinking about what Tommy would say, and how long he'd jail her this time, had him sweating. The front door was shut and, as he discovered, locked.
Aidan moved around the side of the house. His plan was simple. He was going to scare the hell out of her. And then he was going to kiss the hell out of her.
And then...and then he had no idea. Spanking her seemed like a good option.
The sliding back door on Blake's deck was unlocked and opened an inch. This was where she'd entered, and following suit, he slipped inside. The place was dark, but there was a light on upstairs, and he headed in that direction. At a sound behind him, he whipped around just as two hands smacked him in the chest and shoved. As he fell back, he reached out and hauled his assailant with him. He hit his ass on the bottom step and Kenzie landed on him.
_"What are you doing?"_ she demanded.
The stairs biting into his back, her full weight over the top of him, he hissed out a breath of pain. "What am _I_ doing? What are _you_ doing?"
"I'm—" She bit back whatever she'd been about to say, crawled backward off of him and stood up.
"No, it's okay, I'm fine, thanks," he muttered, getting up on his own and brushing himself off. "How did you get in here?"
"Blake gave me his spare key a long time ago."
"Okay, so back to my first question. Why are you here?"
"Looking for clues to Blake's innocence." She glared at him, then pointed to the door. "You need to leave."
"So do you."
"Oh, no. This is my brother's place. I'm his beneficiary. I get to be here."
"Not with the caution tape still blocking the front door, you don't."
She was breathing fast, her voice thick and husky as if she'd been crying. Or maybe she still was. He couldn't see her clearly enough to decide. "Ah, Kenzie. Don't—"
"Go," she said, crossing her arms over her chest.
"Fine. But you're coming with me."
"No, I'm not."
"Yeah, you are." Wrapping his fingers around her arm, he headed toward the sliding door, toting her with him, until she yanked free. Then, lifting her nose, she stalked out in front of him, going willingly but not happily. "Kenzie," he said as she got into her car.
"I don't want to talk right now." She tried to shut the driver's door on him but he stepped closer, holding it open.
"Isn't that convenient."
"Dammit, Aidan. Get out of my way."
"Just tell me where you're going."
For the first time, she hesitated.
"You could try my house," he suggested. "My computer."
She paused another beat. "I wouldn't want to impose."
"Imposing would be getting your pretty ass arrested again, goddammit. Meet me there."
"Fine." Putting the car into gear, she peeled out, leaving him little choice but to hope that she would.
# _CHAPTER 13_
WITH LITTLE TO NO TRAFFIC in the middle of the night, it took only five minutes to get home. Aidan pulled into his driveway next to the little red sports car, watching Kenzie storm up the walk to his front door, looking irritated and frustrated.
Just as irritated and frustrated, he followed. Did she have no clue what she was doing to him?
How could she not?
"Wait," she said, stopping so fast he plowed into her, staring back at the street. "Did you see that car?"
"No."
"It was gray." She chewed on her thumbnail. "Look, I'm not trying to change the subject here, because trust me, I'm pissed and enjoying being pissed, but I think someone's following me."
Reaching past her, he unlocked the door and gestured her in ahead of him, keeping his body in front of her back as he turned to eye the street.
He didn't see the car—at the moment, there were _no_ cars—but he didn't doubt her. "You've seen it before?"
"Yes. Truthfully, I'm beginning to feel sort of stalked." She whirled to face him. "Okay, so back to being pissed off."
Oh, no. Not yet. He'd anticipated her, and was standing so close she bumped into him, squeaking in surprise, but when she tried to take a step back, he held her still. Christ, she smelled good and the way her hair framed her face..."How long have you suspected someone's been following you?"
"Since the boat fire, I guess."
"Have you told anyone? Tommy? The police?"
"I wasn't really sure. I'm still not sure. It's just a feeling."
He let go of her to pull out his cell phone.
"What are you doing?"
"Calling the police."
Kenzie stepped close and shut the phone, stuffing it back into his pocket. "Aidan, listen to me. We both know that you and I don't do _real_ relationships, especially not with each other. Now sex, we do that just fine. And in case you're confused, the biggest difference between the two is that with just sex, there's no sharing of personal information."
He was not liking where this was going. At all. "Meaning?"
"Meaning I don't have to account to you, and you're not responsible for me."
He stared at her, more stung than he'd like to admit. "Well, shit."
"I mean it, Aidan."
"You don't want me to call the police."
"And scare off the guy? No, I don't."
"Fine."
" _Fine._ Now where the hell is your computer? We have some files to access."
"My bedroom."
They were nose-to-nose, now. Breathing in each other's air. He could feel the heat of her radiating into him, and for whatever reason, his hands ran down her arms and then back up again, squeezing a little, more moved by the close proximity than he'd like to admit.
The very tips of her breasts brushed against his shirt. Her thighs bumped into his. Sparks were flying from her eyes, her mouth grim.
A mouth that suddenly he couldn't stop looking at.
Her hands had come up to his chest and she dug her fingers into his pecs, hard enough to have him hissing out a breath. Her eyes were on his, but then they lowered to his mouth.
She was thinking about kissing him.
Leaning in, he took care of that little piece of business for her. Covering her mouth with his, he swallowed her little moan of pleasure and promptly lost himself in her when she melted against him, entwining her arms around his neck so tightly he couldn't breathe. Since breathing was overrated anyway, especially when kissing her, he just hauled her up tighter against him and kept at it. Her hands were in his hair, his molded the length of her body to him, until suddenly, she shoved him clear, turned and stalked off, heading down the hallway and into his bedroom. He stared after her, breathing like a misused race horse, warring with himself. He could go after her. Or _he_ could walk out on _her_ for a change of pace.
_Yeah, right._ He went after her.
* * *
WHEN AIDAN OPENED THE DOOR of his bedroom, Kenzie held her breath. She hadn't turned on the light, so he was silhouetted from behind by the lamp in the living room, looking tall, dark, and so sexy she could hardly stand it.
And attitude-ridden. Don't forget that. Stalking past her, he opened his laptop and hit the power button. While it booted up, she just stared through the dim room at him, wishing...hell.
Wishing things were different. That's all. If only she could call the writers and complain about this particular plotline, and maybe get it adjusted. Or get a new script delivered. Yeah, that would be best. One with a happy ending, please. With a sigh, she moved to the laptop. "Should I download it to your desktop?"
"Yes."
She accessed her mail, and the files she'd saved, clicking on the first of Blake's. "It's going to take a while. It's a big file. And it'll take even longer to flip through it all and see if there's even anything in it that we can use."
"Your caller suggested there was."
"Yes, but how did he know? _What_ did he know?"
"Let's find out. Kenzie—"
"I'm not ready to talk."
He stepped closer, a big, tall, badass outline. "What _are_ you ready for?"
"How about the only thing we're good at?"
With a low sound that might have been an agreeing groan, he came even closer. "Kenzie—"
"No. I mean it." He was hard. She could feel him. Could feel, too, the tension shimmering throughout his entire body. It matched hers. "No talking."
"Fine." With a rough tug, he hauled her up against him. His body was warm and corded with strength, his hands hard and hot on her. And his mouth...
God, his mouth.
He was the most amazing kisser, his lips warm and soft and firm all at the same time, his tongue both talented and greedy and generous.
So generous that she moaned into his mouth and held on for the ride until she couldn't stand it anymore. "Clothes," she muttered, and yanked off her own top, gratified to see him doing the same. She stared through the dark at his bared torso as she worked the buttons on her jeans while simultaneously kicking off her shoes. God, he was gorgeous. Sleek, toned and so damned yummy she wanted to gobble him up on the spot. She shoved down her jeans, watching him do the same, but unlike her, his underwear went bye-bye with his jeans, and her mouth actually went dry.
Riveted by the sight, she stood there in her bra and panties and socks. Staring.
He stood there in nothing. In glorious, mouth-dropping, heart-stopping nothing. Yeah, she'd seen it before, all of it and more— _but, damn._
"You cheated," he said, reaching for her bra.
His erection nudged her belly, and forgetting to finish stripping, she wrapped her fingers around him.
He hissed out a breath.
"Too tight?" she asked as she stroked.
"No, your fingers are frozen."
For some reason that made her laugh. How the hell that was even possible with all the sensations crowding and pushing for space in her brain was beyond her but she stood there, her fingers wrapped around a very impressive erection and laughed.
"Yeah, see, you're not really supposed to hold onto a guy's favorite body part and laugh."
Which, of course, made her laugh harder.
With a shake of his head, he just smiled, clearly not too worried because he remained hard as a rock in her hand...
As his fingers worked their magic and her bra fell to the floor at their feet.
When he stepped even closer, her nipples brushed his chest, and it was her turn to hiss in a breath as they hardened.
And then she couldn't breathe at all because he dropped to his knees, hooked his thumbs in the edge of her panties and tugged.
At the sight he revealed, he gave a low, ragged groan and slid his hands up the backs of her thighs, cupping her bottom in his big palms. "God, look at you."
"Aidan—"
"You're so pretty here." He ran a finger over her. "All wet and glistening. For me." There was a deep, husky satisfaction to his voice that made her thighs quiver.
"Spread your legs," he murmured, skimming hot, wet, openmouthed kisses up an inner thigh. "Yeah, like that." He pulled her forward, and right into his mouth.
At the first unerring stroke of his tongue her knees nearly buckled but he had a grip on her, one hand on her hip, holding her upright, the other exploring between her legs, working with his tongue to drive her out of her mind. "Aidan—"
"You taste like heaven," he whispered against her. _"Heaven."_
And he felt like it. She strained against him, her fingers tunneled into his hair, her head thrown back as he took her exactly where he wanted to her to go, which was to the very edge of a cliff, so high she couldn't see all the way to the bottom, couldn't speak, couldn't do anything but feel.
And she was feeling plenty. Mostly a need for speed at this point, but he purposely slowed her down, dancing his tongue over her as light as a feather. She tightened her fingers in his hair, silently threatening to make him bald if he didn't get back to business. Her business. "Aidan, dammit."
"I could look at you all day."
"Look later. Do now."
"Always in a hurry." He tsked, but obliged.
Oh, God, how he obliged, skimming his hands up the front of her thighs, gently opening her. For a moment he pulled back, admiring the sight before him, wet from his tongue, wet from her own arousal.
Standing there so open and vulnerable, she let out a growl of frustration and need, and he leaned in, this time sucking her into his mouth hard, giving her the rhythm she needed to completely lose it.
When her knees gave out, he let her fall, catching her, rising to his feet, spinning toward the bed, his mouth fastened to hers. His hands moved over her body, thoroughly, ruthlessly, ravenously kissing her as they went, until from somewhere behind them, from the pocket of her pants, her cell phone went off. She couldn't even think about getting it. Hell, the entire place could have gone up in flames right then and there and she doubted she would have thought about it. "In me, in me."
He let out a rough laugh.
_"Now."_
Because now was the only thing that mattered, and this was the only thing that registered, the feel of his hands on her body, molding, sculpting, flaming the wildfire flickering to life inside her.
Aidan crawled up her body. He'd found a condom, and made himself at home between her thighs. Then he stared down into her eyes, his unwavering and fierce. "This is not just sex." His voice was low and rough. "It's not. Not for me."
She blinked, trying to clear her fuzzy head.
"And if that's all it is for you, I want to know it now." He lifted her hips, his strong callused fingers gliding over her flesh, making sure she was ready for him.
She was.
Beyond ready.
"Tell me," he demanded, holding still, waiting on her word. She stared up at him, her heart swelling at the truth. "It's more," she admitted, which— _ding, ding, ding_ —was the right answer because then he spread her thighs wider and drove himself into her, hard and fast, the way she'd wanted, and took her right where she needed to go.
Halfway there, with her breath sobbing in her throat, with their bodies straining with each other, she cupped his jaw and looked into his face.
He was damp with sweat, hard with tension, and so damned sexy she could scarcely speak. "Aidan."
"Don't stop me."
She shook her head at his rough plea. Stop him? Was he kidding? She wanted him to never stop.
Never...a terrifying thought. "Aidan..."
His mouth nuzzled at her ear. "Yeah?"
"I missed you," she whispered, letting him in on her biggest secret, giving it to him without reserve, letting him look deeply into her eyes.
She absorbed both his surprise and his next thrust, and then that was it.
She burst.
And so did he.
# _CHAPTER 14_
AIDAN LAY ON HIS BACK, a hot, naked, still quivering Kenzie in his arms, and let her words soak in.
_She'd missed him._ "Kenz?"
"Mmm." Her face was pressed against his throat, her mouth sending shivers of delight down his spine even now, when his bones had turned into overcooked noodles and he couldn't have moved to save his life.
Well, except a certain part of his anatomy, which appeared to have segregated from his brain. That part moved. That part wanted round two.
And possibly round three, please.
Kenzie lifted her head and looked at him, all sleepyeyed and still glowing. Waiting for him to speak.
He found himself cupping her face, and bringing it in for a kiss that lingered.
And deepened.
"I missed you, too," he whispered against her lips.
She pulled back and closed her eyes.
Staring down at her, he let out a breath. Okay. So she hadn't meant it. It'd just been the heat of the moment talking. He supposed he could understand that. Had to understand that. After all, the moment had gotten pretty damn heated. "It's all right." God, listen to him lie. "I get it."
Across the room sat his laptop, with answers. Or so he hoped. "We'd better get up." He was relieved to note that his voice seemed to sound normal, that he was still breathing and that the heart she'd just stabbed was apparently still in working order.
Even if it was bleeding all over the place. Internal carnage...
But he had no one to blame but himself for opening it up to her in the first place. She'd warned him, hadn't she? She'd warned him and he'd been cocky enough not to believe it possible.
"Aidan?"
He managed to look at her.
"I _did_ miss you. I missed this. But..."
"But life intrudes. I get that, too."
She looked into his eyes, sighed, then slipped from the bed. Gloriously naked, she walked to his computer. Lit only by the glow of the screen, she afforded him a particularly fine view. "Huh," she said, and bent over a little so that her fingers could move over the keyboard.
She was absolutely clueless about the picture she made in green glowing profile, with her hair wild around her head, a whisker burn from his face across a breast and her ribs, and her very sweet ass looking good enough to bite.
"That's odd," she muttered, her fingers moving faster, the furrow between her eyebrows deepening as she frowned.
He opened his mouth to ask what was odd, but she bent a little farther and he couldn't gather enough working brain cells to do anything but stare. Her spine was narrow and pretty, and his gaze followed it down past the indention of her waist and the gentle flare of her hips to one of his favorite parts of a woman's anatomy. Her legs were spread slightly, her thighs taut, allowing him a peek of the treasure between—
"Aidan?"
At the tone, he managed to squelch the lust. _Barely._ Rising, he walked up behind her. Also naked. Curling his body around hers from behind, a good amount of that lust came barreling back, hitting him like a freight train. He couldn't help it. His chest was against her back, her world-class ass pressed into his crotch. His hands went to her hips, one slipping around to her ribs, his fingers just brushing the underside of her breast. Pressing his lips to the side of her neck, he let his hand skim up, gliding over her nipple, which hardened gratifyingly in his fingers.
_Oh, yeah._
His other hand slid to her belly and began a southward descent—
"Look." Catching his hand, she pointed to an opened Excel worksheet. She had brought up an interesting list. "My mysterious caller said to look at the demos," she told him. "I didn't know what he meant, but all the burned buildings have been razed to the ground. I saw the photos in Zach's file—not all of those buildings were severely damaged."
With great difficulty, he frowned at the computer and not at her nude body, his hands still full with warm, sweet, sexy-as-hell woman. "It's true," he said. "But the properties were demolished anyway. Except for the last two."
"On whose orders?"
"The records have been sealed."
"Why?"
"That's the question. Zach tried to get the answer to that and it cost him."
Forcing his concentration from her body, he took in the worksheet in front of him. "Pretty impressive information here." Blake had been busy.
So had he been keeping track of his own handiwork, along with what happened to each property after the fires?
"Who has the power to order a demolition of a burned property?" Kenzie asked him.
"The owner, anyone acting on the behalf of an owner or the fire department, if the property is deemed unstable or unsafe for any reason."
She pulled free and went for her clothes, which were strewn across the room. He watched with great regret as she found the pieces one by one and covered up that gorgeous bod.
With a sigh, he reached for his jeans and slid them on. Back to the grown-up world apparently..."How is it you've never looked through Blake's files before?"
"I never thought to. We regularly sent each other files, just in case. It was our backup system."
"What did you send him?"
She lifted a shoulder. "Rough drafts of stuff."
"Stuff?"
"I've been writing. Scripts." Another lift of her shoulder. "For the day I finally ate too many donuts and didn't get asked to audition anymore."
"I bet you're a great writer."
"Really?"
He thought about how deeply she felt things, how good she was with words, and nodded.
Looking touched, she smiled. "Thanks."
"How long ago did he send you this file?"
"He sent me a backup file every week. We were supposed to keep only the latest version for each other, but I was always too lazy to go back and delete the week before, so I should have them all—" She stared at him for a beat before whipping back to the computer. Her fingers raced over the keys as he bent his head close to hers, looking at what she brought up.
An entire list of arson-related backup files from Blake, starting shortly after the first suspicious fires, until the day before he died.
"So," she said slowly. "Either he was a damned stupid felon, or he was investigating the arsons himself."
Her tone made it clear which she believed.
"Or," he said softly, knowing she was going to hate him. "He's keeping track of the arsons for a partner."
She looked at him again, her eyes cooling to, oh, about thirty-five degrees below zero.
"Open the first file."
Without a word, she clicked on it. It was a Word document, a diary of notes with a running commentary. The first read:
> _Hill Street fire:_
>
> _Second point of origin mysteriously vanished on day of cleanup. Wire metal trash can, unique enough in design that it should be traceable. When I mentioned this to the chief, he said I should stick to fighting fires._
Kenzie read the entry out loud, twice, then scrolled down to the next entry, several weeks later.
> _Blood is thicker than water. I was told that today and apparently need to remember it. If I want to live._
Kenzie whipped her gaze to Aidan. "What the hell does that mean?"
"Sounds like a threat," he said grimly.
"Blood is thicker than water," she repeated. "Who is he talking about? We have no family. At least no family who cares about us, anyway."
He hated the look on her face, the faraway, distant, self-protective look she got whenever she had to talk about her past. There was no doubt, she and Blake had had it rough growing up, being shuffled from one foster home to another. The saving grace was that they'd been kept together. It was what had made their bond so strong—they'd been all each other had had. "Is there possibly a blood relative somewhere?"
"A few, scattered here and there across the country. A great-aunt in Florida, an uncle in Chicago, a cousin in Dallas..." She crossed her arms, closing him out mentally and physically. "Just no one who wanted us."
Gently he turned her to face him. "Could he be talking about you, then?"
"Definitely not. We were in touch all during that time, but we never had a conversation about any of this."
Aidan went back to reading the entries, one of which mentioned employee hours. Copies of the schedules were attached. So was Blake keeping track of _his_ alibi, or someone's whereabouts?
Blake had somehow gotten Tommy's first official reports on the arsons as well. Aidan and Kenzie discovered that he hadn't been on duty at any of the suspicious fires, a fact that Tommy had apparently considered evidence since it left Blake without an alibi for when the fires had been lit. Aidan scrolled down the list.
"Whoa, stop." Kenzie pointed to the second fire. "There. That one can't be right. He had an alibi for that one, he was with me. He'd come to Los Angeles that week. I remember because he was my date for the Emmys. He flew home immediately after, catching a red-eye because he said he had to be back at work for an early shift."
"Okay." Aidan pulled up the employee schedule for that day. "But he's not listed as on duty."
Kenzie stared at the screen, shaking her head. "He wouldn't have lied to me."
She said this with utter sincerity, and Aidan was inclined to absolutely believe because _she_ believed. But if Blake _hadn't_ lied to Kenzie, then there was only one other explanation.
"The schedule got changed?" she asked.
"It could have happened. Someone traded. Or—"
"Or something physically changed the schedule after the fact," she said flatly. "And Blake isn't here to defend himself."
"No, but we are." He was looking at the screen, until he realized that she wasn't. She was staring at him. "What?"
Her eyes were shimmering brilliantly with anger and something else, a deep, gut-wrenching emotion. "I didn't think it was possible." Her voice sounded thick. "I didn't want it to be possible. Oh, God." She covered her face. "This is so stupid."
"What?" He looked at the screen again, trying to figure out what she was talking about. _"What's stupid?"_
"That I could like you more than last time."
The words reached him as little had in all these years. "Kenz." Melting, he pulled down her hands. "I—"
She put a finger in his face. "Don't get excited. I don't want to feel this way, and I'm telling you right now I _am_ going to fight feeling this way."
His heart was squeezed tighter than a bow. "We were just kids, Kenz."
"And now we're not. It doesn't change anything except we're older, and _actually,_ it's going to hurt more." Jaw tight, she shook her head again and looked at the screen. "This first. Blake first. He's far more important than rehashing old emotions that I don't really want to have." She worked the keyboard. "There. He's not on the schedule there, either, but he called me from the station. I know because it was my birthday, see? And he called me at 6:00 a.m. to catch me before work, but I didn't have an early morning shoot that day, and I was irritated that he woke me up. I'd been up late the night before celebrating."
"With Chad?"
She swiveled her eyes in his direction. "Actually, Teddy. Teddy White."
"Wasn't he on _People's_ Most Beautiful list?"
"How do you know that?"
He knew it only because someone had stolen the porn out of the station bathroom, and Cristina had left her _People_ magazine in there in its place, and—And Christ. He was crazy. "Never mind."
"It was just a one-night thing."
_Oh, great._ Even better. Now he could picture them having one-night sex, and—
"He's a friend."
A friend, as in someone who'd pulled her out of a fire? Someone who'd bail her out of jail?
"Yeah," she said softly. "I realize the word _friend_ is a loose term, especially in Hollywood. Not like here."
"Do you miss it? Hollywood?"
She opened her mouth, then closed it and sighed. "I almost said yes, out of habit. The job is fun and the pay is amazing, but..." She lifted a shoulder. "It's empty. And I didn't really get that until I was here, either."
He tried to sort out his feelings regarding this revealing fact.
"And, anyway, it no longer matters." She turned back to the screen. "It's over."
"What do you mean?"
"My soap got cancelled."
"It did?"
"Yeah, and there are auditions for new parts but I've been eating too many donuts, so..."
"So...what?"
"So I'm going to get fat."
He let out a low laugh. "You look great, Kenzie. So great I haven't been able to keep my hands off you, as you might have noticed. But I'm very sorry about your job." He couldn't believe he was going to say this. "You could always stay in Santa Rey."
"I thought about it." She sighed and faced him again. "But staying seems like a comfort thing. You know, like going back to the last place where I was happy. It's a cop-out. And I was only happy here because of Blake."
He held his breath. He'd made her happy, too. Until he hadn't. "Maybe it was more than that."
"I don't know." She sighed without giving away her exact feelings on the matter, although he suspected she didn't know her exact feelings. "I wouldn't be able to get a job here."
"I know they don't film TV or movies anywhere close, but you could do something other than act."
She scoffed, then looked at him with heart-breaking hope. "Like what?"
"You know what. You could write. And eat all the damn donuts you want."
She just looked at him for a long moment, until he nearly squirmed. "What?"
"I'd have thought you'd be holding open the door for me to get the hell out of Dodge."
"Yeah, well, that was the old me."
"Well the new me is here to get Blake's name cleared. That's it."
"And also to stomp on my heart. Don't forget that part."
"I won't." She sighed. "Except I'd really rather get out of here without hurting you at all." With no idea that she'd just stunned him to his core, she leaned in close to see the screen better. A strand of her hair got stuck to the stubble on his jaw. It smelled good.
She smelled good.
It was all he could do not to bury his face in the rest of her hair and say things that would lead her back to his bed but not really get them anywhere. In fact, he'd opened his mouth to do just that when she spoke.
"Look." She pointed to where Blake had entered another note:
> _Not noted in any of the official investigation reports is the fact that the source for the wire mesh trash cans is the hardware store where Tracy works._
Kenzie frowned and turned her head to look at Aidan, who had gone still in sudden shock. "The Tracy who..."
"Died." Aidan managed to find his vocal cords. "Yeah. They dated a couple of times. He really liked her."
"Really? He told me he'd gone out with Tracy, but he never said how much he liked her."
"Maybe he didn't tell you everything."
"He did," she insisted. "We told each other everything."
"Kenzie, you didn't tell him when _we_ were going out. Maybe—"
"No." She shook her head. "You're going to say he kept secrets. That he kept the arsons a secret, but he wouldn't have—He wouldn't have done this, Aidan. Tracy being killed, well that's got to be a terrible coincidence."
"I'm beginning to believe that nothing's a coincidence. Look at the next entry."
> _Tracy's going to get me a list of people who've purchased the trash cans, but she has to wait until the weekend when her boss isn't in._
The next entry didn't clear anything up, but made it all worse.
> _Got the list, and holy shit. Blood_ is _thicker than water. Got to remember that..._
Kenzie's fingers dug into Aidan's arm. "What does that mean, 'blood _is_ thicker than water'? He's written that twice now."
Aidan frowned and shook his head. "I wish I knew."
> _He's onto me. Need to be damn careful now._
" _Who's_ onto him?" Kenzie stood up and paced the length of the bedroom. "God. Whoever he's talking about, do you think...?"
_Yeah._ Yeah, he did. Blake had gotten himself into hot water with someone. And that someone had either been his partner in crime, or, as Aidan was coming to believe, it was the person whom Blake had been privately, quietly, investigating on his own.
And if _that_ was true, and Blake had been a victim, then this other person had not only been an arsonist, but also a murderer.
Aidan's cell phone chirped with a message that he was needed at work, ASAP.
"Go," she murmured. "It's okay. I'm just going to go through all of this and see what else I can find."
"Stay here."
Her gaze slid to his.
"Kenzie..." How to say this without sounding like a complete idiot? There was no way to sugarcoat it, so he decided to just let it out. "I have a bad feeling."
She arched an eyebrow. "You, the most pragmatic, logical, cool person I know, have a bad _feeling?_ "
"Go with me on this."
"You think I'm in danger," she said flatly.
He didn't just think it, he knew it. Only he couldn't explain how or why, and that was going to drive him crazy, along with worrying and wondering where she was and if she was okay.
And safe.
And alive.
"Aidan, I'm not going to hole up here. That's ridiculous. Besides, no one knows what I'm doing."
"You were arrested, Kenzie. Everyone knows what you're doing."
"I'll be fine."
Short of tying her up, which had a _most_ interesting vision popping into his head, what could he do? "Promise me you'll be careful."
She looked at him for a long moment, her hair still crazy from his fingers, her shirt crooked, her feet bare, looking like a hot mess.
A hot mess he wanted in his life.
"I thought we weren't going to do the promise thing," she said. "Not ever again."
"Promise me," he said again.
"Don't worry." She backed away from him, her face so carefully blank. "I intend to be careful and smart, and I intend to get out of here unscathed, on all counts."
What the hell did that mean?
"See you, Aidan."
Okay, that was no simple _"I'll see you later."_ It seemed like a we're-done-doing-the-naked-happy-dance see-you. The get-over-me because I'm-over-you see-you.
Which didn't bode well for his heart, the one that in spite of himself, had gotten attached. Again. More attached, if that was even possible. "I'll be back."
"Okay."
"I will." He paused. "Will you be here?"
She met his gaze. "I don't know."
_Well, hell._ That didn't bode well.
# _CHAPTER 15_
IN BETWEEN CALLS, Aidan slipped into the office of the fire station. He'd never spent much time in there, always preferring to be outside or working, or just about anywhere else.
But he made himself comfortable now. He told whoever gave him a strange look that he was working on his taxes, and given the sympathetic grimaces that got him, it was a genius excuse. Left alone, he went through the daily fire reports and employee schedules, pulling the dates that matched the arsons.
Which is where he discovered that those schedules did not match the ones Blake had saved on his computer.
In fact, according to the office reports, Blake _had_ been scheduled on each of the days of the arsons, whether by coincidence or design, Aidan had no idea. Dispatch didn't always need all available units to go out on the calls. On two of the fires, Blake's unit hadn't been called to respond at all and yet he'd been placed on scene by witnesses.
Had he been the arsonist, or simply trying to stop him?
The door to the office opened and Aidan turned around, the excuse already on his lips about being late getting his receipts together—
"Save it," Tommy said, and dropped a disk on the table.
"What's that?"
"A copy of the surveillance tape I got out of the camera I had at Blake's place."
"You had Blake's place under surveillance?"
"I'm an investigator. It's what I do, investigate."
"What were you looking for?"
"There's a bigger, better question. What was _Kenzie_ looking for?"
"I couldn't tell you."
"Couldn't, or won't?"
Aidan didn't respond to that.
"You're doing a shitty job of keeping her out of my hair."
_Yeah._ He was doing a shitty job keeping Kenzie out of _his_ hair as well.
"Okay, here's how this is going to work," Tommy decided. "You're going to tell me everything you've discovered about these arsons and Blake, and in return, I'm not going to charge you with interfering with my investigation."
Aidan didn't care about the underlying threat in Tommy's voice. What he cared about was discovering the truth. For Blake. For Kenzie. And as big a pain in his ass as Tommy was, Aidan believed them to be on the same side.
"Yes?"
"Yes."
With a nod, Tommy locked the door and pulled up a chair.
* * *
KENZIE HAD NO PROBLEM keeping herself occupied. She spent the day reading Blake's files, poring over them, analyzing each of her brother's entries.
She slept in Aidan's big, wonderful bed all by herself, which wasn't nearly as much fun as sleeping next to the big, wonderful man usually in it. Her dreams were wild, vacillating between nightmares about being trapped in a fire and hearing Blake scream for her, and another type of dream entirely. A dream where Aidan slowly stripped her naked and used his tongue on every inch of her body, a dream she woke up from damp with sweat, panting for air, her own hand between her thighs.
Damn, the man was potent.
In the morning, she went back to _Blake's Girl_. She couldn't help herself. She stood on the end of the dock staring at the shell that used to be Blake's sailboat, a huge lump inside her throat, wondering what the hell she was supposed to do next when her cell phone rang. Her local caller.
"Did you get the backups?"
_"Who is this?"_
"You need to stay away from the boat. There's nothing there for you."
With a gasp, she whirled, searching her immediate area but seeing no one. " _Where are you?_ Are you watching me?"
"Don't be scared."
The parking lot had only three cars in it, no people. No one was on the docks, and the neighboring boats seemed deserted. "Don't be scared? Are you crazy?"
"Listen to me," he said urgently. "It's time for you to back off. Time for you to go home, Kenzie."
The hair at the back of her neck prickled and she once again turned slowly. Behind one of the three cars was another.
Gray. Tinted windows.
Eyes narrowed, she headed toward it, needing to know who the hell she was talking to and why his voice made the hair on her arms stand up, as if she could almost recognize him, but not quite.
"Don't come any closer," he warned.
She kept walking. "Do I know you?"
The car's engine started up.
"No," she cried, breaking into a run. _"Wait—"_
The gray sedan squealed forward and to the right, giving her only the briefest glimpse of the driver behind the wheel. But it was enough to have her gasp in shock as her chest tightened beyond all bearing.
The car ripped out of the lot. She hardly even noticed as she hit her knees on the concrete, her hands fanned over her chest to hold her heart in because she'd have sworn, she'd have laid her life on the line, that the driver of that car had been none other than her dead brother.
Blake.
* * *
SHE SPED ALL THE WAY BACK to Aidan's house before remembering he was at work. Still shaken, she turned around and headed to the station. Zach was there, standing in the middle of the main room. He wore jeans and a T-shirt and a rueful smile as he stuck a pencil down the cast on his arm.
"This thing is driving me crazy." He tossed the pencil to a small desk against a wall. "You looking for Aidan?"
"Yes." Because she wanted to tell him her brother wasn't dead. Or that she was losing her mind. One or the other.
"He's on a call." Zach took a closer look at her and frowned. "Are you okay?"
_No._ "I saw the file you put together on the arsons." The fires had cost Zach his house, which in itself would have given him a good reason to hate her brother. "When Blake died, there wasn't a body."
A shadow crossed his face. "The fire was hot. Nothing survived it."
She begged to differ. "Anything survive? Anything at all?"
"A portion of the shell of the blow torch Blake had been holding, and his hard hat."
"But no physical evidence of _him?_ "
He paused a long moment. "Why?"
_Oh, because maybe he hadn't really died..._ "Do you know when Aidan'll be back?"
"No, but I can have him call you. He was worried about you."
"I'm fine." She smiled to prove it, but truthfully, she was worried, too. She left the station, got into her car and pulled out her cell. Taking a deep breath, she dialed her mysterious caller's number.
"Hello."
Kenzie went utterly still at that voice, still disguised, but it didn't matter. She now knew who she was talking to. "Blake?"
_Click._
_Oh, God._ Heart pounding, she drove straight to Tommy Ramirez's office. He opened his door at her knock, raising a single eyebrow at the sight of her, then simply sighed when she pushed past him and let herself in.
He had three unopened Red Bulls on his desk. She grabbed one, cracked it open and drank deeply. Eyes closed, she stood there until the caffeine kicked in. "God, I needed that."
He shut the door, leaned back against it and just looked at her. "That was my Red Bull."
"Thanks for sharing."
"You know, most people are afraid of me."
"Yes, but most people don't know that once upon a time you paid for my dancing lessons."
"Keep it down, will you? I don't want that to get out."
She shook her head. "Always the tough guy." Back when Blake had been in the academy, she and her brother had made some financial mistakes. Lots of financial mistakes. Tommy had known Blake's situation and had lent him some money to see him through fire school, and Kenzie enough to cover her dance lessons.
Not many knew the investigator had such a soft side; he didn't like to show it. He hadn't shown it to Kenzie since, but she'd never forgotten. Nor had she ever even briefly considered that it could be Tommy framing Blake. Blake had trusted Tommy, and she did, too.
Tommy tossed the files in his hands to his desk and grabbed one of the remaining Red Bulls. "I put you in jail to keep you safe. I didn't intend for you to bail yourself out. I wanted to keep you there until this was over, but it's taking longer than I thought."
"You put me in jail to keep me safe?"
"Trust me, it made sense to me. Look, I know this has been hard on you."
"Yes," she agreed blandly. "It's been hard on me having my brother blamed for something he didn't do. It's been hard on me knowing that all his friends, his coworkers, _everyone,_ believes he committed arson. It's hard on me knowing that he can't defend himself. But it's even harder knowing that you're not."
"You don't understand."
"Then help me to."
He opened his mouth, and then shut it. "I can't."
"Would you like to know what the hardest thing of all is?" she whispered, her throat tight with a sudden need to cry. "I know he's innocent and I know that you believe it, too."
"Kenzie—"
"You can't talk about it, I get it. But I think I saw Blake alive. Can you talk about that?"
He stared at her. _"What?"_
"I think I saw him at the docks, in the parking lot."
Tommy sank to his chair. "What were you doing at the docks?"
"Blake. _Alive._ Did you hear that part?"
His eyes filled with sympathy. "Kenzie—"
"No." She let out a low laugh. "Listen to me. _I saw him._ Plus someone's been calling me, giving me clues. It's him, he—"
"What kind of clues?"
"I don't know, that the key is in the demos, which I don't get. And that blood is thicker than water. I don't get that either, honestly."
Tommy went pale. He came to her, taking her arm and leading her to the door. "I need you to listen to me, okay? Listen very carefully. Go back to Los Angeles. I'll call you—"
"No." She pulled free. "I'm not leaving."
"Yes, you are. If I have to have you arrested again—"
"On what charges?"
"I'll find something."
She looked into his face, where his emotions were clear. "Okay, you're scared for me. I get that. I'll stay back, I'll stay clear."
_"Promise me."_
She took a long look at him. "What did I say? Was it the blood is thicker than water thing?"
"Promise me."
"I promise," she said very quietly. "Now you promise me this. You'll come to me as soon as you can with answers."
"Deal."
* * *
DURING THE SUMMER MONTHS,Santa Rey swelled to upwards of three times its normal population, which was reflected in the increased volume of calls the fire station received. In the past twenty-four hours alone, Aidan had fought a restaurant fire, a storefront fire, a car fire and two house fires, each caused by human stupidity. Then, it happened.
Another explosion.
It thankfully occurred in an empty warehouse this time. No one was injured, except Cristina, who fell off a ladder and hurt her ankle.
Dustin wanted to take her to the E.R. for an X-ray, but in typical Cristina fashion, she wanted to tough it out.
Aidan left them alone to their silent battle of wills, and let himself inside the burned shell of a warehouse.
Tommy was there, with his bag of equipment, his camera out. When he saw Aidan, he jaw ticked. "I've got it from here."
Aidan's eyes went to the wall in front of Tommy, where the burn marks on the wall indicated a hot flash, and most likely, the point of origin. "I never did getonto _Blake's Girl_ after the explosion. But I'm going to take a wild guess that you found something like this there, and also at the hardware explosion that killed Tracy."
Tommy clearly fought with himself, and then finally sighed. "Look, I'm not going to insult your intelligence the way I insulted Zach's, okay? That was a mistake, shutting him out, because it only made him all the more determined to prove he was right—"
"He _was_ right—"
"Yeah, but I was on it. I told him that, but he didn't listen, and then he dug harder and got himself targeted by the arsonist."
"The arsonist? I thought you were so sure it was Blake."
"I'm not going to insult your intelligence," he repeated tightly, "by letting you think what we want the general public to think. So know this. I'm going to nail this guy. So when I say back off, _do it._ Don't pull a Zach and get yourself hurt."
Aidan stared at him. "You know there's someone else."
"I'm close."
"You've always known."
Tommy acknowledged this with a slight nod. "So now all you have to do is stay out of my way. And keep Kenzie out of the way as well. No one else dies."
"Blake's innocent."
"That's one theory."
"Is it the right theory?"
"Jesus, Aidan." Tommy scrubbed a hand over his face. "Are you just playing with that girl?"
"No. And how is this any of your business, anyway? A few days ago you were arresting her."
"Just don't hurt her. You hear me? Don't even think about it."
Aidan let out a low, mirthless laugh. "Trust me, if someone's getting hurt, it's going to be me."
* * *
THE MINUTE AIDAN GOT OFF WORK, he went straight home, hoping Kenzie would be there waiting for him. It was with great relief that he pulled in next to her car. Letting himself in, he called out her name.
No response. Dropping his keys on the small desk in the living room, he moved through the house and heard the shower running. Things were looking up if he had a naked, wet, hot woman in his shower. And at that realization, all the myriad things he'd wanted to say to her flew out the window, replaced by memories of how she looked standing under a stream of water.
She hadn't left...
Weak with relief, he knocked on the bathroom door. "Kenz?"
When she still didn't respond, he cracked open the door and found her sitting in his shower, face to her knees, arms wrapped around herself.
"Kenzie?"
"I'm fine."
_Yeah._ She was fine, he was fine, so they could just all be fine together.
She lifted her head when he opened the shower door but didn't say a word as he stepped into it with her.
"You're dressed," she finally said, inanely.
Yeah, which sucked. "Tell me what's wrong."
"You're not going to like it."
He already didn't like it, or the clothes now sticking to him like a second skin. "Try me."
"I saw Blake."
He blinked away the water in his eyes. "You...saw Blake." He crouched before her. "In a dream?"
"No."
"You saw Blake," he repeated, trying to understand, and failing. "Not in a dream. What does that mean?"
"It means he's alive."
# _CHAPTER 16_
KENZIE WATCHED AIDAN try to absorb her news while the shower rained down over top of him, soaking into his hair, his clothes. "I know, it's a shock," she said.
The water ran in rivulets down his face. His shirt was plastered to his broad shoulders and arms, his pants suctioned to his legs. There was something about the way he'd rushed in there to save her from her own demons that got to her. More than got to her. He devastated her.
She wasn't sure how it'd happened, especially when she'd set out to keep her heart safe, but she'd fallen for him all over again.
"You saw Blake," he repeated.
"He's alive. He's the one who's been calling me." She stood up. "He's been alive and didn't tell me. The men I love suck."
Aidan hissed out a breath and straightened to his feet as well, towering over her, his broad shoulders taking the beating of the water. "The men you love?"
"Go away."
"The men you love?" he asked, staring down at her. "Kenz—"
"No." She shook her head. "Not doing this." She put her hands on his chest to shove him away but somehow ended up fisting her hands in his drenched shirt and yanking. Surprised, he lost his balance as he came toward her, slapping his hands on the tile on either side of her to hold himself upright. "Kenz—"
She stopped whatever he might have said with her mouth. It made no sense, none at all, but she wanted to have him, needed to have him, right there, right then, if only for this one last time before all hell broke loose.
_"God,"_ he managed on a roughly expelled breath as she kissed her way over his jaw while she fumbled with the buttons on his Levi's.
His hands left the tile and squeezed her arms. Water was running down his face. "I thought you'd said good-bye to me."
She'd tried. After all, she had a life to get back to. Too bad she had no idea what that life would entail—but that was a worry for tomorrow. After she figured out the Blake being alive thing. "So I said good-bye. Now I'm saying hello." Still squished between the wall and Aidan, she slid her hands up his chest, her fingers entwining in his hair as she arched back, her breasts sliding along the material of his wet shirt.
Her nipples hardened and she felt the rough grumble of the groan in his chest. Almost as if acting of their own accord, his hands moved down her sides, to her hips, her bottom, which he roughly squeezed while letting out another of those incredibly arousing groans. "Is there another good-bye coming my way after this shower?"
"Maybe not right after," she panted because something was happening to her, something that had nothing to do with lust or hormones or getting an orgasm, but far deeper. Far more dangerous. Tightening her fingers in his hair, she lifted his head from her breast and stared into his eyes. There, she could see the reflection of her own. And in that reflection was her heart and soul, her very life.
She loved him. And if they did this, if she let him inside her body again, she'd never recover. She knew it, but like last time, it wasn't going to stop her. Small wonder when he was against her like a second skin, holding her to the wall. Closing her eyes, she hugged him close, pressing her face to his throat.
Her name tumbled from his lips in a harsh whisper, and then their hands were fighting to get his clothes off, pushing off his shirt, shoving down his jeans. Then he was reaching for those jeans, and the condom in his pocket. He pressed her back against the wall, freeing his hands to skim down her bare, trembling thighs, which opened and wrapped around his waist, bringing him flush to her. In one thrust he was deep inside, and she was...lost?
Not lost.
No, when she was with him, she was found.
* * *
AIDAN'S HEART was still thundering in his ears in tune to the water pounding his back when Kenzie slid free of him. Drained, he watched her lean past him and turn off the water. She tossed him a towel, grabbed one for herself and left him alone in the bathroom.
He had no idea what had just happened.
When he managed to dry himself off and walk out of the bathroom, on legs that still quivered, he found her dressing in his bedroom. "Did you get the license of that truck that just hit me?"
She didn't smile. "I really saw him."
When he just looked at her, she slipped into her shoes. "And I'm going to go find him."
"Kenzie," he said gently. "Blake is—"
"Dead. I know. But he's not." She left the room.
With a sigh, he headed to his dresser for clothes. He'd gotten into a dry pair of jeans when he heard her keys jangling. "Kenzie," he called out. _Dammit._ "Wait." He grabbed a shirt and headed down the hallway just as she opened the front door. She hesitated when her cell phone beeped an incoming text message.
"Is it...him?" he asked.
"Yes, it's him. Texting me from the dead." She opened her phone and let him read over her shoulder.
> _Go home. I'll find you there when this is over, when you're safe._
As they stood there in his open doorway looking down at the screen, a huge trash truck lumbered down the street, making the earth shudder as it went past—
_Boom._
Kenzie's bright red sports car vanished in a cloud of smoke and flames and flying metal as it exploded.
* * *
KENZIE SAT ON AIDAN'S CURB looking out at the street, which was littered with cops and various other official personnel, including Tommy and the chief. And lots of red car parts.
Everyone was trying to figure out what the hell had happened.
Her car had gone boom, just like _Blake's Girl,_ that was what had happened.
"Kenzie." Aidan's athletic shoes appeared in her peripheral vision, and then the rest of him as he sat at her side.
"My insurance company isn't going to be happy," she said. "I blame the trash truck."
"The trash truck saved your life. You car had been rigged to blow when you got into it, but the truck vibrated the street so much it went up early."
"Oh." She winced. "I wish I didn't know that."
"Give me your cell phone."
"Why?"
"So I can call whoever's been calling you."
"Blake. Blake's been calling me."
"Whoever it is." His mouth was grim as some of his clear frustration and fear for her filtered into his words. "I just want him to stay the hell away from you."
"This wasn't him."
"Then who?"
"I'm working on that."
He looked down at her. "By yourself."
"It's how I work best, apparently." She stood up. During the time she'd been gone from Santa Rey, she'd closed herself off, both her heart and soul. It was a hell of a time to realize that. But no matter what happened here—whether she left and went back to Los Angeles, or whether she stayed—whatever she settled on for herself, she couldn't go back to closing herself off.
"Kenzie."
"I didn't mean to get so good at being alone. I didn't realize, living in L.A., the land of pretend, that I'd never built myself any real relationships." She let out a long breath and met his gaze. "But that changed when I got here. When I was with you. I love you, Aidan. Again. Still. I love you."
And while that shocking statement hung in the air, someone called for Aidan. But he just stared at Kenzie. "You—"
"Aidan!"
With a grimace, he looked over his shoulder. "Shit, it's the chief."
"Go."
"Kenzie—"
_"Go."_
A muscle ticked in his jaw. "Don't move, I'll be right back."
Nodding, she watched him walk toward a tall man whose back was to her, stretching out a dark blue shirt that said Chief across the shoulders.
Then she walked away. She didn't have a car, so she had no idea where she thought she was going, but she had to leave.
In her pocket, her cell phone buzzed with an incoming text.
> _Another half block. Gray car._
I LOVE YOU. Aidan muttered the three little words that Kenzie had said to him. She'd said them, and then she'd vanished, and he had no idea where she'd gone. One moment he'd been talking to the chief, and the next...She'd been gone. It'd been hours, and not a word.
He was at the station now, and she still hadn't answered her damn cell phone, and he was starting to lose it. He shouldn't have walked away to talk to the chief, he should have dragged her with him.
"Hey, Mr. 2008." Cristina came into the station kitchen and went straight for the refrigerator. "What are you pouting about?" She helped herself to someone else's lunch.
"You could bring your own."
"I could." Cristina pulled out a thick turkey sandwich. "But I don't."
"Hey, that's mine," Dustin said, joining them from the garage. "What did I tell you about stealing my sandwich?"
Cristina spoke around a huge mouthful. "If I was still sleeping with you, I'd bet you'd _give_ me your sandwich."
Dustin's eyes darkened. "You slept with me once."
"Your point?"
"My point is that if we were _still_ sleeping together, I'd _make_ you your own damn sandwich."
She took another bite, chewing with a moan. "You know, I should give that some thought, because you do make the best sandwiches."
Dustin tossed up his hands and walked back out of the room.
When he was gone, Cristina dropped her tough girl pose, watching him go with a naked look of longing.
"You could just tell him the truth," Aidan said.
"What, that he makes crappy sandwiches?"
"No, that you're scared. He'd understand fear." Hell, he understood it all too well.
"Are you kidding me? I'm not scared." Cristina tossed the sandwich back in the fridge. "I'm not scared of anything." But as she shut the fridge, she pressed her forehead to the door. "Ah, hell. I'm scared. Everything's messed up. Dustin's mad at me. Blake's gone. There's no good food. Blake's gone."
"You still miss him."
"Hell, yeah, I still miss him. He was a great partner. And now even the chief, his own flesh and blood, wants to make him out to be a monster that we know he wasn't."
"Wait." Aidan grabbed her arm. "What?"
"He wasn't a monster."
"The flesh and blood part. What did you mean about that?"
Cristina's lips tightened. "Blake asked me never to tell."
"He asked you never to tell what?"
She sighed. "That the chief's his uncle. They were estranged, though. Blake's parents were—"
"Dead. They died years ago."
"Yeah. But his father was the chief's half brother."
_Blood is thicker than water...Good God._ "If that's true," he asked hoarsely, "why did Blake and Kenzie spend their childhood in foster care?"
"Because the chief didn't want kids. Or something like that." She shrugged. "Not sure on the details."
Neither was he. Except that somehow... _Christ._ Somehow the chief—
His cell phone rang. When he looked down at the screen, his heart skipped a beat. "Thank God," he said to Kenzie in lieu of a greeting. "Listen to me. I just realized—"
"Aidan, I need you. I'm sorry, I know I don't really have the right to say that to you, but I do. Can you come meet me? Now? Please?"
"Just tell me where."
* * *
AIDAN BURST INSIDE the Sunrise Café and looked around the tables.
No Kenzie.
"She's on the roof," Sheila told him, standing behind the bar drying glasses.
"Thanks."
"Something about Tommy being on his way, and having all the answers you need..."
_Aidan_ had the answers. He just didn't have the girl, which he intended to rectify. He headed for the stairs as Sheila turned her attention to someone else. "Hey, there, good-looking," she called out with a smile of greeting. Aidan took the stairs without looking back, coming to a relieved halt on the roof at the sight of Kenzie sitting on the bench.
"Tommy's on his way," she said, standing up. Someone stepped out from the shadows behind her and Aidan's heart stopped.
It was Blake, who by all logical accounts should be dead.
Only there was nothing logical about any of this. Not the arsons, and not the way Aidan knew he loved the woman standing in front of him like he'd never loved anyone before.
"Listen to him," Kenzie said quietly. "Listen to your heart."
He _was_ listening to his heart, which had kicked back to life and was screaming, demanding that he pull Kenzie close and tell her he loved her, too. That he was sorry it'd taken him so long, but like Cristina, he'd been afraid, was in fact _still_ afraid but would no longer run from how he felt.
He'd never again run from her.
But that would have to wait. He looked at Blake, who was thinner than ever. And he walked with a cane. "I know, it's crazy," his old friend said, his voice low and urgent. "You thought I was dead and I'm not. I...faked my own death."
"I'm getting that."
"When I found out who the real arsonist was, I realized no one was safe." Blake's face was twisted in tortured misery. "He killed Tracy right after he blew up my boat."
"I know. I know all of it. I even know _who_ we're talking about. I just don't know why."
"Oh, I can tell you why," said the man who came through the roof door to stand in front of them. The chief nodded in Aidan's direction. "If you really want to know."
_Shit._ Aidan pulled out his cell, hit Tommy's number and put the phone to his ear.
"Nearly there," Tommy said tensely.
"Hurry. Bring backup."
"Oh, it'll be too late," the chief said conversationally.
"Uncle Allan?" Kenzie breathed, staring at the chief. She looked at Aidan. "He's the fire chief? I thought..." She turned back to her uncle. "I thought you were in Chicago."
"I was. I came back here a year ago. A shame we lost touch or you'd have known."
"We lost touch—" Kenzie took a step toward him, or tried to, but Blake grabbed her hand and held her back "—because you didn't want us."
"Now, now. That's not entirely true. I just didn't want to be responsible for raising kids. I never wanted kids."
"But it's okay to be responsible for _killing people?_ "
" _One_ person," he corrected. "Not people. And that was an accident."
"You killed Tracy and that was no accident," Blake ground out. "You murdered her."
"Ah, now, see _murder_ implies intent, and I don't have intent. I have an addiction." He smiled sadly. "It means I can't help it."
Kenzie again tried to charge him, but this time it was Aidan who held her back, not trusting that asshole with her.
"If I was an alcoholic," the chief asked, "would you still be looking at me like that? If I had a drug problem? No, you'd be trying to get me help."
"I _tried_ to get you help," Blake told him. "When I figured out you had started that second fire all those months ago, you begged me to understand. You lied and said it was your first time, and that you'd stop, that you'd get help. Instead a child died and when I tried to turn you in you threatened me."
The chief slowly shook his head. "Tommy was getting close. You wouldn't leave me alone. I had to do something. I had to keep you quiet."
Blake gave Aidan an agonized look, as though pleading for forgiveness. "By then he had implicated me. He'd changed the schedules, he'd planted evidence. He discredited me so that even if I did tell, _I'd_ be the first one they'd lock up. And once I was in jail, he threatened to hurt Kenzie.
"Then Zach started asking questions and the chief tried to kill him by burning down his house. I had followed him, Zach saw me, and I didn't know what to do. I panicked and faked my death. If I was gone, he had no reason to harm Kenzie."
"And I didn't."
"You killed Tracy!"
"But not Kenzie," the chief said calmly. "Look, Tracy was going to put together a list of people who'd purchased those metal trash cans. I would have been on that list."
"You didn't have to kill her," Blake shouted.
"He had to set more fires," Aidan said grimly.
"That's true." The chief nodded emphatically. "I can't help myself. I tried like hell. I couldn't stop, but at least I went for old and dilapidated properties, or overly insured buildings." He paused. "Like this one."
Aidan stared at him. "What?"
"Sheila is getting ready to renovate," the chief said.
"She has to," Aidan said. "The building has structural problems."
"Yes, and now she's over insured to protect it. It's a situation that cries out to an arsonist. It needs to burn."
"Ohmigod," Kenzie breathed, looking horrified. "You're a very sick man."
"Agreed." Her uncle smiled without any mirth. He clapped his hands together. "Well, it's been nice clearing all this up but I've got to end this now."
"You're not walking away," Aidan said. "Not from this. You have to pay for your crimes."
"I'm not paying for anything. You didn't get hurt. None of you died."
"Are you kidding?" Aidan asked incredulously. "Blake nearly died trying to stop you. You nearly killed Kenzie on _Blake's Girl,_ and then again when you blew up her car."
" _Nearly_ won't hold up in a court of law. I was just trying to scare her out of town, anyway. The car was supposed to blow an hour earlier, but a fuse failed me. And the boat was an accident. I was just trying to get rid of Blake's laptop. I didn't know she was there that night."
"There's something else you don't know," Aidan told him. "Blake e-mailed Kenzie backup files."
The chief's mouth tightened. "I'm not going down for this, for any of it. I'm the chief."
"Not for long you're not," Blake said. "You're going to be stripped of that title and put in jail."
"Not happening," the Chief declared. "I won't go to jail—I've made sure of it. I've risked my life to save people for almost thirty years. I _won't_ be remembered as an arsonist."
Aidan's gut clenched. There was only one reason the chief would come out in the open like this and confess his crimes. And that was if he didn't intend for them to live to tell the tale. "Whatever you've planned, _no._ "
"You're too late." The chief looked first to Kenzie, then to Blake. "I'm sorry. Truly sorry."
"What did you do?" Blake demanded. "Oh, Christ, you didn't—" Without finishing that thought, he whirled and limped to the roof door, yelling as he took the stairs, "Evacuate! Everyone out—"
Which was all he got out before a thundering explosion hit. The entire building shook, throwing Aidan and Kenzie to the ground.
# _CHAPTER 17_
AT THE EXPLOSION, the world seemed to stop, or at least go into slow motion. Kenzie managed to lift her head just as Aidan rolled toward her, his face a mask of concern. Her uncle, ten feet away, wasn't moving at all. Pushing to her knees, she stared at the doorway where her brother had just disappeared. "Blake!" she screamed.
He didn't reappear, no one did, nothing except a plume of smoke that struck terror in her heart. "Ohmigod. _Aidan_ —"
"Are you okay?" He was on his knees before her, running his hands down her sides, pushing her hair from her face, looking her over, his expression calm, only his eyes showing his fear. "Are you okay?" he demanded again hoarsely.
Shaken, but all in one piece, she nodded and pointed to the doorway. "Blake—"
His eyes and mouth were grim. "I know. He's down with the others. We'll get to him." He glanced at the chief.
"Is he—"
Aidan checked for a pulse. "Just out cold." He pulled her to her feet, yanking his cell phone out of his pocket. From far below, they could hear screams and yelling over the whooping sound of smoke and car alarms going off.
All of it brought Kenzie back to the night on _Blake's Girl,_ back to that irrational terror. Then they'd been able to jump into the water. Now there was nothing down there except concrete.
Three floors down.
"Call 9-1-1," Aidan said to her, shoving the phone into her hands as he ran past the very still chief to the edge of the building and looked over the side. "Dammit, I can't see if people are getting out of here."
The café hadn't been full to capacity, but there had been at least twenty people inside when they'd entered, and then there was Sheila and her staff.
And Blake. God, Blake. Could she really have found him only to lose him again, for real this time? "Aidan—"
"Listen to me. There's no way off of here except for the stairwell. No outside fire escape or ladder."
They both looked at the dark doorway, emitting smoke now. "Ohmigod." She felt frozen. Logically she knew she had to go down to get to Blake, not to mention to safety. But there was nothing logical about the fear blocking her windpipe. She'd thought Blake had died in a fire. _She'd_ nearly died in the boat fire. Instead of seeing the roof's doorway, she kept flashing back to _Blake's Girl,_ the black night and blacker water. She could feel the heat from that fire prickling her skin even as she could feel the iciness of the water closing around her body—
"Kenzie."
She blinked Aidan into view. He had his hands on her arms and he was frowning into her face.
"I can't go in there," she said, unable to catch her breath. "I just can't."
"Okay." They both looked at the chief, who still wasn't moving. Again Aidan went to the edge of the roof and looked over. Whatever he saw made his jaw go tight and his eyes, grim. Then he backed Kenzie to a corner and gently pushed her down until she was sitting there, her back to the wall, facing the opened door to the only exit. "I'm going—"
"No." She gripped his arms, digging her fingers into the muscles there.
"Kenzie—"
"No!" Icy, terrifying fear overcame her as she stared at the smoke now pouring out through the opened door. "There's a fire down there!"
He didn't say it, he didn't have to.
"I already hear sirens. They're coming to put out the fire. It's going to be okay. But I have to go help. This roof won't be safe to be on for long."
"I know."
With his eyes reflecting the torment he felt at leaving her, he pried her fingers from his arms.
"Come right back," she ordered.
"Okay."
"And stay safe, you hear me?"
"I will."
"And Blake. Bring me Blake."
"I promise." He held her gaze for one beat, letting her see into his heart and soul. He never made promises, never, and yet he did now, to her, which meant more than anything he'd ever done. Pretending to be brave, she nodded and then sagged back, covering her face with her hands so she couldn't see the smoke pouring out of the doorway as he vanished into it.
Dammit, she really needed a new script. Aidan was probably worrying about her instead of completely focusing on the fire—and that was dangerous. She forced her eyes open, glued her gaze to the black doorway. He had saved her life on Blake's boat, and that had been amazing, but she could have saved herself. She knew how to swim.
And she could save herself this time.
All she had to do was get past her fear. Any second now...
The sirens were louder now, and that reached her somehow. Tommy was probably nearly here, too. She got to her feet, wiped the sweat from her eyes and headed to her uncle. He'd hit his head on the A/C vent. Turning her back on him, she headed toward the door. "You're a coward," she told herself. "You're fine, you're fine..." She kept up the mantra as she entered the dark doorway. Unable to breathe through the smoke, she pulled her shirt up over her mouth and took another step.
And then it happened. The floor beneath her rumbled, the walls shimmied and shook, and she froze as a second explosion hit, flinging her against a wall. Then the power flickered and went off, leaving her in complete darkness.
_Oh, God._
Sitting up, she felt for the railing and pushed herself upright. She was okay. Relatively speaking, anyway.
Just as she began heading down again, the stairs beneath her began rumbling, but not with yet another explosion. This time it was pounding footsteps as someone ran up the stairs, and then reached out toward her. "Kenzie?"
" _Blake?_ Ohmigod, Blake, you're okay—"
"Where is he? The chief?" he demanded.
"On the roof."
"Stay here," he commanded. "Stay right here!" And then he rushed up and out.
_Like hell._ She was going to be proactive this time, dammit. She was rewriting this script her way. And when it was over, she was going to write scripts all damn day long to her heart's content. And eat donuts. Yeah, lots of donuts. Heart pounding, she stumbled after her brother. Bursting back out on the roof, she was horrified to see that part of it had begun to cave in, with flames flickering out from underneath. And standing far too close to that area was Blake, facing off with the chief.
"No," she cried, just as Aidan came out the doorway behind her, looking as if he'd been in a car wreck, all torn and bloody, calling her name hoarsely.
"You're hurt," she cried, rushing to his side.
"The explosion kicked me down the stairs." He hugged her tight, not taking his eyes off the chief and Blake. "I'm okay."
It was like a bad movie, playing in slow motion as the chief leaped for the edge of the roof, and Blake leaped for him the best he could, wrestling him to the ground.
Flames shot up through the floor at all of them and Kenzie screamed, trying to get close to her brother, but Aidan had a hold of her, even though _he_ was the one with torn clothing and blood seeping from his various injuries, all covered in soot.
On the ground now, Blake rolled with the chief, the two of them still throwing punches.
"Stay back," Aidan told her, holding onto her. "The flames—"
They were licking at them from all angles now, but suddenly, from below, they were hit with water. Streams of it, coming up from the street.
The fire trucks had arrived, and none too soon as the flames forced Kenzie and Aidan back from yet another cave-in.
"Hold still, you son of a bitch," Blake growled out to the chief, who was trying to crawl free and get to the edge of the roof.
Aidan tried to move around the flames to help Blake with the chief, but suddenly he wavered, then sank to his knees.
"Aidan!"
"Yeah. Think maybe I hit my head before." He blinked at her face as she dropped to her knees in front of him. "There's three of you."
"Oh, God." She touched the gash along his temple, which was bleeding freely. "Hold still!"
"Not a problem."
A ladder and bucket came into view over the roofline, lifted by a crane from below. It held two firefighters, who took one look at Blake and staggered to a shocked halt.
"Later," Blake yelled at them. "I'll explain later! Aidan's down and we need Tommy and some cuffs. Tell me someone has some cuffs!"
* * *
IT ACTUALLY WASN'T THAT EASY, nothing ever was, Kenzie thought. Hours later, they were all sitting around Aidan's hospital bed, where he was being held overnight, thanks to a concussion.
The chief had been taken to jail, which was such a huge town scandal that Tommy had left to prepare for a press conference. Sheila was sitting in a chair, her wrist in a sling. It was her only injury, but the café was a complete loss. Dustin was next to her, his arm around her shoulders. Cristina was there, too, holding a bucket full of money from emergency personnel on the scene who'd already poured some of their support into it for Sheila.
"I could go to Hawaii with all that." Tears were thick in Sheila's voice.
"Or you could rebuild," Aidan said from flat on his back.
At the sound of his voice, Kenzie's heart squeezed. He'd been so damn quiet, and she'd been so damn worried.
On the other side of Aidan's bed, Blake stirred. "The chief's in custody," he told Aidan. "And he's not going to get off easy."
Aidan's gaze tracked to Kenzie. "I don't want to get off easy, either." He reached for her hand. "Not tonight, or any night."
She gripped his fingers tightly and pressed them to her aching heart. He was talking, but not making any sense. She hadn't taken a full breath since they'd taken him for X-rays and she didn't take one now. "I'll go get your nurse—"
"No." His grip was like iron. "I'm not crazy."
"I know—"
"Listen to me. You pulled it off, you broke my damn heart. We're even."
Oh, God, and now he was delirious. "Aidan—"
"Maybe we should give them a moment," Dustin said, guiding Sheila out of the room. Cristina followed.
Blake did not leave. "What's going on?"
"I love you back, Kenzie." Aidan managed a smile, although it was crooked. "But I think you already knew that."
"No." She shook her head, finding herself both laughing and crying. "I didn't. I hoped..."
Blake was staring at the two of them, mouth grim. "Wait. Love?"
Aidan, who still hadn't stopped looking into Kenzie's eyes, nodded. "Definitely love."
And just like that, Kenzie took a full breath. God, it felt good to breathe. Breathe and live and love.
"Okay, somebody talk to me," Blake said.
"Well you've been dead, or I'd have told you before now," Kenzie reminded him. "I've been busy trying to make Aidan pay for breaking my heart all those years ago."
At this, Blake blinked, then sent a glacial stare at Aidan. "You broke my sister's heart?"
Aidan winced. "Yeah, but if it helps, I was an idiot."
"He really was," Kenzie agreed.
"And trust me, she got me back," Aidan said. "Her evil plan worked. I fell hard. I love her, Blake." He broke eye contact with Kenzie and looked right at Blake, his smile gone, eyes dead serious. "I love her with everything I've got."
Blake looked as if a good wind could knock him over. "You put your heart out there? _You?_ "
Bringing his and Kenzie's still joined hands to his chest, Aidan nodded. "Yeah."
"And then she stomped on it?"
"In boots, with spikes on the soles," Aidan assured him.
Blake took this in and considered, then relaxed. "Okay, then. As long as you're even."
"Not even," Kenzie whispered. "Not yet."
Uncertainty twisted Aidan's features. "Kenzie—"
"We're not even until I get my happily-ever-after." Her throat was so tight she could barely speak. "But since I'm going to be writing, I'm pretty sure I can plot it out for myself."
Aidan's eyes registered both surprise and pride. "You're going to be great at writing. But about that ending...Am I in it?"
"I can guarantee it."
He smiled, and right then, Kenzie knew. She didn't need a script for this, her life, not anymore. The real thing was so much better. Taking the first step, she cupped Aidan's gorgeous face and kissed him.
* * * * *
_The game of hockey hasn't exactly been kind to Lainey Harper. In fact, she wouldn't mind forgetting all about it. But when pro defensemen Cooper Mead turns up in her bar, his hard physique and well-practiced charm are hard to resist..._
_Read on for a sneak peak of_ **Playing Dirty** _by Taryn Leigh Taylor._
**_CHAPTER 1_**
"IT'S ABOUT DAMN time you got here, Darius. I know my fa—I know Martin wasn't much for punctuality, but if you want to keep working here, you're going to have to show up on time."
Lainey kicked the beer fridge closed and froze, as though the act had triggered a curse that turned her to stone. In truth, though, her paralysis was directly attributable to the animal magnetism of the man on the other side of the counter.
Black hair just long enough to curl against his collar?
Check.
Dark stubble framing a smirking mouth?
Check.
Muscled arms that could make angels weep and women purr?
Check and check.
"You're..." Cooper Mead, number sixteen, the Portland Storm's latest acquisition, currently tied for highest scoring defenseman in the league. "Not Darius."
"Nope." The single syllable, deep and rough, was enough to detonate an estrogen grenade low in her tummy.
_Dammit_.
Cooper freakin' Mead! Standing in Martin's crappy little sports bar— _her_ crappy little sports bar now, she reminded herself. And boy, was he something to behold. All six feet two inches and 220 pounds of him, per the team stats page. Lainey cursed the lapse in internet browsing judgment that had led to that knowledge. She hadn't watched hockey, talked hockey, _thought_ of hockey in years, but in the three months since she'd come back to Portland, the nadir of all her broken dreams and bad luck, she was already falling into bad habits.
And Cooper Mead was the kind of bad habit that would be hard to break.
With great effort, Lainey beat back the hormonal fallout and cast a wary glance around the bar. Oregon might be a long way from Denmark, but something here was definitely rotten.
The Drunken Sportsman wasn't the type of place that attracted professional athletes. Hell, some weeks it barely attracted enough armchair athletes to keep the lights on and the doors open.
Right now, there were two groups of them, a middle-aged couple sporting his and hers Trail Blazers T-shirts and eating nachos in the booth farthest from the door, and four guys at a table by the window who were stretching a pitcher of beer as far as it could go while staring zombie-like at the basketball pregame coverage on the hulking television above the bar.
She needed to replace it with a couple of flat screens spread around the room for more optimal viewing. She made a mental note to add that to her list and turned back to the defensive juggernaut who stood across from her.
Other than him, there was nothing—and no one—out of place. And yet something about the situation had her on edge. She glanced at Cooper Mead's wicked mouth, the corner quirked up in a grin that did weird things to her insides.
_Maybe I'm allergic to hockey._
Squaring her shoulders, Lainey strove for professionalism in the form of the official bartender's mantra. "So, not-Darius, what'll it be?"
"How about Sex on the Beach and a Screaming Orgasm?"
_No_.
_Don't say it_ , she thought with a desperation that surprised her. _Please don't go there_.
A flicker of indecision crossed his handsome face, one that gave her hope that her telepathy had worked. Then he turned on that easy grin, bracing an arm on the bar and leaning closer.
"But if I'm going to do my best work," he confided in a soft growl that prickled between her shoulder blades, "I'll probably need something to drink first."
_Aaaand he went there_.
"Good one. Very original. You'd think, with me being a bartender and all, I would've heard that one before." She forced herself not to roll her eyes. If getting hit on in bars had taught her anything, it was that derision had more impact when delivered with some restraint. It was important not to cross into "the lady doth protest too much" territory or the playboys and the drunks would never leave you alone.
In response, he upped the wattage of his smile and reached over the bar to liberate a maraschino cherry from the fruit caddy.
"Sarcasm. Nice. You're feisty. I like that." He popped the pointedly sexual fruit in his mouth and chewed. "But in my defense, it's not the small-talk portion of the evening I excel at. Give me your number and I'll prove it to you."
Lainey wanted to be offended, she really did, but damned if his megalomania wasn't working for him, in a basic "the hormones want what the hormones want" kind of way. Still, a woman had to have standards.
"Listen, I appreciate the display of manly bravado, but as much as I'd like to stand here and fend off your advances, I've got a drink quota to maintain. You actually want something, or are you just here to waste my time?" Lainey crossed her arms over her white tank top. Cooper Mead wasn't the only talented defenseman here. Her nickname hadn't been "The Ice Queen" for nothing.
The memory came out of nowhere, like a slap shot to her brain—fast, powerful, and it hurt like a bitch. Her pulse thundered in her right wrist, the one she'd busted in the last hockey game she'd ever played, and she shook her hand to dislodge the sensation. No one had referred to her by her old hockey nickname in ages. The fact that she'd been the one to break that streak said a lot.
One more reason she couldn't let her guard down. She needed to fix up the bar, sell it for a tidy profit, and get the hell out of Portland back to the fabulous, hockey-less life she'd built for herself. The sooner, the better.
It had taken hard work and single-minded focus to become one of the Zenith Advisory Group's top hospitality consultants. And sure, that was just a fancy way of saying that she traveled the country staying in nice hotels and filling out comment cards—but the title came with a generous wage and her choice of locations. Which was why she'd never taken an assignment in Portland before.
Too many ghosts here, and all of them wore skates.
Cooper shot a pointed glance around the almost-deserted bar. "What happens if you don't make drink quota?" He twirled the cherry stem absently between his finger and thumb. He had big hands.
"Oh, you know, swarm of locusts, rain of fire, four guys on horseback."
He nodded, flicking the stem aside. "And what if I guarantee to make any trouble worth your while?"
She didn't like the way her heart sped up at the vow or the way she believed that he could make good on it. "Nice try, Slick, but I wasn't kidding about the drink quota, so you're gonna have to tell me what you want."
Cooper propped an elbow on the bar. "And here I thought I'd been pretty clear about what I want."
"To drink. What do you want to drink?"
"Surprise me."
With a cocked eyebrow, she grabbed a highball glass and turned toward the liquor bottles that lined the shelves. Lainey couldn't help but steal glances at him in the mirrored tiles that stretched from counter to ceiling behind the booze. Damned if she wasn't kind of impressed that a guy who would approach with the lamest of lame pickup lines wasn't standing there ogling her ass. He lifted a hand to rub the back of his neck as he waited, and Lainey noticed for the first time that he looked tired—not like he needed a nap, but like it would be nice to put down the weight of the world for a little while.
She knew _exactly_ how he felt.
"Here's your drink."
She turned to face him and set it on the counter. Despite her earlier pang of empathy, she took great pleasure in the distrustful frown that had overtaken his rugged features.
"Are you sure you didn't grab the wrong glass? Because, and trust me here, I've had some experience ordering drinks and they usually come in liquid form."
Lainey had to admit the congealed glob that came from mixing Bailey's and Sour Puss looked particularly disgusting tonight. The fact that it was floating in Kahlua and Blue Curacao added a previously unsurpassed level of yuck. She lifted one bare shoulder in an offhand shrug. "You're the one who wanted a surprise."
"Yes, I was."
"I call it a Black Widow."
"Of course you do," he said, but she had a feeling the mockery was self-directed. "How much?"
"Twenty."
Straight black brows flicked upward. "As in 'US dollars'?"
"Ten for the drink and the rest is the standard first-time penalty for pickup lines that insult my intelligence."
Cooper's lips twitched with reluctant humor. "Well, just so long as it's not to cover the going rate for arsenic."
"You never know," she warned, nudging the Black Widow toward him with the tip of her red-polished fingernail. "You feelin' lucky, Slick?"
He smiled for real then, a full-fledged, blindingly white smile that kept some dentist's classic Corvette on the road. "I wouldn't mind getting lucky."
Lainey shook off a flash of reignited lust. _Damn, he was good_. "Well, the night is young. Maybe your left hand hasn't made plans yet."
She forced herself not to flinch at the blunder. It was a fatal error to let an egocentric hockey player know you knew anything about him—especially fan-girl minutia, like the fact that Cooper Mead was a south paw.
"Oooooh. So it's gonna be like that, huh? I thought you weren't supposed to start eating me alive until _after_ the sex."
She ignored the black widow reference and held out an expectant hand.
With a self-deprecating nod, Cooper dug out his wallet and handed her a fifty. Her palm tingled where his skin brushed hers. "Would I be wrong to assume you're fresh out of change?" He didn't wait for confirmation before stowing the billfold away.
Lainey tucked Ulysses S. Grant safely into her back pocket. Leaning forward, she rested her elbows on the counter. "You know, you're a much smarter man than first impressions would indicate."
"You like `em brainy, huh?" He mimicked her position, cutting the gap between them. His eyes were dark, like rich espresso, and just as heart-pounding as a jolt of caffeine. The kind of eyes a girl could get lost in if she wasn't careful.
Lucky for her, Lainey was always careful.
"Personally, I find the brain usually gets in the way of all the exciting stuff, but I completely respect alternate lifestyle choices," Cooper continued. "We should hang out sometime. You can help me see the error of my ways. Give me your number and we'll make this happen."
He reached out and tucked a wayward strand of raven hair behind her ear. When his knuckles brushed her cheek, her knees went squishy. And that was _before_ he whispered, "Don't break my heart, gorgeous. Give me your number."
"Wow." Lainey pushed back from the bar, unwillingly impressed and a little woozy from the flare of attraction. "Wow. That was...masterful. Seriously, Slick. You are very, very good."
His slow, self-mocking grin confirmed that the gig was up. "I almost had you at the end there."
"Not even close," she lied.
"Sure I was. But you were a worthy opponent. It's been a long time since someone gave me a run for my money, and considering the number at the bottom of my last bank statement, that's saying something."
Since the Storm had signed him to a two-year, eight-million-dollar contract, she knew his boasting was legit. "Is this the part where I'm supposed to be impressed?"
"It would help," he agreed, down but not out. "I'll give you five hundred bucks for your number."
"Forget it."
"A thousand."
Lainey bit back a grin. "If you'll excuse me, I have a bar to run."
"Fifteen hundred. Final offer."
It was tempting. Not the money, the man himself. She'd been working nonstop for the last few months to put her affairs in order in Portland. And once he'd gotten his dismal approach out of the way, their verbal sparring had been kind of fun.
But she needed to stay far away from hockey—and even farther away from famous men. She'd be better off if Cooper Mead walked out of her bar and just kept walking, no matter what her long-suffering libido had to say on the matter.
"Enjoy your night, Slick. Thanks for the dance." And with that, she shoved a sign that read WAIT STAFF ONLY on the counter and turned her back on him, more determined than ever to unload the bar and blend back into the familiar hustle and bustle of LA by the end of the month.
* * *
HE WAS GETTING too damn old for this.
Coop grabbed his glass from the counter. Revulsion curled his lip as he stared at the sludge he'd just been served while the dust from his spectacular crash and burn settled around him. A post-practice night out with his teammates used to mean a luxurious night in the VIP room of some exclusive New York club, complete with overpriced bottle service, an overhyped DJ, and an underdressed woman. Or two.
Since he'd taken the trade to Portland, there'd been a couple of team dinners, a little charity work and a whole lot of practices. But that's how the Storm had all but guaranteed their spot in the post-season over a month ago. Intense focus.
In fact, it had been so much all-work-and-no-play that his agent, Jared Golden, had called to give Cooper hell. "I can't get endorsement deals for a hermit, Mead. Leaving New York is already hurting your visibility. You know how much harder it is for me to get your picture in a magazine when you're in Portland? At least go out and live a little."
Which was why Cooper had finally relented and accepted one of fellow defenseman Brett Sillinger's relentless requests to "grab a beer and talk hockey." He fully regretted the decision now.
He'd assumed there would be a group of them heading out for one last drink before playoffs got underway. But when he'd asked around the dressing room after practice, it turned out he was on his own. Every player on the team had somewhere else to be—captain Luke Maguire was going to some media shindig with his intrepid reporter girlfriend, centerman Eric Jacobs was meeting some after-hours contractor at the bakery he owned, and goaltender Tyson Mackinaw's kids were performing in some school play.
The rest of the team's excuses followed in those footsteps: wife, wife, girlfriend, kids, girlfriend's kids.
_Jesus_. Everyone on this damn team was—or acted like—an old married guy.
Except for him...and Brett of course.
And for reasons Cooper couldn't possibly explain, the rookie had chosen the worst bar imaginable—a run-down watering hole that probably catered to former high school jocks bent on reliving their glory days through ESPN highlights. And he didn't even have the decency to show up on time.
As if to confirm Cooper's suspicions, the bell on the door dinged and in lumbered a whole flock of washed-up jocks decked out in the finest basketball paraphernalia the mall had to offer.
"Hey there, beautiful lady. Turn up that TV! The game starts in ten minutes."
Coop's fingers tightened on his Black Widow. The bartender's smile was full-bodied and sexy when it wasn't tinged with acid, and he hated that some loudmouth sporting love handles and an ill-fitting Trail Blazers jersey was the recipient and not him.
"Larry, you only think I'm beautiful because I didn't raise the happy hour price of beer." Her admonishment was accompanied by the familiar singsong lilt of sportscasters everywhere as she hit the volume button on the remote.
"Sweetcheeks—" Cooper did his best to stifle a gag at the endearment "—you know that's not true. One word from you and I'd—holy hockey pucks, you're Cooper Mead!"
_So much for laying low_.
"Wow, you're, like, a real athlete! A famous one! Man, you think you could sign something for my kid? He totally idolizes you! And the guys! The whole team! I do, too. I mean, that slap shot of yours? Big fan. We all are! Thanks to you, the Storm might have a real shot in the playoffs." He offered with an expansive gesture. "Guys! Check it out! Cooper Mead! At our bar."
The chorus of greetings and swears of disbelief were accompanied by the materialization of cell phones. Calls were placed. Photos were snapped. The couple from the other side of the bar wandered over. Not exactly how he'd planned to spend his evening, but at least Golden would be happy.
With a resigned sigh, he brought his drink to his lips.
He stopped just in time.
Suicide by toxic sludge was never the answer.
Instead, Cooper turned on his best PR smile and accepted the napkin being thrust in his direction. "Who should I make this out to?"
* * *
"WHAT THE HELL happened here?"
The deep voice ripped into a close inspection of her palm, and Lainey looked up from her crouched position in front of the open beer fridge. From this vantage point, the man fingering the assortment of bottles she'd left on the counter appeared even taller than usual.
Darius Johnson. Prelaw student, smart-ass and not a big fan of hers. Which Lainey figured made sense, seeing as he was her fa— _Martin's_ last hire.
Also, she'd cleaned house when she'd first arrived, firing a dishonest bartender and a couple of slothful waitresses. Despite the months that had passed, Lainey got the impression that the remaining staff were still a little wary that she'd go all "off with their heads" on them at any moment. She didn't bother doing anything to disabuse them of that notion. It didn't matter if Darius was fun to spar with, or that she kind of enjoyed Aggie's no-nonsense wisdom. Lainey was here to sell the bar. She wasn't looking to make friends.
All in all, Darius was a solid bartender and great with the regulars. And Lainey wasn't above exploiting the fact that he was popular with the coeds—they loved his soulful eyes, café au lait complexion and killer smile. Or at least those were some of the giggled compliments she'd heard when they were gathered at the counter, fawning over him on a Friday night. They didn't seem to mind his stupid goatee, either.
She let the flirting stand, because if you could get the ladies into a bar, the guys would follow. And the fact that some of Darius's fellow students were choosing to spend their money in a crappy sports bar instead of a flashy nightclub did good things for the bottom line. And it was a bottom line that needed all the help it could get.
Still, that didn't keep her from imagining firing Darius at least three times per shift, if only for the peace and quiet.
"Give me a hard time for not keeping my workspace clear, but I show up to a mess of bottles on the counter when you're in charge," he muttered, the way he always did when he was trying to get under her skin.
"It was recipe development," she said simply. "It's called a Black Widow."
Darius frowned as he set the Cinnamon Schnapps back on the shelf. "You put all this stuff in the same glass? Whoever he was, he must've really pissed you off."
Embarrassed, Lainey rubbed her fingers against her cheek in a vain attempt to extinguish the lingering prickle where Cooper's knuckles had touched her. "Don't think I didn't notice that you're late." She made sure her voice was as frosty as the draft mugs that rattled when she slammed the cooler door. "For future reference, your shifts are posted in Pacific Time."
Darius glanced over his shoulder as he returned the Kahlua, the Blue Curacao, and some banana liqueur to the appropriate shelves. "He definitely pissed you off."
" _You_ pissed me off," Lainey corrected, standing. "I know Martin let stuff like this slide, but I'm trying to sell this place. I can't afford not to have things running smoothly."
"You keep saying that, but you've been here for three months and counting. I'm starting to think we're never gonna be rid of you."
Lainey pulled a face at his broad back when he turned to clean up her mess.
"You know I can see you in this mirror, right?"
She schooled her features into a neutral expression. "And you know that I have the power to fire you, right?"
"Well, before you let all your authority go to your head and I end up suing you for wrongful termination, you should probably check your phone. I texted you that I was running late. But I'll let it go, because I'm in a stellar mood. Sandra and I shared a hell of a goodbye before her Uber showed up to take her to the airport."
Darius's expression was dripping with satisfaction. "Which is why I got here late, if you know what I mean." He waited a beat. "And what I mean is that we had copious amounts of sexual intercourse."
"Thanks for the clarification, wonder stud." Lainey rolled her eyes at him. "But I'm not sure that's the type of excuse that will stand up in court. As a future lawyer, you'll want to familiarize yourself with labor laws."
The well-timed entrance of Agnes Demille saved Lainey from Darius's retort. The zaftig waitress materialized from the "staff only" door to their right, plopped her massive gold lamé purse on the counter behind the bar, grimaced and slung it back on her shoulder. "Honestly, you two. I've been here for thirty seconds, and there's already a table full of customers with no beer and a sticky counter. This ain't no way to run a business. 'Specially on game night. Let's get a move on, people! Darius, hand me that rag."
Darius peeled the blue rag from the sink and dropped it in front of Aggie, who set to work immediately, scrubbing at the sticky rings on the counter. "So, Lainey," she said, not bothering to look up from her task, "I'm thinkin' the two of us need to have a little chit-chat."
Lainey ignored the resulting shiver down her spine. Aggie could size up a room quicker than anyone Lainey had ever met, and she didn't miss a detail. Especially not a ridiculously handsome one wielding a glass full of sludge. In an attempt to sidestep the conversation, Lainey placed a tray on the counter and systematically loaded it with six frosty bottles of beer from the cooler. "Beers for Larry's table, as requested."
Unfortunately, the announcement didn't faze the formidable woman before her. "They can wait. What you just did to Cooper Mead can't."
"What?" Darius's brows dove into a V as he scanned the customers. A sharp bark of laughter confirmed he'd located his target. "Are you kidding me? The Black Widow was for Cooper Mead? That is so awesome!" He held up an expectant palm in her direction, then thought better of it and aborted the high-five. "Man, it sucks I was late! I would've loved to have seen his face when you handed it over. So what's Mr. Big Shot doing here, anyway?"
"Bible study starts in ten minutes."
Darius shot Lainey a pained smile as she bent to grab a bottle of water from the fridge.
"Well, don't be a moron. It's a bar, for God's sake. What do you think he's doing here?"
"It's a floundering sports bar," he corrected pointedly. "Hardly the preferred scene of professional athletes."
Lainey stiffened at the comment. "Then you should be glad he's here. He shelled out for his drink, so you might actually get paid on time this week."
Darius had the grace to blush. "You know I didn't mean it like that."
"Yeah, I know." Twisting open her water, Lainey took a long swallow and stared blankly at a framed hockey jersey—number 42—on the opposite wall. "I have no idea what he's doing here, either," she confessed.
Lainey took another bracing gulp of water, screwed the lid back on and turned to meet Aggie's unrelenting stare.
"It's no big deal," Lainey assured the carroty-hued waitress. Further proof that cheap self-tanning lotion, like Cooper Mead, was one more on a long list of items to be avoided.
"He fed me a lame line, I gave him a disgusting drink. As you can see, he didn't take it too hard." She gestured toward a smiling Cooper as he posed for a camera phone.
"Just because a man notices you got a nice rack don't mean you need to start handin' out the Black Widows." Agnes shook her frizzy, brassy-hued curls. "I never shoulda told you about those."
"She's right, Lainey," Darius interjected. "You do have a nice rack.
She landed a hard punch on his shoulder. "Back off, pervert."
Lainey turned back to Aggie with "I told you so" plastered all over her expression. "You see? I'm rude to all overbearing jackasses! It's what I do."
Agnes planted a fist on one generous, black-spandex-covered hip. "Yeah, but Cooper Mead ain't every other jackass."
"Oh no? And what makes him so special?"
"That's what I'd like to know," Darius threw in.
"I mean, sure, he's gorgeous," Lainey conceded. "And there's no denying the way that voice rumbles through your chest and trickles down to all the right places, and yeah, okay, I may have almost had an orgasm just looking at him."
Aggie nodded dreamily, and both women shot a wistful look in Cooper's direction. Not that they were bonding or anything. This was strictly physical appreciation of a handsome man, not friendship.
"I can't believe Cooper Mead is signing beer coasters in your sports bar!" Aggie sighed. "It's like a freakin' fairy tale or somethin'."
"Funny. I don't actually remember the part in _Cinderella_ when she had to change her panties."
Lainey grimaced, disgusted out of her aesthetic appreciation. "Ugh. Darius. Seriously. Why do you have to be such a guy?"
"You do realize you're practically forcing me to grab my crotch right now, don't you?"
"All I'm sayin'," Aggie stressed, "is that sometimes you gotta swallow your pride, think of the big picture. Normally when you castrate someone, the fate of your business ain't riding on it."
"What?" Lainey rolled her eyes. "The fate of my business is hardly riding on Cooper Mead's penis."
Darius's snicker earned him two glares. "What? You said _penis_."
"It's resting on my shoulders," Lainey countered, with the pious look of stone angels the world over. "And I can handle it."
"I know you can! But use that big ol' brain of yours. Bein' attentive to a man with fame and money is just good business sense."
Lainey turned her head to hide her frown.
"Cooper Mead is the Pied Piper of cool an' you darn well know it. Where he goes, the puck bunnies and the sports fans follow. I don't think makin' nice with him is too much to ask! You know, most joints would kill to have a pro athlete walk through their door! And you're the one always jabbering about selling this joint."
"You do realize that Mr. Rich and Famous over there was interested in my phone number, not an endorsement deal," Lainey pointed out.
"I think you mean Mr. Sexy, Rich and Famous." Agnes sent an appreciative glance at the object of their discussion, who appeared to be talking to someone's kid via FaceTime. "Emphasis on the sexy."
"Well, Mr. _Sexy_ , Rich and Famous," Lainey amended, "is kind of a shallow, conceited jerk, emphasis on the jerk."
"Who cares? I don't wanna waste time talkin' to him! Man who looks that good could have me anytime, anywhere."
Heat, not unlike the sear of a good shot of whisky, burned in Lainey's stomach at the thought of Cooper and sex, and her mind was seized by an alarmingly vivid vision of him, naked on a king-size battlefield, expertly wielding his...uh, sword.
Luckily the flashing of a disturbingly high number on the "Now Serving" sign above the imaginary bed doused the flame before it reddened her cheeks.
"Listen, your daddy was a good guy, but a so-so businessman. This place can use all the good publicity it can get. 'Specially the free kind." Oblivious to Lainey's inner turmoil, Agnes walked to the other side of the counter and hefted the tray of beer to her shoulder. "I'm gonna deliver these, but I want you to promise me that when you turn around and see that a certain teammate of his is here, you're going to play nice, okay? Take care of things nice and quiet. Don't make a scene."
Aggie's warning tone left little doubt as to the identity of Cooper's teammate, and Lainey's gaze jerked to the newly occupied table in the back corner, near the stage.
With a curse, she stomped out from behind the bar with every intent of telling table seventeen to go to hell, despite Aggie's well-meaning advice.
_Don't miss_ **Playing Dirty** _by Taryn Leigh Taylor! And be sure to look for her other hot romances_
**Kiss and Makeup**
_and_
**Playing to Win**
_Available now from Harlequin Blaze!_
Now, for an exciting look at New York Times bestselling author Jill Shalvis' first Women's Fiction novel,
LOST AND FOUND SISTERS!
_They say life can change in an instant..._
After losing her sister in a devastating car accident, chef Quinn Weller is finally getting her life back on track. She appears to have it all: a loving family, a dream job in one of LA's hottest eateries, and a gorgeous boyfriend dying to slip an engagement ring on her finger. So why does she feel so empty, like she's looking for a missing piece she can't find?
The answer comes when a lawyer tracks down Quinn and reveals a bombshell secret and a mysterious inheritance that only she can claim. This shocking revelation washes over Quinn like a tidal wave. Her whole life has been a lie.
On impulse, Quinn gives up her job, home, and boyfriend. She heads up the coast to the small town of Wildstone, California, which is just a few hours north of Los Angeles, but worlds apart. Though she doesn't exactly fit in right away, she can't help but be drawn to the town's simple pleasures...and the handsome, dark-haired stranger who offers friendship with no questions asked.
As Quinn settles into Wildstone, she discovers the inheritance isn't a house or money, but rather something earth-shattering, something that will make her question everything she thought she knew about herself, about her family. Now with a world of possibilities opening up to her, Quinn must decide if this new life is the one she was always meant to have.
**_CHAPTER 1_**
I walk around like everything is fine but deep down inside my shoe my sock is sliding off.
—from "The Mixed-Up Files of Tilly
Adams's Journal"
Here was the thing: life sucked if you let it. So Quinn Wellers usually worked really hard to not let it. Caffeine helped. For up to thirty-eight blissful minutes it could even trick her into thinking she was in a decent mood. She knew this because it took forty-eight minutes to get from her local coffee shop through L.A. rush-hour traffic to work, and those last ten minutes were never good.
That morning, she got into line for her fix and studied the menu on the wall, even though in the past two years she'd never strayed from her usual.
A woman got in line behind her. "Now _that's_ a nice look on you," she said.
It was Carolyn, a woman Quinn had seen here at the coffee shop maybe three times. "What look?"
"The smile," Carolyn said. "I like it."
Quinn didn't know whether to be flattered or insulted, because she smiled all the time.
Didn't she?
Okay, so maybe not so much lately . . . "I'm looking forward to the caffeine rush."
"Nectar of the gods," Carolyn agreed.
Something about the pleasant woman reminded Quinn of an elementary-school teacher. Maybe it was the gray-streaked hair pulled up in a messy bun, the glasses perpetually slipping down her nose, the expression dialed into sweet but slightly harried.
"You're up, honey," Carolyn said and gestured to the front counter.
Trev, the carefully tousled barista, was an L.A. beach bum and aspiring actor forced to work to support his surfing habit. His hands worked at the speed of light while the rest of him seemed chilled and relaxed. "Hey, darlin', how's life today?"
"Good," Quinn said. She didn't want to brag, but she'd totally gotten out of bed today with only two hits of the snooze button. "How did your audition go?"
"Got the part." Troy beamed. "You're looking at the best fake Thai delivery guy who ever lived. It means my luck has changed, so say you'll finally go out with me."
Quinn smiled—see, she totally _did_ smile!—and shook her head. "I'm not—"
Trev piped in along with her, in perfect sync, "—dating right now," and then shook his head.
"Well, I'm not," she said.
"It's not right, a hot chick like you. You're way too young to be in a rut. You know that, right?"
"I'm not in a rut." She was just . . . not feeling life right now, that's all. "And hey, I didn't tell you my order yet," she said when she realized he was already working on her coffee.
"Has it changed?" he asked. "Ever?"
No, but now she wanted to order something crazy just to throw him off, except she wanted her usual, dammit.
Okay, so maybe she _was_ in a rut. But routine made life simpler and after the complications she'd been through, _simple_ was the key to getting out of bed and putting one foot in front of the other every day.
That and the badly needed caffeine, of course.
"You should go out with him," Carolyn whispered behind her. She smiled kindly when Quinn craned her neck and looked at her. "You only live once, right?"
"Not true," Quinn said. "You live every day. You only die once."
Carolyn's smile slowly faded in understanding. "Then make it count, honey. Go hog wild."
Hog wild, huh? Quinn turned back to Trev.
"I'm all for the hog wild," he said hopefully.
Quinn went for it. "An extra shot and whip."
Trev blinked and then sighed. "Yeah, we really need to work on your idea of hog wild."
* * *
When Quinn finally got to Amuse-Bouche, the trendy, upscale restaurant where she worked, it was to find her fellow sous-chef Marcel already in the kitchen.
He glanced over at her and sniffed disdainfully. Then he went back to yelling at Skye, a good friend of Quinn's she'd brought on board a few months back.
Skye was chopping onions exactly as Quinn had taught her, but apparently Marcel didn't agree. His voice had risen to a pitch designed to split eardrums as he went on and on in a mix of English and German that no one understood but him.
"Leave her alone, Marcel," Quinn said.
He slid her a glacial stare. "Excuse me?"
"I'm the one who taught her how to chop. She's doing it correctly."
"Yes. If you work at a place flipping burgers and asking what size fry you want with your order," he said, dropping the fake German accent as he sometimes did when he lost his temper and forgot to keep it up.
So here was the thing. There were days where Quinn surprised herself with her abilities, and others where she put her keys in the fridge. But she was good at this job. And yes, she understood that at twenty-nine years old and quickly rounding the corner kicking and screaming into thirty, she was young and very lucky to have landed a sous-chef position in such a wildly popular place. But she'd worked her ass off to get here, going to a top-notch culinary school in San Francisco, spending several years burning and cutting her fingers to the bone. She knew what she was doing—and had the tuition debt to prove it.
Oddly, Marcel wasn't that much older than she was—late thirties, maybe. He'd come up the hard way, starting at the age of twelve washing dishes in his uncle's restaurant not all that far from here, but light years away in style and prestige. He was good. Actually he was excellent, but he was hard-core old school, and resented a woman being his equal.
Quinn did her best to let it all bead off her back, telling herself that she believed in karma. What went around came back around. But though she'd waited with pent-up breath, nothing had kicked Marcel in the ass yet.
"You," he said, pointing at her. "Go order our food for the week. And don't forget the pork like last time. Also, your cheese supplier? She's shit, utter shit. Find another."
Quinn bit her tongue as Marcel turned away to browbeat a different kitchen aide who was dicing red peppers, swearing at the guy in German as if that made him more intimidating. Quinn thought it made him more of an idiot. He jerked the bowl away to prove his point and ended up with red pepper all over the front of his carefully starched white uniform shirt.
Ah, karma at last—fashionably late, but better than never.
"I'm sorry about that," she said to Skye.
"You've got nothing to apologize for," Skye said. "If it's got tires or testicles, it's gonna give you trouble."
Wasn't that the truth . . .
* * *
On Sunday, Quinn drove to her parents' place for brunch. A command performance, since she'd managed to skip out on the past two weekends in a row due to working overtime.
She hoped like hell it wasn't an ambush birthday party. Her birthday was still several weeks away, but her mom couldn't keep a secret to save her own life and had let the possibility of a party slip several times. Quinn didn't like birthdays.
Or surprises.
She parked in front of the two-story Tudor cottage that had been her childhood home and felt her heart constrict. She'd learned to ride a bike on this driveway, right alongside her sister, who'd been a far superior bike rider. So much so that Quinn had often ridden on Beth's handlebars instead of riding her own bike. They'd pilfered flowers from the flower gardens lining the walkway. Years later as teens, they'd sneaked out more than a few times from one of the second-story windows, climbing down the oak tree to go to parties that they'd been grounded from attending—only getting caught when Quinn slipped one year and broke her arm.
Beth hadn't spoken to her for weeks.
Once upon a time this house had been Quinn's everything. But now coming here made her feel hollow and empty. Cold. And deep down, she was afraid nothing would ever warm her again.
_It'll get easier._
_Time is your friend._
_She'll stay in your heart._
Quinn had heard every possible well-meaning condolence over the past two years and every single one of them was a lie.
It hadn't gotten easier. Time wasn't her friend. And as much as she tried to hold on to every single memory she had of Beth, it was all fading. Even now she couldn't quite summon up the soft, musical sound of her sister's laugh and it killed her.
Shaking it off the best she could, she slid out of her car and forced a smile on her face. Sometimes you had to fake it to make it.
Actually, more than sometimes.
Late April in Southern California could mean hot or hotter, but today was actually a mild eighty degrees and her mom's flowers were in full, glorious bloom. Quinn ducked a wayward bee—she was allergic—and turned to watch a flashy BMW pull in next to her, relieved to not have to go inside alone.
Brock Holbrook slid out of his car looking camera ready and she couldn't help but both smile and roll her eyes. "Suck-up," she said, gesturing to his suit and tie.
Brock flashed a grin. "I just know where my bread's buttered, that's all."
He worked for her father's finance company and no one could deny that Brock knew how to work a room. He was good looking, charismatic, and when he looked at her appreciatively, her entire body hummed with anticipation. Or it used to.
But these days she didn't feel anticipation for anything. She sighed and Brock tilted his head at her, eyes softer now, understanding.
He knew. He'd been there when she'd found out about Beth's accident. But his understanding didn't help.
She wanted to _feel_ again, dammit. The thrum of blood pounding through her veins in excitement. Happy butterflies in her belly . . .
The front door opened and Quinn glanced over. Both hers and Brock's parents stood in the doorway, the four of them smiling a greeting at their chickens coming home to the roost, where they'd be pecked at for every little detail of their lives.
Quinn loved her parents and they loved her, but brunch promised to be more invasive than a gyno exam on the 405 South at peak traffic hours.
Brock took Quinn's hand and reeled her in, smiling as he planted a kiss on her lips. It wasn't a hardship. He looked good and he knew it. He kissed good too, and he knew that as well.
It had been two years since they'd slept together, two years since she'd felt the zing of sexual awareness or desire, and she didn't feel it now either. Still, the kiss was nice, and normally she'd try to enjoy it—except he was only doing it for show. So she nipped at his bottom lip.
Hard.
Laughing, he pulled back. "Feisty," he murmured. "I like it."
"I'm not going to sleep with you."
"You should."
"Pray tell why."
"It's been so long . . ." He tugged playfully on a strand of her hair. "I'm worried you're depressed."
This was just uncomfortably close enough to the truth to have her defenses slam down. "I'm not depressed."
"Not you," he said. "Your vagina."
She snorted and yanked free. "Shut up."
"Just keep it in mind." He took her hand back and held it as he led the flight-risk chicken up the front path.
"I should've bitten you harder," she whispered, smiling at the parentals.
"Feeling vicious today, I take it?"
" _Annoyed_ ," she corrected.
"Ah. I guess turning old does that to a person."
He was nine months younger than she and for just about all their lives—they'd met in kindergarten when he'd socked a boy for pushing her—he'd been smug about their age difference. She nudged him with her hip and knocked him off balance. He merely hauled her along with him, wrapping both his arms around her so that by all appearances _he'd_ just saved _her_ from a fall. His face close to hers, he gave her a wink.
And suddenly it occurred to her that this wasn't about her at all, but him. His parents must be on him again about giving them grandbabies. And she got it, she did. The truth was, everyone expected them to marry. Brock had been her middle _and_ high school boyfriend, and they'd gone off to college together. During their freshman year, they'd had a wildly dramatic and traumatic breakup involving his inability to be monogamous.
Oh, he'd loved her, she had no doubt of that. But he'd also loved anyone who batted their eyes and smiled at him.
It had taken a few years, but eventually they'd found their way back to each other. He'd grown up a lot and so had she. They were best friends—at times friends with benefits—and at others mortal enemies. But after Beth's death, their physical connection had fallen by the wayside . . . and that was all on her.
They'd eventually had the hard discussion about their different needs, and as a result, they'd gone from lovers to friends. She knew Brock would go back to lovers in an instant if she showed the slightest interest.
But she didn't feel interest, and was starting to be afraid she'd never feel it—or anything—ever again. "You're only making it worse for both of us," she said quietly as they moved toward the mother ship.
"If they think we're working on things, they'll leave me the hell alone."
She had to concede the point as they hit the porch and were enveloped into the fold.
"Still not used to it," her mom murmured to Quinn, clinging to Quinn for an extra minute. "It never feels right, you here without her . . ."
She didn't mean it hurtfully, Quinn knew that. Her mom wouldn't hurt a fly, but as always, a lump the size of Texas stuck in her throat. "I know, Mom."
"I miss her so much. You're so strong, Quinn, the way you've moved on."
Had she? Moved on? Or was she just treading water, staying in place, managing to keep her head above the surface? Burying her feelings deep had been the only way to survive the all-encompassing grief, which at the moment sat like a big fat elephant on her chest. For the most part, she kept it locked away in a dark corner of her heart. She'd even built a wall around it, brick by painstaking brick, to contain the emotions that had nearly taken her down.
But she reminded herself that she was lucky. She had a job she loved, parents who cared, and a best friend/fallback husband if it ever came to that. And yes, she was turning thirty soon, and a surprise party still lay in wait regardless of the fact that she didn't want it. But while she'd like to pretend none of that was happening, it wouldn't derail her. Compared to what she'd been through, nothing could.
* * *
A little over a week later, Quinn was in line for her usual before-work latte when she felt the weight of someone's gaze on her. Turning, she found a guy around her age with tousled black hair and black-rimmed glasses who looked a lot like a grown-up Harry Potter.
He was staring at her with an intensity that caused her to blink and then crane her neck to peek behind her. No one was there, which meant he was staring at her. She shifted away and did her best to ignore him, instead tuning in to the two women in front of her who were chatting.
"Orgasms after the age of fifty suck," one was saying. "No one tells you that, but they totally do."
Her friend agreed with an emphatic head bob. "I know. It's like sandpaper down there in Lady Town. Takes an entire tube of lube and a bottle of gin."
The first woman snorted. "Don't get me started. Alan will spend thirty minutes looking for a golf ball, but he can't give me ten minutes to find the G-spot."
Quinn must have made some sound, because they both turned to her with apologetic laughs. "Sorry," Dry Vagina said. "It's just one of the many, many things you've got to look forward to, along with hot flashes."
"Don't forget the murderous urges," the other woman said. "And we're talking premeditated too."
Yay. Sounded great.
"Excuse me," the man behind Quinn said, tapping her on the shoulder. Harry Potter, her stalker. "I need to speak to you."
"Sorry. I'm not interested."
"Wait, honey. There's no need to make a hasty decision," one of her new friends said. "What if he's suitably employed, with no baggage?"
"Impossible," Dry Vagina said. "That'd be like finding a unicorn."
"Are you a unicorn?" the first woman asked him.
Harry Potter looked at Quinn with more than a little desperation. "Can I please talk to you . . . alone?"
"Not alone," the first woman said. "That sounds like stranger danger. You can do your pickup-line magic right here, or better yet, do it online like the rest of the world."
The guy never took his gaze off Quinn. "You're Quinn Wellers, right?"
Wait a minute. _How did he know her name?_ "Okay," she said. "You're going to need to go first."
"I'm Cliff Porter," he said. "I'm an attorney and I really need a word with you. Privately."
She stared at him, trying to come up with a reason why an attorney would be looking for her.
"Porter or Potter?" Dry Vagina asked. "Because Potter would make more sense."
He looked pained. "I get that a lot, but it's Porter."
"How do you know my name?" Quinn asked.
"Look, can we just . . ." He gestured to a small table off to the side of the line.
Torn between curiosity and a healthy sense of survival, Quinn hesitated. "I'll be late for work."
"This will only take a minute."
Reluctantly, she stepped out of line and moved to the table. "You've got one minute."
He took a deep breath. "As I said, I'm an attorney. I located you through a mutual acquaintance."
"Who's that?"
"I'll get to her in a minute. She let me know I could find you here in the mornings rather than scare you by tracking you down at your place of residence. I'm from Wildstone, a small town about two hundred miles north. I'm here to give you news of an inheritance. It's important we talk about it because—"
"I've never even heard of Wildstone," Quinn said "I certainly don't know anyone from there."
He nodded like he knew this. "We're a small coastal ranching town that sits in a bowl between the Pacific Coast and wine country. Maybe you should sit," he said quietly, and also very kindly she had to admit. "Because the rest of this is going to be a surprise."
"I don't like surprises," she said, "and you have thirty seconds left."
It was clear from his expression that he wasn't happy about having to go into the details in public, but as he was a stranger and maybe also a crackpot, too damn bad. He drew in a deep breath. "The person who left you this inheritance was your birth mother."
She stared at him and then slowly sank into the before-offered chair without looking, grateful it was right behind her. "You're mistaken," she finally managed, shaking her head. "I wasn't adopted."
He gave her a wan smile. "I'm really sorry to have to be the one to tell you this, but you were."
"I have parents," she said. "Lucinda and James Wellers."
"They adopted you when you were two days old."
The shock of that reverberated through her body. "No," she whispered. Heart suddenly racing, palms clammy, she shook her head. "They would've told me. There's absolutely no way . . ."
"Again, I'm very sorry," Cliff said quietly. "But it's true. They adopted you from Carolyn Adams." He pulled a picture from his briefcase and pushed it across the table toward her.
And Quinn's heart stopped. Because Carolyn was the woman she'd met here in this very coffee shop.
**_CHAPTER 2_**
My mom always said that right before she died she wanted to swallow a bunch of popcorn kernels to make her cremation more interesting. She totally would've done it too, if she'd gotten the heads-up that her number had been picked.
—from "The Mixed-Up Files of Tilly
Adams's Journal"
Quinn found herself sitting on the curb outside the coffee shop, staring blindly at her Lexus, the car her parents had given her last year even though she'd wanted something less expensive.
Her parents. Who might not really be her parents . . .
"Here," Cliff said, pushing a cup of cold water into her hands as he sat next to her. "Drink this."
She took the cup in two shaking hands and gulped down the water, wishing a little bit that it was vodka. "You're mistaken," she said again. "Carolyn was just a woman I met here. We spoke only a few times."
"Three." Cliff gazed at her sympathetically. "She told me about the visits. She always came here instead of your condo or work because it was a social setting and she felt she could approach you here. She'd come to get a peek at you whenever she could, born from the desperate curiosity of a woman who had haunting regrets."
Quinn shook her head, unable to descramble her brain. "I don't understand."
"She knew she was terminal," he said. "She had every intention of telling you all of this herself, but she ran out of time. And what she left behind is important because—"
"Wait." Quinn closed her eyes, just now realizing what he was telling her.
Carolyn was dead.
Cliff took the cup of water from her before she could drop it. "The funeral was a few days ago," he said quietly. "We really need to talk, Quinn. In Wildstone. There are things you don't know that you need to."
Quinn let out a sound that might have been a mirthless laugh or a half sob, she wasn't sure. She shook her head for what felt like the hundredth time in the past few minutes, but the cobwebby feeling didn't clear.
It couldn't be true, any of it. Harry Potter here was just a stalker, a good one. Or maybe a scammer. She hated to think that the nice woman she'd known could be a part of some kind of con, but she simply couldn't accept that her parents wouldn't have told her she'd been adopted. "I don't want any part of this." She stood up and a wave of dizziness hit her.
Cliff rose to his feet too and put his hand on her arm to steady her, looking at her with nothing but kindness and concern in his gaze. "Take my wand."
She focused in on him, expecting to see a lightning-bolt scar on his forehead. "What?"
"My card," he said, the furrow between his brows deepening with concern. "Take my card. Think about it and give me a call tomorrow so we can talk about the inheritance. We _really_ need to talk about the inheritance, Quinn." He paused. "Are you going to be all right?"
"Yeah. Sure," she said and drove to work on autopilot, where she proceeded to spill things, plate the wrong entrées, make silly mistakes like using shallots instead of onions—
"What the hell's wrong with you?" Marcel demanded. "Get out of my kitchen until your head's screwed on straight!"
For once he was right. Her head was most definitely not screwed on straight.
_They adopted you when you were two days old . . ._
"Are you even listening to me?" Marcel yelled up at her. Up, because he was five feet two to her five feet seven, something that normally gave her great pleasure. " _Du flittchen,_ " he muttered in disgust beneath his breath and the entire staff froze in the kitchen like deer in the headlights.
Slut.
Quinn set down her knife so she wasn't tempted to run him through as she turned to him. " _Schiebe ex,_ " she said, which meant "shove it." It was the best she could do, at least in German. Pushing past him, she walked out of the kitchen.
"Where are you going?" he screamed after her. "You can't just leave!"
But leaving was exactly what she was doing.
Skye followed her outside. "Quinn? You okay? What's going on?"
"You've got to go back in there before he gets mad at you too," Quinn said.
Skye shrugged. "He was born mad. Talk to me."
So Quinn told her what had happened at the coffee shop, and Skye just stared at her. "Shut the front door."
"I've got to go. I need to talk to my parents," Quinn said.
"Uh, yeah you do."
From inside they could hear Marcel yelling for Skye, who squeezed Quinn's hand. "Call me."
Quinn promised she would and gave her a quick hug. Then she headed toward her car, pulling out her cell phone to call her boss, Chef Wade.
Chef Wade never wasted words. He answered with, "Talk."
"I need to leave early," Quinn said. "I'm so sorry for the short notice, but there's . . . an emergency. Marcel's here. He's got things under control." By being a tyrannical asshole, but that was another story.
"What's wrong?" he asked, voice softening with concern, as he was a longtime friend of her parents.
"Nothing I can't handle," she promised and hoped that was true. She disconnected and drove straight to her parents' house.
Her mom and dad were in the living room in front of their lit gas fireplace, sharing a drink. Yes, it was 3 P.M. in April in L.A., which meant the air conditioner was on full blast, but her mom liked her alcohol with ambience.
"Darling," her mom said, smiling as she stood in welcome. "Such a lovely surprise. Where's Brock?"
"I'm alone." Quinn didn't bother to address—for the thousandth time—that she didn't spend nearly as much time with Brock as they seemed to hope. "I met someone today."
Her mom looked dismayed. "Other than Brock? But what will people think?"
"Mom . . ." Quinn pressed her fingertips into her eye sockets to ward off an eye twitch. "I keep telling you, Brock and I aren't together like that anymore."
"Right _now_ you mean," she said. "Right?"
A conversation she didn't have the strength for. "The man I met today had an interesting story to tell me." Her breath caught. "He said that I'm adopted."
Twin looks of shock and guilt slid over her parents' faces like matching masks and reality hit Quinn smack in the face. "Oh my God." She staggered to the couch opposite them and sank to it, staring at them. "Oh my God, it's not a story. It's true."
At the awkward beat of utter silence, Quinn stood back up and headed straight to the kitchen. She needed alcohol or sugar, stat. Thank Toll House, she found some ready-made cookie dough in the fridge. Her mom didn't bake. For that matter, neither did Quinn. She loved to cook and she was good at it, but for whatever reason, baking skills eluded her.
She was stuffing spoonfuls of dough into her mouth when her parents—who were apparently _not_ her parents at all—appeared in the doorway. It was the most disorienting thing she'd ever experienced, looking at them and realizing her life was forever changed, that the very foundation of her entire world had crumbled. "It's day one of my new raw food diet," she said inanely.
Her parents exchanged a concerned look. "We need to talk," her dad said solemnly.
Little late for that . . .
"Honey," her mom said earnestly. Quinn turned to her hopefully.
"If you eat that whole package, it's the equivalent of forty-eight cookies."
Quinn blinked. "Are you kidding me?"
Her dad sighed and leaned onto the island between them. He nudged the block of knives out of her reach and said, "We never expected you to find out."
"Okay," she said, absorbing that with a nod. "Wow." She scooped up the last of the dough.
Her mom opened her mouth but nothing came out of it because Quinn jabbed a finger at her. Then she popped the last bite in, chewed, and licked her thumb before taking a deep breath. "Why?" she finally asked, suitably sugared up. "Why didn't you just tell me? People are adopted all the time. What possible reason could you have for keeping it a secret?"
"Because I wanted you to be mine," her mom whispered, her eyes soft and, dammit, a little damp.
Her dad slid an arm around her mom's waist. "It wasn't important to us _how_ we got you," he said. "We wanted a baby, and we couldn't have our own."
Quinn sucked in a breath. "Beth," she said through a throat blocked by what felt like a regulation football. " _Was she adopted too?_ "
Her mom shook her head. "No. We'd been trying for years before we were told we couldn't have our own. So we set an adoption in motion. When you came along, we were so happy, just completely over the moon." She stopped and drew a deep breath, as if reliving the joy. "But then the unbelievable happened. When you were four months old, I learned I was pregnant." She put a hand to her chest. "I'm more grateful to Carolyn for you than you could ever know," she said fiercely. "Because we were twice blessed. But . . ." She looked to Quinn's dad.
"But," he continued on for her, "Carolyn signed a confidentiality agreement. We could sue her for discussing the adoption. She had no right."
"Too late," Quinn said quietly. "She's dead. And apparently she left me some sort of an inheritance."
"That doesn't make any sense," her mom said. "She had nothing of worth to speak of."
"I was so shocked I didn't ask for details," Quinn said. Details Cliff had tried to give her. She hugged herself, feeling a little sick from the dough.
Or her life. "So . . . were you sorry you'd adopted me once Beth came along?" she asked.
"Oh my God, no." Her mom came around the island and took Quinn's hands in her own. " _No_ ," she said again more firmly. "It was a happy accident. The truth is, we didn't want to take away from _either_ of you, so we just kept it quiet. It didn't matter to us, and I know this is asking a lot, but I wish it wouldn't matter to you."
Her dad nodded his agreement on that.
But Quinn didn't know how to make it not matter. She didn't know what to feel, not about the adoption, the devastating betrayal, or the fact that she and Beth had never been sisters at all. She let out a breath and took a step away from them. "I need to think."
"It doesn't matter," her mom said. "None of this matters."
"Mom, how can you say that?"
"Because we love you. Maybe we were wrong to not have told you about being adopted, and I'm sorry you found out in such a shocking manner, but we've never thought of you as anything but a real daughter. Ever."
This brought a huge lump to Quinn's throat so all she could do was nod.
"Now," her mom said, tears shimmering brilliantly in her own eyes as well as she patted Quinn on the arm. "Let's just look forward, to you marrying Brock and getting on with your lovely life."
Quinn closed her eyes. "I'm not getting married to Brock. And even if I wanted to, how could I?" she asked. "I don't even know who I am anymore."
"Okay," her dad said. "That seems a little dramatic."
Quinn let out a low laugh. "You're right. It is. And now I'm going to take my dramatic ass home. I need some time."
"Time?" Her mom exchanged another worried look with her dad. "But you're still coming over next weekend for dinner, right? Say Saturday night . . . seven o'clock? On the dot? And you'll text me once you get here, before you come in?"
Quinn had gotten to the door. She turned around to find them standing in the same position at the island, looking shocked at her unusual temper tantrum. "Let me get this straight. You can't keep my surprise party a secret, but you were able to keep my adoption one?"
Her mom bit her lower lip. "I don't know what you mean about a surprise party."
With another low, mirthless laugh, Quinn walked out. She drove home to the quiet little condo she was mortgaged to the eyeballs for and stared at herself in the bathroom mirror. She was in shock. And adrift. And . . . sad. Angry too . . . and so much more.
It was shocking for more than one reason, not the least of which was that she felt more emotion right this minute than she'd felt in two years.
She'd meant it when she'd told Cliff that she didn't want anything to do with any inheritance, especially not from someone who'd apparently thrown her away without so much as looking back.
Not that she was happy with her parents right now either. They should've told her the truth a long time ago. Instead they'd hidden it and even now had tried to underplay everything, encouraging her to get on with her nice, comfortable life.
But it suddenly didn't feel so nice or comfortable at all.
Feeling shockingly alone, she looked at her phone. She wanted to call Beth. God, how she wanted that, but instead she called Brock.
"Hey," he said when he picked up, his voice brisk and rushed. "I'm in a meeting. Leave a message and I'll get right back to you."
His voice mail. Disappointment washing over her, she tried to tell herself she was fine, she didn't need anyone. But her heart was racing and it didn't seem to fit in her rib cage anymore. _Everything_ felt tight and she couldn't breathe because she had no one else left to call.
Well, except one person.
Harry Potter, aka Cliff Porter.
Lost and Found Sisters
will be available June 20th, 2017,
wherever books are sold!
LOST AND FOUND SISTERS. Copyright © 2017 by Jill Shalvis.
ISBN-13: 978-1-48803221-9
FLASHBACK
Copyright © 2008 by Jill Shalvis.
All rights reserved. By payment of the required fees, you have been granted the non-exclusive, non-transferable right to access and read the text of this e-book on-screen. No part of this text 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 publisher, Harlequin Enterprises Limited, 225 Duncan Mill Road, Don Mills, Ontario, Canada M3B 3K9.
All characters in this book have no existence outside the imagination of the author and have no relation whatsoever to anyone bearing the same name or names. They are not even distantly inspired by any individual known or unknown to the author, and all incidents are pure invention.
This edition published by arrangement with Harlequin Books S.A.
® and ™ are trademarks of the publisher. Trademarks indicated with ® are registered in the United States Patent and Trademark Office, the Canadian Intellectual Property Office and in other countries.
www.Harlequin.com
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Q: Android strace in Real device I have the following situation, I want to monitor the system calls on Android phones so
I made an script to do that. With Android Emulator works perfectly (writes the traces of the application in a specific file on my Ubuntu).
The problem is when I attach a real phone to analyze it, it says the following in the result file:
ptrace attach failed: Operation not permitted
I'm using this code to get it, but I don't understand why it works on the emulator and not in the rooted real device.
This is the comand I use in perl:
system("$dirTools/adb -s $Device shell strace -p $PID[1]>$dirRecordDataSet/$Date/$appName &");
Any suggestion? Thanks in advance
A: Maye be the adb daemon runs with limited privilege. You can try to install strace with setuid roo. Here is manpage advice about it :
Setuid Installation
If strace is installed setuid to root
then the invoking user will be able to
attach to and trace processes owned by
any user. In addition setuid and
setgid programs will be executed and
traced with the correct effective
privileges. Since only users trusted
with full root privileges should be
allowed to do these things, it only
makes sense to install strace as
setuid to root when the users who can
execute it are restricted to those
users who have this trust. For
example, it makes sense to install a
special version of strace with mode
'rwsr-xr--', user root and group
trace, where members of the trace
group are trusted users. If you do use
this feature, please remember to
install a non-setuid version of strace
for ordinary lusers to use.
A: You can install your APK file to Android emulator by following answer of this question. After installation,
Go to Eclipse, Open DDMS view from Windows => Open Perspective.
Go to File Browser tab, click on any folder, and Click on 'Push a file to a device'. You can upload any empty trace.txt file here. I would recommend you put your file in sdcard folder.
Go to CMD as given in the above link. Type adb shell. Go to sdcard by typing cd sdcard.
Now type ps on the command line and find out PID of your APK file via its package name.
Now type strace -p <pid> -o trace.txt. All system calls will be placed in this file. When you want to finish stracing your APK, you can download this trace.txt file to your computer and see the traces.
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{
"redpajama_set_name": "RedPajamaStackExchange"
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The house is on the same block as McKinley Elementary school. The oversized lot (including the attached outlot) is over 11,000 square feet and backs onto a small patch of woods.
The house is 1 mile from the East Falls Church metro and 1 mile from the Westover shops and library. Westover has the Lost Dog, Stray Cat, Lebanese Taverna and Thai Noy restaurants; a dry cleaner, hardware store and library. The Willston Shopping Center is also within a mile and includes Safeway, Target, CVS pharmacy and much more. And Home Depot is also within a mile. Route 66 is close by as is the bike path. And McKinley Elementary school is just a short walk.
Some photos are old, but provide a reasonable view of what the house. I will update the photos but for now they are from several years ago.
The house is located in Arlington County.
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"redpajama_set_name": "RedPajamaC4"
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Fora as músicas incluídas no disco A Complete Demonstration, há uma variedade de outras músicas do VAST que não estão (ou ainda não estão) incluídas em algum disco oficial.
Segue abaixo uma lista de músicas não oficiais, não incluídas em algum disco oficial:
Músicas demonstrativas de Jon Crosby em sua fase inicial
Neon Muzzle
Pretend
Sacred
Schizophrenia
The Queen Is My Slave
Tycoon
Ao Vivo
Everything Passing By (foi lançada no disco April (Retail Version), mas tocada anteriormente nas apresentações da turnê de 2007)
Punish Me (Versão anterior de "Sunday I'll Be Gone" do disco April (Versão Online) e April (Retail Version) - também lançada em uma nova versão no disco solo de Jon Crosby chamado Generica Vol. 1)
I Thought By Now (Prevista para aparecer no disco Closed Romantic Realism - apresentada ao vivo em uma estação de rádio Ianque)
Outside of My Dreams (Tocada ao vivo numa apresentação em Cleveland em 23/02/2008)
You´re the Same (Música do extinto Jon Crosby and The Resonator Band, antigo projeto folk de Jon Crosby. A música vem sendo apresentada nos shows do VAST)
I Can´t Buy You (Tocada ao vivo em Cleveland, porém não se sabe se o nome da música é este mesmo)
You Destroyed Me (Tocada ao vivo em Cleveland, porém o nome não é certo)
VAST Fan Club (2005 - até hoje)
Frog (Lançada em uma versão diferente no disco April (Retail Version))
My TV And You (Versão Acústica)
When We First Met (Lançada no projeto solo de Jon Crosby de nome Generica Vol. 2)
Like God (Demo) (Expectativa de aparecer no disco Closed Romantic Realism)
VAST
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Smart phones and other new gadgets may be functional and attractive, but can sometimes come with hidden costs. Ask lots of questions about plans and charges, what features are included in the monthly bills, what is not included and how much you have to pay for extras. Shop around, and make sure you're making an informed decision before buying a new phone, especially for your children.
Before travelling overseas, we encourage consumers to contact their telco and make sure they are fully informed about all potential costs and charges of international roaming and other international telephone services.
The TIO regularly receives complaints from consumers who return to Australia after travelling overseas to large bills caused by using mobile roaming.
The TIO has seen a sharp decline in new complaints, beginning in April 2012 and continuing into the new financial year. Between 1 July and 30 September 2012 we received 37,777 new complaints, 10.8 per cent fewer than in the previous quarter, when we received 42,353 new complaints. Complaints are down 26.1 per cent decrease from the same period last year.
Outside the TIO, the importance of the Telecommunications Consumer Protections (TCP) Code in our day-to-day work may not be fully understood. The TIO has the power to receive, investigate, and help resolve consumer complaints that involve code-related matters. At all stages of a complaint, the TIO uses codes to help understand and resolve complaints.
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Bachem ist ein südöstlicher Stadtteil von Frechen im Rhein-Erft-Kreis. Er ist nach der Innenstadt und Königsdorf drittgrößter Stadtteil Frechens mit rund 5400 Einwohnern.
Lage
Bachem liegt entlang der B 264. Die nächsten Autobahnanschlussstellen zur A1 sind "Frechen" an der B 264 und "Gleuel" unmittelbar an der Grenze zum gleichnamigen Hürther Stadtteil. Der alte Ort lag südlich der B 264 langgestreckt im Bachtal des Bachemer Baches, der mit zwei Quellen am Villehang beim Feldhof entsprang und im weiteren Lauf die Burgweiher von Burg Bachem, Haus Bitz und Burg Hemmerich und dazu noch zwei Mühlen speiste, bevor er sich mit dem Frechener Bach vereinigte.
Geschichte
Der Ortsname Bachem leitet sich von Bach-Heim ab. Im Prümer Urbar, dem Güterverzeichnis der Abtei Prüm von 893, ist der Ort zum ersten Mal erwähnt. Im Mittelalter gehörte Oberbachem mit der Burg Bachem und Unterbachem mit Hemmerich zu verschiedenen Territorien, konkret Herzogtum Jülich und Kurköln. Bachem hatte deshalb auch zwei Kirchen, die jeweils in der Nähe der Burgen standen. Im Bereich, wo einst die zu Hemmerich gehörende Marienkirche stand, wurde 1960 die Heilig-Geist-Kirche errichtet. 1794 wurde der Ort von französischen Revolutionstruppen besetzt. 1815, nach den Koalitionskriegen, fiel Bachem an das Königreich Preußen. Bachem bildete im Kreis Köln, Regierungsbezirk Köln, eine eigene Gemeinde, die zur Bürgermeisterei Frechen gehörte. Am 1. Dezember 1910 hatte Bachem 1702 Einwohner.
Am 2. März 1927 schloss sich Bachem mit den Gemeinden Frechen und Buschbell zusammen.
Bachem wurde im 20. Jahrhundert lange durch die Industrie des Rheinischen Braunkohlereviers geprägt. Die Grube und Brikettfabrik Schallmauer, benannt nach der Wasserburg Schallmauer an der Grenze zu den Hürther Ortsteilen Gleuel und Berrenrath, bestand von 1901 bis 1963. Heute erinnert nur ein Magazingebäude in der Schallmauerstraße, in dem eine Schule untergebracht ist, an diese Vergangenheit.
Sehenswürdigkeiten
Sehenswert sind das Haus Bitz, die Burg Bachem, die Burgruine Hemmerich (nicht frei zugänglich; nur noch Grundmauern vorhanden) im privaten Garten des Gutes Neuhemmerich und die St. Mauritiuskirche.
Das alte Pfarrhaus neben der neuen ab 1960 errichteten und 1962 geweihten Heilig-Geist-Kirche datiert aus dem Jahre 1738. Der Steinzeugboden im Flur des Hauses besteht aus senkrecht gestellten Frechener Tonplättchen, die während des Brandes von Gefäßen zum Stapeln des Brenngutes im Steinzeugofen verwendet worden sind; daher sind sie hart versintert. Ein gleicher Fußboden befindet sich in der St. Ägidiuskapelle in Frechen-Hücheln.
Die Kirche St. Mauritius ist die alte Bachemer Pfarrkirche und wurde als Burgkapelle der Freiherrn von Geldern 1721 errichtet. Ihr Turm stammt aus dem Jahre 1935.
In der Hubert Prott-Straße steht das Kriegerdenkmal für die im Ersten Weltkrieg gefallenen Sohne der Gemeinde. Das Denkmal ist insofern sehenswert, als hier Christus auf dem Sockel sitzt. Das Kriegerdenkmal wurde nach einem Entwurf des Architekten Julius Gatzen gemeinsam mit Franz Albermann (1877–1959) (Sitzbild Christi) geschaffen.
Bildung
In Bachem sind zwei Kindergärten und eine Grundschule zu finden sowie die Albert-Einstein-Schule als Förderschule des Rhein-Erft-Kreises im Sekundarbereich. Der Heiliggeist-Kindergarten gehört zu der Heiliggeistgemeinde und ist wie die Mauritiusschule katholisch orientiert. Die beiden weiteren Kindergärten sind der AWO-Kindergarten Villa Kunterbunt und die Kindertagesstätte (Hort) Knöpfchenhaus.
In Bachem ist auch ein Berufsbildungswerk des CJD beheimatet.
Verkehr
Die VRS-Buslinien 145 der KVB sowie 910 und 960 der REVG verbinden den Ort mit Frechen Mitte, Bocklemünd, Gleuel, Hürth und Bergheim.
Sport
Der Fußballverein VfR Bachem 1932 e.V. stieg 2008 in die Landesliga auf und spielt seit 2010 wieder in der Bezirksliga. Diesem Verein angegliedert sind die RFC Royal Flush Cheerleader.
Literatur
Helmut Wirges: Bachem – Der Geschichte eines Dorfes auf der Spur. Rheinland-Verlag GmbH, Köln 1991, ISBN 3-7927-1210-5.
Helmut Wirges: Bachem – Der Spurensuche zweiter Teil. 2021
Weblinks
Drei Kirchen und drei Kneipen
Einzelnachweise
Ortsteil von Frechen
Ehemalige Gemeinde (Rhein-Erft-Kreis)
Ersterwähnung 893
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{"url":"http:\/\/odin-osiris.usask.ca\/ReST\/SasktranIF\/hr_details.html","text":"# Limb Geometry Overview and Setup\u00b6\n\nA typical limb-viewing geometry is shown in the figure below where a satellite is orbiting the Earth at locations P1, P2 and P3. At each location an instrument looks towards the atmospheric limb in directions L1, L2, L3 observing tangent points T1, T2 and T3.\n\nThe HR engine is used to calculate the vector or scalar radiance observed by the instrument at each location. There is a huge computational saving if we assume all measurements are sufficiently close together that they essentially occur at the same instant with the same solar illumination conditions. If we can make this assumption then we only have to solve the diffuse radiative field once and can quickly integrate through the field for each line of sight. If the assumption is not valid then the calculation must be broken down into smaller subsets and the HR engine re-configured for each separate illumination condition.\n\nWe generally use 2 minutes as a reasonable period to consider measurements instantaneous. This period is the time taken for the Sun to move one solar disk, or 0.5 degrees, across the sky (assuming 15 degrees\/hour). This seems a reasonable limit given that the 0.5 degree, finite size of the Sun is not generally accounted for in the HR engine.\n\n## Lines of Sight\u00b6\n\nAll sasktranif engines assume you can define each line of sight with:\n\n1. the position (P1, P2, P3) of the observer as (X,Y,Z )vectors from the center of the oblate Earth.\n2. the look direction (L1, L2, L3) away from the observer as unit vectors.\n3. the time of the measurements as a modified julian date. This condition is relaxed if you know the direction of the Sun and all climatologies are time independent.\n\nX is a horizontal unit vector in the equatorial plane of the oblate Earth from the centre of the Earth towards 0 degrees longitude, Z is a unit vector parallel to the Earth\u2019s spin axis and Y forms the third vector of a right-handed system. This system works well for many observational data sets but if you have a different way of specifying the same information then class ISKGeodetic has been written to help you convert to the required system. Let us know if ISKGeodetic is lacking the functionality you need.\n\n## Sun Direction\u00b6\n\nYou will normally not specify the unit vector of the Sun as the HR engine by default will calculate the Sun\u2019s position using the average of the modified julian dates (mjd) of the lines of sight. You must override the default setting if you are creating test lines of sight and\/or want to explicitly set the position of the sun. The sun position is specified as a unit vector from the centre of the Earth toward the Sun using the same X,Y,Z corodinate system as the observer coordinates.\n\n## Reference Point and Osculating Sphere\u00b6\n\nThe HR model approximates the shape of the Earth to be a sphere. The radius of the sphere is chosen so it matches the radius of curvature of the Earth at a Reference Point. This sphere is called the osculating sphere as it kisses the true oblate Earth at the reference point, see figure below. The Reference Point is normally chosen from the average of the tangent point or ground intersection of each line of sight.\n\nThe centre of the osculating sphere is offset from the centre of the true oblate Earth and the HR engine creates an internal coordinate transformation that maps positions from one system to the other and vice-versa. The transformation from true oblate Earth to osculating sphere coordinates generates a small height registration difference for locations not directly above the reference point. This is important as lines of sight not tangent at the reference point will have a tangent height slightly higher in the osculating system than in the oblate system. By default the HR engine adjusts the look vector of lines of sight which are tangential within the true oblate atmosphere so they look at the same tangent height in the osculating system.\n\n## Surface Height and TOA\u00b6\n\nBy default the HR engine will assume the surface of the Earth is at 0 meters above mean sea-level and the top of the atmosphere is at 100 kilometers (100,000 m). Both the lower and upper altitudes can be set to other values using properties TOAHeight and SurfaceHeight. Lowering the upper altitude can have a significant positive impact upon performance as it reduces the size of the diffuse field.\n\n## Polarization\u00b6\n\nThe HR engine supports calculation of either scalar radiances or full stokes vectors, see property SetPolarizationMode(). The polarization calculation is done through an approximation\u2013for low orders of scatter the full calculation is performed, while higher orders of scatter are assumed to have a specific polarization state. The level of approximation used is controlled through calling SetPolarizationMode().\n\nThere are two functions that perform the radiative transfer calculation, CalculateRadiance() and CalculateStokesVector(). These do not determine whether or not a polarized calculation is performed, they only determine the final return type: CalculateRadiance() returns a two dimensional array of scalar values representing $$I$$ while CalculateStokesVector() returns a two dimensional array of ISKStokesVector objects. Whether or not a polarization calculation is done or not is determined entirely by SetPolarizationMode().\n\n## Number of Scatter Orders\u00b6\n\nThe successive orders algorithm successively calculates higher orders of scatter using the previous order of scatter. The maximum number of scatter orders calculated is set through:\n\nok = engine.SetProperty('numordersofscatter', n)\n\n\nwhere n is the number of scatter orders. The default value is n=50.\n\nCaching is done in the HR engine which makes the second order of scatter take the longest time to run. All orders after the second are relatively quick. Furthermore, convergence checking is done internally in the engine which stops the calculation when rays stop contributing to the final answer. For these reasons it is recommended to use either n=1 (for a quick not-accurate calculation) or n=50 (for the full calculation).\n\n## Diffuse Profiles\u00b6\n\nThe primary option that controls the accuracy of the HR radiative transfer calculation is the number of diffuse profiles. Each diffuse profile represents a single solar zenith angle where the multiple scattering source function is calculated. By default, only one diffuse profile is used in the calculation and is placed at the solar zenith angle of the average tangent point. Whether or not this is appropriate is heavily dependant on a variety of factors\n\n\u2022 Wavelength \u2013 Wavelengths where the atmosphere is thick and there is little absorption (e.g. around 350 nm) have a large amount of multiple scatter and are more sensitive to the number of diffuse profiles.\n\u2022 Solar zenith angle span \u2013 Since the multiple scattering source function varies with solar zenith angle, lines of sight which have a large solar zenith angle span will be sensitive to the number of diffuse profiles. Solar scattering angle is a rough proxy of this effect. Forward and backward scatter geometries will be most sensitive to the number of diffuse profiles, while scattering angles near $$90^\\circ$$ will see no effect.\n\u2022 Solar zenith angle \u2013 The absolute solar zenith angle also effects the optimal number of diffuse profiles. The multiple scattering source function is slowly varying for low solar zenith angles (sun high in the sky) and varies rapidly at high solar zenith angles near the solar terminator. Generally, more diffuse profiles are required the closer you get to the terminator.\n\u2022 What the radiance is being used for \u2013 Often times the absolute radiance is not the quantity of interest. For example, in many retrievals radiances are normalized by high altitude measurements. This normalization process removes a lot of the dependence on the number of diffuse profiles.\n\nThe number of diffuse profiles can be changed in the HR option through:\n\nok = engine.SetProperty('numdiffuseprofilesinplane', n)\n\n\nwhere n is the number of diffuse profiles. Calculation time scales roughly linearly with the number of diffuse profiles; a calculation with 5 diffuse profiles will take approximately 5 times as long as a calculation with 1 diffuse profile. RAM usage scales the same way.\n\nFiguring out the optimal number of diffuse profiles for a specific situation is an important and difficult problem. As a first step, we recommend switching between n=1 and n=5 and seeing how the results change. As an additional tool, radiances can be benchmarked against the MC engine to verify that the number of diffuse profiles used is sufficient.\n\n## Weighting Functions\u00b6\n\nThe HR engine supports analytical computation of weighting functions for absorbing and scattering species. The theoretical basis for the method is described in [Zawada2015]. Prior to calculating weighting functions it is necessary to both tell the engine to calculate weighting functions, and specify the species we are calculating weighting functions for.\n\nExample:\n\nimport sasktranif.sasktranif as skif\nengine = skif.ISKEngine('HR')\n\n# Set up lines of sight\/wavelengths\/atmosphere\n\noptprop_o3 = skif.ISKOpticalProperty('O3_DBM')\nok = engine.SetProperty('WFSpecies', optprop_o3)\nok = engine.SetProperty('calcwf', 2)\nok, wf = engine.GetWeightingFunctions()\n\n\nThe wf object that is returned is a three dimensional numpy.ndarray with dimensions corresponding to [wavelength, line of sight, volume],\n\n$\\texttt{wf[i, j, k]} = \\frac{\\partial I(\\lambda_i, \\text{LOS}_j)}{\\partial x_k},$\n\nand has units of [radiance\/cm^{-3}]. In our example, the quantity $$x_k$$ is the ozone number density over a finite volume. Since we set calcwf = 2 the finite volumes are uniform spherical shells spaced 1 km apart evenly from 0.5 km to 99.5 km. Therefore, the quantity wf[i, j, 10] can be thought of as the derivative of the radiance (for wavelength i and line of sight j) with respect to changing ozone number density in the 10.5 km shell.\n\nNotes\n\nCurrently the model is limited to calculating weighting functions for one species at a time. This is normally not an issue, since typically wavelengths used for retrievals are sensitive to only one species.\n\nThe weighting function species does not have to be a species that was added to the model.\n\nWeighting functions are not calculated by assuming a constant perturbation within the specified finite volume. Rather, the returned quantity should be thought of as the derivative of radiance with respect to how HR responds to changes in that volume. For example, when the finite volume is spherical shells, linear interpolation is done between the shell directly above and below the specified shell to better represent how the radiative transfer calculation is done.\n\n### Standard Mode\u00b6\n\nThis is the recommended mode for weighting function calculations using a one dimensional (spherically symmetric) atmosphere, and is set through:\n\nok = engine.SetProperty('calcwf', 2)\n\n\nWeighting functions are calculated on a grid of spherical shells. By default this is an evenly spaced grid from 0.5 km to 99.5 km in steps of 1 km; i.e. the returned wf object will have dimensions (len(wavel), len(lineofsight), 100), and the quantity wf[i, j, 10] can be thought of as the derivative of the radiance (for wavelength i and line of sight j) with respect to changing number density in the 10.5 km shell.\n\nThe shell locations and widths can be changed through the properties wfheights and wfwidths:\n\nwfwidths = [2000, 2000, 2000, 2000]\nwfheights = [10000, 20000, 30000, 40000]\n\nengine.SetProperty('wfwidths', wfwidths)\nengine.SetProperty('wfheights', wfheights)\n\n\nWeighting functions will now be calculated for spherical shells centred at [10 km, 20 km, 30 km, 40 km] with widths of 2 km.\n\n### Line of Sight Mode\u00b6\n\nThe line of sight weighting function mode is a special mode where weighting functions are calculated on the tangent altitude grid of the lines of sight. This can be useful for some retrieval algorithms and is set through:\n\nok = engine.SetProperty('calcwf', 1)\n\n\nThe weighting functions returned has dimensions (len(wavel), len(linesofsight), len(linesofsight)) and are centred on the line of sight tangent altitudes. The widths are chosen such that the spherical shell extends between the line of sight directly below and the line of sight directly above.\n\nNot yet written.\n\n### Other Options\u00b6\n\nWhile the computation time of weighting functions for a standard multiple scatter calculation is negligible compared to the actual calculation, for some situations (e.g. single scatter in the IR region) the majority of time could be spent calculating weighting functions. For this reason we have added the option wfprecision.\n\nBy default, weighting functions are calculated considering three effects:\n\n\u2022 Line of sight\n\u2022 Solar\n\u2022 Higher Order\n\nBy setting:\n\nok = engine.SetProperty('wfprecision', 1)\n\n\nThe solar and higher order effects are turned off, thus only line of sight contributions are accounted for. This degrades the quality of the weighting functions, but often they are still usable for retrieval purposes.\n\n## Three Dimensional Atmospheres\u00b6\n\nThe HR model supports calculations where atmospheric constituents are allowed to vary three dimensionally.","date":"2017-09-26 16:30:37","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7152999639511108, \"perplexity\": 938.2429315659181}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-39\/segments\/1505818696653.69\/warc\/CC-MAIN-20170926160416-20170926180416-00381.warc.gz\"}"}
| null | null |
Q: Dynamic casting to a generic type I'm trying to get to the Cases.Values property of System.Activities.Statements.Switch object. The problem is that I cannot cast Activity object at runtime to Switch<> type (which derives from it).
var switchType = activity.GetType();
bool isSwitch = (switchType.IsGenericType && switchType.GetGenericTypeDefinition() == typeof(Switch<>));
if (isSwitch)
{
Type[] genericArgumentTypes = switchType.GetGenericArguments();
if (genericArgumentTypes.Length > 0)
{
var switchStatement = (Switch<genericArgumentTypes[0]>) activity; //that's incorrect
foreach (var aCase in switchStatement.Cases.Values)
{
ProcessActivity(aCase, dataSets, context);
}
}
}
Also,
dynamic switchStatement = activity;
foreach (var aCase in switchStatement.Cases.Values)
{
ProcessActivity(aCase, dataSets, context);
}
throws an error, that the property is not there, while debugger is showing it's not true. The T is irrelevant for me - I need only the Cases collection.
EDIT
Actually, I've found even cleaner solution, than the one I set as an answer.
dynamic switchStatement = activity;
var cases = switchStatement.Cases as IEnumerable;
if (cases != null)
{
foreach (dynamic aCase in cases)
{
ProcessActivity(aCase.Value);
}
}
A: You can't.
But instead of your loop, put:
var process = typeof(CurrentHelperClass).GetMethod("ProcessSwitch`1").MakeGenericMethod(typeof(genericArgumentTypes[0]));
process.Invoke(null,new object[]{activity});
and define a new method in the same class:
static void ProcessSwitch<T>(Switch<T> switchStatement)
{
foreach (var aCase in switchStatement.Cases.Values)
{
ProcessActivity(aCase, dataSets, context);
}
}
A: var switchType = activity.GetType();
var prop = switchType.GetProperty("Cases", System.Reflection.BindingFlags.Public
| System.Reflection.BindingFlags.Instance); //move it outside of the method and use the same property for every time you call the method since it's performance expansive.
bool isSwitch = (switchType.IsGenericType && switchType.GetGenericTypeDefinition() == typeof(Switch<>));
if (isSwitch)
{
IEnumerable dictionary = prop.GetValue(activity,null) as IDictionary;
foreach (var item in dictionary.Values)
{
ProcessActivity(item, dataSets, context);
}
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,030
|
L'Œil et l'Esprit est le dernier ouvrage du philosophe Maurice Merleau-Ponty, entièrement achevé de son vivant et publié en 1960. Dans cet essai, le philosophe français s'intéresse à l'expérience de la vision à travers la peinture de Cézanne.
Analyse
Dans cet écrit, Merleau-Ponty commence par distinguer l'art de la science. L'art porte sur le monde vécu et vivant, le corps et l'existence, alors que la science prend le monde comme objet de connaissance "dissocié" du sujet existant, pour dégager des lois par-delà les phénomènes. La science manque le primat de la perception, le fait que nous sommes d'abord au monde et dans le monde, avec un corps et que l'expérience perceptive constitue la connaissance première.
La vision est une certaine expérience du monde et de l'espace de ce monde, dont le peintre, en particulier Cézanne, tente de rendre compte. Ce rapport au monde et à l'espace est celui de la distance, que le peintre tente de ramener dans sa toile, non par une technique, mais par un sentir particulier, dont celui des couleurs, qui ne sont pas un ornement contingent, mais participent du monde visible.
Notes et références
Essai français des années 1960
Essai paru en 1960
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,640
|
{"url":"https:\/\/cs.stackexchange.com\/questions\/55611\/ideal-value-of-d-in-a-d-ary-heap-for-dijkstras-algorithm\/55612?r=SearchResults","text":"# Ideal value of d in a d-ary heap for Dijkstra's algorithm\n\nI stumbled upon the following statement:\n\nBy using a $d$-ary heap with $d = m\/n$, the total times for these two types of operations may be balanced against each other, leading to a total time of $O(m \\log_{m\/n} n)$ for the algorithm, an improvement over the $O(m \\log n)$ running time of binary heap versions of these algorithms whenever the number of edges is significantly larger than the number of vertices\n\nApplications of d-ary heaps\n\nI don't understand why we choose to have a heap where nodes have exactly $m\/n$ children to speed up Dijkstra's algorithm. Remove min takes overall $O(n \\log_d n)$ time and decrease takes $O(m \\log_d n)$, so total runtime is $O(m \\log_d n)$.\n\nWhat I don't understand is say we have $m=3, n=1$, $m\/n$ gives 3, but $O(m \\log_3 n)$ is slower than $O(m \\log_4 n)$, so why not choose 4 as value of $d$ instead? What motivates taking $m\/n$?\n\nThanks!\n\n\u2022 Dijkstra is not a greedy algorithm! [tag removed] \u2013\u00a0Raphael Apr 7 '16 at 8:08\n\nwhenever the number of edges is significantly larger than the number of vertices\n\nWhat they probably mean is that $m \\in \\omega(n)$. Then, $m\/n \\in \\omega(1)$ -- the base of the logarithm grows with $n$, that is the resulting sequence of values grows more slowly than with a fixed base.\n\nFor dense graphs in the sense that $m \\in \\Theta(n^2)$, the effect is most pronounced. For the sake of simplicity, say $m = cn^2$ for some $c \\in (0,1)$. Then, $m\/n = cn$ and therewith $\\log_{m\/n} n = \\log_{cn} n \\in O(1)$ -- the logarithmic factor has vanished, the algorithm has running-time in $O(m)$.\n\nNote that all of this is discussing only the upper Landau-bound. In which way the true running-time cost is affected by this change is not per se clear, and the Wikipedia article is bold in claiming that it's an \"improvement\".\n\n\u2022 The vanishing log if $m=n^{1+\\epsilon}$, with $0<\\epsilon<1$ is what I was looking for, thanks \u2013\u00a0user1354784 Apr 8 '16 at 22:53\n\nPlease note that $m \\leq n(n-1)\/2$, thus $m\/n < n\/2$. So the example $m=3, n=1$ does not happen.\n\nFirst, the article clearly claims that this improvement is achieved whenever $m$ is larger than $n$,\n\nBy using a $d$-ary heap with $d = m\/n$, ..., an improvement over the O(m log n) running time of binary heap versions of these algorithms whenever the number of edges is significantly larger than the number of vertices.\n\nSecondly, the article claims that $d$ must create a balance between $m$ and $n$ in order to improve the running time,\n\nBy using a $d$-ary heap with $d = m\/n$, the total times for these two types of operations may be balanced against each other...\n\nGenerally, we need $n$ min-extract and $m$ decrease-key operations. In the case of $d=2$, each of these operations takes $O(\\log n)$ time. However, when you increase $d$, the decrease-key operation becomes faster, but min-extract can become slower. Because, within each min-extract, we also need to run min-heapify that depends on the number of children of the parent in each level.\n\nIn the extreme case, if you consider $d=n$ then $m \\log_n n < m \\log_{m\/n} n$, but the running time won't be better in practice. Because you need at most $n$ min-extract operations, each includes a min-heapify. Therefore, you need $n$ min-heapify operations, that will cost $O(n)$ in this case (because $d=n$ and there are $n$ numbers in the next level of your tree). Thus the running time of your algorithms becomes $O(n^2)$ ($n$ times running min-heapify).\n\nRuntime of Dijkstra with $n$ nodes and $m$ edges and using a $d$-ary heap for the priority queue will be $O((nd+m)\\log_d n)$:\n\n\u2022 $O(nd\\log_d n)$ from ExtractMin operations and\n\u2022 $O(m\\log_d n)$ from ChangePriority (strictly DecreaseKey) operations.\n\nWe want to balance $nd$ against $m$, given $m\\gg n$. I.e. $nd=m$, or, $d=m\/n$. If we have a constant $d$, asymptotic time will lean towards $O(m)$ [$m\\gg n$].\n\nIn practice you'll need to play around a little to know what value of $d$ works for you.","date":"2019-10-17 13:20:34","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7998101115226746, \"perplexity\": 382.18017140527246}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570986675316.51\/warc\/CC-MAIN-20191017122657-20191017150157-00151.warc.gz\"}"}
| null | null |
\section{Background and motivation}
\label{sec:intro}
Glasses feature distinct properties compared to their crystalline counterparts, where the differences are commonly attributed to the lack of long-range order in their characteristic disordered structures. Such differences manifest themselves in a broad range of physical properties, including those related to energy/heat transport. A crucial step in developing predictive theories of such transport properties is understanding how long-wavelength phonons (waves) --- the energy carriers --- are attenuated in glasses, and in particular elucidating the dependence of the phononic attenuation rate $\Gamma(k)$ on the wavenumber $k$ in the long-wavelength limit. In perfect crystals, phonons are expected to propagate indefinitely without attenuation. In this case, finite energy transport emerges due to anharmonicity, e.g.~phonon-phonon interactions~\cite{ashcroft_mermin}. In glasses, structural disorder can lead to the attenuation (scattering) of phonons even in the purely harmonic regime, which is expected to be realistic for low temperatures.
Consequently, the attenuation of phonons in glasses is intimately linked to the nature of glassy disorder. For a long time, it has been assumed --- with some inconclusive experimental support~\cite{experiments_300K_vSiO2,experiments_1620K_vSiO2,experiments_570K_dSiO2} --- that the phononic attenuation rate in glasses follows Rayleigh scattering scaling~\cite{Rayleigh}, $\Gamma(k)\!\sim\!k^{{\,\mathchar'26\mkern-12mu d}+1}$, in the long wavelength limit $k\!\to\!0$ (${\,\mathchar'26\mkern-12mu d}$ is the spatial dimension). In this physical picture, glassy disorder plays a similar role to that of small, randomly distributed impurities that are responsible for wave scattering as originally described by Rayleigh~\cite{Rayleigh}. That is, at least as far as the attenuation of long-wavelength phonons is concerned, glassy disorder in this picture is basically equivalent to generic randomness without distinct features~\cite{ganter2010rayleigh,schirmacher2011comments,Maurer2004,mw_EM_epl,eric_boson_peak_emt}. On the other hand, recent developments in studying computer glass-formers have revealed distinct properties of glassy disorder and their manifestations on the physics of glasses.
An attenuation rate that follows a generalized-Rayleigh scaling of the form $\Gamma(k)\!\sim\!k^{{\,\mathchar'26\mkern-12mu d}+1}\log\!{(k_0/k)}$ ($k_0$ is some characteristic wavenumber) has been recently observed in computer glass simulations~\cite{lemaitre_tanaka_2016}. The logarithmic enhancement of wave attenuation, relative to Rayleigh scaling, has been therein attributed to long-range spatial correlations in the local elastic constants of glasses~\cite{lemaitre_tanaka_2016}. Subsequent work on jammed harmonic sphere packings reproduced the logarithmic enhancement, but also demonstrated that the removal of frustration-induced internal-stresses leads to its disappearance (i.e.~Rayleigh scaling is recovered)~\cite{Ikeda_2018}. Roughly at the same time, it was shown that frustration-induced internal-stresses, which are a generic property of glasses~\cite{shlomo}, are responsible for the existence of low-frequency quasilocalized nonphononic excitations in glasses~\cite{inst_note, ikeda_pnas}.
These nonphononic excitations have been hypothesized to exist for a long time~\cite{soft_potential_model_1991,Gurevich2003}, also in relation to the low-temperature anomalies mentioned above, but only recently their statistical and spatial properties have been directly revealed using computer glass simulations. It has been shown that the nonphononic excitations follow a universal density of states (DOS) $D_{\rm G}(\omega)\!\sim\!\omega^4$ ($\omega$ is the vibrational frequency), which is different from Debye's phononic DOS $D_{\rm D}(\omega)\!\sim\!\omega^{{\,\mathchar'26\mkern-12mu d}-1}$, and that unlike low-frequency phonons (which are spatially extended) they are quasilocalized in space (featuring a disordered core decorated with a power-law decay)~\cite{modes_prl_2016,modes_prl_2018}. Finally, recent computational advances~\cite{itamar_swap,berthier_prx} allow to obtain glassy states that are comparable to or even more stable than laboratory ones (while conventional computer glasses are generated using cooling rates that are orders of magnitude larger than laboratory ones). It has been shown that the universal DOS $D_{\rm G}(\omega)\!\sim\!\omega^4$ of nonphononic excitations remains valid for such deeply supercooled, experimentally-relevant glasses, but that their number (quantified by the prefactor in the DOS) is severely reduced~\cite{LB_modes_2019}.
Despite this progress, the possible implications of the existence and number of nonphononic excitations on glassy transport, and on wave attenuation in particular, have not be elucidated. Moreover, it remains unclear whether macroscopic glasses, generated by realistic laboratory protocols, feature Rayleigh or generalized-Rayleigh scaling in the long-wavelength limit. In this paper we use a combination of theory and extensive computer simulations to address the following fundamental, intrinsically-related questions: (i) What is the scaling of $\Gamma(k)$ in macroscopic glasses generated through conventional laboratory protocols? Addressing this question using computer glasses requires understanding finite size effects that should be systematically eliminated to expose the macroscopic limit. (ii) What is the physical origin of the generalized-Rayleigh scaling, when it is observed?
\section{Theoretical considerations: from finite size effects to Rayleigh and generalized-Rayleigh scaling}
\label{sec:theory}
To set the stage for addressing these questions, we discuss in this section the mathematical formulation that allows calculating the attenuation rate $\Gamma(k)$ and several theoretical predictions regarding $\Gamma(k)$.
{\em Mathematical formulation ---} We are interested in the attenuation rate of long-wavelength phononic excitations (plane-waves) that are introduced at time $t\=0$ as perturbations to metastable equilibrium glass states (inherent structures) characterized by particle positions ${\bm r}_i$, and take the form
\begin{equation}
\dot{\bm u}_i(t\=0) = {\bm a}_p\,\sin({\bm k}\!\cdot\!{\bm r}_i) \ ,
\label{eq:phonon}
\end{equation}
where $\dot{\bm u}_i$ are the particle velocities (hence ${\bm u}_i$ are the displacements), expressed in terms of the relevant natural units, as explained below. Here ${\bm a}_p$ is a unit polarization vector ($|{\bm a}_p|\=1$), $p\={\rm T}$ corresponds to transverse phonons (shear plane-waves) ${\bm a}_{\rm T}\!\cdot\!{\bm k}\!=\!0$, $p\={\rm L}$ corresponds to longitudinal phonons (dilatational plane-waves/sound) ${\bm a}_{\rm L}\!\cdot\!{\bm k}\=k$, and $k\=|{\bm k}|$ is the wavenumber corresponding to the wavevector ${\bm k}$. The wavevector is given by ${\bm k}\=(2\pi/L)\sum_{j=1}^{\,\mathchar'26\mkern-12mu d} n_j {\bm e}_j$, where $L$ is the linear size of the glass (hence its volume is $L^{\,\mathchar'26\mkern-12mu d}$), $j$ represents a Cartesian direction, ${{\bm e}_j}$ are orthonormal Cartesian vectors and ${n_j}$ are integers that are solutions to the sum of squares problem in ${\,\mathchar'26\mkern-12mu d}$-dimensions $\sum_{j=1}^{\,\mathchar'26\mkern-12mu d} n_j^2\=q$, where $q$ is an integer. The number of solutions to this problem for a given $q$ will be denoted hereafter by $n_q$, where we are mainly interested in relatively small $q$ values that correspond to long-wavelength excitations, $k\!\to\!0$. The glass-forming models and preparation protocols used in this work are presented in Appendix~\ref{sec:numerics_appendix}.
The only information about the glass that is incorporated into Eq.~\eqref{eq:phonon}, except for the particle positions ${\bm r}_i$, is its linear size $L$. As such, Eq.~\eqref{eq:phonon} represents long-wavelength phononic excitations that correspond to a perfectly ordered system, which are not eigenmodes of the glass Hessian ${\calBold M}$; the eigenmodes are determined by ${\calBold M}\cdot{{\bm \psi}_m}\=\omega^2_m {\bm \psi}_m$, where ${\bm \psi}_m$ is an eigenmode with an eigenvalue $\omega^2_m$, and $\omega_m$ is a vibrational frequency (assuming unity masses). The difference between the pure phononic excitation of Eq.~\eqref{eq:phonon} and the set of the glass eigenmodes is at the heart of wave scattering in glasses.
To see this, we follow \cite{lemaitre_tanaka_2016} and define the ${\,\mathchar'26\mkern-12mu d} N$-dimensional velocity vector $\dot{\bm u}(t)$, composed of the velocity vectors $\dot{\bm u}_i(t)$ of all of the
particles that satisfy for $t\!>\!0$ the harmonic dynamics $\ddot{\bm u}_i(t)\=-\calBold{M}_{ij}\!\cdot\!{\bm u}_j(t)+\dot{\bm u}_i(t\=0)\,\delta(t)$, where here and in what follows repeated indices are summed over. The attenuation rate is probed through the time-evolution of the velocity autocorrelation function \cite{lemaitre_tanaka_2016}
\begin{equation}
\label{eq:corr}
C(t) = \frac{\dot{\bm u}_i(t)\!\cdot\!\dot{\bm u}_i(t\=0)}{\dot{\bm u}_i(t\=0)\!\cdot\!\dot{\bm u}_i(t\=0)}=\sum_m \xi_m^2 \cos(\omega_m t) \ ,
\end{equation}
with
\begin{equation}
\label{eq:projections}
\xi_m = \frac{{\bm \psi}_{m,i}\!\cdot\!\dot{\bm u}_i(t\=0)}{\sqrt{\dot{\bm u}_i(t\=0)\!\cdot\!\dot{\bm u}_i(t\=0)}} \ ,
\end{equation}
where ${\bm \psi}_{m,i}$ is the ${\,\mathchar'26\mkern-12mu d}$-dimensional (Cartesian) vector component of ${\bm \psi}_{m}$ at the $i$th particle.
Equations \eqref{eq:corr}-\eqref{eq:projections} clearly show that wave scattering in glasses is of structural, not dynamical, origin; and that it is fully encapsulated in the normalized projections $\xi_m$ of the initial phononic excitation $\dot{\bm u}_i(t\=0)$ on the eigenmodes ${\bm \psi}_m$ of the glass Hessian. As $\Gamma(k)$ quantifies the characteristic decay rate of $C(t)$ in Eq.~\eqref{eq:corr}, it is intrinsically related to the spectral width over which the projections $\xi_m$ are sizable for a given $k$, as will be clarified below. Finally, $C(t)$ --- measured in simulations by averaging over independent glassy samples --- is commonly described by the damped oscillator model of the form
\begin{equation}
\label{eq:damped_oscillator}
C(t)\simeq\exp\left(-\tfrac{1}{2}\Gamma_{\rm T,L}(k)t\right)\cos\left(\Omega_{\rm T,L}(k)t \right) \ ,
\end{equation}
for both transverse and longitudinal phononic excitations. Equation~\eqref{eq:damped_oscillator} allows (if the damped oscillator model is valid, see discussion below) to extract the attenuation rate $\Gamma_{\rm T,L}(k)$ and wave speed $c_{\rm T,L}(k)\!\equiv\!\Omega_{\rm T,L}(k)/k$. In the remainder of the paper we will focus on the attenuation rate $\Gamma(k)$ \cite{footnote2}.
{\em Finite-size scaling theory of $\Gamma(k)$ ---} Our major goal, as stated above, is to understand the scaling properties of $\Gamma(k)$ in macroscopic glasses in the small $k$ limit. Experiments, using measurement techniques such as high-resolution inelastic x-ray scattering (IXS)~\cite{Ruocco1996,ixs_review_arXiv}, are insightful, but also suffer from some limitations (see Sect.~\ref{sec:exp}). Moreover, controlling and manipulating glassy disorder is highly challenging from an experimental perspective. Here we address the posed questions using computer glass simulations, which offer powerful opportunities to explore the physics of glasses in general, and in particular in the context of glassy disorder and structures.
One obvious limitation of present-day computer glass simulations is that computationally feasible system sizes are significantly smaller than the sizes of laboratory glasses. In the present context, this limitation may make it difficult to probe the macroscopic low-frequency/wavenumber regime using computer glasses. More specifically, it is clear that the lowest frequency/wavenumber response of computer glasses is dominated by finite-size effects; consequently, we need to understand these finite-size effects in order to systematically eliminate them, allowing to cleanly probe the macroscopic response. As will be shown below, understanding finite-size effects also provides basic insight into the general scattering problem.
The finite-size scaling theory of $\Gamma(k)$ has been developed very recently in~\cite{phonon_widths}. It starts with the observation that phonons of low-frequency $\omega$ in finite-size systems are quantized into discrete bands with a degeneracy $n_q(\omega)$, defined above through solutions of the integer sum of squares problem. The basic idea is that the presence of disorder lifts the degeneracy of the bands such that they are composed of disordered phonons of close, but not identical, frequencies. Consequently, the lowest frequency phononic bands feature finite spectral widths $\Delta\omega(\omega)$ that are smaller than the gaps between adjacent bands.
It is therefore clear that a pure phononic excitation that belongs to one of these finite width bands features sizable projections $\xi_m$ on disordered phonons within the band, but negligible projections on disordered phonons outside of the band. Hence, we expect $\Gamma(\omega)\!\sim\!\Delta\omega(\omega)$. Very recently, the spectral widths $\Delta\omega(\omega)$ have been theoretically derived using degenerate perturbation theory, random matrix theory and simple statistical considerations~\cite{phonon_widths}. The theory, which is described in great detail in~\cite{phonon_widths}, yields
\begin{equation}
\label{eq:finite-size}
\Gamma(\omega) \sim \Delta\omega(\omega) \sim \frac{\omega\sqrt{n_q(\omega)}}{\sqrt{N}} \ ,
\end{equation}
where $N\!\sim\!L^{\,\mathchar'26\mkern-12mu d}$ is the number of particles in the system. The explicit appearance of $N$, and of the degeneracy level $n_q(\omega)$ of discrete phononic bands of index $q$, make the finite-size nature of this result evident. The theoretical prediction in Eq.~\eqref{eq:finite-size} has been verified for disordered lattices, but not yet for glasses. We extensively test this prediction for glasses below and use it to eliminate finite-size effects, thus exposing the macroscopic limit. Finally, note that as the disordered phononic modes in the finite-size regime satisfy wave dispersion $\omega\!\propto\! k$ (where the proportionality factor is either the transverse or the longitudinal wave-speed, with transverse phonons populating the lowest frequencies), Eq.~\eqref{eq:finite-size} is equally valid for $\Gamma(k)$ once $\omega$ is everywhere replaced by $k$.
{\em Derivation of a relation between $\Gamma(k)$ and the phononic DOS $D_{\rm D}(\omega)$, and of Rayleigh scattering scaling ---} Equation~\eqref{eq:finite-size} predicts that the width of disordered phononic bands increases with $\omega$ for a fixed $N$. On the other hand, the spectral gap between adjacent phononic bands decreases with increasing $\omega$. Consequently, there exists a characteristic frequency $\omega_\dagger$ for which the band width $\Delta\omega$ becomes comparable to the gap between adjacent bands and above which discrete phononic bands cease to be well-defined. The crossover frequency $\omega_\dagger$ has been shown to take the form~\cite{phonon_widths}
\begin{equation}
\label{eq:omega_dagger}
\omega_\dagger(N) \sim L^{-2/({\,\mathchar'26\mkern-12mu d} + 2)} \sim N^{-2/{\,\mathchar'26\mkern-12mu d}({\,\mathchar'26\mkern-12mu d} + 2)} \sim k_\dagger(N) \ .
\end{equation}
It is clear that the finite-size scaling theory of Eq.~\eqref{eq:finite-size} is valid for $\omega\!<\!\omega_\dagger$. What happens for $\omega\!>\!\omega_\dagger$? As explained above, discrete phononic bands do not exist anymore for $\omega\!>\!\omega_\dagger$, rather disordered phonons form a continuum that is expected to follow Debye's DOS $D_{\rm D}(\omega)$. In order to understand wave attenuation in this regime, we are interested in the spectral width $\Delta\omega(\omega)$ of bands comprised of disordered phonons, on which a pure phononic excitation of frequency $\omega$ has sizable projections $\xi_m$. The number of such disordered phonons is simply $N D_{\rm D}(\omega)\Delta\omega(\omega)$. While these disordered phonons do not necessarily emerge from strictly degenerate phonons in the corresponding ordered problem (as in the case of discrete bands), they are clearly nearly degenerate and hence their number plays the role of $n_q(\omega)$ of Eq.~\eqref{eq:finite-size}.
With this identification, we can substitute $n_q(\omega)\!\sim\!N D_{\rm D}(\omega)\Delta\omega(\omega)$ in Eq.~\eqref{eq:finite-size} and treat it as a self-consistency equation for the spectral width $\Delta\omega(\omega)$ in the $\omega\!>\!\omega_\dagger$ regime. Solving for $\Delta\omega(\omega)$, and using $\Gamma(\omega)\!\sim\!\Delta\omega(\omega)$, we readily obtain
\begin{equation}
\label{eq:Gamma_DOS}
\Gamma(\omega) \sim \omega^2 D_{\rm D}(\omega)\qquad\hbox{for}\qquad \omega>\omega_\dagger\ ,
\end{equation}
which is a fundamental relation between the wave attenuation rate $\Gamma(\omega)$ and Debye's phononic DOS $D_{\rm D}(\omega)$. This relation is known as the ``Rayleigh-Klemens law"~\cite{Klemens1951}, and has been obtained in the past using quite different considerations and mathematical tools~\cite{mw_EM_epl,eric_boson_peak_emt,john1990localization, Maurer2004}. We believe that the derivation just presented, based on the analogy with the finite-size scaling theory in Eq.~\eqref{eq:finite-size}, provides basic and transparent understanding of the physical content of Eq.~\eqref{eq:Gamma_DOS}.
An immediate corollary of Eq.~\eqref{eq:Gamma_DOS}, simply by recalling that $D_{\rm D}(\omega)\!\sim\!\omega^{{\,\mathchar'26\mkern-12mu d}-1}$ and $\omega\!\sim\!k$, reads $\Gamma(k)\!\sim\! k^{{\,\mathchar'26\mkern-12mu d}+1}$ for $k\!>\! k_\dagger$, which is nothing but the well-known Rayleigh scattering scaling~\cite{Rayleigh}. We summarize this discussion with the prediction
\begin{equation}
\label{eq:scattering_summary}
\Gamma(k) \sim \left\{ \begin{array}{cc}k\sqrt{n_q(k)}/\sqrt{N}\,,\vspace{0.3cm}&k<k_\dagger\sim N^{-2/{\,\mathchar'26\mkern-12mu d}({\,\mathchar'26\mkern-12mu d} + 2)}\\k^{{\,\mathchar'26\mkern-12mu d}+1}\,,&k>k_\dagger\sim N^{-2/{\,\mathchar'26\mkern-12mu d}({\,\mathchar'26\mkern-12mu d} + 2)}\end{array}\right.\,,
\end{equation}
which we will verify using numerical simulations in the subsequent sections.
{\em Quasilocalized nonphononic excitations ---} The theoretical considerations culminating in Eq.~\eqref{eq:scattering_summary} suggest that had the elementary low-frequency excitations in glasses been exclusively comprised of disordered phonons, we would expect the wave attenuation rate of macroscopic glasses $\Gamma(k)$, in the small wavenumber limit, to exhibit Rayleigh scattering scaling. Moreover, had it been the case, we expect finite-size computer glasses to reveal this scaling in the range $k\!>\!k_\dagger$, once the finite-size effects in the range $k\!<\!k_\dagger$ are properly identified and eliminated. However, as was already stated in Sect.~\ref{sec:intro}, it is now well-established that glasses feature in addition to disordered phonons also low-frequency nonphononic excitations, which are characterized by quasilocalized spatial structures and follow a universal non-Debye DOS $D_{\rm G}(\omega)\!\sim\!\omega^4$.
These nonphononic modes --- sometimes termed glassy modes --- are known to hybridize with disordered phonons when they both coexist in the same frequency range \cite{SciPost2016,phonon_widths}. As such, and in light of the basic understanding that $\Gamma(k)$ is encapsulated in the projections $\xi_m$ of a pure phononic excitation of wavenumber $k$ on the eigenmodes of the glass Hessian, one expects glassy modes to affect wave attenuation in glasses. In fact, preliminary evidence in the literature indicates that glassy modes might be related to the generalized-Rayleigh scaling $\Gamma(k)\!\sim\!k^{{\,\mathchar'26\mkern-12mu d}+1}\log\!{(k_0/k)}$ (where $k_0$ is some characteristic wavenumber) recently observed in~\cite{lemaitre_tanaka_2016}; this connection may be implied by the observation of~\cite{Ikeda_2018} that the generalized-Rayleigh scaling disappears in favor of Rayleigh scaling once frustration-induced internal-stresses, which are known to be responsible for low-frequency nonphononic excitations in glasses~\cite{inst_note, ikeda_pnas}, are removed from jammed packings of harmonic spheres~\cite{footnote}.
In the subsequent sections we test the theoretical picture discussed here and the outstanding questions that accompany it. We will first test the finite-size scaling theory of $\Gamma(k)$ in the range $k\!<\!k_\dagger$ and use it to identify the macroscopic regime in the range $k\!>\!k_\dagger$. We will then explore the scaling of $\Gamma(k)$ in the range $k\!>\!k_\dagger$, focusing on the existence of Rayleigh and/or generalized-Rayleigh scaling regimes and on their possible relation to low-frequency nonphononic excitations. We note that macroscopic glasses are also known to feature a higher-$k$ scaling regime $\Gamma(k)\!\sim\!k^2$ above a characteristic wavenumber $k_*$, which is related to the Boson peak~\cite{experiments_1620K_vSiO2,Ruta_JNCS_2011,eric_boson_peak_emt}. This scaling is consistent with Eq.~(\ref{eq:Gamma_DOS}) for a flat $D(\omega)$ as observed in the vicinity of the Boson peak frequency in studies of the unjamming transition \cite{ohern2003,liu_review,van_hecke_review}. While some of our data sets will demonstrate this quadratic scaling regime as well, our major focus is on the range $k_\dagger\!<\!k\!<\!k_*$.
In what follows, simulational observables are made dimensionless: lengths are reported in terms of $a_0\!\equiv\! (V/N)^{1/{\,\mathchar'26\mkern-12mu d}}$ where $V,N,{\,\mathchar'26\mkern-12mu d}$ denote the volume, number of particles, and spatial dimension, respectively; times (rates) are reported in units of $a_0/c_{\rm T}$ ($c_{\rm T}/a_0$), where $c_{\rm T}\!\equiv\!\sqrt{G/\rho}$ denotes the shear wave speed, $G$ denotes the athermal shear modulus and $\rho$ denotes the mass density.
\section{Eliminating finite-size effects reveals the macroscopic low-frequency scaling regime}
\label{sec:2dipl}
As explained in the previous section, wave attenuation rates in finite-size glasses are expected to be affected by the quantization of disordered phonons into discrete bands at frequencies $\omega\!<\!\omega_\dagger$ (cf.~Eq.~(\ref{eq:finite-size})), or, alternatively, at wavenumbers $k\!<\!k_\dagger$ (cf.~Eq.~(\ref{eq:scattering_summary})). This expectation is verified in our simulational data presented in Fig.~\ref{fig:Fig1} below, generated using a generic computer glass forming model, referred to hereafter as the 2DIPL model. A comprehensive description of this model, and of other employed models and measurement methods, is provided in Appendix~\ref{sec:numerics_appendix}; here we briefly note that 2DIPL glassy samples were quenched from the equilibrium liquid phase using a constant and relatively large quench rate. This results in glasses that possess an abundance of soft quasilocalized modes, as discussed previously (e.g.~in \cite{modes_prl_2018}) and in what follows.
\begin{figure*}[ht!]
\centering
\includegraphics[width=1.9\columnwidth]{Fig_1_resub}
\caption{Finite-size effects in wave attenuation rates in computer glasses and their systematic elimination. Results are shown for the 2DIPL model (see Appendix~\ref{sec:numerics_appendix} for model details and methods). (a) The attenuation rate $\Gamma_{\rm T}$ of transverse waves vs.~wavenumber $k$ for various system sizes (see legend). The inset zooms in on the meandering away from the $N$-independent curve at low $k$. (b) The same data as in panel (a), but rescaled following the finite-size predictions for $\Gamma(k,n_q,N)$ and $k_\dagger(N)$ given in Eq.~(\ref{eq:scattering_summary}). The data collapse (see horizontal dashed line) at low $k$ allows to estimate the crossover wavenumber $k_\dagger(N)$ (vertical dashed line). The insets show the accompanying crossover in the functional form of the envelopes $C(t)$ of the velocity autocorrelation functions from exponential (bottom-right inset) to compressed-exponential (upper-left inset). (c) The exponent $\alpha(k)$ obtained by fitting the envelope of $C(t)$ to a compressed-exponential form (here $N\!=\!409600$) vs.~wavenumber $k$. A transition from $\alpha\!=\!1$ for $k\!>\! k_\dagger$ to $\alpha\!\approx2$ for $k\!<\! k_\dagger$ is observed. (d) The same data as in panel (a), once finite-size effects are eliminated, and $\Gamma_{\rm T}$ is rescaled by $k^3$. The dashed line corresponds to a generalized-Rayleigh scaling of attenuation rates, see text for additional discussion.}\label{fig:Fig1}
\end{figure*}
In Fig.~\ref{fig:Fig1}a we show the bare attenuation rates of transverse waves $\Gamma_{\rm T}(k)$ vs.~wavenumber $k$, measured in 2DIPL glassy samples for various system sizes, as indicated in the legend. A regime of wavenumbers in which the attenuation rates are \emph{independent} of system size is clearly identified. This independence holds down to an $N$-dependent crossover wavenumber $\omega_\dagger(N)$, as expected from our theoretical considerations. Below this crossover wavenumber, attenuation rates begin to meander away from the $N$-independent curve (see inset of Fig.~\ref{fig:Fig1}a). Similar finite-size-induced deviations were discussed in~\cite{lemaitre_tanaka_2016}.
According to our theoretical considerations (cf.~Eq.~(\ref{eq:omega_dagger}) and preceding discussions), the crossover wavenumber $k_\dagger$, below which finite-size effects on attenuation rates should be observed, decreases with system size as $N^{-1/4}$ in 2D glasses. Below this crossover, attenuation rates are expected to follow the finite-size scaling given by Eq.~(\ref{eq:scattering_summary}), i.e.~$\Gamma\!\sim\! k\sqrt{n_q(k)}/\sqrt{N}$. To test these predictions, we plot in Fig.~\ref{fig:Fig1}b the attenuation rates $\Gamma(k)$ rescaled by $k\sqrt{n_q(k)}/\sqrt{N}$, against the rescaled wavenumber $N^{1/4}k$. The data collapse to a constant curve below the crossover wavenumber $k_\dagger\!\sim\! N^{-1/4}$ (vertical dashed line) validates both the prediction for the scaling of the crossover wavenumber $k_\dagger(N)$ given by Eq.~(\ref{eq:omega_dagger}), and the prediction for the wavenumber $k$, phonon-band degeneracy $n_q$, and system size $N$ dependencies of attenuation rates for $k\!<\!k_\dagger$, given by Eq.~(\ref{eq:scattering_summary}).
The change in the scaling behaviour of attenuation rates at $k\!\!\!\!<\!\!\!\! k_\dagger$ is accompanied by a qualitative change in the \emph{functional form} of the envelope of the correlation function $C(t)$, as shown in Fig.~\ref{fig:Fig1}b (insets). While exponentially-decaying oscillations, consistent with Eq.~\eqref{eq:damped_oscillator}, are observed for $k\!\!>\!\! k_\dagger$, compressed-exponentially decaying oscillations are observed for~$k\!<\! k_\dagger$. To quantify this qualitative change in functional form, we fitted the envelope of the autocorrelation function $C(t)$ to a compressed-exponential $\exp\left[-\tfrac{1}{2}(\Gamma(k)t)^{\alpha(k)}\right]$, where the exponent $\alpha(k)$ is shown in Fig.~\ref{fig:Fig1}c. It is observed that while the conventional assumption of exponential decay (cf.~Eq.~\eqref{eq:damped_oscillator}) is valid above $k_\dagger$, it breaks down below it, where $\alpha$ is larger and appears to approach $2$ at the lowest $k$'s. Consequently, care should be taken when extracting attenuation rates in the finite-size regime $k\!<\! k_\dagger$. We expand on this point and on our fitting procedure in Appendix~\ref{sec:fitting_procedure}.
In order to understand the crossover from the exponential decay of oscillations at $k\!>\!k_\dagger$ to compressed exponential decay of oscillations for $k\!<\! k_\dagger$, one should reexamine Eqs.~\eqref{eq:corr}-\eqref{eq:projections}; from these relations it is clear that $C(t)$ (which becomes an even function of time by replacing $t\!\to\!|t|$) is approximately given by the Fourier transform of the squared overlaps $\xi_m^2$ weighed by the DOS, which in turn depends on the wavenumber $k$ (or, alternatively, on the frequency $\omega$). In the finite-size regime, in which phonon bands are well-isolated from each other, the DOS-weighed $\xi_m^2$ is sharply peaked (as shown e.g.~in \cite{lemaitre_tanaka_2016}), with a highly confined support on the frequency axis. This leads to the expectation that the Fourier transform of the DOS-weighed $\xi_m^2$ should approximately feature a Gaussian envelope (as holds e.g.~for the Fourier transform of a Gaussian), as we indeed observe. Once the support of $\xi_m^2$ becomes unconfined, as occurs for $k\!>\! k_\dagger$, $C(t)$ features exponentially-decaying oscillations. Finally, note that in recent work~\cite{Grzegorz_scattering_arXiv2} it has been claimed that at very short times $C(t)$ features an exponential decay in the finite-size regime $k\!<\! k_\dagger$; we comment on this claim in Appendix~\ref{sec:networks_appendix}.
Identifying the finite-size effects on the attenuation rates, made apparent by the presentation scheme of Fig.~\ref{fig:Fig1}b, provides us with a solid handle on systematically eliminating them from simulation measurements. Consequently, we follow below the very same procedure to identify the crossover wavenumber $k_\dagger$ for each data set. In order to properly probe the scaling behaviour of macroscopic glasses, we shall only consider attenuation rates associated with wavenumbers $k$ sufficiently \emph{larger} than the identified crossover $k_\dagger$. Applying this scheme to our 2DIPL wave attenuation rates data, we replot $\Gamma(k)$ in Fig.~\ref{fig:Fig1}d, this time rescaled by $k^3$ (Rayleigh scaling prediction for ${\,\mathchar'26\mkern-12mu d}\=2$), keeping only data points that fall \emph{above} the crossover wavenumber $k_\dagger$. We find that $\Gamma(k)$ in this system follows a generalized-Rayleigh law $\Gamma(k)\!\sim\!k^{{\,\mathchar'26\mkern-12mu d}+1}\log\!{(k_0/k)}$ with $k_0\!\approx\!1.07$. This scaling, first observed by Gelin et al.~\cite{lemaitre_tanaka_2016} and subsequently in~\cite{Ikeda_2018}, appears to persist down to the lowest wavenumber accessible above the finite-size scaling wavenumber regime $k\!>\! k_\dagger$. We note that $k_0$ close to unity corresponds to a length of about $6$ interparticle distances, very similar to the size of the core of quasilocalized modes observed in computer glass models \cite{modes_prl_2016,SciPost2016,modes_prl_2018}. Indeed, we next establish that the abundance of quasilocalized modes of frequencies comparable to $\omega_\dagger$ --- which depends in turn on glass stability \cite{protocol_prerc, cge_paper, LB_modes_2019} --- gives rise to the observed generalized-Rayleigh scaling of attenuation rates.
\section{The generalized-Rayleigh scaling depends on glass stability}
\label{sec:2d3dSWAP}
As mentioned in Sect.~\ref{sec:theory}, recent computer simulations by Mizuno and Ikeda~\cite{Ikeda_2018} have shown that wave attenuation rates measured in packings of harmonic spheres both in 2D and 3D follow Rayleigh scattering scaling $\Gamma\!\sim\! k^{{\,\mathchar'26\mkern-12mu d}+1}$ once the internal-stresses/forces between particles are artificially turned off \cite{footnote}, while the generalized-Rayleigh scaling $\Gamma\!\sim\! k^{{\,\mathchar'26\mkern-12mu d}+1}\log\!{(k_0/k)}$ is observed when these internal-stresses/forces are left intact. In other work \cite{inst_note, ikeda_pnas} it was shown that the very same procedure of turning off forces between particles suppresses the occurrence of soft quasilocalized modes, whose universal statistics have been recently established \cite{modes_prl_2016,cge_paper, modes_prl_2018}. These two observations lead together to the hypothesis that the emergence of generalized-Rayleigh scaling depends on the abundance of soft quasilocalized modes, and possibly on their characteristic localization length, both of which have been shown to depend, in turn, on glass stability \cite{protocol_prerc, cge_paper, LB_modes_2019}.
To test this hypothesis we seek to substantially increase the stability of our computer glasses, and by doing so to suppress the occurrence and spatial extent of soft quasilocalized modes in a physical manner. To this aim we employ the Swap Monte Carlo method and an optimized glass forming model~\cite{berthier_prx}, which was recently shown to be very efficient in 2D~\cite{two_dimensions_swap}, allowing for extremely deep supercooling. Details about our simulations and employed methods are provided in Appendix~\ref{sec:numerics_appendix}.
\begin{figure}[ht!]
\centering
\includegraphics[width=\columnwidth]{Fig_2_resub}
\caption{Sound wave attenuation rates in a stable 2D computer glass of $N\!=\!16384$. Results are shown for the 2DSWAP model (see Appendix~\ref{sec:numerics_appendix} for model details and methods). (a)~The attenuation rate $\Gamma_{\rm L}$ of longitudinal waves vs.~wavenumber $k$, rescaled according to the finite-size prediction for $\Gamma(k,n_q)$ given in Eq.~(\ref{eq:scattering_summary}). The predicted finite-size effect is verified (horizontal dashed line), along with an estimate of the accompanying crossover wavenumber $k_\dagger$ (vertical dashed line). (b) Rayleigh scattering scaling $\Gamma_{\rm L}\!\sim\! k^3$ is observed (dashed line, $1\!:\!3$ triangle) once finite-size effects are eliminated.}\label{fig:Fig2}
\end{figure}
In Fig.~\ref{fig:Fig2} we present the attenuation rates of longitudinal (sound) waves $\Gamma_{\rm L}(k)$ measured in very stable 2D computer glasses of $N\!=\!16384$ particles, referred to as the 2DSWAP model. Figure~\ref{fig:Fig2}a employs the same presentation scheme discussed in the preceding section, namely we plot $\Gamma_{\rm L}/k\sqrt{n_q}$ vs.~the wavenumber $k$ (here we do not vary $N$), in order to identify the crossover wavenumber $k_\dagger$. As expected, our data show that at the lowest wavenumbers $\Gamma_{\rm L}\!\sim\! k\sqrt{n_q(k)}$, apparently at odds with the results of~\cite{Grzegorz_scattering_arXiv}, where sound (longitudinal) wave attenuation rates in stable computer glasses in 3D appeared to follow Rayleigh scaling at the lowest accessible wavenumbers. This discrepancy stems from the differences in the location of the crossover frequency $\omega_\dagger$ relative to the lowest sound waves' frequencies: in Appendix~\ref{sec:soundwaves_appendix} we show that in our 2D stable glasses several discrete phonon bands that pertain to sound waves exist \emph{below} the crossover frequency $\omega_\dagger$, whereas in stable 3D glasses only a single sound wave band resides below the crossover frequency $\omega_\dagger$. We therefore suggest that if larger 3D stable glasses could have been generated, more sound waves would appear as discrete phonon bands in the vibrational density of states, resulting in sound wave attenuation rates that follow the finite-size scaling theory at the lowest accessible wavenumbers, as shown in Fig.~\ref{fig:Fig2} for 2D.
Having identified the crossover wavenumber $k_\dagger$, we next plot in Fig.~\ref{fig:Fig2}b the attenuation rates of longitudinal waves $\Gamma_{\rm L}(k)$ vs.~wavenumber $k$, keeping only data points that pertain to wavenumbers $k\!>\! k_\dagger$. We find that the enhanced stability of the studied model glasses results in a Rayleigh-like scaling $\sim\!k^3$ (in 2D) of attenuation rates of longitudinal waves, rather than the generalized-Rayleigh scaling as seen in less stable glasses in the preceding section. This observation supports the idea that the abundance of soft quasilocalized excitations gives rise to generalized-Rayleigh scaling of wave attenuation rates, and that Rayleigh scattering scaling is observed in their absence.
To further establish the generality of this conclusion, we also created an ensemble of stable glasses in 3D (see details in Appendix~\ref{sec:numerics_appendix}), referred to as the 3DSWAP system, and measured the attenuation rates $\Gamma_{\rm T}(k)$ of transverse waves. In Fig.~\ref{fig:Fig3} we employ the presentation scheme of Figs.~\ref{fig:Fig1}b and \ref{fig:Fig2}a for shear wave attenuation rates, to identify the onset of the finite size regime for each system size. The data collapse below the crossover wavenumber $\sim\! N^{-2/15}$ to a constant curve validates once again our theoretical predictions given by Eqs.~(\ref{eq:omega_dagger}) and (\ref{eq:scattering_summary}), this time in 3D.
\begin{figure}[ht!]
\centering
\includegraphics[width=\columnwidth]{Fig_3_resub}
\caption{Transverse (shear) wave attenuation rates in stable 3D computer glasses. Results are shown for the 3DSWAP model (see Appendix~\ref{sec:numerics_appendix} for model details and methods). (a) The attenuation rate $\Gamma_{\rm T}$ of transverse waves vs.~wavenumber $k$, rescaled according to the finite-size predictions for $\Gamma(k,n_q,N)$ and $k_\dagger(N)$ given in Eq.~(\ref{eq:scattering_summary}). The predicted finite-size effect is verified (horizontal dashed line), along with an estimate of the accompanying crossover wavenumber $k_\dagger$ (vertical dashed line). (b) Rayleigh scattering scaling $\Gamma_{\rm T}\!\sim\! k^4$ is observed (dashed line, $1\!:\!4$ triangle) at low $k$ once finite-size effects are eliminated. A quadratic scaling regime $\Gamma_{\rm T}\!\sim\! k^2$ is observed above $k\!\approx\!1$ (dashed line, $1\!:\!2$ triangle).}\label{fig:Fig3}
\end{figure}
In Fig.~\ref{fig:Fig3}b we plot the shear wave attenuation rates measured in the 3DSWAP system, after identifying and eliminating finite-size effects as described above. At the lowest wave numbers above $k_\dagger$ we find a very clean Rayleigh scattering scaling $\Gamma_{\rm T}\!\sim\! k^4$. A similar observation was recently made in~\cite{Grzegorz_scattering_arXiv}. These results reinforce our assertion that the abundance of soft quasilocalized modes gives rise to the generalized-Rayleigh scaling, and that, in their absence, Rayleigh scattering scaling is observed.
We note that our 3D data show the onset of a quadratic scaling of wave attenuation rates $\Gamma_{\rm T}\!\sim\! k^2$ above $k\!\approx\!1$, consistent with the Effective Medium Theory~\cite{eric_boson_peak_emt} and experimental observations~\cite{experiments_1620K_vSiO2,Ruta_JNCS_2011}. We further note, however, that wave attenuation rates are of the order of unity in this regime, indicating that the decay time of the velocity autocorrelation function becomes comparable to the wave oscillation time. In turn, this implies that the wavenumber window in which quadratic scaling is observed is quite limited in stable glasses.
\section{Numerical evidence for a crossover from Rayleigh to generalized-Rayleigh scaling}
\label{sec:crossover}
In the previous sections we established that if soft quasilocalized modes are too scarce, wave attenuation rates at low $k$ (but still above $k_\dagger$) follow Rayleigh scattering scaling $\Gamma\!\sim\! k^{{\,\mathchar'26\mkern-12mu d}+1}$, whereas generalized-Rayleigh scaling $\Gamma\!\sim\! k^{{\,\mathchar'26\mkern-12mu d}+1}\log\!{(k_0/k)}$ is observed in less stable glasses, in which soft quasilocalized modes are abundant. Recalling that the universal non-Debye DOS $D_{\rm G}(\omega)\!\sim\!\omega^4$ decays rather rapidly as $\omega\!\to\!0$, it is reasonable to expect that at frequencies corresponding to phonons/waves with very low wavenumbers, the abundance of soft quasilocalized modes would become so low that attenuation rates would cross over from the generalized-Rayleigh scaling to Rayleigh scattering scaling. In other words, based on the evidence presented in preceding sections we hypothesize that even in poorly-annealed glasses in which soft quasilocalized modes are abundant, an onset wavenumber $k_{\rm on}$, below which wave attenuation rates should cross over from $\Gamma\!\sim\! k^{{\,\mathchar'26\mkern-12mu d}+1}\log\!{(k_0/k)}$ at $k\!>\!k_{\rm on}$ to $\Gamma\!\sim\! k^{{\,\mathchar'26\mkern-12mu d}+1}$ at $k_\dagger(N)\!<\! k\!<\! k_{\rm on}$, should exist.
Testing this hypothesis requires creating glasses for which the crossover frequency $\omega_\dagger\!=\! c_{\rm T}k_\dagger$ is very low, achievable by increasing the system size, as evident from Eq.~(\ref{eq:omega_dagger}). While larger lengths and longer simulation runs are generally more feasible in 2D, we do not use 2D glasses in the present context. The reason is that the characteristic frequency of soft quasilocalized modes in 2D also decreases with system size, as shown in~\cite{cge_paper, modes_prl_2018}. This leads to a competition between soft quasilocalized modes' frequencies and the crossover frequency $\omega_\dagger$, both decreasing with system size, thus making it difficult to observe the hypothesized crossover in the scaling form of wave attenuation rates in 2D.
In 3D, the characteristic frequencies of soft quasilocalized modes is system-size independent, and we therefore opt to test the aforementioned crossover hypothesis in 3D. We employ a generic computer glass forming model --- referred to as the 3DIPL model --- and generate very large glassy samples of up to $4$M particles using a continuous, rapid quench from equilibrium liquid states (see Appendix~\ref{sec:numerics_appendix} for further details about the model and protocols). These glassy samples have been shown to feature an abundance of soft quasilocalized modes~\cite{phonon_widths, cge_paper, modes_prl_2018}.
In Fig.~\ref{fig:Fig4} we present the shear wave attenuation rates as measured in the 3DIPL model. As was the case with previous data sets shown above and as explained in Sect.~\ref{sec:2dipl}, here too we identified and eliminated finite-size effects, showing only data for $k\!>\! k_\dagger(N)$. Figure~\ref{fig:Fig4}a shows the attenuation rates $\Gamma_{\rm T}(k)$ vs.~wavenumber $k$; $\Gamma_{\rm T}(k)$ appears to approach Rayleigh scattering at the lowest wavenumbers, and transitions at sufficiently high wavenumbers to the expected quadratic scaling $\Gamma_{\rm T}\!\sim\! k^2$, both marked by dashed lines.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.85\columnwidth]{Fig_4_resub}
\caption{(a) Transverse (shear) wave attenuation rates $\Gamma_{\rm T}(k)$ measured in 3DIPL computer glasses of various sizes (see legend), plotted against wavenumber $k$. Data shown only for $k\!>\! k_\dagger$, as described in Sect.~\ref{sec:2dipl}. Rayleigh scattering scaling $\Gamma_{\rm T}\!\sim\! k^4$ is observed (dashed line, $1\!:\!4$ triangle) at low $k$ and a quadratic scaling regime $\Gamma_{\rm T}\!\sim\! k^2$ is observed at higher $k$ (dashed line, $1\!:\!2$ triangle). (b) Plotting $\Gamma(k)/k^4$ reveals that attenuation rates cross over from generalized-Rayleigh scaling at intermediate $k$ (tilted dashed line) to Rayleigh scaling at low $k$ (horizontal dashed line). The crossover, whose wavenumber $k_{\rm on}$ is roughly estimated by the intersection of the two dashed lines, occurs well within the regime in which the universal non-Debye DOS $D_{\rm G}(\omega)\!\sim\!\omega^4$ of soft quasilocalized modes is observed, as shown in Appendix~\ref{sec:DOS_appendix}.}\label{fig:Fig4}
\end{figure}
To better assess the scaling form of attenuation rates at intermediate wavenumbers, we plot in Fig.~\ref{fig:Fig4}b the rescaled rates $\Gamma_{\rm T}(k)/k^4$ against $k$; this presentation reveals that at intermediate wavenumbers the generalized-Rayleigh scaling $\Gamma_{\rm T}\!\sim\! k^{{\,\mathchar'26\mkern-12mu d}+1}\log\!{(k_0/k)}$ emerges, consistent with previous observations~\cite{lemaitre_tanaka_2016,Ikeda_2018}. Our key result, however, is the crossover at low $k$ from generalized-Rayleigh scaling to Rayleigh scattering scaling $\Gamma\!\sim\! k^4$, which is hinted at by our data for systems of 1M particles (orange circles), but robustly observed for our largest systems of 4M particles (brown squares). This result strongly suggests that macroscopic glasses always exhibit Rayleigh scattering scaling at low wavenumber, \emph{regardless of glass stability}. This suggestion is further supported by experimental data from recent literature, discussed at length in Sect.~\ref{sec:exp} below. We stress that the crossover observed in Fig.~\ref{fig:Fig4}b occurs well within the regime in which the $D_{\rm G}(\omega)\!\sim\!\omega^4$ DOS is realized in our calculations, as shown in Appendix~\ref{sec:DOS_appendix}.
A similar crossover was recently announced by Mizuno and Ikeda~\cite{Ikeda_2018}. These authors performed measurements of wave attenuation rates in packings of particles interacting via one-sided harmonic potentials both in 2D and 3D, a well-studied model in the context of the unjamming transition~\cite{ohern2003,liu_review,van_hecke_review}. For the largest systems employed in~\cite{Ikeda_2018}, of over 4M particles in 3D, a clear finite-size quantization of the lowest frequency phonon bands is observed, see e.g.~Fig.~1A of~\cite{ikeda_pnas}. Despite this inevitable low wavenumber quantization, it was found in~\cite{Ikeda_2018} that Rayleigh scattering scaling appears to persist all the way down to the lowest accessible wavenumbers --- and thus necessarily for $k\!<\!k_\dagger$ --- where we expect the finite-size scaling theory Eq.~(\ref{eq:scattering_summary}) to hold. While one can speculate that this discrepancy may arise from not carefully taking into account the change in the functional form of $C(t)$ below $k_\dagger$ (cf.~Fig.~\ref{fig:Fig2}c) when extracting attenuation rates, we presently cannot fully explain this striking disagreement with our theoretical arguments and numerical results.
\section{The generalized-Rayleigh scaling originates from quasilocalized nonphononic excitations}
\label{sec:log}
In previous sections we provided substantial evidence that the abundance of quasilocalized modes at frequencies in the vicinity of and above the crossover frequency $\omega_\dagger$ gives rise to the generalized-Rayleigh scaling of wave attenuation rates. In order to further cement this conclusion, we seek to study model computer glasses in which the abundance of quasilocalized modes at low frequencies is controllable by tuning an external model parameter. By the controlled depletion of soft quasilocalized modes, we expect to be able to systematically shift the crossover wavenumber $k_{\rm on}$ that separates the generalized-Rayleigh scaling of attenuation rates to Rayleigh scattering scaling.
To this aim we first consider glassy samples of the 2DIPL model; in simple models of computer glasses such as the 2DIPL model, the Hessian matrix can be decomposed as $\calBold{M}\!=\!\calBold{M}_\kappa + \calBold{M}_f$, where $\calBold{M}_\kappa$ accounts for pairwise stiffnesses, and $\calBold{M}_f$ accounts for pairwise internal-stresses/forces~\cite{eric_boson_peak_emt, inst_note}. To control the abundance of quasilocalized glassy modes, we employed an augmented Hessian that reads
\begin{equation}
\label{eq:augmented_hessian}
\calBold{M}(\delta) = \calBold{M}_\kappa + (1-\delta)\calBold{M}_f\,,
\end{equation}
where $\delta$ is a dimensionless control parameter. In previous work~\cite{eric_boson_peak_emt,inst_note} it was argued that increasing $\delta$ from zero creates a gap in the nonphononic density of states $D_{\rm G}(\omega)$ at zero frequency, which grows at small $\delta$ as $\sim\!\sqrt{\delta}$. We thus expect wave attenuation rates measured in a system governed by a Hessian of the form of Eq.~(\ref{eq:augmented_hessian}) to feature a crossover from Rayleigh scattering scaling to generalized-Rayleigh scaling at $k_{\rm on}(\delta)$, which is now expected to increase as $\delta$ is increased. This expectation is indeed verified in Fig.~\ref{fig:Fig5}a.
\begin{figure}[ht!]
\centering
\includegraphics[width=\columnwidth]{Fig_5_resub}
\caption{(a) Rescaled transverse (shear) wave attenuation rates $\Gamma_{\rm T}(k;\delta)/k^3$ measured for dynamics governed by the Hessian as given by Eq.~(\ref{eq:augmented_hessian}), for various values of the parameter $\delta$, and $N\!=\!25600$. As predicted, increasing $\delta$ shifts the crossover wavenumber $k_{\rm on}(\delta)$, from Rayleigh to generalized-Rayleigh scaling, to higher values. (b) The same as (a), but measured in computer glasses of $N\!=\!1$M particles created using a model that allows particle sizes to fluctuate during glass formation (but not afterwards, see~\cite{fsp_pre} and text for details). Particle-size fluctuations during glass formation are governed by the parameter $\kappa$, with $\kappa\!=\!\infty$ meaning no fluctuations. Similar to (a), here we find that $k_{\rm on}(\kappa)$ shifts to higher wavenumbers when $\kappa$ is decreased, as predicted.}\label{fig:Fig5}
\end{figure}
We next consider the 2D variant of the model computer glass introduced in~\cite{fsp_pre}, referred to as the 2D Fluctuating-Size Particles (2DFSP) model. The 2DFSP model consists of particles whose sizes are considered as degrees of freedom that are subjected to a potential of the form of a well, characterized by a stiffness $\kappa$. Within the framework of this model, particle sizes are allowed to fluctuate during glass formation, but, upon completion of glass formation, particle sizes are fixed for all subsequent analyses. In~\cite{fsp_pre} it was shown that tuning $\kappa$ (denoted by $k_\lambda$ there) has a dramatic effect on the elastoplastic properties of the resulting glasses. In particular, it was shown that the nonphononic density of states features a gap at zero frequency that scales as $1/\sqrt{\kappa}$ at large $\kappa$. We therefore expect wave attenuation rates to show a crossover from Rayleigh scattering scaling to generalized-Rayleigh scaling at a crossover wavenumber $k_{\rm on}(\kappa)$ that should increase with decreasing $\kappa$. This expectation is indeed verified in Fig.~\ref{fig:Fig5}b.
The nature of the nonphononic DOS of the two models investigated in this section is anomalous: generic glass formers, and even very well-annealed model glasses~\cite{LB_modes_2019}, feature the universal $D_{\rm G}(\omega)\!\sim\!\omega^4$. Notwithstanding, by exploiting the control offered by computer models, we are able to firmly establish that soft quasilocalized modes are the origin of the generalized-Rayleigh scaling of wave attenuation rates, and not correlations in local elastic moduli fields (which are present for all $\kappa$ values in the FSP model~\cite{fsp_pre}) as suggested in~\cite{lemaitre_tanaka_2016}.
\section{Experimental evidence}
\label{sec:exp}
As recently pointed out~\cite{lemaitre_tanaka_2016}, several experimental data sets support that wave attenuation rates in laboratory glasses follow the generalized-Rayleigh scaling over some wavenumber range. In Fig.~\ref{fig:Fig6}a we present digitized data from~\cite{experiments_300K_vSiO2,experiments_1620K_vSiO2} of wave attenuation rates in vitreous silica (silicon dioxide, SiO$_2$) measured down to $k\!\simeq\!1$nm$^{-1}$ using high-resolution inelastic x-ray scattering (IXS)~\cite{Ruocco1996, ixs_review_arXiv}. Note that the experimental data correspond to two different temperatures (see legend), with no noticeable temperature dependence, indicating the absence of significant anharmonic effects (additional evidence for the existence of a sizable temperature range that is dominated by harmonic effects is presented in~\cite{Ruffle2006,Ruta2010}). Consequently, our predictions in the harmonic regime are expected to be directly relevant for these experimental data. While these data --- presented and discussed also in~\cite{lemaitre_tanaka_2016} --- are consistent with generalized-Rayleigh scaling, the results extensively discussed above predict that a crossover to Rayleigh scaling should be observed at yet smaller wavenumbers $k$. However, present-day IXS spectrometers operate at momenta transfer corresponding to $k\!>\!1$nm$^{-1}$, so this lower $k$ regime is currently inaccessible using this experimental technique. Yet, as shown above, suppressing the density of soft quasilocalized modes pushes the crossover to Rayleigh scaling to higher wavenumbers, which in principle could make the crossover observable using present-day IXS. This might be achieved using a permanent densification procedure~\cite{densification}.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.93\columnwidth]{Fig_6_resub}
\caption{Wave attenuation rates, rescaled by $k^4$, measured in experiments on vitreous (panel~(a), data digitized from~\cite{experiments_300K_vSiO2,experiments_1620K_vSiO2}) and densified (panel~(b), data digitized from~\cite{experiments_570K_dSiO2}) SiO$_2$, using the high-resolution inelastic x-ray scattering technique, see text for discussion. Panel (b) exhibits a crossover from generalized-Rayleigh to Rayleigh scaling (dashed lines), as predicted.}\label{fig:Fig6}
\end{figure}
Permanent densification of glassy materials has been previously shown to greatly suppress the magnitude of the Boson peak in the DOS of glasses~\cite{Effect_of_Densification_prl_2006}. We hypothesize that permanent densification likewise leads to the annihilation and stiffening of soft quasilocalized modes, thus suppressing their DOS at low wavenumbers. In turn, this densification process potentially pushes the crossover wavenumber $k_{\rm on}$ into the wavenumber regime accessible to IXS. To test this idea, we present in Fig.~\ref{fig:Fig6}b digitized data from~\cite{experiments_570K_dSiO2} of wave attenuation rates measured (by the same experimental group that measured the data presented in Fig.~\ref{fig:Fig6}a) in silica that was subjected to permanent densification under a pressure of $8$GPa (see~\cite{experiments_570K_dSiO2} for the details of the densification procedure). Strikingly, a clear crossover from generalized-Rayleigh to Rayleigh scaling at $k_{\rm on}\!\approx\!2$nm$^{-1}$ is observed, strongly supporting the physical picture discussed in this paper. We note that according to a recent study~\cite{Deschamps2014}, densification of vitreous silica at $8$GPa does not lead to significant changes in the elastic moduli. Consequently, the differences observed in attenuation rates between vitreous and densified silica could not be attributed to shifts in characteristic phononic frequencies; rather, we propose that they emerge from the suppression of soft nonphononic quasilocalized modes.
The experimental data set shown in Fig.~\ref{fig:Fig6}b supports our assertion that attenuation rates in macroscopic glasses should feature Rayleigh scaling at the lowest wavenumbers $k$ and a crossover to generalized-Rayleigh scaling at higher $k$. Clearly, additional measurements of attenuation rates at sufficiently lower wavenumbers in glasses are needed to further test our predictions. Moreover, improved control over structural disorder in glasses, e.g.~as recently done in~\cite{Ketkaew2018}, can affect the crossover to Rayleigh scaling and make it accessible to present-day measurement techniques.
\section{Concluding remarks and open questions}
\label{sec:sum}
In this paper, we have theoretically and computationally established three basic results in glass physics. First, we have shown that the finite-size quantization of phonons into discrete bands in finite glassy samples gives rise to qualitative changes in the physics of wave attenuation: for wavenumbers $k$ below the $N$-dependent crossover wavenumber $k_\dagger(N)$, attenuation rates follow the finite-size scaling theory Eq.~(\ref{eq:scattering_summary}), meaning that they depend explicitly on system size, and on the number of phononic modes of the corresponding discrete phonon band. Above the crossover $k_\dagger$, attenuation rates become independent of system size, and only depend on the wavenumber. In addition, the functional form of the velocity autocorrelation function, from which the attenuation rates are extracted, changes qualitatively between above and below the crossover.
The second key result we have established is that the recently-observed generalized-Rayleigh scaling of attenuation rates originates from the presence and number of soft quasilocalized modes emerging from glassy disorder/microstructures, whose universal statistical and structural properties have recently been revealed~\cite{modes_prl_2016, modes_prl_2018, ikeda_pnas}. The generalized-Rayleigh regime corresponds thus to a distinct frequency regime in the vibrational DOS --- below the boson peak frequency --- in which phonons and an abundance of soft quasilocalized modes coexist (in hybridized form, see discussions in e.g.~\cite{SciPost2016,phonon_widths}). It therefore constitutes a distinct wave attenuation regime, and not merely a crossover between the Rayleigh ($\sim\!k^4$) and the `boson peak' ($\sim\!k^2$) regimes.
Finally, we have shown that at very low wavenumbers corresponding to frequencies at which soft quasilocalized modes are scarce, wave attenuation rates follow Rayleigh scattering scaling. This implies that in macroscopic glasses Rayleigh scaling is the expected asymptotic low-wavenumber behavior of wave attenuation rates, and not generalized-Rayleigh scaling as recently claimed~\cite{lemaitre_tanaka_2016}. Generalized-Rayleigh scaling is expected to be observed at higher wavenumbers. These predictions are further supported by experimental data from recent literature~\cite{experiments_300K_vSiO2,experiments_1620K_vSiO2,experiments_570K_dSiO2}.
These results raise various interesting questions, both theoretical and experimental; among these, we would like to highlight the most prominent and pressing ones, in our view. First, we need to understand how quasilocalized nonphononic modes give rise to the generalized-Rayleigh scaling of attenuation rates once they are sufficiently abundant. That is, we need to develop a first principles theory of the apparently universal scaling relation $\Gamma(k)\!\sim\!k^{{\,\mathchar'26\mkern-12mu d}+1}\log\!{(k_0/k)}$ for $k\!>\!k_{\rm on}$ and of the crossover wavenumber $k_{\rm on}$, based on the physical insight gained in this work.
Second, we need to explore the implications of the obtained results for $\Gamma(k)$ at small wavenumbers $k$ for the low temperature energy diffusivity and heat transport in glasses~\cite{Kittel_1949,Zeller_and_Pohl_prb_1971}. In addition, we need improved experiments, guided by theoretical estimates of $k_{\rm on}$, which aim at directly measuring the crossover from Rayleigh to generalized-Rayleigh scaling of long wavelength wave attenuation rates in glasses. We hope that these important challenges will be addressed in future work.
Finally, in a broader context, it would be interesting to understand whether the ideas developed in this paper might be related to universal low-temperature thermodynamic and transport anomalies in glasses. It is well-established that the specific heat $C(T)$ of glasses scales linearly with the temperature $T$ below $1$K, instead of Debye's $T^3$ prediction for crystals~\cite{Zeller_and_Pohl_prb_1971}, and exhibits a distinct hump around $10$K, when normalized by $T^3$~\cite{soft_potential_model_1991}. In the very same temperature range, $\sim\!1\!-\!10$K, the thermal conductivity $\kappa(T)$ of glasses exhibits a plateau that is entirely absent in their crystalline counterparts~\cite{Eucken_1911,Zeller_and_Pohl_prb_1971,Freeman_Anderson_prb_1986}. Despite extensive efforts~\cite{Kittel_1949,Anderson,Phillips,Tanguy_pre_2018,moshe_prb_2013,Schober_prb_1992,energy_transport_jamming,Gurevich2003,ganter2010rayleigh,schirmacher2011comments,Maurer2004,mw_EM_epl,eric_boson_peak_emt}, a complete and fundamental understanding of these universal low-temperature glass anomalies is currently missing. Future work should clarify whether the physics behind sound attenuation discussed in this paper might be related to these universal low-temperature anomalies.
\textit{Acknowledgments.--} E.~L.~acknowledges support from the Netherlands Organisation for Scientific Research (NWO) (Vidi grant no.~680-47-554/3259). E.~B.~acknowledges support from the Minerva Foundation with funding from the Federal German Ministry for Education and Research, the William Z.~and Eda Bess Novick Young Scientist Fund and the Harold Perlman Family.
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{"url":"http:\/\/slideplayer.com\/slide\/3872369\/","text":"# Slope Is a Rate of Change Section 2.4. Lehmann, Intermediate Algebra, 4ed Section 2.4 The ratio of a to b is the fraction A unit ratio is a ratio written.\n\n## Presentation on theme: \"Slope Is a Rate of Change Section 2.4. Lehmann, Intermediate Algebra, 4ed Section 2.4 The ratio of a to b is the fraction A unit ratio is a ratio written.\"\u2014 Presentation transcript:\n\nSlope Is a Rate of Change Section 2.4\n\nLehmann, Intermediate Algebra, 4ed Section 2.4 The ratio of a to b is the fraction A unit ratio is a ratio written as with Suppose the sea level increases steadily by 12 inches in the past 4 hours as it approaches high tide. We can compute how much sea level change per hour by finding the unit ratio of the change in sea level to the change in time: Slide 2 Definition Calculate the Rate of Change Definition Example\n\nLehmann, Intermediate Algebra, 4ed Section 2.4 So, sea level increases by 3 inches per hours. This is an example of rate of change We say, \u201cThe rate of change of sea level with respect to time is 3 inches per hour.\u201d The rate of change is constant because sea level increases steadily Slide 3 Definition Calculate the Rate of Change Solution\n\nLehmann, Intermediate Algebra, 4ed Section 2.4 Examples of rates of changes: The number of members of a club increases by five people per month. The value of a stock decreases by \\$2 per week. The cost of a gallon of gasoline increases by 10\u00a2 per month. Slide 4 Examples of Rates of Change Calculate the Rate of Change Examples\n\nLehmann, Intermediate Algebra, 4ed Section 2.4 Suppose that a quality y changes steadily form y 1 to y 2 as a quality x changes steadily from x 1 to x 2. Then the rate of change of y with respect to x is the ratio of the change in y to the change in x: If either quantity does not change steadily, then this formula is the average rate of change of y with respect to x. Slide 5 Formula for Rate of Change and Average Rate of Change Calculate the Rate of Change Definition\n\nLehmann, Intermediate Algebra, 4ed Section 2.4 1.The number of fires in U.S. hotels declined approximately steadily from 7100 fires in 1990 to 4200 in 2002. Find the average rate of change of the number of hotel fires per year between 1990 and 2002. Slide 6 Finding Rates of Change Calculate the Rate of Change Example Solution\n\nLehmann, Intermediate Algebra, 4ed Section 2.4 The average rate of change of the number of fires per year was about \u2013241.67 fires per year. So, on average, the number of fires declined yearly by about 242 fires. Slide 7 Finding Rates of Change Calculate the Rate of Change Solution Continued\n\nLehmann, Intermediate Algebra, 4ed Section 2.4 2.In San Bruno, California, the average value of a two-bedroom home is \\$543 thousand, and the average value of a five-bedroom home is \\$793. Find the average rate of change of the average value of a home with respect to the number of bedrooms. Slide 8 Finding Rates of Change Calculate the Rate of Change Example Continued\n\nLehmann, Intermediate Algebra, 4ed Section 2.4 Consistent in finding signs of the changes Assume that number of bedrooms increases form two to five Assume that the average value increases from \\$543 thousand to \\$793 thousand Slide 9 Finding Rates of Change Calculate the Rate of Change Solution\n\nLehmann, Intermediate Algebra, 4ed Section 2.4 Average rate of change of the average value with respect to the number of bedrooms is about \\$83.33 thousand per bedroom Average value increases by about \\$83.33 thousand per bedroom Slide 10 Finding Rates of Change Calculate the Rate of Change Solution Continued\n\nLehmann, Intermediate Algebra, 4ed Section 2.4 Suppose that a quantity p depends on a quantity t: If p increases steadily as t increases steadily, then the rate of change of p with respect to t is positive If p decreases steadily as t increases steadily, then the rate of change of p with respect to t is negative Slide 11 Increasing and Decreasing Quantities Calculate the Rate of Change Properties\n\nLehmann, Intermediate Algebra, 4ed Section 2.4 Suppose that a student drives at a constant rate. Let d be the distance (in miles) that the student can drive in t hours. Some values of t and d are shown in the table. Slide 12 Comparing Slope with a Rate of Change Slope Is a Rate of Change Example 1.Create a scattergram. Then draw a linear model.\n\nLehmann, Intermediate Algebra, 4ed Section 2.4 Draw a scattergraph that contains the points 2.Find the slope of the linear model. Slide 13 Comparing Slope with a Rate of Change Slope Is a Rate of Change Solution Slope formula is, replacing y and x with d and t, respectively, we have: Example Continued Solution\n\nLehmann, Intermediate Algebra, 4ed Section 2.4 Arbitrarily use the points (2, 120) and (3, 180) to calculate the slope: Slide 14 Comparing Slope with a Rate of Change Slope Is a Rate of Change Solution Continued The slope is 60 Checks with calculations shown in the scattergraph\n\nLehmann, Intermediate Algebra, 4ed Section 2.4 3.Find the rate of change of distance per hour for each given period. Compare each result with the slope of the linear model. a.From b.From Slide 15 Comparing Slope with a Rate of Change Slope Is a Rate of Change Example Continued Calculate rate of change of the distance per hour from Solution\n\nLehmann, Intermediate Algebra, 4ed Section 2.4 The rate of change (60 miles per hour) is equal to the slope (60) For part b., calculate the rate of change of distance per hour from Slide 16 Comparing Slope with a Rate of Change Slope Is a Rate of Change Solution Continued\n\nLehmann, Intermediate Algebra, 4ed Section 2.4 The rate of change (60 miles per hour) is equal to the slope (60) Slide 17 Comparing Slope with a Rate of Change Slope Is a Rate of Change Solution Continued\n\nLehmann, Intermediate Algebra, 4ed Section 2.4 If there is a linear relationship between quantities t and p, and if p depends on t, then the slope of the linear model is equal to the rate of change of p with respect to t. Slide 18 Slope is a Rate of Change Slope Is a Rate of Change Property\n\nLehmann, Intermediate Algebra, 4ed Section 2.4 Suppose that a quantity p depends on a quantity t: If there is a linear relationship between t and p, then the rate of change of p with respect to t is constant. If the rate of change of p with respect to t is constant, then there is a liner relationship between t and p. Slide 19 Constant Rate of Change Slope Is a Rate of Change Property\n\nDownload ppt \"Slope Is a Rate of Change Section 2.4. Lehmann, Intermediate Algebra, 4ed Section 2.4 The ratio of a to b is the fraction A unit ratio is a ratio written.\"\n\nSimilar presentations","date":"2018-01-22 01:36:36","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8220283389091492, \"perplexity\": 486.446474668851}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-05\/segments\/1516084890928.82\/warc\/CC-MAIN-20180121234728-20180122014728-00530.warc.gz\"}"}
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Q: Check for status of subprocess to kill audio playback in python Im currently using sox to apply effects to an audio file and play the file back. Currently I have the code set to create a new subprocess to play the file when button 1 is pressed and to kill the process when button 2 is pressed.
What I want to do is change it so that if Button 1 is pressed multiple times it will check to see if a previous process is active and if so kill it and if not then just play as normal.
My code as of now is:
def Button_1():
p = subprocess.Popen(['play','playback.wav'])
def Button_2():
p.send_signal(signal.SIGNIT)
A: Use p.poll() to check if the process is still running.
The code will look something like this (untested):
def Button_1():
if p.poll():
p.send_signal(signal.SIGNIT)
else:
p = subprocess.Popen(['play','playback.wav'])
def Button_2():
p.send_signal(signal.SIGNIT)
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Tuesday through Friday, 10 a.m.–5 p.m.
Saturday and Sunday, 12 noon–5 p.m.
Closed Mondays; all College holidays; July 4, 2019; August 12–September 2, 2019; November 28–29, 2019; and December 9, 2019–January 6, 2020.
Middlebury College is located in the town of Middlebury, midway between Rutland and Burlington, Vermont, at the intersections of Vermont Routes 7, 30, and 125. The museum is in the Kevin P. Mahaney '84 Center for the Arts on the southeastern edge of the Middlebury College Campus.
For directions from other colleges and nearby cities, check the Middlebury Campus page, or see below.
From Route 7, coming from the North: Stay on Route 7 South. Follow signs for Route 30. The Center for the Arts is about 4/10 mile down Route 30 to the left. To park, turn at sign: Parking, Center for the Arts.
From Route 7, coming from the South: Stay on Route 7 North. Follow signs for Route 125 West, then signs for Route 30. The Center for the Arts is about 4/10 mile down Route 30 to the left. To park, turn at sign: Parking, Center for the Arts.
From Route 30, coming from the South: Coming up the hill toward the College, you will pass (on right) the College golf course, football stadium, and several field houses. The Center for the Arts is the next building on the right. To park, turn right onto the next road (after passing two more College buildings on the right) at sign: Parking, Center for the Arts.
From Route 125, coming from the East: Follow Route 125 through Hancock and over the Middlebury Gap, through Ripton. Stay on Route 125 through East Middlebury to Route 7. Turn right onto Route 7 North. Follow directions (above) for "Route 7, coming from the south," after junction with Route 125.
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{"url":"http:\/\/newsinfoguide.com\/download\/category\/science-mathematics\/page\/4","text":"# Posts in category: Science Mathematics\n\n\u2022 Home\n\u2022 Science Mathematics\n\n## Download Continued Fractions. Volume 1: Convergence Theory by Lisa Lorentzen PDF\n\nApril 4, 2017 | | By admin |\n\nBy Lisa Lorentzen\n\nEndured Fractions comprises volumes -- quantity 1: Convergence concept; and quantity 2: illustration of features (tentative title), that's anticipated in 2011. quantity 1 is devoted to the convergence and computation of persisted fractions, whereas quantity 2 will deal with representations of meromorphic features by means of endured fractions. Taken jointly, the 2 volumes will current the elemental endured fractions idea with no requiring an excessive amount of prior wisdom; a few simple wisdom of complicated features will suffice. either new and complicated graduate scholars of persevered fractions shall get a complete figuring out of ways those endless constructions paintings in a couple of functions, and why they paintings so good. A diverse buffet of attainable purposes to whet the urge for food is gifted first, earlier than the extra simple yet modernized idea is given. This re-creation is the results of an expanding curiosity in computing designated capabilities by way of endured fractions. The tools defined intimately are, in lots of circumstances, extremely simple, but trustworthy and effective. Contents: Introductory Examples; fundamentals; Convergence standards; Periodic and restrict Periodic persisted Fractions; Numerical Computation of persevered Fractions, a few persevered Fraction Expansions.\n\n## Download Elementary computability, formal languages, and automata by Robert McNaughton PDF\n\nApril 4, 2017 | | By admin |\n\nBy Robert McNaughton\n\n## Download Imbeddings of three-manifold groups by Francisco Gonzalez-Acuna, Wilbur C. Whitten PDF\n\nApril 4, 2017 | | By admin |\n\nBy Francisco Gonzalez-Acuna, Wilbur C. 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## OUT OF THE DEEP
A MYSTERY IN ACADIA NATIONAL PARK
## GLORIA SKURZYNSKI AND ALANE FERGUSON
Text copyright © 2002 Gloria Skurzynski and Alane Ferguson
Cover illustration copyright © 2008 Jeffrey Mangiat
All rights reserved.
Reproduction of the whole or any part of the contents is prohibited without written permission from the National Geographic Society, 1145 17th Street N.W., Washington, D.C. 20036.
Map by Carl Mehler, Director of Maps. Map research and production by Joseph F. Ochlak and Martin S. Walz Humpback whale art by Joan Wolbier
This is a work of fiction. Any resemblance to living persons or events other than descriptions of natural phenomena is purely coincidental.
Library of Congress Cataloging-in-Publication Data
Skurzynski, Gloria.
Out of the deep / by Gloria Skurzynski and Alane Ferguson.
p. cm.—(Mysteries in our national parks; #10)
Summary: Jack, Ashley, and their unreliable new foster sister set out to solve the mystery of why whales are beaching themselves at Acadia National Park.
ISBN: 978-1-4263-0973-1
[1. Whales—Fiction. 2. Acadia National Park (Me.)—Fiction. 3. Foster home care—Fiction. 4. National parks and reserves—Fiction. 5. Mystery and detective stories.] I. Ferguson, Alane. II. Title. III. Series.
PZ7.S6287 Ou 2002
[Fic]—dc21
2002005547
For Stephanie Alm,
a rising star
## ACKNOWLEDGMENTS
The authors want to thank the following people for their wonderful help. At Acadia National Park: David A. Manski, Biologist and Chief of Resources Management; David Buccello, Chief Park Ranger; Deborah Wade, Interpretive Ranger. At Allied Whale, we're extremely grateful to Sean Todd, Senior Researcher. Sean is also Professor of Science Resource at College of the Atlantic. Many thanks also to Rosemary Seton, Whale Biologist, Director of Stranding Response Program. We're grateful to District Court Judge Kevin Sidel for his suggestions and to GenAnn Keller, Librarian. Very special appreciation goes to Vicki Lockard, editor of Canku Ota (Many Paths), an online newsletter celebrating Native America, for granting us permission to use the legend about the Great Spirit and the bowhead whale. Visit Canku Ota at http://www.turtletrack.org/
## CONTENTS
CHAPTER ONE
CHAPTER TWO
CHAPTER THREE
CHAPTER FOUR
CHAPTER FIVE
CHAPTER SIX
CHAPTER SEVEN
CHAPTER EIGHT
CHAPTER NINE
CHAPTER TEN
CHAPTER ELEVEN
AFTERWORD
ABOUT THE AUTHORS
##
"Of course I know what's at stake," the man said gruffly, pressing the receiver against his ear. "Millions. A huge international deal. Don't worry, I won't screw up. I'll do whatever it takes. You know that."
He took a drag from his cigarette and looked around to make sure his conversation had gone unnoticed. Through the haze, he saw a couple huddled over a small table, while a grizzled man stared vacantly into his glass.
It was then that he noticed the top of a head rising from a nearby booth and two round eyes staring at him. Anger surged through him. What was a kid doing in a place like this? How much had she heard?
"Something just came up. I'll call you back," he said, slamming the pay phone into its cradle. He couldn't let some kid ruin his plan. Not now—not when they were about to cash in!
He turned quickly to make his way toward the girl. Whatever it takes, he told himself. Whatever it takes....
## CHAPTER ONE
When Jack saw his mother's face, he knew the news on the other end of the phone was bad.
"Two more are dead?" Olivia cried into the receiver. Squeezing her eyes shut, she let out a long sigh. "This is nothing short of bizarre. I truly don't understand why they're dying this way. Where did you find the bodies? Uh-huh," she nodded, peering at a map she'd laid out on her desk, tracing it lightly with her fingertip. "Yes, I can see where that is—right at the edge of the peninsula. How badly decomposed?" Another pause, and then, "I'm sure that will make your job harder. The stench can be overpowering."
"What's up?" Jack's 11-year-old sister, Ashley, asked as she walked into the Landons' study. Their newest temporary foster child, Bindy Callister, trailed behind, a bowl of popcorn perched on her round hip. Munching noisily, she shoveled another fistful into her mouth, her cheeks bulging out like a chipmunk's. Although Bindy had been at the Landons' home for only three days, she already knew where all the food was kept and didn't seem the least bit shy about foraging through the cupboards, helping herself to whatever she found. The strangest foster kid we've ever taken in, Jack decided the day Bindy arrived.
It wasn't about the way she looked, although that had been odd enough. Bindy's tie-dyed T-shirt was wildly bright, with fluorescent swirls that splashed across her in neon constellations. Mousy brown hair had been pulled into a limp ponytail, and her too-tight jeans looked as though they'd fused onto her skin. Loud and boisterous, Bindy seemed to think she knew something about absolutely everything. When Ms. Lopez, the social worker, tried to speak, Bindy talked right over her, waving her arms as though she were on stage.
"Be patient with her," Jack's father, Steven, had told him later. "I know she can be a bit—overpowering—but she's been through a lot."
"'Cause her own parents don't want her," Ashley told Jack. "I heard Ms. Lopez tell Mom about it."
"Please don't say anything about that to Bindy!" Steven urged.
"Oh, I won't. It's just really sad. I don't know what I'd do if you and Mom didn't want me."
Now, as Bindy settled into a chair next to Jack, he tried to imagine what it would be like to be 14 years old and dumped into a foster home, waiting to hear what the judge ruled about your life. How would it be to have your family reject you? How would it be to have your whole future decided by someone you'd never even met before? No matter how annoying she was, Jack knew he'd have to give Bindy some space. It was the least he could do.
"Ashley, sit there," Bindy directed, pointing to a spot on the floor. "I get the chair because I'm older than you. Age before beauty!"
Ashley shot Jack a look, shrugged her shoulders, then dropped onto the carpet.
"OK. But we have to be quiet, Bindy," Ashley whispered. "Mom's on the phone."
Though Olivia Landon normally worked at the Elk Refuge at Jackson Hole, she'd converted a corner room into a home office. The large oak desk was piled high with papers stacked into a double helix. A tall coffee mug and a glass of water sat next to the computer keyboard, something their father said was dangerous, but Olivia insisted she was careful enough to handle things in her own space. Books wrapped in every color of the rainbow filled an oversize bookshelf, all of them bearing scientific titles that twisted Jack's tongue when he tried to read them out loud. The pale blue walls had been peppered with pictures of every kind of wildlife, from soaring eagles to bright-eyed foxes to coiled snakes, all photographed by their father, who dreamed of becoming a full-time photographer. Jack loved the clutter of it all. "Ideas ferment in here," Olivia always told them.
"So there's no sign of disease?" she was asking into the phone. She twisted her chair from side to side, paused, then asked, "When will the results be in?"
As Bindy noisily sucked the butter off each finger, Jack felt his teeth clench, but he willed himself to be patient. It sounded as though there might be bigger problems than an annoying foster kid.
"Hey, Jack-o, you want some popcorn?" Bindy asked, extending the bowl in his direction. "I made it the real way, with a pan and oil and real butter instead of that imitation-powder-microwave junk—"
He shook his head. "Shhhh. Mom's talking to a biologist at Acadia National Park. They found more bodies on the beach, which brings the total to 12."
"Twelve people? No way!" Bindy bellowed, slapping a thick thigh.
"Don't be stupid," Jack hissed. "A whale and some seals and stuff from the ocean. They've washed up dead, and the park people can't figure out why. Nothing quite like this has ever happened before. Try to be quiet, Bindy. My mom's talking, and I want to listen."
"Can you cut off the heads and put them on ice?" he heard his mother ask. Nodding tersely, she scratched notes on a yellow pad. Her reading glasses rested on her thin nose like half-moons, while her hair swirled to her shoulders in dark, smoky curls. Olivia, a wildlife veterinarian, was frequently called in by the parks to solve animal mysteries. There was a good possibility that she'd now be asked to Acadia National Park in Maine, about as far as you could go from their home in Jackson Hole, Wyoming, and still be in the continental United States. And maybe, Jack hoped, the rest of them would get to go, too. He'd never been to Maine.
Sighing, Olivia said, "It's a big job, but you'll need to cut off the whale's head and cool it down fast. Decomposition hides details. Heads can yield valuable clues."
Wrinkling her nose, Bindy cried, "Cutting off heads? That's so gross."
Ashley placed her finger to her lips. "Shhhh."
"So, who's Acadia?" Bindy asked Jack, not bothering to keep her voice low.
"Acadia is a park," he answered softly.
Olivia gave the three of them a look and extended her hand in a signal that meant "be quiet or get out." Flipping the page, she scrawled more notes.
"I still don't get it," Bindy pressed. "Why are the people in Acadia calling your mom? She's a vet in Jackson Hole. Doesn't she just deal with elk and other four-footed creatures?"
"Mom knows all about whales. She did a whole seminar on them when she was at College of the Atlantic," Ashley whispered. "Now hush—"
It was too late. "Excuse me, Sean. I'm sorry to interrupt, but my kids are chattering and I can't hear a word you're saying." Olivia covered the mouthpiece of the phone and waved them away. "You kids go outside for awhile. Better yet, start packing. Something's really wrong in Maine, and I need to get out there fast. We'll all go."
"Me too?" Bindy asked, wide-eyed.
"You too. We're going to Acadia!"
To Jack, it seemed that Bindy never stopped blabbing the whole trip. Luckily, the airlines provided earphones that could be plugged in to recorded music. Jack turned the volume as loud as he could, trying to drown out Bindy. The only time she kept silent was when she munched on the pretzels the flight attendants brought. Bindy never settled for one bag of pretzels; she always demanded three or four.
For just a little while, when they reached their motel, Bindy stayed silent. She stood on the little deck outside the room she was to share with Ashley, awestruck at the beautiful Atlantic Ocean. Gulls swooped down into the waves, picking up shellfish and dropping them onto the rocks on the beach—when the shells broke open, the gulls feasted on the critters inside.
A long wooden pier stretched from the shore, reaching like a bony finger about 60 feet into the ocean. It looked rickety, as though its support pilings had been eroded by decades of salt water. Halfway along its length there was a No Trespassing sign hanging from a chain that stretched between two posts. Except for one small rowboat near the shore, no other boats were tied to the pier.
"This is my first look at the Atlantic," Bindy murmured. "It looks greener than the Pacific."
"You've seen the Pacific?" Ashley asked.
"I used to live in California," she answered. "In Hollywood, actually."
Yeah, sure, Jack thought. In Hollywood, with movie stars, no doubt. He'd had enough of Bindy. The deck outside the girls' room was connected to the deck outside his parents' room, where Jack would be sleeping on a cot. With one hand on the banister, Jack vaulted over the railing onto his parents' deck. Then he felt like a fool, because the sliding door to his parents' room was locked. He was stuck out there, while the girls laughed at him.
Even though he'd been trying to avoid Bindy, later that evening Jack found himself knocking on the girls' door. He'd had enough of watching waves lap the shore, and his parents weren't being much company right then.
"Mom's reading a bunch of research papers about whales and Dad's going through his camera stuff, so I came to see what you guys are doing," he told Ashley when she opened the door.
"Not much. We're just flipping around the different channels." With her arm straight out, Ashley clicked the channel changer button on the remote control again and again. Bindy had spread herself on one of the queen-size beds with a book propped under her chin. She didn't bother to look up.
Suddenly Ashley yelled, "Hey wait! Look there—it's one of my favorite movies. Melissa's Dream."
"You've already seen that about ten times," Jack told her, wrestling her for the remote. "You don't need to watch it again."
Ashley struggled to keep the channel changer out of Jack's reach, but his arms were longer than hers. "Jack! The movie's almost over anyway—just let me finish watching it 'cause the end's the best part."
Lifting the changer so high that Ashley couldn't reach it—considering that Jack was a good head taller than his puny little sister—he said, "OK, we'll let Bindy decide. Bindy, do you want to watch the end of this dumb movie or...."
"It's not dumb," Bindy answered. "I was in that movie."
There she goes again, Jack thought. "You mean you were into the movie," he said sarcastically. "Like if you go, 'I'm really into stock-car racing.' Or 'I'm really into extreme sports.' Or 'I'm really into potato chips.'"
Bindy shook her head. "I mean I was in the movie. I acted in it. I didn't have the leading role, but I was the cute little girl next door."
Staring at the screen, Ashley asked, "You mean Amanda? That was you? No way."
"Amanda's a redhead," Jack protested.
Scornfully, Bindy slapped her book onto her bed. "Well, duh! You've heard of hair dye, haven't you? I told the set's hairdresser not to make me so red, but she wouldn't listen because she said red was what the script called for, so red I would be. I told her it made me look like a pumpkin head. She got all mad when I said that, and then told me I didn't know anything about the business, and the only words I should speak in her presence were the lines from my script. What a grouch!" Pointing a ring-clad finger, Bindy said, "See, there I am—right there. I just walked into Melissa's kitchen. That's me."
Jack studied the girl on the small screen. If he squinted, maybe that girl did look a little bit like Bindy, but she was a lot younger and she was—thin!
"Oh, come on. You're just teasing...," Ashley began.
"No. It's me. I swear!" Flopping onto her stomach, Bindy crossed her ankles and propped her chin in her hand. Then, amazingly, she began to recite the lines at the same time the girl in the movie was saying them. Word for word, without hesitation, her lips moved in perfect sync with the dialogue on the screen. Even if she was faking it, Jack had to give her credit for a good memory.
When Bindy finished, she shot them a triumphant look. Ashley stared at her. "So that really was you!"
"Of course," Bindy said matter-of-factly. "I don't lie. OK, now it's over. Here come the credits. There's my name—Belinda Taylor—that's me."
"But your name's Bindy Callister," Jack broke in.
Rolling her eyes, Bindy sighed loudly. "Bindy is short for Belinda. And my name used to be Taylor until I was adopted. Before my real mother died."
"Oh," Ashley murmured. "I'm sorry...."
"Yeah, well, it was a long time ago, and I don't like to talk about it." Jerking her fingers through her thin hair, Bindy seemed to shift gears. "So anyway, I acted in seven TV commercials and one sitcom, and I had parts in two movies. The first movie was just a small part, but in Melissa's Dream my role was a lot bigger."
"Wow!" Ashley blurted. "That is way cool. Tell us about it. Tell us everything. Did you meet famous stars? What were they—"
Throwing up her hand like a traffic cop, Bindy demanded, "Wait! First things first. Is there anything to eat in this room? 'Cause I'm starved. If I'm going to do any talking, I need something to eat. And a can of something to drink—anything but diet. I hate diet soda."
"There's a candy machine at the end of the hall," Jack told her. "I have some change."
"So get me two Butterfingers and a can of orange soda," Bindy ordered. "Thanks, Jack-o. You're a real pal."
## CHAPTER TWO
Sheesh! She was so bossy! Jack hurried down the hall to the candy machine, halfway eager to hear Bindy's story, but three-fourths of the way doubting that whatever she told them would be true. After all, Bindy was a known liar.
The evening Bindy had arrived at the Landon home, Jack had overheard his parents talking about the reason she'd been placed into temporary foster care. Olivia and Steven were at the kitchen table in terrycloth robes, sipping mugs of hot tea, their voices barely above whispers as they discussed Bindy's situation. Jack had hung back in the hallway, just until they were finished talking. It wasn't exactly eavesdropping, he'd told himself. He just didn't want to interrupt.
"...Bindy's brother Cole," Olivia was telling Steven. "According to Ms. Lopez, the tension between Bindy and Cole goes way back. Apparently he's some kind of a football star."
"More like a superstar," Steven countered. "Ms. Lopez told me Cole has already been offered full scholarships from colleges all around the country."
"Did you know that after Bindy's accusation, the football coach and his teachers all wrote letters of support for Cole, saying he was an honest and decent kid who couldn't possibly do such a thing? Ms. Lopez said no one came forward to defend Bindy."
When he craned his neck ever so slightly, Jack could catch a glimpse of his father. Steven shook his head and took another sip from his mug. "So, Bindy's adoptive parents believe Cole is telling the truth and Bindy is flat out lying. How sad for Bindy."
"I know. Still, it's possible she invented the whole thing, Steven. Ms. Lopez says Bindy comes up with one fantastic story after another. Even Ms. Lopez isn't sure how much of what Bindy says is true."
Steven set down his mug. "Having said that, it's still no excuse for what the parents are doing. I mean, how could anyone try to get rid of their own child, even if she's adopted?" Suddenly, his head jerked up and he looked toward where Jack was standing. "Wait a minute—Jack! Why are you lurking out there in the hall?"
Shuffling his feet, Jack emerged from the shadows. For the next ten minutes, his parents gave him a verbal going-over. Jack should never listen in on their conversation. They respected Jack's privacy, and he should do the same for them. They told him to keep everything he'd heard to himself because Bindy's private affairs were just that—her own private affairs. Since the social workers and the therapists didn't know what to make of Bindy's story, Jack shouldn't judge it, either. Instead he should give Bindy the benefit of the doubt.
He was not to tell any of this to Ashley. Finally, if Bindy wanted to share her own story with Jack, that was fine, but he should in no way ask Bindy about her court case. Let her come to you, was how his mom had put it.
Now, when Jack returned to the girls' motel room, he found his sister working Bindy's hair into stubby braids. Smiling brightly, Bindy held out her hand for the Butterfingers.
"Here," Jack told her. "Catch!"
She caught one neatly and tore off the wrapper, then took a big bite. With her mouth full, she said, "Ashley's been asking questions about my so-called career, but I told her to wait until you got back. That way I can double my audience, ha ha." Pulling herself into a seated position, she proceeded to tell them about her life B.C.—before the Callisters.
She could hardly remember her father. He was killed in a speedboat accident off the coast of California when she was only three.
Back then, Bindy was cute—everyone said so—with nice round cheeks, big blue eyes, and curly blond hair. Her mother signed her with a casting agent who arranged for Bindy to make a few commercials, but the jobs were few and far between. Bindy and her mother lived in a tiny apartment over a garage two blocks from Hollywood Boulevard. "Then I got my first movie," she said.
"And after that you were rich and famous," Ashley stated, believing it.
"Ha! I wish! I had about a dozen lines to say in that movie. Mostly I had to jump rope to 'Down by the river, down by the sea, Johnny broke a milk bottle, blamed it on me....' We must have shot that scene 20 times, and when I got so tired I started to cry, my mother took me on her lap and told me this was my big chance, and I had to be brave. So I kept doing it, over and over."
"How old were you?" Ashley asked.
"Seven. I didn't get the part in Melissa's Dream till I was nine. By then, my mother already had cancer."
Each day, Bindy said, her mother managed to take her to the studio where the movie was being filmed; each night they returned to the cramped apartment over the garage and rehearsed the script again and again until Bindy learned her lines. She had to be perfect; she couldn't lose that job, because they had no other income and no hospital insurance.
"That must have been awful for you," Ashley sympathized.
"No it wasn't, because we were a team. My mother loved me!" Bindy said fiercely. "We were always together—she stayed with me on the set every minute. It was months before Melissa's Dream was released in the theaters, and my mother kept getting sicker, but finally we went together to see the movie. Two days later, she died."
Jack felt his throat tighten as he thought of what Bindy had been through. If Bindy Callister—or Belinda Taylor—wasn't telling the truth, she was one fabulous actress. But she stayed dry eyed as she sat there telling them the rest of her story, which only got worse.
"So Aunt Marian came to Hollywood to take me home with her—she was my mother's sister. She'd seen Melissa's Dream, too, and she thought if she adopted me, she'd get a pretty, talented little girl to be part of her family, along with her handsome, smart, athletic son, Cole. A perfect Barbie to go with her perfect Ken doll. Hey, throw me that other Butterfinger, would you, Ashley?"
"So...so what happened?" Ashley whispered.
"Well, I didn't want to be part of Aunt Marian's perfect family. I hated Cole on sight, and he hated me, too. So...I ate. And the more I ate, the more upset Aunt Marian got. Twice she dragged me back to Hollywood to get me into another movie, but the casting director took one look at me and said no film needed a prepubescent girl with weight issues. That's how they talk in Hollywood." Bindy threw back her head and laughed a laugh so full of anger it made Jack feel creepy.
He wanted out of there. Reading his mother's thick books on whales would be better than hearing Bindy talk about her awful life, even if it was all made up.
Someone was moving around in the room. Jack opened one eye to stare at the digital clock on the lamp table. 12:35. Barely past midnight. He'd been asleep for only one uncomfortable hour, because the cot he was on felt lumpy.
Hair stood up on his arms as he watched the deep shadow glide silently across the floor. He could make out the outline of his parents in their bed, so it wasn't one of them who'd gotten up to go to the bathroom or anything. The shadowy shape, moving so soundlessly through the room, had to be an intruder. A thief! His fingers trembled as he watched the shape move closer. Should he call out and wake his parents or just keep quiet and let the thief take whatever he wanted?
His heart thumping so loudly the thief might hear it, Jack opened his other eye. Whoever the intruder was, he seemed awfully small. Then the shadowy figure bumped into Jack's cot and muttered, "Ouch!"
That was Jack's chance. He leaped up and grabbed the person, who wiggled and yelled, "Let go, you dork!"
"Ashley?"
"Who'd you think it was? Freddy Krueger?"
By then Jack was feeling pretty stupid—for the second time in the past six hours—so he grumbled, "What the heck are you doing sneaking around in the dark?"
"What's going on?" Olivia asked, turning on the bedside lamp. Her dark hair sprang from her head in wild curls. Blinking hard, she asked, "Where's Bindy?"
"That's what I came to tell you," Ashley answered calmly. "She's gone."
Steven sat upright. "Gone? Gone where?"
Plunking herself down on the end of Jack's cot, Ashley replied, "I have no idea. I heard a door close, and at first I thought it was the bathroom door and Bindy had just gone in to...you know. I was kinda sleepy, so I don't know how much time went by, but then I looked over at her bed and it was empty. I got up and looked into the bathroom, and that was empty, too."
Both Steven and Olivia were on their feet so fast it was as though they'd been shot out of a cannon.
They practically dove into their jeans and then pulled sweatshirts over their heads, yanking them into place as they ran through the door that connected Ashley's room to theirs. In less than a minute they were back, looking grim.
"I'll check at the front desk," Steven said.
"It's after hours. I doubt anyone will still be there," Olivia said.
Jack suggested, "Maybe she just couldn't sleep. She could have gone out on the beach to look at the waves." Before he even spoke the last word, Steven rushed out the door. Jack listened for the clatter of his father's footsteps going from the front deck down the wooden stairs to the parking lot, then remembered that his dad hadn't bothered to put on shoes. Steven's feet were going to get awfully sore clambering barefoot over the rocky beach.
Olivia had begun to page through a phone book, muttering, "I'm calling the police."
"Don't you think you ought to wait a little while?" Jack asked her. "At least until we look around the motel. Maybe she just went for a Coke in the drink machine." Or a couple more candy bars, he thought.
Olivia slammed down the receiver, saying, "You're right. Kids, get dressed. We'll do a thorough search. Then I'll call the police. And grab your dad's shoes on your way out. He's going to need them."
After catching up to Steven and handing him the shoes, Olivia and Ashley left to scour the grounds of the motel while Steven and Jack walked along the shore, peering inside weathered boats and searching protruding rock formations that seemed to bubble up from the water's edge. Lights from a few distant buildings twinkled in the darkness. Jack would have felt cheered if it hadn't been such a serious situation. Olivia's faint voice wafted to them. "Anything?"
Steven called back, "No!" Then, to Jack, he grumbled, "This is ridiculous. Where could she be?"
Suddenly, Jack snapped his fingers. "Hey, wait a minute—did you check the pier?"
"Of course I checked the pier."
"But did you go past the No Trespassing sign? There are steps way at the end of the pier—I could see them from our balcony. Bindy might be sitting at the bottom of the steps just looking at the waves. She said she likes the Atlantic," he finished lamely.
Steven sighed and ran his fingers through his thinning blond hair. "I went to the point where the chain blocks it off. I couldn't see too well, but I called for her. Nothing. She's not there."
"Except she might not have answered you. She's really weird. Maybe she'll answer if it's me. Dad, you just keep looking around. I'll be right back."
"All right. I'll give you two minutes. Then we're going back to the motel to get help."
"Right. Two minutes!" Broken seashells crunched underfoot as Jack made his way to the pier. Tall enough for larger boats to load and unload, the pier had a row of rickety stairs that descended from the far end to the water's surface. At the halfway point, a metal chain had been strung across to prevent access, with the No Trespassing sign hanging from the links and a smaller sign reading "Enter at your own risk" beneath that. But no signs would keep Bindy out. She did what she wanted to do and went where she wanted to go.
Moving along the creaking, splintery slats, Jack called out her name into the night sky. Only the sound of waves and the groan of the wooden pier echoed back. As he squinted into the darkness, he saw what he thought was a dark shape, a deep patch of black turned toward the sea. "Hey, are you there?" Jack cried. The shape seemed to move farther away, hovering at the pier's end, then disappearing.
Glancing around quickly to see if his dad was watching, Jack easily climbed over the chain. A sudden wind whipped his face, ballooning out his shirt as though it were a shroud and rocking the pier like a trapeze. For a moment Jack wondered whether there might be any missing boards underfoot that he couldn't see in the dark. He didn't want to fall through onto the rocky beach beneath, where waves could grab him and soak him to the skin. With every motion the sun-bleached boards creaked under his feet, as the cold and insistent wind tried to push him backward. But someone was down there. It had to be Bindy.
He cupped his hands again against the wind and called out, "Is that you?" After waiting a beat, he shouted again. The boardwalk stretched into darkness. Jack could hear, rather than see, the water beneath him, rushing against the timbers before receding back to the sea.
The inky night at the pier's end seemed denser now. As the shape blocked his view of whitecaps on the dark waves, Jack noticed a pale, orange glow. It illuminated the shadowy figure's head. The shape was bigger than Bindy, taller, broader in the shoulders—or was Jack being deceived by the darkness? He took another step. "Bindy?"
When the shape turned, Jack's breath sucked into his throat. This wasn't Bindy. It was a man, dressed in black, with a black wool coat that skimmed the tops of his boots. A rectangular metal box—a suitcase?—rested inches from his feet.
"What do you want?" the man growled. His knit cap had been pulled down onto his thickly featured face. A cigarette hung from his lips, the lit end dancing in the night. The light from the cigarette let Jack see the man's expression, and the look made his mouth go dry.
"I asked you a question. Are you going to answer me?" The man took a drag from the cigarette, then flicked it into water.
"I want—nothing," Jack stammered. "I'm looking for a girl. Have you seen her?"
"I dunno. What's she look like?"
"She's 14, she has light brown hair, and she's...uh...." Jack made a half-hearted gesture.
"Kind of chunky?" the man finished.
"Yes! So you saw her?"
"No, I didn't see anyone like that. I came down here for a private smoke," he answered, lighting up another cigarette. "No one indulges anymore, so I have to find places where I won't bother anyone, and no one will bother me. OK?"
"It's just that she—the girl—is missing. Have you been up here long?"
"No." Taking another drag, the man blew it between his teeth and asked, "Why?"
"I'm asking just in case maybe you saw her walking along the beach. We're really worried about her." Smoke curled toward Jack, and the smell hit him, acrid and pungent. How could anyone suck that stuff into their lungs? It was gross.
Suddenly, the heel of the man's boot struck hard on the boardwalk as he took a step forward. "Where are you staying?"
The question caught Jack off guard. "At the Seaside Motel. Up there." He gestured.
"Yeah? What's your name?"
"I—I don't think you need to know my name. Anyway, I'd better go." There was something wrong here, something Jack couldn't quite put his finger on. The man had only taken a single step toward him, and yet Jack felt his muscles tense in a "flight or fight" reaction.
He was relieved when he saw his father halfway down the beach. Steven spotted him and waved his arms in the air. "Jack!" he yelled. "I told you not to go past the chain. Come back here right now!"
The man snorted. "So it's Jack, is it? Well, Jack, I guess it's time for you to go. To answer your question, I didn't see anything, I didn't hear anything. And Jack—it'd be smart if you did the same."
What did that mean? Spinning on the toes of his sneakers, Jack began to climb the stairs. With his back toward the man, he felt exposed, as if something might hit him between the shoulder blades at any moment. Don't be stupid, he chided himself. The man's just weird. With Bindy gone, the Landons had bigger problems. Swinging himself over the chain, he hurried along the pier to join Steven, who had a look of panic on his face.
"Dad—there's this guy up on the pier—"
"Did he see Bindy?"
"No."
"I can't find her anywhere. Let's move it. We need to look around the motel."
The four Landons checked all the halls, which were strangely empty. "What if she's gone into someone's room?" Steven worried.
Olivia groaned, "I can't even deal with that possibility. I'm calling the police right now!"
## CHAPTER THREE
Jack could hear only one side of the conversation as his mother stated, "Her name is Bindy Callister. B-I-N-D-Y. Short for Belinda. Fourteen, blondish hair, a bit overweight." With her hand over the mouthpiece, she asked Ashley, "Do you know what she had on?"
Ashley shrugged. "The last time I saw her, she was wearing a sleep shirt. She was reading in bed with the light on. Then I fell asleep."
Olivia had turned all her attention to the phone again, concentrating so hard it looked like she might shoot through the phone lines, like Trinity in The Matrix. "Yes," she was saying. "Yes, that's right. Fourteen. She is? You do? Oh thank—We'll be right there. Uh...where is the police station? We just arrived this afternoon, and we don't know anything about Bar Harbor." Grabbing a ballpoint pen from the desk drawer, Olivia began to scribble directions. Then, slowly, she returned the phone to its cradle.
"Good news or bad news?" Steven asked.
"Both. The police have her. But they picked her up in a bar."
Had Jack heard that right? "Did you say they picked her up in Bar Harbor?" he asked.
"No, I said in a bar. A place that serves alcohol. Oh, Steven," Olivia cried, reaching for his hand, "maybe we're in way over our heads with this girl. She was able to sneak out right under our noses. When I imagine what could have happened—maybe she's too much for us to handle. We've never dealt with anything like this before."
"Now calm down," he said. "Let's all pile into the car and find the police station."
That's what they did, heading onto the highway that led to Bar Harbor, since the Seaside Motel was located about five miles from the town proper. In the back seat, Ashley held a flashlight while Jack tried to follow the street map of Bar Harbor, and Olivia studied the directions she'd scribbled. The town wasn't all that big, but it had a lot of quirky little side streets that confused Jack. "I can't really tell...," he muttered. "Wait, turn here," he told his father, who was driving. After a couple more turns they found the police station, a pale brick building, squat and square and plain, as if it, like the state of Maine, would tolerate no nonsense. Lights radiated from inside the building, casting a greenish glow onto the street. What a scary place for Bindy to be! Steven must have been thinking the same thing, because he didn't even bother to parallel park. He left the car sitting with one tire on the curb and the headlights still on, as the family hurried into the station.
The first thing Jack saw when he walked in was Bindy. She sat alone on a wooden bench, elbows resting on her knees and her head in her hands. Her mousy hair had fallen forward to cover her features. When she looked up, Jack could see fear in her eyes.
A policewoman, stifling a yawn, stood up from behind her desk to approach the Landons. "Sorry to drag you folks in here in the middle of the night. I'm Officer Bartlett. Is this the girl you phoned about?"
"She's the one," Steven answered grimly.
"Officer Wilson picked her up in Smokey's Bar about an hour ago—the bar's up the hill, not too far from your motel. Anyway, the bartender had called us, saying he had a minor on his premises. He said she was a lot more underage than what he usually gets—which is, you know, 17-or 18-year-olds. That's why he didn't want to throw her out alone into the night. So we told him to just leave her there and not say or do anything until we sent an officer."
Olivia's brows knit together as she asked, "Is she being charged with a crime?"
"No. She didn't try to order any alcohol; she said she just went into the bar to use the pay phone. We could charge her with breaking curfew, but...let's just say she convinced us all that she'll never do it again. Your girl can be very persuasive."
Steven and Olivia sat down on either side of Bindy. Jack could tell that his mother was trying to keep her voice calm as she said, "That sounds pretty lame, Bindy. The pay phone? If you wanted to make a call, why didn't you just use the phone in your room?"
Squeezing her eyes tight, Bindy answered, "I didn't want Ashley to hear. It was a private call."
"To whom?" Steven demanded. "Who were you trying to call at midnight?"
"Why should I even answer? I know you won't believe me. Nobody ever believes me. Except these kind officers here. They listened."
"Try us," Steven said. It was Olivia, though, who reached out to cover Bindy's hand with her own. Maybe she'd noticed the tears welling up in the girl's eyes. Even from across the room, Jack had noticed that. Real tears? Or part of an act?
Her words came out in a rush. "I wanted to call Aunt Marian, but I never even got to use the pay phone because this jerky man was on it and he wouldn't hang up—he kept talking to someone about a boat and he was going on and on and on. I was in a booth right behind him, and I waited and waited, and then he turned and looked at me and said—" She stopped for breath, then muttered, "Forget it—it doesn't matter what he said.
So I went to ask the bartender if I could use his private phone and I'd pay him for the call, but before I could, the policeman came in and arrested me."
"Why did you want to call your aunt?" Ashley broke in. "You told us she was really mean to you."
Olivia shook her head, trying to cue Ashley to keep quiet, but too late—Bindy dissolved into tears as she wailed, "Because I want to go home. When we were watching Melissa's Dream, I started thinking about my mom, and—and I started to miss having a family. Aunt Marian and Uncle Jim and Cole—they're the only family I've got left." Her voice quivered as she spoke, but she seemed to will herself to go on. "OK, so she loves Cole way more than me, but I can live with that. At least with them I had a home. Now I don't know what's going to happen to me. No one wants me. The only one whoever really loved me is dead."
Both Olivia and Steven put their arms around Bindy and raised her to her feet. "It's all right," they were telling her. "You're with us now. Let's get back to the motel. It's late, and we have to check on a dead whale tomorrow." To the officer, Steven said, "I guess it's all right for us to take her with us, isn't it, since she's not being charged with anything."
"You have to sign some papers," Officer Bartlett answered, "and then she can go. Technically, we could charge her with theft, but we'll let it go—at least this time."
"Theft!" Steven exclaimed.
"I needed money for the pay phone, so I borrowed a bunch of quarters off a table," Bindy cried. "I had three dollars in my pocket—I was going to put the bills back on the table to replace the quarters. Honest!" When Olivia looked skeptical, Bindy added quickly, "I just didn't have time before I was arrested."
The ride back to the motel was silent, except for Bindy's sniffles. Jack couldn't tell if she was still crying or if she was pretending. With Bindy, the actress, it was hard to separate truth from fiction. Yet her tears in the police station, when she'd sobbed that nobody wanted her, had seemed real enough.
Jack was ready to agree with his mother. Bindy Callister might be more than the Landons could handle.
Everyone in the rental car stayed quiet. They'd had less than five hours' sleep from the time they got back from the police station until the alarm clocks buzzed in both their motel rooms at 7:30 a.m.
That is, everyone but Bindy, who chattered just as much as usual. "...so when I found out they were shooting the movie in New Zealand, I thought maybe I could get a role as a hobbit, just to get away from my aunt. After all, kids at school kept telling me I looked like a hobbit—short and wide. One guy even asked me to take off my shoes so he could see if I had hairy feet. So I did. I took off one shoe and hit him over the head with it. Too bad it wasn't a spike heel...." And on and on.
If Bindy hadn't yapped so much, Jack could have enjoyed the scenery more. The park covered 35,000 acres of much larger Mount Desert Island, named by the French explorer Samuel de Champlain, who landed there in 1604. They hadn't reached the park boundary yet; instead, they drove on a winding two-lane road through hills bedecked with greenery—beautiful but impossible to appreciate because Bindy the Blabber showed no signs of winding down.
Finally, to shut her up, Jack asked, "Mom, what about these marine mammals that are stranding?"
Before Olivia had a chance to reply, Bindy said, "Mammals. That must be where the word 'mamma' comes from. Mammals, mamma. Mamma, mammals."
Olivia answered, "Those words aren't connected, Bindy. 'Ma' is one of the easiest sounds for a baby to make. Proud mothers tell you, 'Oh, she's so smart. She's only four months old, and she's already saying 'Mamma,' but it's only baby babble. It doesn't mean anything."
Bindy smacked her forehead and cried dramatically, "Oh dear! Another illusion smashed!"
Sheesh! Tired and cranky, Jack decided he'd had enough of Bindy's theatrics. "Will you please keep quiet long enough for my mother to answer my question about the strandings?" he demanded.
"I do talk a lot, don't I. When I was making movies—"
"Just—shut—up!"
"Jack!" his father warned, frowning at him in the rear view mirror—the three kids were in the back seat of the rented Ford Taurus, crowded tight because of Bindy's width.
"Sorry," Jack mumbled. "Mom, please tell us about the strandings."
His mother twisted around from the front seat to face him. "First, Jack, I don't like you being rude to Bindy. Second, I want to finish what I was explaining. The word 'mammal' comes from the Latin word—"
Oh, crud! Jack knew where the word "mammal" came from, and he knew exactly what the Latin word meant—it had to do with how female animals fed their babies. It would be so embarrassing to listen to an explanation of mammary glands while he was jammed thigh to thigh beside Bindy. "Let her look it up in the dictionary," he muttered, but his mother ignored him. He covered his ears with his hands and started making soft na-na-na noises inside his throat until Olivia finished her lecture, but he could still feel his cheeks growing hot.
"You are such a dork, Jack," Ashley told him, reaching across Bindy to smack him on the knee. "You just acted like you were about three years old."
For once, Bindy said nothing, but Jack could see that she looked a little embarrassed, too.
"Now about strandings," Olivia went on. "As you know, Bindy—or maybe you don't know—marine mammals like whales and dolphins and porpoises and seals live in the water, but they have to breathe air.
They stay submerged for a while, then every so often they surface to take a breath. If they didn't, they'd suffocate, just as you or I would drown underwater if we couldn't breathe."
Sitting twisted around like that must have made Olivia uncomfortable, because she turned to face forward again. Since she never missed a chance to teach something to kids, she pulled down the car's sun visor and spoke into its mirror, looking at the kids' reflections while she talked.
"To answer your question about the strandings, Jack, marine mammals strand for a variety of reasons—injury or disease or harassment from humans or pollution in the water or getting tangled in nets. And if baby whales become separated from their mothers, they'll often strand because they can't find food by themselves."
Steven added, "Sometimes stranded marine mammals are already dead when they wash ashore. Other times they wash ashore first. And then they die."
"Do they always have to die? Can't anyone save them?" Ashley pleaded.
Olivia hesitated. "Seals are easiest to save; dolphins and porpoises, maybe half the time. Whales are harder to save. Very hard."
She paused then, as though she didn't know whether to get specific.
"They can be pushed back to sea, can't they?" Jack asked. "I've read about that. And then they'll make it OK, won't they? They'll live?"
Olivia was shaking her head again, more slowly this time. "Rescuers do try to haul them back into the water, and sometimes it works, especially with the smaller whales. But usually they're just too huge to move. Time is really critical when a whale is stranded. If it can't be refloated quickly...."
In a very small voice that didn't even sound like her, Bindy asked, "What happens then?"
"Well, nobody likes to see a whale die an agonizing death, its body crushed under its own weight on a beach. So they're often euthanized—put to death as humanely as possible. They're so huge, it takes massive doses of euthanizing agent."
Bindy gasped. Jack guessed she didn't know about the bad things that could happen in the animal world, the way he and Ashley did. They'd traveled with their mother and father to a number of national parks where species were in trouble, and sometimes animals died—the condors at Grand Canyon; the manatees at Everglades; the cougar at Mesa Verde that had to be put down because it had attacked a child. Nature could be brutal, yet all too often the damage to animals was caused by humans. Usually it happened because people were just careless, but other times it was because they were criminals, like the men in Glacier National Park who kidnapped bear cubs.
"I think we're here, guys," Steven announced after turning onto a side road. He swung into a parking lot and pulled up near a building marked Visitor Center Acadia National Park. "OK, everybody out!" he ordered, but even before he said it, Jack had flung open the car door to escape, glad to get some space again.
Bindy got out more slowly. From the curb, she pointed to the Visitor Center and asked, "Do they sell candy bars in there? I'm starved."
"Well, if you hadn't caused so much trouble last night," Jack snapped, "we could have had time for a real restaurant breakfast." Instead, they'd settled for oatmeal bars Olivia had brought from home.
"Everything's always my fault," Bindy muttered.
"Never mind," Steven told her. "When it's time for lunch, I'll see that you get a decent meal."
## CHAPTER FOUR
The park resource manager, Greg, impressed Jack. He was tall and muscular, with thick salt-and-pepper gray hair, and he looked good in his National Park Service uniform. At every national park the Landons had visited, Jack had felt admiration for the rangers, biologists, naturalists, and law enforcement people who seemed to care so much about the jobs they were doing to preserve the best part of America—its wildlands, history, and natural beauty. He'd started to think he might like to work for the Park Service himself, after he grew up and finished college.
"My office is pretty small," Greg was apologizing, "so I thought we'd better meet here in the conference room. I didn't know there'd be so many of you Landons," he added, laughing.
"I'm not a Landon," Bindy announced, shaking Greg's outstretched hand. "I'm Bindy Callister, a problem child the Landons are stuck with for a while."
Greg looked a little surprised, but he smiled and said, "Well, have a chair then. You can sit next to me, Bindy Callister, problem child." When he pulled out the chair, Bindy plunked onto it and grinned up at him.
The Landons seated themselves around the table, with Olivia opposite Greg. Immediately getting down to business, she said, "Just to review the facts, you had 12 marine mammals strand at Isle au Haut a week ago. All of them were dead, or died shortly after stranding, correct? There were 5 seals, including 3 mature animals and 2 pups; 6 porpoises, all mature; and one humpback whale, a mature female 40 feet long and weighing approximately 37 tons." Dropping her professional manner, Olivia exclaimed, "You must have had some job getting that body off the beach."
Greg nodded. "Even after the head was cut off and sent to Harvard Medical School for examination, that still left a lot of whale carcass to remove. Fortunately, we had a large group of volunteers helping us do the job, and the weather has been cool for May. If it had been hot and sunny, the smell would have made us all gag. It was bad enough as it was."
"Where are the other carcasses?" Steven asked.
"In an ice storage unit in Bar Harbor. We'll drive over there later, Olivia, so you can examine them."
Bindy wrinkled her nose as though the thought of examining dead animals was disgusting. She was just about to say something when Steven gave her "the look," a forbidding expression he'd perfected with his own kids. It worked on Bindy, too. She kept quiet.
"The first thing that crossed my mind was sonar testing," Olivia continued. "After that case in the Bahamas where 16 whales and a dolphin beached...."
"Olivia, refresh my memory about that case, will you?" Greg asked. "I know that acute auditory trauma and the intense pain connected with it can really mess up a whale's navigation system. Maybe you didn't hear about it, but just north of here, off the coast of Newfoundland, there were some explosions from an underwater drilling operation that could have interfered with the navigational skills of a bunch of humpback whales. They blundered into fishing nets."
Steven commented, "Getting tangled up in nets can be as bad for whales as strandings."
"Definitely. Still, in the 11 years I've been here," Greg went on, "we've had a couple of whales that were already dead come floating ashore, but we've never had anything like the mass stranding that happened this week. It sounds more like the case in the Bahamas."
Olivia shuffled some papers before she said, "I'll tell you what I've been able to research so far, Greg. The Bahamas stranding involved 16 whales and 1 dolphin." She went on to explain that just before the whales washed up on the beach, the U.S. Navy had been testing mid-frequency sonar in the ocean, not too far away. The stranded whales were a smaller species that weighed only about 2 tons each, so volunteers could push most of them back into the water. "But 7 of the whales died right on the beach, and none of the others have been seen since then."
"So they died, too," Ashley whispered softly. "Except they died out in the ocean."
Greg asked, "Did you read the necropsy report? I managed to get a copy of it. Biologists examined the tissue and bones around the whales' ears and found that they'd hemorrhaged."
"Right. Since whales live in a world of sound, they need their hearing for communicating with each other—and for finding their way around, locating food, and avoiding predators. In other words," she said, explaining it for Bindy's benefit, "they make sounds and listen to the sounds echoing back to them. That's how they tell where objects are in the water."
Steven added, "It's the same principle as the sonar the Navy was testing—they send out a signal and listen for its echo."
"Right. Anyway, it seemed pretty certain that the Navy's sonar testing confused the whales and caused the stranding in the Bahamas." Olivia picked up the report and waved it. "As things turned out, the necropsies—that means autopsies on animals, Bindy—proved that the sonar had done more than just confuse them, it had actually damaged the whales' ears. Three of them showed signs of bleeding in their inner ears, and one showed signs of bleeding around the brain."
"That's bad," Jack declared. "I hope the Navy stopped doing the sonar testing after that."
Greg looked a bit uncomfortable. Drumming his fingers on the tabletop, he said, "You know, I was in the military myself before I joined the Park Service. Quite often, the military has to walk a fine line between defense measures and environmental harm."
"Yeah, I read about the decision in this particular case," Steven agreed. "The Navy said it would protect marine mammals as much as possible—during peacetime. But they also said that national security comes first. And right now, since the war against terrorism began, this is no longer considered peacetime."
"Correct. And the sonar they were testing is used to detect enemy submarines," Greg added. Standing up for emphasis, he declared, "However, I am absolutely sure that the Navy has not been testing low-frequency, mid-frequency, or any range frequency of sonar in these waters this week or the week before. I personally spoke to a high-ranking officer in the Navy Department, and he assured me of this. I believe him."
"Then what?" Olivia asked. "What caused it? Has anyone performed a necropsy so far on any of the dead marine mammals?"
Greg sat down again, looking grim. "Yes. On one of the dolphins."
"And the results?" Steven asked softly.
"Hemorrhage of the inner ear."
The room was silent. Sonar, Jack thought.
"So the Navy lies," Bindy blurted. "Hey, everybody calls me a liar, but my little exaggerations are puny compared with this coverup!"
"Bindy!" Steven cried sharply. "That's enough."
With her forefinger, Bindy pretended she was sewing her mouth shut.
Looking angry, Olivia murmured, "Steven, why don't you take the kids for a drive around the island. Greg and I have a few things to go over, and then I want to examine the dead marine mammals. We'll meet later at the motel—you can take the kids back there after they've had a chance to see the park."
"Right." Steven had them on their feet and out of the meeting room so fast, Jack felt like he was being herded by a sheepdog.
Jack had jumped into the front seat of the car next to his dad. He didn't want to get stuck with Bindy, who sat in the back with Ashley. Heading south from the Visitor Center, they'd done the tour the way most tourists did—from the inside of their car. Steven drove silently, his mouth clamped in a firm line. Jack was afraid to ask him to stop so they could get out to enjoy the view.
With nothing better to do, Jack started reading the roadside signs. One said, "Wild Gardens of Acadia," but they passed that turnoff. The next read "Abbe Museum," and they passed that one, too. The next sign said "Bridge Clearance 12 feet 2 inches." With interest, Jack studied the bridge that arched above the road. Made entirely of slabs of sand-colored stone, it seemed to have been put together as intricately as a jigsaw puzzle.
He wondered if his dad was going to keep driving aimlessly, or if he had a place in mind to stop and take pictures. They'd already passed a lot of scenic spots. At last he slowed the Taurus and pulled into a small parking place. "Everybody out," he told them. "This is the place I wanted you to see."
Red spruce trees towered overhead. As they walked, Jack could smell the salty Atlantic mixed with the pungent tang of evergreen. The sound of waves crashing against rock filled him with anticipation, yet he couldn't see beyond the army of trees, which frustrated him. Jackson Hole had towering mountains, but there was something almost mystical about the ocean, and the sooner he could touch the waves, the better. Ashley, too, seemed excited. She kept dancing on ahead until she disappeared behind a bend in the trail. Jack, Steven, and Bindy trotted along behind. Silently, Jack willed Bindy to keep quiet and move faster.
"Hey—how much farther?" Bindy asked, sucking in a gulp of air.
"Not much," Steven replied. "It's just down that path."
"Good. I'm not much of a hiker, although I've played one on TV. Hah!" Bindy laughed at her own joke. "Did I ever tell you about how I did this one show where I was supposed to take a fish off a hook, but I said to the director, 'I can't do it 'cause its little fishy eyes are staring at me!' and the director told me I had to for the scene, you know, so then I go, 'Well, it's against my religion because I'm a vegan, and vegans don't touch flesh.' Technically, I wasn't actually a vegan, but my mom was, so I figured it half counted. So then the director yells to the prop guy to get a rubber fish and that's what I ended up pulling off the hook—a wet fake trout that still grossed me out, 'cause one of its glass eyes fell right into my hand. Whew! How long is this trail?"
"Just a little farther," Steven replied. "We're almost there. I wanted to take you through the back trail so you could get the full effect."
"So you might want to save your breath," Jack added.
"Hey Jack," Ashley called out, "turn around and look at that mountain."
Jack glanced behind him, then twisted to the left and the right. Puzzled, he asked, "What mountain? I don't see any mountain."
Ashley grinned. "Back home in Jackson Hole, we'd call it a bump in the road. Here, it's named The Beehive. I saw a sign."
After a minute Jack saw what Ashley was pointing to. "That?" he exclaimed. In Wyoming, the Landons lived in the shadow of the 13,770-foot-tall highest peak of the Grand Tetons. Back there, this little Maine "mountain" would hardly qualify as a molehill. "How high is it?" he asked his dad.
Steven checked his guidebook before he answered, "520 feet above sea level. And that one over there, Gorham Mountain—" He pointed. "The book says it's 525 feet high. But the really big one here on Mount Desert Island is called Cadillac Mountain. It stands all of 1,530 feet high."
"Wow!" Jack exclaimed sarcastically.
When all three Landons began to laugh, Bindy scolded," You guys are mountain snobs. Mountains have feelings, you know."
"Oh, come on...," Jack began, but Bindy broke in with, "And I have feelings, too, like hunger. I don't suppose there are any restaurants up ahead?"
"Hello—it's a national park," Jack snapped. "There are no restaurants on the beach. Besides, how could you be all that hungry when you had five oatmeal bars less than two hours ago?"
"Sheesh, it was just a question. Chill out, Jack-o. If I get too hungry, I'll bite my nails." A beat later she added, "That was a joke, in case you didn't catch it. Besides, I can't help it if I have a healthy appetite."
"If you exercised more you'd—"
"Jack, could I talk to you for a minute?" Steven broke in.
Sighing, Jack fell back into step next to his father. Steven didn't have to say a word, since Jack already knew the drill: Be nice, be supportive, and above all, don't provoke the foster kid. But what about when the foster kid provoked him?
It took a moment for Steven to speak. A layer of pine needles muffled their footsteps, as if they were walking on blankets. The two of them swung into an easy rhythm, pulling low-hanging branches back in tandem. "You doin' OK, Son?" his father finally asked.
"Yeah," Jack shrugged. "Just tired, I guess."
"It was a late night for us all. Don't let your tiredness get the better of you, though. Understand?" When he reached out and ruffed Jack's hair, Jack managed a weak smile. It wasn't his dad's fault that they'd gotten "The Mouth" for a foster child. Well, no matter what, the one thing Jack could take comfort in was that foster kids didn't stay in the Landon home forever.
From behind, he noticed that Bindy didn't move anything like Ashley, who hopped over rocks and ducked boughs as nimbly as a deer. Bindy's gait, in contrast, seemed almost awkward, as if her round arms and legs couldn't quite swing in rhythm. It was hard to believe she had ever been in movies. Jack suddenly remembered that the girl in Melissa's Dream hadn't moved like that—the character named Amanda was a gymnast who could walk on a balance beam while talking to Melissa. One more clue that Bindy was faking. Could he ever believe a word out of her mouth?
Steven hitched his backpack full of camera equipment farther up on his shoulders. "Hang on, Bindy, it's just about a hundred yards more. One thing you should know—even though Sand Beach will look like it has real sand, it's actually quite different from what you'd find anywhere else in the park. You can see that on either end, Sand Beach has rocks like the rest of the shoreline of Acadia, but along the crescent—"
"Uh-uh," Bindy interrupted. "Can't get a tan on those rocks, that's for sure. You know, a tan is mandatory on the West Coast. Everyone in Hollywood keeps their skin bronzed and their teeth white, although I think their smiles look like a bunch of piano keys with lips. So fake. Plus, everyone in Hollywood has to be stick thin. Even the guys eat salad. I'm telling you, it's a weird place."
Steven nodded. "Yes, well, as I was saying, the sand on this particular beach isn't really sand. It's made up of ground-up shells that've been bashed by the waves until they turned flaky and gritty. Olivia and I came to this park when we were first married, and I found out just how different Sand Beach was. That shell dust stuck to my wet feet like a second skin, and it was all but impossible to brush off. Look," he pointed, "there are the steps that lead to the beach. I see Ashley's already found her way down."
"If there's no food, can I at least have some water?" Bindy begged. "You have some water in your backpack, don't you Mr. Landon?"
"I do. Jack, I know you're eager to get to the beach. You go on ahead while I dig out the water bottle."
Jack didn't need to be told twice. The army of spruce had broken apart to reveal a stone stairway descending to the shoreline. Bolting down the steps, he emerged onto a crescent-moon beach of pinkish beige sand that stretched the width of more than two football fields.
On either end of the crescent, slabs of granite rose out of the waves, as though a race of giants had been making sand castles that suddenly turned to stone. Ashley stood at the water's edge, hugging herself as the ocean breeze whipped her hair into tight ringlets. Except for his sister, the beach was empty. The sand gave way under his feet as he walked over to join her.
"Where's Bindy?" she asked, her eyes still focused on the waves.
"Back with Dad. It'll take her a minute."
"Good. No offense, but this place needs to be enjoyed in silence. Isn't this awesome?"
Jack nodded as he soaked up the view around him. Dozens of tidal pools shimmered like liquid glass in the dim sunlight, the largest of them surrounded by angry gulls quarreling over what was left of a fish. Beyond the tide, seawater pounded huge boulders, roiling and foaming white before retreating to the sea, only to surge again. It was the mix of colors that mesmerized Jack.
The Virgin Islands had been painted with pastels; Acadia's palette was forged of grays, greens, blues, and pearl.
"I've hardly had a chance to talk to you," Ashley began. She stood unmoving, her eyes on the waves. "What do you make of Bindy?"
Jack shrugged. "I don't know. What's your opinion?"
"I'm the one asking you, Jack."
"Well, I don't know what to say. She's...different."
"It's just, last night, she—she told me some things. Do you believe all the stuff she tells you?"
He hesitated just long enough to give his answer. No. The more time he spent with Bindy, the more he was convinced that everything, from her brother to Hollywood and everything in between, had been embroidered with untruths stitched upon exaggerations so that nothing real remained. Maybe the fabric underneath was true, but that was all.
Placing his hand on his sister's elbow, he asked, "What's going on?"
"After we got back and Mom moved into our room to make sure Bindy didn't leave again," Ashley began, her words rushing on top of each other, "Bindy started telling me this strange story about this guy in the bar that wouldn't let her use the phone."
"Did Mom hear?"
"No, she was asleep. Bindy said the guy threatened her. The whole thing really weirded me out."
"So did you ever tell Mom or Dad?"
"No! Bindy made me promise to keep it a secret. Besides, Mom's so stressed with the whale thing, and Dad was all upset about losing Bindy, and I didn't want to give them any more problems, especially if none of it's true. Do you think it's true?"
"I don't know," Jack shrugged. "It all sounds pretty bizarre. Besides, if some guy really did threaten her, why didn't she tell the police? She was in the police station for over an hour, right? It doesn't make any sense she wouldn't tell them."
A wave rolled in, this time licking the toes of their shoes with foam. Neither one of them moved, and Jack felt dampness seep into his shoes. The sea suddenly seemed a deeper gray, colder and unforgiving, as if the sun were hidden behind a cloud. Glancing up, he saw that the sun was shining, just as it had been moments before.
"Bindy said this guy told her he knew where to find her and that he would hurt her. Jack, if he could hurt her, he could hurt Mom or you or me."
Jack pushed his hands deep into his pockets. "I wouldn't worry about it," he said slowly.
"Why not?"
"Bindy makes up stuff. She's talking, that's all."
"How do you know?"
"I know."
"But how?"
"Because I overheard something back home, when Mom and Dad were talking in the kitchen. I'm not supposed to tell."
"Wait a minute! That's not fair!" Ashley blazed. "I told you my secret! What do you know that I don't? Come on, Jack, tell me!"
Jack took a breath, then let it out slowly. His parents had told him not to speak of what he'd overheard, but this situation had changed in a way no one could have guessed. Bindy had graduated to spinning lies that spooked his sister, and that wasn't right. Some things were bigger than rules.
"OK, but don't ever say I told you! It was about why Bindy was put into foster care."
"Why?" Ashley asked, wide-eyed. "I asked, but she never told me."
"Basically, it's because her own family says she's a liar."
"What?"
"I didn't hear what she said her brother did to her, but whatever it was, everyone in the school came to Cole's defense and nobody, not one person—not her aunt or her uncle or anybody—believed Bindy. What does that tell you? Plus, Bindy took that money right off the table at the bar—remember? If she really was going to replace it, she would have put those dollar bills down when she picked up the quarters. I don't believe her. It's like she spins everything, twists every story to make herself look good, but I think it's all just that—stories! I mean, just listen to her talk on and on about all the stuff she says she did. She says she's an actress, but was she really?"
"She had to be!" Ashley protested. "She knew all of Amanda's lines—"
"So what? You've seen that movie so many times you know the lines, too. It doesn't mean anything. The truth is—" Jack hesitated before blurting—"The truth is, Ms. Lopez said it was Bindy's own aunt and uncle, the ones who adopted her, who are trying to get rid of her now because she lies."
"You mean they're the ones who stuck her in foster care? That's awful!"
"Yeah. I know they're not her birth parents, but when you adopt someone, you're supposed to become their real parents, aren't you? Like, be their real mom and dad? Anyway, now they're trying to give her back. How many parents would do that?"
"Jack, shhhhh!" Ashley hissed.
No! It couldn't be! In an instant Jack saw a third shadow darkening the sand, and as that realization slammed into his brain, he felt his insides turn upside down. Bindy was behind him! Bindy must have heard everything he and Ashley had been saying! Whipping around, Jack practically ran into Bindy's thick body. A deep flush had crept across her cheeks, and tears glittered at the edges of her lashes, but her eyes were on fire. The chattering Bindy was gone; a smoldering, angry person stood in her place.
"Where—where's my dad?" Jack stammered.
"Back on the path, shooting some pictures of an eagle. I kept wanting to call Aunt Marian, so he let me borrow his cell phone. So, you want proof that I was an actress? You call her."
"Uh—I don't—" Jack stammered.
"Go ahead! I know her work number. Call her and ask her if I played Amanda in Melissa's Dream. That was the whole reason she took me in. I was a perfect, ready-made, talented little girl to match her perfectly golden son. Only I didn't turn out the way she wanted, did I? Guess that made me disposable. Go on! Hear it from her own mouth. She'll tell you I was in Melissa's Dream and on television. I do not lie, Jack! Everything I talked about was true!"
"Then why are they trying to get rid of you?" Ashley asked softly.
Bindy shook her head, her expression condemning Ashley for her foolishness. "Because my so-called brother Cole used to use me for a punching bag, and I finally tried to stop it. He's smacked me around ever since I moved in with them, and for years I took it and took it—I thought I had to be grateful. Remember, I'm just an orphan!" She spat out the word as if it were poison. "Then one day a speaker came to my school and said, 'The worst secret is the one you hold inside. If someone hurts you, you must tell.' So I told. Look what it got me!" She laughed harshly. "There are things worse than bruises."
"I'm—I'm sorry, Bindy—" Ashley began, but Bindy cut her off. "Forget it. You're just as bad, Ashley. You're just like everyone else. If I were still pretty, you'd believe me. Pretty people don't lie, right? Only ugly ones. The truth is, I don't care anymore what either one of you thinks." Her eyes were gray now, a distant, cold gray that matched the ocean. "Here, let me dial it for you. Ask my aunt. She'll tell you."
Silence suddenly enveloped them, a chill silence as clouds hid the sun. It was as if all the emotion had boiled over into the sea, churning the water and shooting it over the rocks like geysers, and now the very waters reflected Jack's insides. He wished he could take back his words, the same way the waves retreated back into the depths of the ocean. Too late for that. He didn't know how to make it right.
The phone's keyboard glowed orange as Bindy held it out to him, but when Jack shook his head, she slowly let her hand drop to her side.
Suddenly Ashley sprang forward, shielding her eyes. "Jack—Bindy—look at the water!"
"Ashley, we have enough problems here—"
"No, look. I see something. Over there by those big rocks. You see it too, don't you, Bindy? That shape—like a boat—but it's floating upside down or something. It's a dark color."
Bindy rocked onto her toes. "Yeah. I see it."
"Where?" Jack asked, and Bindy pointed the way. Jack squinted until a shape he thought was a rock rolled forward, then was drawn back by the tide. Whatever was out there, it looked at least as big as a rowboat, only thicker.
"Is it a person?" Ashley asked.
"If it is, he must be dead," Bindy answered.
Ashley took a step into the water, and then a second step and a third until she was standing knee-deep in foam. "Jack, I know what it is. Call Dad!" she screamed.
## CHAPTER FIVE
"It's a whale!" Ashley yelled, splashing into the surf.
"I think it's stranding!"
The large, rounded mound rolled closer, but this time it seemed to flounder on the rocks as the waves sucked back to the sea, leaving the animal lodged in shallow water. Its top half was exposed to the fitful sun. If the clouds parted and the sun shone through, its heat could harm the whale in a matter of hours.
The whale didn't move at all on its own but seemed to rock with the rhythm of the sea. For a moment Jack had the sickening feeling it might already be dead. Another wave crashed around it, sending a spray of foam into the air.
"We've got to get Dad—" Jack cried, but Ashley had already plunged through the first swell. Now thigh-deep in seawater, she dashed toward the whale, instinctively slowing down before getting too close. A beat later, Jack plunged in, with Bindy close behind. "Wait, guys, don't rush at him," Ashley ordered when they caught up to her. "You'll only scare him! Move slow."
"Yeah, Bindy, quit splashing."
"I'm not! The water—it freaks me out."
Ignoring them both, Ashley gingerly moved forward with Jack right at her heels, while Bindy hovered behind. He'd never seen a whale up close before. The hide was slick and gray, and the grooves below its bow-shaped mouth looked like an accordion-pleated bowl. Fourteen feet long and four feet high, it had beached itself less than ten yards from shore. Another wave swelled around it, rocking its thick body forward like a boat tied to a slip.
"It's OK, we're not going to hurt you," Ashley cooed, inching closer. "We're going to get you back in the ocean, where you belong. Don't be scared. It'll be all right."
"Aren't you guys getting too close? Can't it bite?" Bindy asked.
"I'm not worried about that, but the tail could whip around and hit us," Jack warned. "It would be like getting knocked over by a truck."
When Ashley carefully placed her hand on the whale's back, Jack did the same. The hide felt like wet rubber beneath his fingers. Reacting to their touch, the whale shuddered.
"Look, it's just a baby," Ashley wailed. "Oh my gosh, I can't believe another whale has beached itself! What is going on?"
Bindy's voice seemed too thin as she asked, "Is it dead?"
"No," Jack said, pointing. "Look at his eye." He wasn't prepared for how human the whale's eye looked. A sliver of white showed at the bottom lid, and the pupil, liquid and brown, expressed plain, raw fear. This animal was scared to death.
Suddenly it let out a wheezy noise that made Jack jump back in fright. "What was that?" he yelled.
"Sounds like it has asthma," Bindy answered.
"Don't be silly. Whales don't get asthma."
"How would you know?" she asked him.
That was true. Jack knew next to nothing about real whales that stranded themselves, then made little thrashing movements and funny noises like this one did. He was amazed to find, on the top center of the whale's head, not one but two blowholes—crescent-shaped slits close together like nostrils.
A large wave swelled forward, pushing with cold force before it curled past to lose its energy at the shoreline. The three of them were suddenly soaked to the middle of their chests. Although Jack could feel the sun on his face, the water itself was frigid. They couldn't stay out here long. His feet were already beginning to go numb.
"Man, I hate these waves!" Bindy cried.
"Why don't you go back to the shore and find my dad?" Jack demanded.
"No—I want to help. If we could just push him back into the water.... We could grab him by the flippers and pull—"
"No!" Jack caught her as she put her hands against the animal's side. "Not the flippers!" Even though he was a baby, the humpback's armlike pectoral flippers, several feet long, were more delicate than they looked. To pull on them, Jack knew, could really hurt the whale. "Mom said a lot of people try pushing or pulling stranded whales back into the water and just end up hurting them. Anyway, this whale must weigh about two tons. It's too heavy for us to move."
"Well, then, what's your idea? We can't just stand here staring at him! We've got to do something!" Bindy declared.
Ashley looked worried. "Mom said that when a whale's out of the water it can't deal with gravity, remember? She said a whale's insides can be crushed by the weight of its own body."
"Yeah, I remember," Jack answered grimly. "Not only that, but the way those clouds look overhead, there might be a storm coming. And if the waves start to whip up real hard, they might dash him against these rocks. That would really be bad."
As if in response, another wave swelled forward, rocking the whale toward Ashley. The bottom three inches of her hair dripped salt water, flattening into tendrils that looked like black seaweed against her yellow Gore-Tex jacket. Staring out at the ocean, she cried, "Oh my gosh, I think I see another whale! Out there, in the bay, straight ahead. Do you see it?"
Shading his eyes, Jack strained to look. A huge, dark shape moved against the horizon, barely above the waterline, creating a slice that seemed to move against the ocean current. A small puff of water shot into the sky from the blowhole, and then the shape disappeared from view until the scalloped end of its tail flipped into the sky. The mother searching for her baby? Or just a lone whale gliding through the waters? Maybe it, too, had its sonar scrambled by some strange phenomenon. What if it was getting ready to beach itself? The thought spread a chill through Jack, colder than the Atlantic waters. Another animal might wash onto shore. It seemed impossible, yet nothing about these strandings could be considered normal. His sister must have been thinking the same thing, because her mouth grew tight. "Bindy, you've got Dad's cell phone. Call 911."
Bindy visibly paled. "Uh-oh," she gulped.
"What do you mean, 'uh-oh'?"
Jack watched as Bindy reached into her back pocket to retrieve the cell phone. Water trickled out of it in a tiny stream. "I'm sorry. I forgot. I'll get him a new one."
Great! Jack screamed inside his head. Just great! Now they had no way to call. Quickly scanning the beach, Jack looked for any flicker of movement that would let him know someone was there. Other than the gulls, the white sand gleamed empty. Where was his dad?
"Hey, it's not my fault your phone's not waterproof. Maybe it'll still work." Shaking the receiver, Bindy punched the numbers, then held it to her ear. She jiggled it again, then pushed it into her sweatshirt pocket. "Maybe not. I'll go find your dad. Don't worry, we're not that far from the car. He'll find a phone somehow."
"While you're gone, Jack and I will keep the back of the baby whale as wet as we can," Ashley told her. "I remember that's important. I just hope the mamma whale doesn't decide to beach, too. I don't want 40 tons of whale on top of me."
"She's just looking for her baby," Bindy said. "Is that your mamma out there, searching for you? Huh? Oh, you poor little thing." Gingerly, Bindy stretched out her hand, letting it hover over the baby whale as if feeling some energy force emanating from its skin. Then, almost imperceptively, she touched the back of the whale with her fingertips. "I hate to see any animal suffer. They're the only creatures on Earth that don't care if you're fat or thin. Only if you're nice."
"Bindy," Jack began, trying to keep the impatience out of his voice, "you've got to go now!"
"But I think I've figured it out. No matter what, I won't let them get away with this," Bindy said softly. "Look at what they're doing. They're liars who hurt innocent whales. It's not right."
"Who? Bindy, what are you talking about?"
Jack stared at her, and she looked right back at him, her eyebrows arched like two half moons. "What would you say if I told you I think I know why all this is happening?"
Jack held on to his temper. "We don't have time for this—"
"Listen to me! I think the government is testing that sonar. I didn't put it together before, when we were in that meeting with your mom—you know, the one with Greg. But I've been thinking about it, rattling the whole thing 'round in my head ever since, until I have put it all together. I know, Jack. Our wonderful government is lying through its teeth."
Another one of Bindy's wild stories? Why was she doing this now, when he couldn't possibly have time to pay attention to her ramblings? Jack clenched his teeth so hard he could feel sand grit between his molars. A baby whale was dying, another whale might come sailing on top of them at any second, and Bindy was up to her old headline-grabbing tricks. A government conspiracy? Bindy, possible child star and maybe abused adopted sister, had cracked the case while staying in Acadia less than 24 hours. Right. It was too ridiculous to waste his energy on—he had bigger problems than spies and aliens and all the other bizarre stuff that fermented in Bindy's strange mind. For now, he needed her, though. Keeping his face as smooth as he could, he said, "Make sure you tell my dad about the government thing when you find him."
"Why should I tell him? He won't believe me." Bindy's voice was flat. "No one ever believes me. It's obvious that you don't believe me. Even if I know the truth."
From the corner of his eye he saw the whale in the ocean make another pass, closer this time. "Jack," Ashley cried, "that big one's getting near. And we've got to get some water on the back of the baby before it dries out. Start bailing."
"I'm sorry, Bindy." Jack's words came out in a rush as he began splashing at the whale's side. "You'll have to let my dad handle your government conspiracy theory. I've got to keep this guy wet."
"Fine. I'll go now."
Another large wave crashed, but this time Bindy rode it to the shore. She stumbled only once before righting herself to stagger onto the sandy part of the beach. Waddling awkwardly, she made her way toward the steps until she finally disappeared from view. What a wacko, Jack thought.
Ashley started singing an Irish song to the whale, soft and melodic, but Jack could see how stressed the animal was. The whale's eyes rolled back, and every few moments his flukes strained or his tail would thrash helplessly in the surf as he made those wheezy noises. "Shhh," Ashley cooed softly. "You're going to be all right. But why did you come up here, little one? Don't you know you should never come out of the deep water?"
"Did you hear what Bindy said about the government and the sonar?" Jack asked. "There is something really wrong with that girl."
"Can't worry about that now," Ashley replied. "Feel his skin—it's getting warm and dry on top. You've got to throw water on him. Cup your hands and sling water over his back!"
"Remember to keep the water away from his blowholes," Jack told her. "Mom said they can drown from that."
"Right. We'll start near his tail and work up."
Jack splashed as much as he could onto the baby whale's back, but the tiny scoopfuls seemed as useful as a single raindrop on a garden. There had to be a better way. But what? His shoes? They'd hold little more than his hands did. He began to peel off his jacket, shivering as a wave washed over him.
"What the heck are you doing?" Ashley cried.
"Making a bucket. Here, take two corners of my jacket. We'll scoop the water over him."
Without a word Ashley grabbed the ends and held them taut. Dipping the jacket as deep as they could, they counted to three and pulled up hard. The ends of the jacket ripped out of Ashley's hands and fell into the water, useless.
"I can't—it's too heavy."
"We're scooping too much water. Don't go down so far this time. Wrap the sleeves around your wrists before you pull; it'll give you better leverage."
They dipped his jacket again, this time being careful to go no farther than a foot beneath the waves. "One, two, three!" An arc of water sailed through the air and smacked onto the baby whale's back, trickling down his sides in a smooth sheet of liquid.
"It's working!" Ashley said gleefully. "We got ten times more water on him than we did using our hands. Don't be scared, baby whale, this is going to help you stay well."
"We'll need a system so we can get all of him. Move down one step, and then we'll dip it again. We've got to keep his back wet without hitting the blowholes—we'll need to do his head, but we can do that by hand. Spud needs to stay hydrated until help comes."
"Spud?"
"He's got to have a name, doesn't he? He looks like a Spud to me."
"How long's it been since Bindy left?" Ashley asked.
"I don't know. Ten minutes, I guess. Maybe 15."
Biting the corner of her lip, his sister looked out into the ocean. "The other whale's still out there."
"I know it," Jack replied. "And I hope it stays out there. The big ones almost never survive. It's going to be rough enough for Spud, and he's a baby."
"But we're keeping him wet!"
"You know that's only half the problem. If he's stuck on the rocks when the tide goes out, he'll crush under his own weight. We've got to get him back into the ocean."
"The rescue team will know what to do," Ashley insisted. With a circular motion, she rubbed her hand over the whale. "Spud'll make it."
"I hope they hurry because I'm starting to freeze. This water is so cold! We've got to keep moving." Muscles strained on his back as Jack dipped, hurled, stepped, then dipped again. If it was tough for Ashley, she didn't let on. Every few minutes a wave would knock her off balance, but she'd right herself, take another step, then fling another arc of water onto the whale. When they threw the water, drops flew back onto the two of them, which meant they were both soaked from head to foot.
Time crawled. The waves seemed colder now, turning Jack's feet into blocks of ice and the skin on his bare arms a deep red. He saw but did not feel a strand of kelp wind around his leg. Pulling it free, he flung it away from him like a snake. Where were they? Mentally, he calculated how long it would take Bindy to get to his dad. Ten minutes max for Bindy to get back to the trail. If Steven had gone very far chasing the eagle, it was possible Bindy wouldn't find him easily. Those pine trees were thick, with a tangle of foliage underneath that made walking difficult. No, if she couldn't see him, Bindy would be sure to call out, and Steven would drop everything to come running. The most he should add would be five minutes. That would leave another seven minutes for them to get to the car, followed by another ten to reach a phone. Would he go straight to the Visitor Center, or stop anyone he could find to ask for a cell phone?
Thirty-two minutes there and at least as many back. With a sinking feeling, Jack realized this wouldn't be over any time soon. Another thought, unbidden, crept into the corners of his mind. If Bindy even went for help. What if she just took off and left them all? She was acting so strange, nothing seemed impossible. The color of Ashley's lips was deepening at the edges, as if she'd sucked a blue Popsicle, and he could see her teeth chattering. She had to get out of the water.
"Go to the shore and get warm, Ashley. Then I'll go, and we'll switch places."
"N-n-no. It takes both of us to get water on S-Spud."
"He won't dry out that fast. Listen to me, you're turning blue. You can't help Spud if you freeze to death."
"I'm OK."
"I'm telling you to do it!"
"But—"
"Ash-ley! Ja-ack," a faint voice cried. "We're coming."
Even from a distance Jack could recognize his mother's voice. Soon his father and a group of five rangers swarmed onto the beach like ants, some carrying coolers while others had blankets and buckets.
"They're here!" Jack yelled. "They got to the beach pretty darn quick."
Relief flooded Ashley's face. "I knew Bindy would pull through."
"Yeah," Jack agreed. "I've got to admit I wasn't sure. She was acting so weird, I didn't know what she'd do."
The two of them barely got to shore before they were shrouded with blankets that felt as warm as toast to Jack. His mother hugged him tight for just a moment, whispering how proud she was into his ear before reaching over to give Ashley a squeeze and saying, "You guys stay here and warm up. I've got to see about saving that baby humpback."
"Sp-Spud," Ashley chattered. "J-Jack named him."
"I'm sure Spud is very grateful. The humpbacks are endangered. We need every single one of them."
"Where's Dad?" Jack asked.
"He and Bindy followed me in his car. I couldn't believe it—the two of them burst into my meeting at the Visitor Center, and then the rangers jumped up and got gear while I called Allied Whale to tell them to send their rescue team. Then we raced down here. The rescue team should be here any minute."
"There's another whale out there, Mom," Jack told her. "I'm afraid it might beach, too."
Olivia peered into the ocean, then shook her head in disbelief. "What is going on? Not once before this has there been an incident of a whale beaching in Acadia National Park. This is nothing short of disaster. I just don't understand any of it!"
She was zipping herself into a wet suit when the eight-member Whale Rescue Team arrived. In minutes every one of them had put on wet suits, too—blue or red or black. Each person on the beach seemed intent on only one mission now, to save the baby humpback.
Special blankets for keeping his back wet were dipped in water and applied to Spud's hide. Olivia quietly waded in with a long needle to take a blood sample, which would be analyzed at the local hospital. Someone smeared zinc oxide on Spud's blowholes to keep them from drying out, while two other team members used a suction cup to fasten a platelike device onto the baby whale's back.
"That's a TDR—a time-depth recorder," Olivia explained. "When we get Spud back into the ocean, that will let us know whether or not he's swimming out to where he's supposed to be. It'll fall off after a couple of days."
"How are you going to get him back into the ocean?" Jack asked.
"Look over there." Steven, who'd been taking pictures nonstop, pointed to a man and a woman. The two rescue-team members were carrying a huge harness toward Spud. By the time they dropped it onto the rocks next to him, the tide had begun to come in a bit stronger. Jack could hear the rescue-team workers talking about what they planned: As soon as the tidewaters became deep enough, the eight of them would work together to refloat Spud, gently wrapping the harness around his pecs and tail to guide him back to the water. It wouldn't be easy.
Even though Jack felt cold and wet, one sight he didn't want to miss was Spud being returned to the ocean where the mamma whale waited for him. He felt excited for Steven, too, because photographing this whale rescue would be the chance of a lifetime for his father. His photos might be printed in USA Today, or even—and this was Steven's dream—in National Geographic!
As Jack wrapped the blanket tighter around himself, he noticed Ashley shivering. "Pretend you're not cold," he whispered to her. "I don't want Mom or Dad to send us up to the Visitor Center, because then we wouldn't get to see everything. Anyway, I'm not that cold." And that was true. He'd felt the blood return to his feet and then his hands. At first the tingles hurt, but soon they subsided, and he felt as if he could last for as long as the rescue took—hours, if necessary.
"I'm cold, but I can stand it. What about Bindy, though?" Ashley asked. "You know how she complains about everything."
Jack turned all the way around, searching as far as he could see across the crescent of Sand Beach. "I don't see Bindy. Where is she, Dad?"
"She should be around here somewhere. She said she'd be driving down from the Visitor Center with your mother."
"Well, I can't see her."
At that moment, Olivia waded out of the surf, looking pleased by the progress being made by the rescue team. "Mom, where's Bindy?" Ashley asked.
"Your father brought her," Olivia answered.
"No I didn't. She told me she was going to ride with you," Steven said. "You mean she wasn't with—"
"No!"
"Then where—"
Jack dropped the blanket and looked wildly around the beach. Every single person milling about was a park ranger or wore a jacket marked "Allied Whale Rescue Team."
Bindy was gone. Again.
## CHAPTER SIX
"I'm sorry, Dad," was all Jack could think to say.
"It's not your fault. It's mine." The sun streamed into the interior of the car, shadowing the deep frown lines between Steven's eyebrows. "I should never have believed Bindy when she told me she was going with your mother," he said. "I should have put her in the back seat of the car myself and buckled her in with my own hands. Or I should have made sure Olivia knew I was leaving Bindy with her. She told me she wanted to watch the rescue squad save the whale, but, then again, Bindy is not to be trusted. Obviously." He slammed the heel of his palm into the steering wheel. "Now I'll have to go back to the police and tell them she got away from us again. This time they might take her away from us for good."
"Not if we find her first," Ashley said.
"I hope we do, kids. I surely hope we do."
The scratchy gray blanket the rangers had given Jack rubbed damply against the backs of his legs. The park rangers had said both he and Ashley could return the blankets tomorrow, which was a relief, since standing on the shoreline had chilled him thoroughly, and their car was parked a distance away. Sniffing, he realized they'd have to wash his blanket before returning it. It smelled dank, like seaweed mixed with salt. He licked his lips, tasting salt on them, too.
Ashley, who had her blanket wrapped around her papoose style, leaned as far forward as her seat belt would allow to announce, "I think Bindy's gone after the conspiracy people."
They'd already told Steven the story about the government cover-up, which he didn't believe any more than Jack did. "That's just too ridiculous to even consider," he said now. "Bindy just made up another tall tale to impress you."
"Then where do you think she went?" Jack asked.
"My guess? She's probably trying to get back to her aunt. Maybe she thinks running away from us will prove to her family how much she wants to stay with them. Who knows what goes on in that girl's mind?"
"I don't even care why she ran away," Ashley fumed. "It's just plain selfish—she never thinks about anyone but herself. Now you can't take pictures of Spud, Dad, all because of her."
Steven drew a sharp breath. "And I'll have to contact Ms. Lopez. Boy, I'm not looking forward to making that call. How do I explain to her that I lost the girl—twice? Ms. Lopez will have to let Bindy's family know, of course. This could get ugly."
For as long as Jack could remember, his father had lectured him about obeying the posted speed limit, but this time Steven was pushing it a lot faster than he should. Oak trees and pines streaked by in a silver-green blur. Jack could feel centrifugal force pressing him against the side of the door as Steven sailed around a bend. Ashley shot Jack a worried look; Jack shrugged in reply.
"Dad," Ashley cried, "slow down."
When Steven hit the brakes, Jack snapped forward. "Kids, I want you both to look for Bindy while I drive," he said. "It's a long shot, but it's possible she decided for some strange reason to walk back to our motel. Although she doesn't have a key to the room. But then again, the lady at the front desk would probably let her in if she asked."
"We're watching for her, Dad," Jack assured him. "Ashley's taking the left side, and I'm looking out the right. If she's out there on the road, we'll spot her."
Not that checking the roadside would do much good. Bindy would never have made it this far, at least not under her own steam. A few cars passed by as visitors entered the park, but not many cars were leaving. Jack saw a biker in an apple-green helmet whizzing along on razor-thin tires, plus one woman on a motorcycle, then another trickle of cars and a Hummer. But no Bindy. She'd just vanished.
Pressing his forehead against the glass, Jack watched the spruce and oak trees fly by. Acadia was breathtaking. The whole park seemed to have been gilded with yellow-gold, not the autumn kind that meant the approach of winter, but the bright sun gold of early summer. Boulders appeared every now and then, like large stone turtles, and every so often the trees would part to reveal a slice of coastline. The ocean looked more gentle from a distance, with tiny whitecaps that laced the ocean like bits of frosting, but Jack wasn't fooled by its benign appearance. His skin was still cold from standing in the 50-degree waves that had chilled him to the bone. Tightening the blanket underneath his chin, he pictured a hot shower, dreaming of how amazingly good it would feel to steam himself half raw. That was the first thing he'd do. The second thing would be to wash out his running shoes, since they were beginning to smell like fish.
"Kids, I know you're cold and wet, but could you bear with me while I take a quick check of the town? The chances are pretty slim that we'll spot Bindy on the streets, but I'd like to give it one last shot before I have to—" His voice turned grim. "To go to the police."
"No problem, Dad," Jack and Ashley both agreed.
Moments later, Steven turned down a road that led him to the main street of Bar Harbor, their first chance to see the town in daylight. Brick storefronts were shaded by scalloped awnings, wedged between small wooden buildings that had been painted the colors of gourmet jelly beans.
"Wow, this town is so cool! When can we come back for a real visit?" Ashley asked.
"I have no idea," Steven told her. "This trip has turned into a nightmare. You'd better take it all in now."
Knots of tourists milled along the sidewalks, peering into windows or walking hand in hand, some with peppermint-striped paper bags from the candy shop, others carrying shopping bags filled to overflowing. Wooden signs with all kinds of pictures decked the shops: One had a blue fish, another a lobster, and a third showed a family of bears carved into wood. There was no sign of Bindy, and no way, Jack decided, they could find her by driving aimlessly. Steven must have figured it out, too; without a word, he headed for the Seaside Motel. After he dropped them off, he'd go to the police.
"You have your key, Ashley?" Steven asked as they pulled into the parking lot.
She held it up. "I've got it."
"Good. Would you please run up to the room and check and see if by some wild chance Bindy is there?"
"Sure, Dad." Ashley unbuckled her seat belt, saying, "Be right back."
Since the steps leading to the upstairs rooms were outside the building, Jack could see his sister quickly scramble up and run to her door, which was only three down from where the steps ended. A moment later she disappeared into the room.
Steven sighed and pushed a lock of blond hair off his face. He seemed in no hurry to move, and Jack didn't want to rush him, so he sat perfectly still and waited. Glancing at his father's profile, Jack noticed that Steven looked different than he had just a few days before.
His jawline had softened, and there were pouches under his eyes that seemed smudged with gray. Bindy was wearing him out.
Ashley appeared at the door and shrugged her shoulders, hands spread apart, palms up.
"Well, as I suspected, she's not there," Steven said, his voice tense. "All right, Jack, you're in charge. I want you two to stay put in the room until I get back. If Bindy shows up or phones the room or you hear from her in any way, I want you to immediately call your mother's cell phone. Mine's still with Bindy. It's not working, of course. Seawater will do that. I tried to dial the number countless times, just in case it started working again and she'd perhaps answer, but...nothing." Raising his sunglasses, he rubbed the bridge of his nose. "Remember, I'll be checking with your mother every 15 minutes to pick up messages."
"We'll call if we see or hear anything. Promise."
"Good." He gave a quick wave as Jack scrambled out of the car, and then, tires squealing, he drove away.
Ashley was already using the shower in the connecting room when Jack arrived. He turned on his own shower, standing in the hot steam until his skin turned the color of the lobster he'd seen on the Bar Harbor sign. Lathering up his hair, he rinsed, then lathered again, letting the bubbles run down him in foaming sheets. He could think in here.
The Landons had lost Bindy twice, which meant his family could be in legal trouble—being a foster parent involved a lot of responsibility. The Bar Harbor police already had a record of Bindy escaping once, which could be bad. Two times might make them think of negligence. What if Marian, Bindy's adoptive mother, decided to sue them? Even if she herself didn't want Bindy, she might be the type to cause trouble if she thought the Landons hadn't done their job properly.
Jack stuck his head under the water to wash away the thought. He was most likely borrowing trouble. No one would go after his parents, not when they were trying so hard to help. His dad had been a foster child himself, shuttled from one family to another before living his teen years on a ranch for boys. Over and over again, Steven had told Jack to reach out to those who weren't as fortunate.
"The easiest thing in the world is to slip-slide into an attitude of entitlement, especially living in this country," his father had told him. "I want you to see real heroes—kids who've been knocked around by life, yet still rise up to take on the world. I want you to see what it is to have grit. Your mother and I want to make a difference in the lives of unfortunate kids." Now it was Bindy who was making a difference in the Landons' lives. Except it wasn't for the better.
When he came out of the bathroom, he found Ashley spread-eagled on one of the beds, her wet hair spiraling onto the comforter.
"Took you long enough. I was out ten minutes ago, and I'm a girl."
"I didn't know we were racing."
"Doesn't matter," she said, grinning. "I still won."
"Yeah, right."
"Jack?"
Jack sat on the other bed, looking at his sister. "What?"
"I have an idea, but I don't know if it's an OK thing to do. Remember how I went through Lucky's stuff in Mesa Verde and everybody yelled at me for doing it? You said her things were private, and I was totally wrong for going through them."
Jack nodded. He remembered. Lucky was one foster child he would always remember.
Winding a lock of hair around her finger, Ashley asked, "I was wondering if this time, with Bindy, I mean, if this time is different enough to check out her things.
We might be able to figure out where she's gone. It's just, Mom and Dad are totally freaking out, and—"
"I think it's a great idea," Jack jumped in, mad that he hadn't thought of it first. "You look in her dresser drawer, and I'll see if there's anything in her suitcase."
Ashley bounded off the bed, hurrying into the room she'd shared with Bindy, while Jack went to the corner where the suitcases were stacked. Bindy's battered blue one was at the very bottom of the pile. Pulling it free, he snapped open the locks and opened it. Nothing. Even the pockets were empty.
"Jack, come here!" Ashley cried. "Wait'll you see this!"
Rushing through the connecting door, Jack saw Ashley standing in the middle of the room, holding up a crumpled towel and rolled-up clothes. Bindy's clothes.
"These were in the bottom drawer. She was really here, Jack."
"How? When?" Jack stammered.
"I don't know. But I'm positive this is the outfit she had on when we were with Spud. The clothes are wet, and they smell like seawater. So she got back here to the room somehow to change." She held up the towel, as wilted as a lettuce leaf. "Looks like she showered, too.
I didn't see any of it before because I didn't look in her drawer. What a little sneak!"
"Is the rest of her stuff still here? Her other clothes? Toothbrush?"
Nodding vigorously, she said, "Yep, all of it—well, I don't actually know about her toothbrush." She went to their bathroom and pushed open the door. "Her toothbrush's here, and her earrings, too. I don't get it—if she was actually running away, wouldn't she have packed up her stuff?"
"You'd think so," Jack agreed.
"We need to call Mom."
Jack crossed the room to the square table that held the phone. Once he had an outside line, he punched in the numbers and waited. After three rings, a message kicked in:
You have reached Dr. Olivia Landon at the Elk Refuge in Jackson Hole. If this is an emergency, please call the Jackson Hole Veterinary Hospital. Or, if you'd like to leave a message, please....
"She must be on the phone," Jack mouthed to Ashley. When he heard the beep, he said, "Mom, it's me. Bindy was here, at the motel. We found her wet clothes and a towel stuffed into a drawer. Call me when you get this."
He dropped the receiver into the cradle, then sat on the edge of Ashley's bed. So, Bindy had been here. What on Earth was she up to? If she'd been running away, it stood to reason she would have wanted her things—it would have taken only minutes to pack. Was she trying to fool them, keep them all off balance? Or was there something worse going on?
Ashley looked as confused as Jack felt. Sliding into one of the green corner chairs, she stared out the glass door that led to their balcony. Her fist rested on her cheek so that Jack could not see her face. "I don't get it," she mumbled, more to herself than to Jack. "I just don't get it."
Jack called his mom again, and got another recording. After trying again he said, "I'm going to go out on the balcony. Maybe the air will help me think."
He opened the sliding glass door that led to the balcony, with its Astro-Turf carpeting and white plastic chairs. A four-foot-high metal railing, once painted white but speckled with rust, caged him in. Beneath him was a rolling lawn, green and trimmed, and then the rocky shoreline that led to the water's edge. To the left stood the long pier with its chain at midpoint.
Leaning on the railing, he studied the long pier. That's where he'd been last night, at the end of it, where the weird stranger had wanted to know his name. Because of the darkness, Jack had never gotten a good look at the stranger, but in daylight, the pier looked innocent. The same small rowboat, tied to the pier near the shore, still bobbed gently in the water. Gulls wheeled through the air while a band of ducks waddled out from a cluster of trees.
Jack's thoughts needed sorting out. What did he really know about Bindy? No one at her school believed she was telling the truth about her brother Cole beating her up, again and again. Her adoptive parents didn't like her—that much had come from Ms. Lopez. But did that actually prove anything? An idea nibbled at the edges of his mind. What if she'd been telling the truth? What if the problem was that Marian and her husband didn't want to take Bindy's word—Bindy, the adopted girl who'd turned out to be such a disappointment—against the word of their own flesh-and-blood son? What was it Bindy had said? Jack searched his memory. Only ugly people lie; that's what everyone thinks. Bindy wasn't ugly, but Jack could tell she believed she was.
The phone rang, causing Jack to jump. Finally! Ashley was still sitting in the chair as he hurried inside.
"It must be Mom," he told her as he reached for the receiver. "She must have gotten my message." Placing the receiver to his ear, he said, "Hello?"
"Hi, Jack. Surprise, surprise." Although there was a lot of static, the voice on the other end of the line was unmistakable.
"Bindy?"
"It's me."
"Where the heck are you?" Jack shouted. "Do you have any idea how worried—"
"Just quit talking, Jack. I baptized your dad's phone in the Atlantic, remember? Even though it dried out, it's still not working too well. I don't know how long it will hold out."
"Where are you?" he asked again, softer this time. Ashley had run over to where he stood, watching him with wide eyes.
"I like that red shirt you have on now. You should wear red a lot—the color looks better on you than that yellow jacket you wore this morning."
"What?"
"I saw you on the balcony, Jack. Like, two minutes ago."
"Where are you? Why did you run?"
"Because you wouldn't believe me. No one ever believes me. But now I have proof. So listen carefully. I want you to walk toward the pier, but not all the way. Stay to the left. You and Ashley go to that jagged place in the rocks, the one covered with all the barnacles and stuff. Do you know which place I mean?"
"Yes."
"And then I want you to walk around the shoreline. There's no one over here. Meet me in ten minutes."
"We're not allowed to leave!" Jack protested.
"Ten minutes," Bindy told him.
The line went dead.
## CHAPTER SEVEN
"Sh-she's out there," Jack stammered. "Bindy—she said she wants us to meet her on the shore.
Right now!"
Ashley's dark eyebrows shot halfway up her forehead, and her voice came out even higher. "You know we're not allowed to leave this room.
We can't go."
"I tried to tell her, but she didn't listen. She said she could see me standing on the balcony, which really creeps me out. She said we had ten minutes to meet her, and then she hung up."
Ashley didn't wait for Jack to finish. Banging open the sliding door, she ran to the balcony and waved wildly, calling, "Bindy! Bin-dy!" When she leaned over the railing so far that her head seemed to disappear, Jack grabbed her by her belt loops, yanked her back, and set her down with a thud. "She can't hear you. She's way too far away." He pointed to the rocky coast where she'd told them to meet her, slicked wet with ocean water and kelp. The tree line began just beyond the rocks—tall, cone-shaped pines, dark and full of shadow, that seemed to guard the land from the sea. Was she hiding in there, watching both of them now?
"We've got to call Mom and tell her what's happened," Jack said grimly.
"Bindy said ten minutes. How much is left?"
Jack glanced at his watch. "Eight minutes."
Hurrying back inside, he grabbed the phone and punched his mother's cell number. Come on, Mom, be there. Answer this time. Come on!
You have reached Dr. Olivia Landon at the Elk Refuge in Jackson Hole. If this is an emergency, please call the Jackson Hole Veterinary Hospital. Or, if you'd like to leave a message, please....
Slamming the receiver into its cradle, he yelled, "I don't believe it! Who the heck is she talking to?"
"The park people are probably using it."
"Yeah. Maybe you're right."
"Should we call the police?"
"Bindy will be gone before they can get here."
He began to pace the room, stopping at the sliding glass doors, looping back to the bathroom, then back to the balcony. Ashley sat on a bed, cross-legged, her dark eyes following him. For a second time he called his mother, this time leaving a message. After a third attempt, he found himself fidgeting at the motel door; he could feel Ashley staring at him.
"You're not seriously thinking of going out there."
"Got to," he snapped. "Time's past up. If I could ever get through to Mom, she'd probably tell us to go after Bindy. I've got to find her and make her come back."
"Why?" Ashley thundered. "Just let her go. She's nothing but trouble."
Jack's voice was just as hot. "You don't get it, do you? Just think about how bad it could be for Mom and Dad if Bindy disappears. They're the ones responsible for her! It could get really, really ugly, and I'm not going to let that happen."
With her thin arms crossed tightly over her chest, Ashley said through tight lips, "What if Dad comes back and finds us gone? It could get really, really ugly for you and me. I don't want to get grounded for the rest of my life because of that—troublemaker!"
But Jack wasn't listening. He'd been over all of it in his mind, and there was no other way. He had to at least talk to Bindy, convince her to come back. If that failed he could try to force her, but the truth was, Bindy was a year older than Jack and 40 pounds heavier. The thought of trying to wrestle her into submission was ludicrous.
Yanking on his wet shoes, he gave instructions to Ashley about what to say when she finally got through to their mother, but when he looked up, Ashley was already pulling a navy blue sweatshirt over her head. "I'm going, so don't even start," she announced. When he hesitated, she said, "It's my family, too, Jack. I'll write a note for Dad and tell him what's happened."
"Well, write it fast. We gotta go."
Bits of seashell crunched underfoot at they made their way to the cove, and his shoes still squished from their dunking in the Atlantic. The rocks had some sort of sea plant growing all over them, slimy and thick and brown; it looked like the whiskers on a walrus's face. It was easy to slip here, and it stank. Flies swarmed at him, then vanished, only to reappear moments later, buzzing his ears like mosquitoes.
"Man, these flies are nasty," he told his sister.
"I think they're hanging around those tide pools in the rocks—there's a lot of dead stuff floating in them. Let's get up where the trees are. It'll be better there." Ashley, who had always been nimble, scrambled over the rocks as easily as a cat. It wasn't so easy for Jack. With his arms outstretched like airplane wings, he tried to keep his balance but slipped at every turn.
Bindy was nowhere to be seen. He knew there was no use calling out to her. If she'd watched them clamber up the rocks, she would know exactly where they were. No, in her own good time, Bindy would come to them. Ashley had already settled onto a rotting log. He joined her, scooting to the side when he discovered he'd sat squarely on a knot.
"How long now since she called?" Ashley asked.
"Twenty-eight minutes."
"She said we had to get here in ten. Do you think she left?"
"Maybe. But I don't think so." Straining for any human sound, Jack looked overhead into the thick criss-cross pattern of spruce branches. Wind rustled the pines, causing them to shiver in the perpetual dusk of their shade. Sounds were muffled by a thick layer of needles that carpeted the ground. Where was Bindy?
"So you came," was the way she greeted them.
Jack whirled around to see Bindy standing five feet behind him, making her usual dramatic entrance, leaning against a tree with one hand high on the trunk and the other on her hip. She wore a pair of faded jeans and a blue-and-green plaid flannel shirt that hung past her hips in loose folds. Her hair fell in damp strings. If she was worried about running away, she didn't show it.
"Yeah, we're here," Jack nodded. "And what about you? How'd you get back to the motel? You couldn't have walked."
"I hitched," she answered. "This sweet old lady stopped to give me a ride. I told her that my car had broken down a ways back, and I needed to get to the motel where I was staying."
"Your car! Like you were driving your own car?" Jack scoffed.
"You think I can't play the part of a 16-year-old? Watch me."
Right before their eyes, Bindy seemed to grow older. She straightened up, flipped back her hair, pulled in her cheeks and looked—16.
"OK," Jack said, "but now you have to come back with us to the room. Bindy, do you have any idea how much trouble you've caused? My dad's driving all over Bar Harbor, frantic! He thinks you've run away. He's gone to the police, and he'll have to call Social Services in Wyoming to report that you're gone.
Do you know what that means? You could get taken away from us."
"Not that you'd care. Now maybe someone will listen to me. That'd be a first!" Bindy laughed harshly.
Ashley rolled to her feet, facing Bindy squarely, demanding, "Are you coming back with us or not?"
"What about what I found out? Don't you care about that?" When neither Jack nor Ashley answered, she said, "I know why the whales and all the other marine animals are dying. I can prove it."
"Come on, Bindy," Jack answered scornfully. "Is this that government conspiracy thing? That was pretty lame—"
"Shut up and listen to me!" she blazed. "I heard a man talking in the bar." She closed her eyes as if trying to remember—or maybe she was just rehearsing. When she opened them, she began, "I was waiting for the pay phone. At first I wasn't really listening to what he was saying because I had other things on my mind—like what I was going to tell Aunt Marian. Should I say that even though I told about Cole beating on me, I wasn't actually trying to get him into trouble, wasn't trying to make him lose his scholarship or anything? Cole has this big football scholarship to Duke University, did I tell you that?" She didn't wait for any acknowledgment, but went on so fast it seemed one word touched the next. "Or, should I tell her I know they love him more than me, but now I'm OK with that? He's their own son, right? Anyone would love their own child more than their adopted niece. Don't you think?" Bindy's question was tinged with sadness; how was Jack supposed to answer it? He remained silent.
"So all that was going through my head. Then I started to get impatient, because this man kept talking to Alex, whoever he was. He kept saying, 'You better have the money, Alex. These techie guys want to get paid right away. They're yammering that sonar components are pricey to build and the suppliers need to be paid off.
The price is up 'cause it's a rush job.' I didn't know what 'sonar components' meant, and right then I didn't care. But the man said it a couple more times—sonar components—and then he goes, 'I got the parts right here at my feet.'"
Now Bindy appeared more intense, leaning forward as though she really wanted to convince them. "I looked over to see what he was talking about, and there was this square metal case, like a suitcase, only silvery. He said, 'I'll wait at the pier till midnight.' Whatever this Alex answered got him upset, because he said, 'I thought you were the one in the big rush. If you don't show by midnight, I'm gonna leave it with the bartender. You can get it from him tomorrow. Yeah, the bartender's my partner. He'll be counting the cash, so don't try any games. God bless America.'"
"God bless America?" Jack questioned.
"Don't interrupt. So all of a sudden he saw me, and he slammed down the phone. He said, 'How long you been sitting there?' and I told him, 'Not long.' He stared at me with the coldest eyes, and he asked me, 'You know what I'm doing here?' I shook my head no, because really, I had no idea what was going on, and I didn't care—I just wanted to use the phone. But then, after about a minute, he goes, 'What I just did on the phone is my business. Anyone who messes with my business messes with me. I don't think you'd want to do that. If a person repeated my phone conversation—to anybody—I'd consider that messing with me. People who mess with me end up dead.'"
"Oh, come on, Bindy," Jack scoffed. She had to be making that up.
Confused, Ashley looked at Jack, and asked, "What does any of this have to do with the whales?"
"Sonar components!" Bindy cried. "Weren't you listening? That ranger said the government had sonar that could blast out a mammal's ears, but the government wasn't using it. Right? Well this guy was obviously selling sonar to the Navy—he even said God bless America."
"Why does it have to be people in our government?" Jack asked.
"Because they're the only ones buying. Who else would even want sonar? That has to be the answer!"
"Bindy, you're totally guessing. You don't know if anything you heard has anything to do with the sonar that blows out the mammals' ears," Jack protested. "You don't have any proof."
"I know it in here!" She pushed her finger into her gut.
"Here," Jack said, pointing to his own stomach, "isn't worth much. The whole thing makes no sense. If the Navy needed sonar components, why would it be getting them from some sleazebag who left them in a bar in Bar Harbor?"
"Conspiracy, conspiracy, conspiracy!" Bindy practically danced as she hissed out the words. "Think about the facts. Fact: A whole bunch of animals are washing up onto Acadia's shore. Fact: That's never happened before. Fact: Sonar blew out the eardrums of the whales in the Bahamas. Fact: The animals in Acadia have the same broken eardrums. Face it—everyone says they're not sure what's causing it, but we know it has to be the sonar.
We can prove it." She jerked her fingers through her hair, pushing her locks apart in damp rows. "We have to."
If Jack hadn't known it before, he knew it now. She really was crazy. Bindy, with her mousy hair and thick, plain features, had somehow blurred the line between television and reality. It was as if she thought they were in some spy show, with bad guys and techno-gadgets and a possible happy ending all wrapped up into a tightly written script. But this wasn't television, this was reality, and the only thing he cared about was getting her back to the room. Let his parents deal with Bindy's lunacy. He just wanted out of the middle of it all.
A couple of gulls flapped through the air to land beyond the swells, feet first, onto the slate gray water. They began to squabble and bicker loudly before settling down, bobbing on the waves. That's what Jack wanted to do. He'd like to float for a while, instead of struggling to keep his family's head above water. For a moment he was half tempted to walk away from it all. Then he thought of his parents.
"Bindy, my folks'll get in big, massive trouble if you leave. Will you please come back with us?" Even Jack could hear how weary the question sounded.
"Yes."
The answer took Jack by surprise.
"I'm so glad," Ashley breathed. "I mean, I thought you were going to—I'm just so glad you'll go with us. Dad's been so worried, driving all over, trying to find you. This is great. Mom and Dad can maybe investigate that sonar thing for you." Ashley stood, brushing flakes of bark from her jeans.
"Wait—not so fast." Bindy held up her hand. "I'll come back to your folks and be a good, obedient foster child, but on one condition."
"A condition?" Jack frowned. "What condition?"
"You have to come with me first. I have a plan. See, I watched your face when I told you my story, and it was the same old thing. You didn't believe a word I said."
"That's not true," Jack lied, embarrassed that he'd been so transparent.
"It's just like with Cole. No one would believe that my gorgeous adoptive brother slapped me around, not even when I showed them the bruises. But now you guys have got a problem because your folks are on the hook with me takin' off like I did. So here's the way it'll be." Leaning an elbow on her thrust-out hip, she wagged a finger at them and said, "You help me, and I'll help you. With this deal, there are no negotiations."
Ashley looked at her quizzically. "Where do you want us to go?"
"Back to the bar." Bindy grinned defiantly. "We're going to steal us a silver box."
"What!" Jack couldn't believe he'd heard her right.
"Actually, not me, Jack-o. You! I can't go bopping back into that bar—the bartender knows me. It's gotta be you. Unless Ashley—"
"You leave my sister out of this," Jack stormed.
"Well then—" Bindy tossed her head at him as she declared, "It's you, you, you. Say yes, or I take off this minute."
Talk about being between a rock and a hard place! He was standing on a slab of shore rock that suddenly felt as big as the whole state of Maine, while this lunatic demanded that he sneak into a bar and steal some imaginary silver suitcase. If he didn't humor her, his parents would get in a whole heap of trouble. If he did, he might get hit over the head with a baseball bat. In movies, bartenders always seemed to keep baseball bats behind their bars.
"OK," he said, giving in. "Tell me what you want me to do."
"Jack!" Ashley protested. "You're not going to go along with this—"
"Yeah. I am. Me and me alone," he answered. "You stay out of it. Go back to the room and—"
"Forget it!" Ashley had that look on her face again, the stubborn look that meant Jack might as well save his breath, because Ashley would do what she wanted. He groaned. Now, on top of everything else, he had to worry about keeping his sister safe.
"Here's what'll happen first," Bindy was saying. "We'll put together the whole scene like a movie script—dialogue, action, and me as the director." As she began to outline her plot, Jack felt his stomach sink to his toes. This was a role he definitely did not want to play!
## CHAPTER EIGHT
Smokey's Bar looked like the kind of place for hard drinks and harder people. Situated less than half a mile down the shore from the Seaside Motel, Smokey's sat squat in a cluster of trees 50 feet from the shore, a plain rectangle with a flat roof and tin chimney. The building was made of wooden logs stained tobacco brown, with one small shuttered window that reminded Jack of a patched eye. The only decoration of any kind was a neon sign that read "Beer and Spirits," which Jack could hear humming faintly in the distance, like mosquitoes at dusk. This was not like the other buildings that dotted the shoreline. Smokey's looked rough.
"Go around to the back, where no one can see us," Bindy hissed. "Keep your heads down, but don't act weird, like you're trying to hide something. Just be natural."
"We are," Ashley retorted.
"I'm just telling you that as an actress, I've learned how to get into character."
Ashley rolled her eyes at Jack, who shook his head back. There was nothing either one of them could do. Bindy had them caught in her web, and the only strategy Jack had left was to get it over with as fast as he could. Feet crunched on gravel as the three of them made their way through the back parking lot, stopping behind a garbage bin to make sure no one had seen them. He did not want to go into a bar. He did not want to steal a metal suitcase. He did not want to get caught and face the same officer that had arrested Bindy. The whole plan was nothing short of insane.
"OK, do you remember what you're supposed to do?" Bindy whispered for the 70th time.
"Yes. Create a diversion—"
"—and make sure you believe what you're telling them, or they'll see right through you and know you're lying," Bindy interrupted. "Make your story good. Act. Once you start this, you can't back out."
"You do realize I'll be stealing," Jack reminded her.
"No you're not. You'll be preventing a crime. Fight fire with fire, that's what I say."
He tried again. "If I get caught, I go to jail."
"Then remember two points," Bindy said, holding up her fist. "Number one"—her index finger punched the air—"it's not really stealing if, by taking the object, you stop a much bigger crime, like the killing of the whales. Number two—" She held up her middle finger and grinned, "Don't get caught."
"Gee, thanks for the great advice."
"What if we get the suitcase out, and there's nothing in it?" Ashley demanded. "Then what?"
"Then we sneak the case back to Smokey's, and I go home with you. Case closed. Agreed?"
Jack nodded, while Ashley stood frowning, with her lips pursed.
"Once you get inside, Jack, you'll see the bar on your left and the pay phone straight back at the far end of the wall. It's still pretty early, so there won't be that many people in there."
"That's good, isn't it?" Jack asked.
"Yes and no. The downside is you'll really stand out. That means you need to go straight to the bar to do your thing. You ready?"
Jack wasn't, but he nodded again.
"All right. It's show time!"
Taking a deep breath, Jack squared his shoulders. He could do this. Think of his parents and how much they needed this, he told himself. And don't get caught. For sure, don't get caught.
Since it was only five o'clock, the parking lot stood almost empty except for an old van and a rusted BMW. Acting as if he belonged there, he crunched up the gravel path, pulled open the door and stepped inside. The bar was dim and hazy, as if Smokey's had settled for a kind of twilight. It took a moment for Jack's eyes to adjust. He blinked, trying to pull the scene into focus. The tables sat empty except for a couple of men hunched in a small booth in the back. The wooden floorboards had pathways worn where the veneer had been scuffed away. A bartender had been wiping down the counter with a towel, but when Jack approached he stopped in mid-wipe. He stared at Jack coldly.
"Hi," Jack said, giving a halfhearted wave.
"Hey, kid, you can't come in here," the bartender growled, "unless you're 21. You don't look 21. Unless you're some kind of a dwarf. That what you are?
A dwarf?" He laughed at his own joke and said, "I already got in trouble with a kid in here, and I don't need no repeat performance. Beat it." He was an older man with a face grizzled by sun and salt. Thick gray eyebrows cast deep shadows over hooded eyes. "You deaf?" he demanded, rubbing the countertop once again.
When Jack didn't answer, he barked, "I said get outta here, kid. You're in the wrong place."
"There's...there's...." Jack could hardly get the words past the lump in his throat. His parents, Sunday school, the Boy Scouts—every institution he believed in taught that his word was his bond. Now he was going to lie outright. Well, once Bindy kept her promise, he told himself, he could sort out the rest. For now he had no choice.
"There's water shooting out of the back of your building," Jack told him.
The bartender's eyes grew wide. "Where out back?"
"I don't know. I was walking by and I saw it and I thought you might have broken a pipe or something. I thought you should know."
"All I need is a stinking plumber bill...." The bartender smacked the towel down on the counter and ran along the hallway toward the back exit, leaving the bar unattended.
Jack tensed. This was it! No one was watching him. Feeling a thousand pricks of adrenaline, he hurried behind the bar and looked underneath the counter.
At first all he saw were cabinets and row upon row of liquor bottles, tall and short, fat and thin, with all kinds of foil labels in brilliant colors. Then—a flash of silver! So, at least that much of Bindy's story was true. Looking around one last time, Jack saw no one. As quickly as that, he grabbed the suitcase and began to sprint to the door. It was heavy! His knees almost buckled beneath him.
"...don't see no water," the bartender's voice shot back. "So what is this, a trick?" Jack could hear the anger rising in his voice. "You some kind of prankster? Hey, kid—what are you—" His voice turned deadly. "Give me that case!"
Jack ignored him. Yanking open the door, he bolted out into the parking lot and raced toward the trees where he was to meet Bindy and Ashley.
"Drop it, kid," the bartender bellowed. "Now!"
His heartbeats hammered in his ears as Jack dove into the trees. Where were Bindy and Ashley? For a crazy moment, he thought Bindy might have set him up, but then he saw them hovering behind a knotted pine.
"Go! Go!" Jack shouted. He couldn't waste breath explaining anything—carrying the heavy case took a lot of energy, and they all needed to run. Tree limbs snapped at his face as he took the lead into the woods, the leaves churning beneath his feet like dust. He could hear Ashley and Bindy crashing behind him—or was that the bartender giving chase? Did the bartender have a gun? The thought, which he hadn't considered before, scared Jack. Would he get shot for stealing a suitcase? Curling his back as he ran, he thrust forward like a runner breaking through a finish line. He could almost feel the bullet in his spine.
"Jack! Wait!" Ashley cried.
Jack whirled around, and when he did, the suitcase banged against a tree with a thud. "I better not break whatever's in there," he muttered. "Not after all this."
"Jack, I said wait! It's Bindy—we're losing her." Ashley had stayed right at Jack's heels, but Bindy trailed behind. Scanning the woods, pushing panic down, Jack forced himself to wait. A moment later he saw Bindy's round figure struggling up the hill, but nothing else. The bartender seemed to have vanished.
When Bindy finally caught up to them, she bent over, hands on her knees, breathing hard. She panted, "Good work, Jack-o. You got the case. And we lost that old dude. He'll never find us now."
"Are you sure he's not still back there?" Jack asked.
"Might be. But I kept turning around—he gave up a long time ago. He won't find us now."
"Don't be too sure. We need to get back to our room and stay there."
"What about the case?" Ashley asked.
Jack tightened his fingers around the handle. "What about it?"
"Should we open it to see what's inside?"
"Not here."
"Why not? Let me see it," Bindy demanded.
Jack clutched the heavy suitcase to his chest. "No. We don't open it. Not until we're in our room."
Bindy's eyes narrowed. "What are you saying, Jack?"
"I'm making sure you keep a bargain. I did my part, now you have to keep your end of the deal."
"So you still don't believe me."
Jack didn't answer.
"OK, fine," Bindy said, sweeping out her arm. "Lead the way."
They kept to the trees, winding their way toward the Seaside Motel. They could tell they were heading south by watching for glimpses of the bay. Jack didn't want to get too far afield, and yet he still wasn't sure the bartender wasn't back there, lurking among the trees. Finally he saw the roof of their motel, slate-colored and flat. He could feel himself relax. They were almost there. One road, then a parking lot, then the steps, and they would be home free.
"Just walk natural," Bindy instructed. "Act like you own the road. No one will notice you if you act like you belong."
She was right. No one seemed to pay the slightest bit of attention to the three of them as they sauntered across the road. By the time they hit the parking lot, Jack could almost taste the safety of his room. Up the stairs the three of them clambered, not even bothering to be quiet. When he finally opened the door to his room, Jack collapsed on his bed and threw the suitcase down beside him.
"We made it!" Ashley cried, dancing around the room on her toes. "We got Bindy back. We need to call Mom and tell her! I hope she can find Dad. Whew!"
"So open the case," Bindy urged. "You'll see that I was right. It's some kind of sonar thing to kill whales. I'm not making it up. Open it."
Jack heaved himself to his feet to stand over the case. Ashley and Bindy hovered behind him. It was an expensive piece, like nothing he'd ever seen before. Brushed to a satin finish, it reflected his face in a warped blur, his blond head featureless, distorted. It reflected Bindy, too, who stood so close he could feel her breath on his neck. He inched away from her.
"Come on! Do it!" Bindy cried.
He hooked his thumbs on the tabs on either side. "Well, here goes." He pushed on the tabs. Nothing happened. Hooking his thumbs harder, he tried again.
Ashley frowned. "What's wrong?"
"I don't know. I think it's locked."
"Locked?" Pushing Jack aside with her hip, Bindy yanked the tabs so hard her face reddened. "Uh," she said, slamming her fist into the top. "I don't believe it—the jerk locked it. We need to get the key! Maybe it's under the counter. Maybe it's in the cash register."
Despite the warm, stale air in the room, Jack felt a cold chill spread through him. "You're not serious."
Bindy shoved her fists into her pockets and stared at Jack. She nodded decisively.
"We are not going back, no way. Look, I've already stolen for you. I'll be giving this thing back to that bartender guy, but that doesn't change the fact that I took it in the first place. That's trouble enough. Now you're talking about a stupid key?"
"But Jack, how can I prove—"
"Forget about that. I did what you asked, it didn't work, and that's the end of it. I'm through breaking the law. And if you really want us to believe you, you'll keep your word and stay here and wait for my mom and dad. You promised."
Bindy stood with her head bowed. Silence filled every corner of the room as Jack held his breath and waited. Different scenarios seemed to flash across her face, one after the other as she weighed her options, her fists still jammed into her jeans so that the pockets bulged, her brows knit tightly together. It was Ashley who seemed to know what to do. Gently, as if she were coaxing one of her baby animals, Ashley guided Bindy toward the glass double doors, saying, "Why don't you get some fresh air? Just sit out there and watch the waves and think. I need to let my folks know you're back. They're going to be really happy about it."
Without a word, Bindy sank into one of the plastic chairs on the deck. On the ground beneath the deck, ducks waddled happily, quacking loudly at nothing, but Bindy didn't seem interested. She kept her eyes focused on her shoes. Jack couldn't help noticing she looked utterly defeated.
When Ashley stepped back into the room, she put her finger to her lips, signaling to Jack to be quiet. "I'm going to close these curtains a bit. Jack, we need to talk."
They crossed to the bathroom, the farthest point from the balcony, so they could talk without Bindy overhearing. Ashley's dark, anxious eyes searched Jack's face. "I'm worried we can't keep her here," she whispered. "I'm not so sure she'll stay. Then we'll be right back where we started."
"At least we're better off than we were two hours ago," Jack told her.
"Except we have to get the suitcase back to the bar. I still can't believe you had the nerve to steal it."
"I couldn't think of any other way to help Mom and Dad. I'll just have to take it back and explain and hope that bartender doesn't kill me. If that works out, then I think we'll all be home free."
"Yeah," Ashley smiled. "Home free."
Suddenly a fist pounded against the door so loudly it made Jack's teeth chatter. "This is the police," a deep voice boomed. "Open up!"
## CHAPTER NINE
"The police?" Ashley cried.
"I already know you stole from Smokey's Bar," the voice barked. "My name is Officer Norton, and I'm with the Bar Harbor Police Force." Pounding again, he said, "Don't make it any harder on yourself. Open the door! I've got my badge right here—look through the peephole, and you'll see it."
Jack felt his insides turn to ice. So the police already knew. Now he couldn't go to the bartender and beg for mercy and smooth it all over—no, thanks to Bindy, it was too late for that. He'd been caught stealing, something he'd never done before, something he'd never do again. What would his parents say? He stood there, numb, as Officer Norton banged again. "Kid, I know you're in there! Give me that suitcase!"
"I don't see any badge," Ashley said, peering out.
Pound, pound, pound.
"Jack, we can't just stand here," Ashley told him.
Pound, pound, pound!
Officer Norton's voice became deeper, more insistent. "If you cooperate, I might forget the whole thing. Smokey just wants his case back." There was a pause, and then, "Did you hear me? Open the door!"
"Should I let him in?"
"I don't know!" Jack cried. "I don't know what to do!" His mind had gone blank. Hide the case and deny it all? Get it back to the bar on his own? Call his parents? Or just confess and take his punishment? Ashley moved to the door and waited, her hand hovering over the doorknob.
"You're only making it worse! Open up!" Officer Norton demanded.
"Jack, we've got to let him in. He's a policeman." Opening the door a crack, Ashley peered outside. Suddenly thrust backward, she let out a cry as the door almost knocked her off her feet. The man had pushed himself inside. His eyes quickly scanned the room.
"That's a smart girl," he told her, slamming the door shut. It made a sickening thud behind him. "So there's two of you. Anyone else?"
He looked nothing like the police officers Jack had seen. Tall and thickly muscled, with a large, bulbous nose, he stood with legs planted far apart. Greasy hair had been slicked back, and he smelled of cigarettes and diesel fuel. He wore no uniform, just a long black coat that skimmed the top of heavy boots. Just like—just like—the man on the pier! Jack was almost sure of it! This was no detective, it was the man from last night! And they'd let him waltz right in. Jack's palms got wet with sweat.
"I asked you kids a question. Is anyone else here?"
"My parents. They'll be back in a minute," Jack lied. "They just went down to get a newspaper."
Ashley glanced at him sharply, then looked away. Wrapping her arms around her chest, she held herself so tightly her shoulders seemed to swallow her neck.
"That so?" the man answered coolly. He opened the bathroom door and peered inside. "There's a newspaper in here. How many newspapers do your folks need?"
Jack didn't answer.
"Don't try to play me, kid. I get real angry when someone jerks me around. And you don't want me to be angry, do you?"
Jack tried to keep his face smooth as he asked, "Why aren't you wearing a uniform?"
"I'm working undercover."
"Could I see your badge?"
The man reached inside his coat and pulled out a leather wallet. He flipped it open quickly, but when Jack moved closer, he snapped it shut and slipped it back inside his pocket.
"We didn't get a chance to look—" Ashley began.
The man ignored her. "Where's the case?"
She pointed. "Right there, Officer Norton."
"Go get it for me."
Grunting at the weight, Ashley snatched up the case and handed it to the man. He tried to open it, realized the tabs were still locked, and grinned. "So, you didn't mess with it. That's good. Smokey'll be real happy to hear you weren't diddling with his stuff."
"We didn't touch it," Ashley assured him. "Are you going to arrest us?"
"Now that depends. Smokey's a friend of mine. I was just two blocks from his bar when he calls me all panicked. He says some blond kid told him there was a flood out back of his place—busted pipe or something. Told me the boy walked right in and lied to him, then ran off with his case. That was you, wasn't it, Jack?"
"How'd you know my brother's name?" Ashley asked.
"Oh, I'm a very good detective."
Jack wondered how long the man would keep up the officer myth. Ashley still seemed to believe it. Well, it might be better that way. Don't challenge him, and maybe he'll take the case and leave, Jack hoped. Right now, that was the only goal in his mind. Get this man out of their room.
"See, I'm smart," the man said, tapping his temple. "I remember all kinds of things. Like, for instance, last night. I met young Jack on the pier. He told me he was staying right here, in this motel. I remember that. And I remember telling him to mind his own business. Too bad you don't listen, huh, Jack?"
He pulled out a knife, a long switchblade, and flipped it open. Calmly, he began to run the tip beneath his fingernail as if he were cleaning it. Jack felt hairs rise on the back of his neck.
"But what I don't understand is why you took the case in the first place. It troubles me. How'd you know about Smokey's? How'd you know anything?" He leaned against the door, blocking it with his heavy body, and continued to flick the knife's tip around his nails. "So, why'd you take it, Jack?"
"I don't know."
He shook his thick head. "See, I think you're lying again. I warned you not to do that. Looking at Ashley, he asked, "What's your name?"
"Ashley. I'm his sister."
"Ashley, I think you need to tell brother Jack here that he's got some explaining to do. There's booze under Smokey's bar. And money in the cash register. But your brother Jack goes right for the silver suitcase. Can you tell me why?"
"I—I...."
"Give me a reason, Ashley, and I won't arrest either one of you. Tell me why he wanted the suitcase. That's all I need—a simple explanation."
"Ashley—don't!" Jack cried, but it was too late. His sister's words were already tumbling out of her mouth.
"We thought the silver case had some sonar stuff in it—there are whales dying and we thought some Navy guy named Alex was using sonar and we were trying to get the sonar so we could prove that the whales were being killed—"
"Alex?"
"I think that was his name. Anyway, we didn't know if it was true, but we thought we could check and then put the case back—"
"How did you hear about Alex?" The man's eyes suddenly grew charged with fury. "Tell me!" he thundered. "How did you know? It was that fat girl, wasn't it? The one in Smokey's last night. She must have been the one you were looking for on the pier, right, Jack? Where is she?"
"Gone," Jack lied. Taking a deep breath, he tried to remember all the instructions Bindy had given him about acting. Look confident. Make eye contact—don't let your glance shift away from the person's face. "She never came back last night," he said, staring at the man. "She's a foster child, and she ran away before I saw you on the pier. We never found her. We looked all over the place, and then my parents reported to the police that the girl was gone."
"Jack...."
"Ashley—shut up!"
Ashley stared at Jack, ready to say something. Then, all at once, she got it. This was no policeman standing in front of them!
"Does anyone else know about this? Does anyone else know about Alex?" he shouted.
"No one. She only told me," Jack said, not wanting to reveal Bindy's name, if the man didn't remember it from the night before. "She told me right before she ran away. I—I might have mentioned a little bit to Ashley, but she doesn't understand anything. She just heard the words, but they don't mean anything to her. I swear!" He was shaking inside, but he prayed it didn't show, prayed that the man would believe him. Jack, who'd hardly ever lied in his life, was making up for it now—big time! "You said if we gave you an explanation, you would let us go. We told you what happened. We're supposed to meet our parents by the front desk. They went to buy a newspaper."
The man laughed at this. "Well, now, Jack, looks like you and me have a lot more in common than you thought." His eyes widened so that Jack could see tiny red veins, and when he leaned close, Jack could smell stale cigarettes on his breath. "You're lying about where your parents are, and—guess what!
I lied, too."
With a motion as fast as a snake, he caught Ashley, pulled her back against his chest and jerked her upward, the blade within an inch of her throat. Ashley recoiled as she tried to pull her neck away from the knife point. For one nauseating second Jack realized how small his sister was. Her legs dangled above the floor like a rag doll's.
"Jack, I want you to listen carefully. When we walk out that door, we will be one big happy family. Understand? We will all take a stroll to the pier. Then the three of us are going on a boat ride."
"Boat ride?" Jack asked hoarsely.
"Yeah. You messed up the pick-up arrangements, so now I have to deliver it myself, and you two are going with me. If you try anything, Jack, anything at all...."
He pulled the knife away from Ashley's neck and set her back onto the floor. "I'm not a violent man, not unless someone makes me that way. Don't make me hurt your little sister." He touched her cheek with the blade of the knife, the other hand still clamped onto her shoulder. "I'd hate to have to teach you a lesson, Jack. But I would. Remember that. Now pick up the suitcase." He waved the knife toward the door. "You go first. My little girl Ashley will be right by me—" Suddenly, he froze. "What was that?"
Jack heard it too. A scraping noise came from the balcony. Bindy! He'd half forgotten she was out there! Had she listened in? Did she know what was happening? Jack's heart felt as if it would explode right out of his chest as the man zeroed in on the glass door. The curtains, which were half shut, hid most of the view. There was no place to hide out there. Jack could feel himself stiffen as he pictured the man finding Bindy.
"Are you still playing games with me, Jack? Is someone out there?"
"No!"
"Well, now, this will be your first test. Let's see how well you do. If you are lying, your sister pays the price." He narrowed his eyes at Jack. "Don't move a muscle." Clamping Ashley around the neck, the man dragged her toward the balcony, yanking open the glass door with the hand that held the knife. As the man's head swiveled from side to side, Jack held his breath until the blood pounded furiously in his ears, as loud as drums. He waited for Bindy to be discovered, but a moment later, the man pulled his head back inside.
"There's a bunch of ducks out there, that's all you heard," Ashley babbled. "No one's out there. Let me go! You're hurting me."
"Shut up and move," was all the man said.
Bindy had escaped! The scene flashed through Jack's mind like a neon sign—Bindy must have climbed over the balcony into the adjoining room, the one she shared with Ashley. She would call for help, and the police would catch up to them before they had to enter the boat! Call, Bindy, call! Jack willed her. Hurry!
Herded through the door, along the front deck and down the motel stairs, he tried to buy time by pretending that the heavy suitcase slowed him down even more than it did. The man had tucked Ashley beneath his left arm, his overcoat practically swallowing her like a blanket.
His right hand, still clutching the knife, was hidden in his pocket. Anyone watching would have thought it all looked perfectly normal. A father, daughter, and son enjoying one last view of the ocean before they checked out of the motel. Jack looked around, desperate to see anyone who might help. He heard wheels clacking along a walkway, and moments later saw a mother pushing a blue stroller, her child's round head bobbing as the woman maneuvered around a corner, disappearing without a glance in their direction.
"Go to the pier," the man ordered. "If anyone looks your way, just smile and nod. Smile and nod."
They followed a narrow path strewn with crushed shells and gravel all the way to the water's edge. An elderly couple returning to the motel strolled past them on the path, close enough to touch, but they were deep in conversation and didn't look their way even once.
The rest of the beach was empty except for the gulls. A steel-gray wave curled, then crashed onto the beach, licking the sand with foam before slipping back to the sea. It smelled like fish and damp wood here. Jack strained to listen for the sound of a siren but heard nothing save the screeching gulls. They were utterly alone.
"Now climb onto the pier. You remember the pier, don't you Jack? It holds a special memory for me."
Without a word of protest, Jack stumbled along the boards, followed by Ashley and the man. Shadows from the railing made a pattern on the bleached wood pier, black on gray.
"Move it!" the man said, shoving Jack in the middle of his back.
The pier seemed different in late afternoon light. At its end, he saw the wooden steps—gallows steps, he thought now—and something else that hadn't been there the night before. Tied to a wooden piling, a speedboat bobbed like a seagull in the black water, only inches from the landing.
The man hurried them down the rickety steps, gestured to the boat, and ordered Jack, "Put the suitcase in there first. Then you get in."
Where were the police? Surely Bindy had called them by now! Jack didn't want to climb into the boat, but when he hesitated, the man moved his arm under his overcoat, and Ashley cried out in pain.
"I'm getting in," Jack cried. "Leave my sister alone."
"Do exactly what you're told. We're going for a ride."
Jack felt helpless—totally and completely helpless. No one could see them from where they were, no boats were in the marina, no people sauntered along the beach, no police had come to save them. There was nothing to do but obey orders. Stepping into the boat, he felt it rock beneath him. The man maneuvered himself on board with Ashley still clamped to his side. When a wave swelled, the boat thrust up and then down again.
"Sit," the man ordered Jack. "There." He pointed to a seat with a red cushion. Jack sat, his spine as straight as if it had been lined with steel. He looked around but saw nothing he could use as a weapon.
"Put your hand on that rail. Do it!"
The metal of the boat's railing felt cold against Jack's wrist as he laid his hand on it. From a pocket the man took handcuffs that he snapped first onto Jack's wrist, then onto the railing, tethering Jack to the boat. The vessel itself was his anchor now, and there was no way out. If it sinks, he thought, I'll drown.
Moments later, as the motor roared to life, Ashley sprawled onto the floor, sliding on her stomach toward the back of the boat. The man had secured the suitcase so it wouldn't slide, but he didn't care about Ashley.
As they sped into the waves, wind churned against them, making Jack's eyes blur with tears. He could make out Ashley crawling toward him on all fours, her hair whipping into a black cloud, her shirt puffed out like a sail. The motel shrank in the distance to the size of a child's toy, then to a dime, then to a single piece of confetti. The wake sprayed into his face, chilling him. Turning in a wide arc, the boat headed for the open sea, and the man, shielded by glass, lit a cigarette. Jack felt hate surge through him as he frantically pulled against the handcuff. The metal bit into his skin.
Ashley rocked to her knees, grabbed Jack's waist, caught her balance, then placed her mouth as close as she could to Jack's ear. "What are we going to do?" she cried. Her eyes looked terrified.
"You need to—"
"What?"
The noise was deafening. There was no way the man could hear them over the motor's roar. Louder this time, Jack yelled, "Take this cushion and jump overboard. It'll float." The words seemed to fly back into his throat as the wind beat against him.
"No—no way. Anyway, he'd just come after me."
They were moving so fast the nose of the boat rose above the water until it pointed to the sky. Ashley held on to Jack with a viselike grip. "I'm so scared."
"Me too. But there's—" Even with the noise of the motor and the wind, he didn't want to say the name out loud, so he mouthed "Bindy." "She'll call the police. She heard. She knows!"
"Jack—they might not believe her!"
The thought chilled him more than the spray from the wake, slamming into him like a fist. It was true. Bindy had told so many fantastic tales in the past! He knew how unbelievable the story would sound to his parents, to the police, to anyone who would listen as Bindy tried to convince them: The three of us stole a case full of secret sonar equipment, part of the government conspiracy that hurt the whales, and a strange man in black kidnapped Jack and Ashley and made them get into a boat.
It sounded utterly preposterous. Just another one of Bindy's lies added to the pile.
Jack closed his eyes, and in the darkness behind his lids, thought of what lay ahead. Their only hope rested on Bindy.
## CHAPTER TEN
Wind cut against Jack, salt stung his cheeks, and the roar of the motor nearly deafened him. When would this ride be over? Still kneeling, Ashley clung to him, burying her face against his arm. Whenever the boat lurched, tossed by a wave, she hung on him even tighter. His left wrist, handcuffed to the rail, grew numb; his right arm, clutched by Ashley, began to ache. How much longer? And where were they being taken?
Jack squinted at the sky. In Maine in the month of May the sun didn't set until around 8:00, he had noticed the night before. Jack tried to read his watch, but the face of it was so covered with moisture from ocean spray that he couldn't make out the digital numbers. The sun was behind him, which meant they were heading east. He couldn't turn around to get a look at the sun's position because the handcuff, as well as Ashley's clinging, trapped him into a cramped position. Taking a guess, he figured it must be somewhere around six in the evening now, a full hour since the man—whose name Jack still didn't know—had come bursting into their room.
The roar of the motor suddenly changed pitch, dropping lower as the boat began to slow down. A thousand feet ahead of them, a much bigger boat sat at anchor—they were heading straight toward it! The pilothouse on the top deck had been painted white, but a dull red covered the walls of the cabin on the lower deck. Attached to that deck was a huge spool wound with cable, probably used to raise and lower the anchor, or maybe it had something to do with sonar.
As they got closer, Jack estimated that the boat had to be about a hundred feet long. It dwarfed the speedboat they were riding in. When they pulled alongside, the man secured the boat then jerked Ashley roughly by the arm and forced her to climb ten feet up a rope-and-wood ladder to the boat deck. Next, he hurried back down to grab the suitcase, leaving Jack behind, still handcuffed to the railing.
Scared, Jack wondered how long he was going to be left there, a captive in the speedboat. After what seemed a long time but was probably only a few minutes, the man returned for Jack. "Now you," he said, after unlocking the handcuff. "Get up there with your sister."
Jack quickly climbed the ladder because the man was right behind him, prodding him hard on the back. When he reached the deck, he found Ashley shivering there, all alone, looking cold and small and fragile.
Suddenly a voice assaulted them over a loudspeaker. "Scully, what the crud have you done? Where'd you get these kids? What are they doing here?"
Scully! So that was the name of the man who'd kidnapped them! Scully looked up toward the pilothouse and called back, "I'll explain everything when you get down here, Alex."
In seconds a door burst open above them. Descending as nimbly as a monkey, Alex clattered down a ladder and dropped to the deck, saying, "OK, Scully, start talking." Thin, wearing black pants, a short black jacket, and a black stocking cap, Alex had a high voice for a man almost six feet tall. "Explain this to me, and it better be good if you expect to get the rest of your money." In a gesture of angry frustration, off came the black stocking cap, revealing spiked blond hair and a smooth pale forehead. Jack noticed long, thick lashes and—he couldn't believe his eyes! Alex was a woman! A pretty woman, who looked younger than his mother!
"Did you bring the device?" she demanded.
"It's over there." Scully pointed to the suitcase. "About the kids, it's a long story, Alex. To make it short, they found out about the sonar."
"How? How could they possibly find out?"
"Like I said, it's a long story."
"You!" She strode toward Scully, her finger pointing accusingly. "You screwed up big, and your timing stinks.
I don't know what we're going to do with these kids because Hashim's on board, demanding to witness one more test."
"Where is he?"
"In the galley with the crew, eating supper, but he'll be up here soon. Give me the key to the suitcase."
Scully mumbled, "I don't have it."
"What! Oh, for crud—" Crouching down so that her feet stayed flat on the deck while her long legs bent completely in half, Alex yanked a screwdriver from her pocket and forced open the suitcase. "We need to do a test as soon as I install this, because a chopper will pick up Hashim in a couple of hours. We gotta prove to him that this new sonar will work in the shallow waters of the Persian Gulf."
Jack kept his face perfectly still, trying not to react to the words "Hashim" and "Persian Gulf." He was smart enough to know that learning names and places could be dangerous. How many times had he heard on TV, "If I tell you that, I'll have to kill you!" And there he stood, getting an earful of what surely must be secret information.
Ashley had no such concerns—she spoke right up. "You're not going to test that sonar again! You've already killed too many whales with it. You promised you wouldn't do it again."
Surprised, Alex asked, "Who promised?"
"You! The U.S. Navy!"
Scully guffawed, while Alex shot him a dirty look.
"Ashley," Jack muttered, "these people aren't part of the U.S. Navy."
Still scowling, Alex said, "Well, I was once. I worked on sonar technology."
"Yeah, until she got kicked out," Scully said, still grinning. "Dishonorably discharged for conduct unbecoming an officer and a gentleman."
"Gentleman!" Alex spat. "That's what's wrong with—hey, I don't have time to deal with garbage right now, so just shut up, Scully!" She turned her attention to the suitcase, lifting the lid.
At last Jack got to see what was inside. At first all he noticed was thick Styrofoam, layered to hold tight whatever the object was so it wouldn't shift around.
Alex tore off the Styrofoam and threw it on the deck. Then, bending forward, she lifted—a plate?
It looked like a plate, but as she picked it up Jack saw that it was a heavy steel rim painted yellow, with bolts all around the edge and in the center, a sheet of thin, shiny, crinkled black metal. The whole thing measured about 18 inches in diameter. What it might be was impossible to guess.
"It ticks me off that those geeks charged me a quarter of a million to rebuild this part after the experimental one broke," Alex was complaining. "That's as much as I paid for my whole boat."
Scully answered, "Yeah, but this boat is an old tug from back when they shoveled coal into the boilers in the engine room."
"Right, and why'd I buy an old tug? Guess! Because it doesn't arouse suspicion. A year ago, this 'old tug' was fitted with an 800-horsepower diesel engine, get the picture? And now it'll have a phenomenal sonar system, once I ditch the defective part and install this state-of-the-art acoustics transducer. Nobody else will have anything this good for the next five years. Anyway, I gotta go tell Hashim the part's here," she said, hoisting it. "So what do you plan to do with these two kids, Scully?"
"Throw them overboard."
Ashley gasped and Jack jerked backward, ready to run, but where could he run to? It was like being on a tiny island, with no escape.
"You're joking, right?" Alex asked Scully.
Coldly, he answered, "I'm not gonna screw up a ten-million-dollar operation because of a couple of nosy brats. You got any better ideas?"
"Me?" she cried. "Hey man, they're your problem, not mine. But let me tell you, I'd prefer not to have any kids murdered on my boat." Alex spoke calmly, as though she were discussing a maintenance problem. "So whatever you do with them, I don't wanna know about it."
What did that mean? If Scully murdered Jack and Ashley on the boat, it would be OK as long as Alex didn't find out?
Trust Ashley to always spout off at the wrong time. "I guess if you're the kind of person who murders whales, Alex, you won't worry about killing us. But, listen, if you let us go, I promise we won't tell anything." She held up her right hand as though taking an oath in court.
"Like I believe that," Alex scoffed. "I can't waste any more time on this, Scully. Do whatever you have to." Cradling the sonar device in her arms as if it were a baby, Alex turned her back on them and walked through the cabin door.
Jack and Scully eyed each other. Ten feet of deck separated them, enough to give Jack and Ashley a slight head start if they ran. Jack had no intention of standing still and letting Scully kill them—not without a fight. They'd run first, and maybe find someplace to hide.
As Scully reached for his knife, Jack yelled, "Go," as he shoved Ashley away from him; whatever direction she ran, he'd move the opposite way, hoping Scully would chase him and not Ashley.
She sprinted toward a ladder that reached from the deck they were on to an upper deck behind the pilothouse. Jack hesitated for a split second until he was sure Scully was focusing on him, then ran toward the giant spool wound with inch-thick cable, scrambling behind it and crouching down. He heard Scully's heavy boots slamming on the deck, then a few seconds of silence while Scully tried to see where Jack had gone. Those seconds would give Ashley time to locate a hiding place on the upper deck, he hoped.
When Scully spotted him, Jack leaped to the top of the huge spool, grabbed the base of the flagpole, and hoisted himself onto the upper deck. He couldn't see Ashley, but that was good. It meant she'd found shelter.
Swearing, because he was too big and heavy to swing himself up the way Jack had done, Scully ran back around the deck to the ladder and started to climb. Looking around, Jack saw two freestanding structures, one of them a shed about the size of his bedroom closet, and the other one was—a smokestack! Left over from when this old boat burned coal in its engine room.
He heard her then, a soft call, "Jack! Up here."
It was Ashley, inside the smokestack, clutching the edge of it, white-knuckled, with both hands. Ashley, always nimble, had somehow scaled the smokestack, even though its top stood a good ten feet above the deck. How did she do it? Jack realized she must have climbed to the top of the shed and from there jumped over to the smokestack, slithering inside it.
Scully's swearing sounded closer, so Jack thought he'd better try the same escape. Looking for toeholds in the rough wood, he dug in with the rubber tips of his sneakers and made it onto the shed's roof, but not before Scully saw him. Knowing that Scully, who was already panting, couldn't climb up there very fast, if at all, Jack stood poised to jump—but not to the smokestack. That would show Scully where Ashley was hiding. Instead, if Scully started to climb, Jack would jump straight down on him, hoping to knock the big man onto the deck and kick the knife out of his hand.
"Jack, look!" Ashley screamed. That was when Jack heard it—the thump of rotor blades on a helicopter. Alex had said a chopper would be coming to pick up Hashim—that must be it. Jack wondered where it was going to land, since it looked too big for the deck. It was about half as long as the whole hull of the boat.
And then he spotted the most beautiful words he'd ever seen: "U.S. Coast Guard."
Painted bright red, with a broad white stripe near the tail, the helicopter hovered over Jack like an angel. He stood up and raised his arms, waving wildly. Beneath him, he saw Scully running, probably to warn Alex. In the distance, two Coast Guard vessels skimmed the waves, moving fast toward the boat.
While the chopper hung overhead, wind from its rotor nearly blew Jack off the roof of the shed. Then, amazingly, the door of the chopper opened and a man wearing a helmet and a harness was lowered on a hoist. Toward Jack!
"Get my sister!" Jack yelled as loudly as he could to be heard above the roar of the rotors. He gestured to the smokestack where Ashley was hiding, her head now poking out like a jack-in-the-box. The man gave Jack a thumbs-up, then maneuvered the hoist toward Ashley. After he reached down to grab her around the waist, he slipped a rescue sling over her head and under her arms, then signaled someone in the helicopter to raise the hoist. With her rescuer hanging beside her on the hoist, they were safely inside the chopper in less than a minute.
The helmeted man, still on the hoist, pointed to Jack and then pointed to one of the Coast Guard vessels that had almost reached the tug. So Jack was going to be picked up by boat! As thrilled as he felt about the arrival of rescuers, he also felt a pang of disappointment. Ashley got lifted on a hoist into a helicopter, while Jack would be rescued by boat. It wasn't fair.
Suddenly an amplified voice boomed at Jack, "Wanna do it her way?"
Jack nodded furiously. Immediately the hoist snaked down toward him, with the helmeted man soon dangling above Jack's head. Thrusting his arms quickly through the rescue sling, Jack felt the hoist move upward, and then he was flying! Instead of pulling him up into the chopper, the pilot moved slowly over the nearest Coast Guard vessel. Gently, as though Jack were a spider on a strand of silk, the chopper lowered him to the deck, where waiting arms caught him.
## CHAPTER ELEVEN
"It looks just like a white castle," Ashley exclaimed.
"No, it looks like a lighthouse," Bindy corrected her. "'Cause that's what it is—Bass Harbor Head Lighthouse. It was built in 1858, and, according to this brochure, it stands 56 feet above the mean high water mark."
When Ashley murmured, "Well, it looks like a castle to me," Bindy put an arm around her and teased, "You're such a romantic. Maybe you should take up acting!"
"No way! I'd rather save whales."
Steven's voice floated up to them, "Kids, come down the stairway—you've got to see the ocean from here."
As Ashley and Bindy followed the path toward the stairs, Jack called after them, "Tell Mom and Dad I'll be right behind you guys. First I want to get a few shots up here." He'd brought his camera to Acadia, but with all the exciting things that had happened, he'd hardly had a chance to take pictures until now.
He checked his camera settings as he moved around to get the best view of the lighthouse. Ashley was right, Jack decided—Bass Harbor Head Lighthouse did look like a miniature castle, rising from a cliff that overlooked the ocean's endless blue-green water. A small roof of red tile capped the big lights at the top of the tower. Once the sky darkened, those lights would come alive, sending out beams like Star Wars light sabers. Jack could imagine himself inside the lighthouse, scanning the ocean for tall, masted ships; it would be like living in a postcard. He kept clicking the camera's shutter until he ran out of film.
After their exploration of the lighthouse, they drove to the nearby town of Bass Harbor and found a small restaurant huddled in a row of brightly painted stores. Inside, Jack and Ashley sat opposite their parents while Bindy perched on a chair at the end of the table. Since this was Bindy's celebration party, she'd been allowed to order anything she wanted from the menu.
She set down her spoon and stared ruefully at half of a chocolate fudge nut sundae still in her dish. "Whew, that's enough!" she exclaimed. "Even I have a limit. But I hate to waste it."
"It's all right, Bindy, you can waste it," Jack told her. "After all, it's your third one."
To stifle a giggle, Ashley covered her mouth with her hand, getting a bit of chocolate on the tip of her nose. That made Bindy giggle, too.
"Wasting a fudge sundae isn't important," Olivia commented. "Wasting a life—that's tragic."
"You're thinking about Alex?" Steven asked.
"Yes, former Lieutenant Commander Alex Turner, the technical wizard with no moral principles. Greed got in the way of that brilliant mind. What a waste!"
"And she's so pretty, too," Ashley said. "She could have been anything she wanted to be. A model!" When Bindy glared, Ashley stammered, "Not that good looks mean anything! Anyway, Bindy, you said everyone believes you if you're beautiful. Well, no one believes Alex now. She's in jail. Maybe forever."
"For selling technology secrets to the enemy," Bindy agreed.
"And that punk Scully is in jail too," Jack added vehemently. "I hope he stays there till he rots."
"Take it easy," Steven told him. "They'll both get what they deserve."
Olivia reached over to wipe the chocolate off Ashley's nose just as Ashley asked, "Dad, I still don't understand everything about how you found us. Tell me one more time."
Steven answered, "Luckily, I got back to the room within minutes after you two were abducted. Bindy had been trying to call the police, but they weren't paying any attention to her, and neither would I, at first, and I feel really bad about that now. Anyway, she dragged me out onto the balcony to show me where the boat had gone." Steven paused. "It just happened that I already had my strongest telescopic lens on my camera, so I was able to see the boat and take pictures of the direction it was headed. After that, everything fell into place."
"That's the part I want to know about. Keep going, Dad," Jack urged.
"It all worked out because of our nation's homeland security program—that, and our incredible communications system. I talked to the Bar Harbor police; they called the Navy. The Navy was keeping an active file on Alex, so they knew about the boat she'd bought." As he spoke, Steven traced circular patterns and lines on a paper napkin with his fork, explaining, "They plugged into a surveillance satellite that within minutes located the tug in the Atlantic. It even spotted the speedboat just when it was docking alongside the tug. The Coast Guard was contacted next, because they already had units patrolling the area. That's it!"
"Amazing!" Olivia breathed.
"And they caught Alex before she could put the replacement part in the new sonar she was selling to the bad guys. That means whales' lives are going to be saved. All because of Bindy!" Ashley exclaimed. "Let's all celebrate Bindy, the hero!"
Looking a little sheepish, Steven said, "I apologize for not believing you at first, Bindy."
She raised her eyebrows. "Yeah, getting believed. That's always the hard part."
"You're right," Steven said, covering her hand with his. "Now I understand what it's like not to be trusted. When I tried to tell the police what you'd said, they just brushed me off. I knew what they were thinking—I was some kind of incompetent who couldn't keep track of a foster child. After all, I'd already been to them twice about you running away, Bindy."
"And then you called them with this wild story about your two children missing, too," Bindy said. "So they figured this guy's a nut case, trying to get some attention. Yeah, I've been there."
All of them fell silent, probably thinking about Bindy's problems with Cole. Ashley asked, "How'd you get them to listen to you, Dad?"
"I made them call Harvard Medical School for the necropsy report on the dead whale. That's when they started to pay attention."
"Yay!" Ashley cheered, clapping her hands. "What a smart dad I have!"
Steven still looked a bit unhappy, and Jack wondered whether he was regretting all those great pictures he didn't get to take of Spud returning to the ocean. Maybe Olivia was thinking the same thing, because she said, "I know Steven's whale rescue photographs would have been the best, but there was plenty of media coverage about Spud. As the day went on, all the television stations for hundreds of miles around sent reporters and cameramen. The whole rescue is on videotape, so you kids will get to see Spud swimming back to his mamma."
"And you'll see me, too," Bindy said.
"What? Swimming back to your mamma?" Jack was trying to be funny, but as soon as he said it he wished he could bite his tongue. Bindy looked crestfallen.
"No. I meant I was on camera. Three of the TV reporters interviewed me about how I got you two rescued." Suddenly she brightened. "You know what?
I liked being back on camera. I decided I really miss it."
Olivia nodded. "I can understand that, Bindy. You're a natural-born actress."
The stone steps leading down to Thunder Hole were slick with seawater. Careful to keep his hand on the railing, Jack could feel the condensation as he slid his palm down the cool metal. Since his parents had run into Greg in the parking lot, they'd told the kids to go on ahead to Thunder Hole, promising to catch up soon.
Before they left, Greg asked, "You kids know much about Thunder Hole?"
The three of them shook their heads no.
"It's a true natural wonder. Every once in a while a really big wave hits, and the force of it smacking the air inside the hollow cavern creates a clap so loud it's almost like a sonic boom. But it gets cold down there," he warned them. "Thunder Hole sprays out a lot of water, and with the breeze up like it is now, well, you'll feel the chill. You might want to grab your jackets."
"We didn't bring any," Ashley told him.
Olivia brightened. "Bindy, why don't you get those blankets we're going to return to the rangers? I left them in the car."
"Sure thing," Bindy answered, backtracking quickly.
Now, as the three of them made their way down the steps, Bindy threw one of the blankets around her shoulders while Ashley draped a second blanket over her head, clutching it under her chin. Jack had already decided that wearing a blanket would look too wimpy. He preferred to tough it out in his T-shirt.
The end of the pathway led to an enormous slab of rock enclosed by a steel railing. Its shape reminded Jack of the bow of a ship. Beneath him, he saw a natural cavern that had been carved, over eons, by the force of waves crashing against rock. The bullet-shaped hollow was deep enough that when the ocean rushed into it, water shot into the air like a geyser, before curling back on itself.
As Jack leaned over the railing, a spent wave retreated; the blue-gray water roiled as if boiling in a cauldron.
"How come I don't hear anything but splashing?" Ashley asked. "Where's the boom?"
"Remember what Greg said. We have to wait for a really big wave before the sound happens. You have to be patient."
Another wave hit the hole, sending up a spray of water that seemed to burst into a million water crystals, but there was no boom. Jack watched the ocean, shivered, and waited. More swells hit Thunder Hole and retreated. A mist chilled his bare skin, and he could feel gooseflesh rising on his arms. Maybe he should have accepted a blanket.
Suddenly, Bindy grabbed Jack's arm. "Look! Here comes a huge one!" she cried, pointing. "Whoa—this ought to do it!"
He watched the line of a wave swell and curl, pushing toward the shore with mounting force until it hit the hole with tremendous power. A sound like thunder exploded around them, causing spray to shoot so high it seemed to touch the clouds. The water rained back down onto Thunder Hole before receding to the ocean once more. In the silence that followed, Jack heard droplets drizzle off the rocks in tiny waterfalls. "Man, that explosion sounded like a cannon," he exclaimed. "It was loud!"
"Hence, the name," Bindy told him, grinning. Pulling the blanket around her, she squinted into the sea, the wind tousling her hair into languid wisps. She seemed to be thinking about something.
"Hey, Bindy," Ashley commented, "what's wrong? It's like you got sad all of a sudden."
"I'm not sad. It's just—oh, I don't know. I was looking out there in the ocean for Spud and his mamma, hoping the two of them would be all right. I guess that made me think of my own mom. I...I miss her. So much." A flush crept across her cheeks, and Jack could hear her voice tighten. "Sometimes I'm OK, and then I see real families and I feel so cheated. It's not fair. None of it's fair."
Ashley moved closer, huddling against Bindy. "I heard that you got a call from your aunt today. Dad told me."
"Yeah." Bindy's face contorted. "So?"
In a voice barely heard above the waves, Ashley asked, "Is everything all right?"
Bindy shrugged. "Everything's fine. Couldn't be better." She pulled a strand of hair from her mouth, then looked back at the sea. Neither Jack nor Ashley said a word. Another huge wave rolled in, exploding into Thunder Hole and then showering the rocks in a furious torrent. After what felt like forever, Bindy finally spoke. "I might as well tell you what happened. I didn't tell your folks because I didn't feel like talking about it, but...." She took a deep breath. "Cole got into more trouble."
Jack leaned forward, alert. "What trouble?"
"I don't know all the details. I guess he beat up some player after a football game the other team won. This time a coach caught him doing it. Cole outweighed the other guy by about 60 pounds and hurt him pretty bad."
"Wow!" Jack breathed.
"Aunt Marian told me Cole has to go for psychological counseling. It's like a court order or something." Bindy frowned. "Cole tried to lie his way out of it, but when the coach told him he might lose his scholarship if he didn't complete the counseling, he broke down and confessed everything. He even told the coach how he used to hit me."
Ashley touched Bindy's arm and said, "That's good, isn't it?"
Waving her off, Bindy answered, "So now Cole's in this anger management class and Aunt Marian said...." She swallowed, then went on, "She said she...she wants me back. She said she realized she was wrong. She promised that Cole will be better now." Bindy laughed, but it wasn't a happy sound. "Like that fixes everything. All of a sudden I'm supposed to be grateful when she tells me she understands now. The part I don't get is—" Her eyes filled with tears as she wailed, "Why wouldn't she believe me to begin with? I told her what happened—I told her and told her and told her. How come my words weren't good enough? Why wasn't I good enough?"
Jack didn't know what to say. Bindy was right—what happened to her wasn't fair. But one thing he learned about having foster kids in his home was that sometimes bad things happened and there was no justice to it, but in the end you just had to go on, somehow. He rubbed his arms and searched for something to say, but everything that came into his mind sounded hollow, even to him.
Quietly, Ashley said, "You know, Bindy, when you were talking about Spud, I thought of a story I heard. It's an Inuit legend. Would you like to hear it?"
"I thought I was supposed to be the performing artist," Bindy said. "But go ahead, Ashley. Tell your story."
Pulling her blanket tightly around her shoulders, Ashley closed her eyes and began.
Many, many years ago, when the Earth was born, the Great Spirit created the land. Everything he made was good—he placed the sun in the sky to give warmth by day and the moon in the heavens to give light by night. He placed fish in the sea and filled the air with every kind of bird. He made the great bear, and the walrus, and the seal. Then, the Great Spirit made the Inuit people. And because the Great Spirit had a special love for the Inuit people, he became their teacher, showing them how to live by using everything around them.
Then the Great Spirit decided to make one thing more, the very best of his creation—the bowhead whale. This was his most beautiful creature. He gave it a song, and as it sailed though the waters, sharing its melody, the whale was in perfect balance with all of creation.
But the Great Spirit saw something else. His people needed the whale to survive the bitter-cold winters. Without the muktuk, the flesh of the whale, the people could not stay warm and healthy during the frigid nights. They needed the bones of the whale to create their homes.
In short, they needed every part of the bowhead whale in order to live.
And so the Great Spirit gave the bowhead to the Inuit.
With spring comes the thaw. The ocean ice breaks apart, creating a water road called the Open Lead. It is on the Open Lead that the bowheads swim, right into the harpoons of the Inuit. Every year the whales sing, and every year they come, waiting patiently for their death.
But, the Great Spirit decided this also. At the same time every year, when the Open Lead is formed, the Great Spirit sends a cloud of heavy mist to hover just above the ice, above the whales, and above the Inuit. The heavy cloud hangs in the air between the sky and the sea.
"Though I gave you permission to kill my most perfect creature, the whale," the Great Spirit said, "I do not wish to watch it happen."
Opening her eyes, Ashley searched Bindy's face.
"What are you trying to say?" Bindy asked. "Is that story supposed to mean something to me?"
Ashley nodded. "Maybe," she answered, "your aunt didn't want to believe what her son was doing, so she made a cloud to hide the truth. But now that she knows, she's trying to tell you she's sorry."
For a long moment Bindy stayed silent. Then she murmured, "Thanks, Ashley."
"Are you going to go back to her?"
"I don't know. Maybe. Maybe not." Turning to Jack, Bindy struggled to clear the emotion from her face. "You're wet. Look at you, you've got goose bumps all over. Here, take my blanket."
"I'm fine," Jack told her, shivering.
"No offense, but you've always been a bad actor. Come on, Jack-o, you're freezing. Take it!"
When he refused again, she said, "All right, then, we'll share. Ashley, you throw your blanket around him on one side, and I'll throw mine on the other, and if we huddle up together...." Jack felt himself enveloped in warm blankets. "Maybe we'll all survive."
Jack could hear Bindy's breathing, feel her rounded arm pressed against his, and on his other side he felt Ashley's knob of an elbow drill his ribs. They sat, the three of them, separate in their thoughts, yet united by what they'd been through together.
"We make a whale of a team," Bindy murmured.
She was right.
## AFTERWORD
The evil scientist is caught. Jack, Bindy, and Ashley save the day, and Olivia and Steven save the whale. All is well. But while this story didn't actually happen, many of the details are based on fact. There are some important themes in Out of the Deep that are worth returning to.
First, it is important to understand there are two types of whales—those that have teeth and those that don't. Toothed whales (odontocetes) are typically smaller, and include dolphins, porpoises, beaked whales, sperm whales, and killer whales. Baleen whales (mysticetes) lack teeth. Instead they have rows of baleen plates that hang from the roof of the mouth. These plates strain microscopic food from the water. Common mysticetes include the humpback, fin, and minke whales.
Do whales really beach themselves? Yes, but not frequently. In Maine, where Acadia National Park is situated, we respond to 5–10 toothed whale strandings each year. Baleen-whale strandings are much rarer and happen perhaps once a year. In most cases the animal washes ashore already dead. Finding Spud alive in Out of the Deep meant people still had a chance to save him.
Why do whales strand? We really don't know. There probably isn't just one cause. In some cases whales may deliberately swim toward a beach and strand. If rescue teams are unable to re-float the whale and push it back to sea, then it will likely die of asphyxiation (meaning it will suffocate) or from massive internal trauma. Out of water, a whale cannot support its own weight. Gravity will crush its organs. In some cases, a whale will repeatedly strand. This apparently suicidal behavior is difficult to understand. Even more puzzling are mass strandings, in which several animals strand within a short period of time.
Scientists are careful to examine all possible causes of any kind of stranding. If a stranded whale dies, we quickly perform a necropsy to see if we can understand more about what happened. These examinations don't always answer our questions, as we still know very little about a whale's physiology. From personal experience I can tell you that performing a necropsy on an animal as large as a whale is a smelly and gory business, not for the faint of heart!
In this story, Olivia remembers a mass stranding that actually happened in the Bahamas. Necropsies revealed that the ear structure in some of the whales had been destroyed. Eventually, the stranding was linked to local sonar testing by the Navy. I was one of the researchers at the Newfoundland stranding Greg mentions in the book. Examination of that humpback revealed that powerful sound waves had sheared the whale's earbone. That investigation was one of the first to suggest that exposure to loud sound could lead to a whale's death.
You may be wondering how sound can be so deadly to a whale. The answer lies in the way these animals perceive their world.
Think for a moment about how you take in the space around you. Vision is the sense we humans rely on most. This is because the air that surrounds us is transparent. But whales live in a watery environment where vision, at best, can only be used at short range. (Picture how far you can see underwater, even if you wear a mask and snorkel.) In contrast, sound travels very long distances in water—much farther than it does in air. Whales most likely "see" their surroundings as pictures painted not in color but in sound. When intense sound waves harm a whale's inner ear, they rob the animal of its ability to accurately sense its environment. It loses its ability to find its way.
Humans are but a small blip in the history of life on this planet. In contrast, whales have existed for about 60 million years. For most of that time, their environment was relatively quiet. But steadily over the past 200 years, humans have made the ocean a noisy place—first with the roar of boat engines, then with sonar and other types of sound devices. Whales have no way to protect themselves against this noise pollution. As a result, these magnificent animals have begun to suffer.
It is humbling to reflect upon how little we humans know about whales. We must learn about our impact on these mystical animals—not only about the effects of noise pollution or the use of sonar but also about the devastation of hunting that has brought many species to the brink of extinction. We must protect these creatures of the deep. Only knowledge combined with action ultimately will lead to a solution.
May it come in our lifetimes.
Sean Todd, Ph.D.
Professor, College of the Atlantic
Senior Researcher, Allied Whale
http://www.coa.edu/alliedwhale
## ABOUT THE AUTHORS
An award-winning mystery writer and an
award-winning science writer—who are also mother and
daughter—are working together on
Mysteries in Our National Parks!
ALANE (LANIE) FERGUSON'S first mystery, Show Me the
Evidence, won the Edgar Award, given by the
Mystery Writers of America.
GLORIA SKURZYNSKI'S Almost the Real Thing won the
American Institute of Physics Science Writing Award.
Lanie lives in Elizabeth, Colorado. Gloria lives in Boise,
Idaho. To work together on a novel, they
connect by phone, fax, and e-mail and "often forget which
one of us wrote a particular line."
Gloria's e-mail: gloriabooks@qwest.net
Her Web site: www.gloriabooks.com
Lanie's e-mail: aferguson@alaneferguson.com
Her Web site: www.alaneferguson.com
##
Founded in 1888, the National Geographic Society is one of the largest nonprofit scientific and educational organizations in the world. It reaches more than 285 million people worldwide each month through its official journal, NATIONAL GEOGRAPHIC, and its four other magazines; the National Geographic Channel; television documentaries; radio programs; films; books; videos and DVDs; maps; and interactive media. National Geographic has funded more than 8,000 scientific research projects and supports an education program combating geographic illiteracy.
For more information,
please call 1-800-NGS LINE (647-5463) or write to the following address:
NATIONAL GEOGRAPHIC SOCIETY
1145 17th Street N.W.
Washington, D.C. 20036-4688
U.S.A.
Visit us online at www.nationalgeographic.com/books
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Should she have spoken out more strongly?
margin, but they'll voted to leave nonetheless. -- they voted to leave.
which is a legacy of the past executive and is appalling.
you're going to back It is back to that word democracy.
internal exile in the rest of the UK. These are fundamental issues.
governments. Minor modifications and amendments - to you except that?
the juggernaut. That's the phrase your part the Lido. -- party leader.
Ireland is recognised as being most affected but is also most prepared.
times between now and the 2nd of March.
people at the meeting tomorrow take very different views on this issue.
the assembly ended in low-key fashion. The assembly is adjourned.
Gareth Gordon looking back at another busy week.
With me is the Fine Gael TD for Louth, Fergus O'Dowd.
How important is that meeting between the two premiers?
island and in the United Kingdom as well.
Does that suggest Ireland is not a priority for her?
have everything in each other's countries, so we must work hard.
perspective that Theresa May is driving a hard Brexit juggernaut.
with Theresa May is critical to progress on all of these islands.
with the UK than he is over the border between the north and South?
relationships on this island with each other are the priority.
so clumsily footed? It is up to him whether he should leave the office.
my hopes are not so great.
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MIUI will it ever bless the Vibrant?
I'm searching up and down for threads asking this same question, and unfortunately none.
Well what are the reasons barring a port to the Vibrant?
IS it a CDMA/GSM thing?
Someone that has a vibrant needs to know how to translate it. the people who translate and port miui usually have the phone they are doing it for.
A. Need a person with the know how and knowledge to do the above and continuously update it.
Thanks for the quick response WiiNeedMore much appreciated. maybe I can post something about it on XDA and hope.
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A Picture of Guilt
A Picture of Guilt
Libby Fischer Hellmann
www.libbyhellmann.com
Poisoned Pen Press
Copyright (C) 2003 by Libby Fischer Hellmann
First Edition 2003
Library of Congress Catalog Card Number: 2003104004
ISBN: 1-59058-073-7 Hardcover
ISBN: 9781615950942 ePub
_A Picture of Guilt_ is a Berkeley original paperback publication and this hardcover edition has been published with the kind permission and cooperation, and is reprinted by arrangement with The Berkeley Publishing Group, a member of Penguin Group (USA) Inc.
All rights reserved. No part of this publication may be reproduced, stored in, or introduced into a retrieval system, or transmitted in any form, or by any means (electronic, mechanical, photocopying, recording, or otherwise) without the prior written permission of both the copyright owner and the publisher of this book.
Poisoned Pen Press
6962 E. First Ave., Ste. 103
Scottsdale, AZ 85251
www.poisonedpenpress.com
info@poisonedpenpress.com
Dedication
For Michael and Robin
Acknowledgments
This book would not exist without the input of many people, all of whom I owe a tremendous debt of gratitude. Especially Don Whiteman, whose knowledge, patience, and careful reading of the manuscript were a godsend. Also FBI Special Agent James Whitmer, Bob Egan, David Wechsler, Gerry Kessler, Leon Guaquil, Northbrook Deputy Commander Mike Green, Northfield Chief of Police Bill Lustig, Dave and Jean-Marie Case, and the Red Herrings.
To Jacky Sach and Samantha Mandor: your insights and support are unparalleled--thanks for believing in Ellie. To Nora Cavin, I will always depend on your ear. And special thanks to Barbara Peters, TEE, who is really an angel in disguise.
Epigraph
Whoever destroys a single life is as guilty as though he had destroyed the entire world; and whoever rescues a single life earns as much merit as though he had rescued the entire world.
-- _The Talmud_ , Mishna. Sanhedrin
Contents
Dedication
Acknowledgments
Epigraph
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
Chapter 18
Chapter 19
Chapter 20
Chapter 21
Chapter 22
Chapter 23
Chapter 24
Chapter 25
Chapter 26
Chapter 27
Chapter 28
Chapter 29
Chapter 30
Chapter 31
Chapter 32
Chapter 33
Chapter 34
Chapter 35
Chapter 36
Chapter 37
Chapter 38
Chapter 39
Chapter 40
Chapter 41
Chapter 42
Chapter 43
Chapter 44
Chapter 45
Chapter 46
Chapter 47
More from this Author
Contact Us
Chapter One
The raft plunged straight down and slammed into a wall of water. It flew up at a ninety-degree angle, propelling me up and out into the river. The rapids spun me around and threw me from side to side, burying me under a blanket of waves. I tried to right myself so I could rise to the surface, but I couldn't tell which way was up. My lungs were on fire; my eyesight grew dim. Then a powerful force shoved me up, and I broke the surface, gasping for air.
Feet up, legs straight. That's what the guide said. I tried to stretch my legs out, but a fresh torrent wrenched me under. I tumbled over like a flimsy rag doll. Then, as if taunting me with the promise of release, the river raised me up again. Two giant boulders loomed ahead. I gulped down air and squeezed my eyes shut, certain my last sensation on earth would be the excruciating pain of bone splintering against rock.
When I opened my eyes, the boulders were behind me. A surge of foam and spray had pushed me through a narrow channel between them. Above the roar of the rapids, I heard screams. I twisted around. Rachel! Twenty yards away, my thirteen-year-old's yellow helmet bobbed in the churning waves.
My stomach clenched. I lunged and thrashed my arms, trying to swim to her, but the rapids carried me in the opposite direction. Just before I went under, I saw the guide throw her a lifeline. It landed short. When I came up, she was gone.
The river carried me another quarter mile. Then, as if underscoring its absolute control, it quieted. A hush descended. Tiny bubbles eddied across the surface. Gentle waves rippled peacefully. Hot sun struck my face.
I didn't care. Rachel was gone. I wanted to cry, but the tears wouldn't come. I wanted to scream, but the sound died in my throat. I stopped struggling. The water had won.
***
"I should have known, when they said the river was kicking butt." I drained my third glass of wine.
"It was up two feet from the big storm the other day," Rachel added. She stirred her Coke with her swizzle stick.
"But you survived," Abdul, our dinner companion, said to Rachel.
Her blond curls framed her face like a soft, golden cloud. They bounced as she nodded. "Another raft picked me up."
Rachel inherited her coloring from her father, but we share the same gray eyes and feisty ways. I pushed a strand of thick, black hair behind my ears, thinking how close I'd come to never seeing those curls again.
"The guide said you weren't in any real danger, with the life preservers and helmets," David said.
I glared. "What did you expect him to say?"
We were finishing cocktails in the main dining room at the Greenbrier. Nestled in the backwoods of West Virginia, the resort is one of the plushest in the world. With its graceful columns, sculpted gardens, and antebellum buildings, it fairly drips Southern gentility.
Which is why it sounded like a perfect idea when David asked us to meet him there for the Labor Day weekend. It would be a grand finale to summer, an elegant start to fall. It was also a chance for the three of us to spend time together, since Rachel and I live in Chicago. To bond, in that trying-each-other-out-as-family kind of way. I'd even started to look forward to the trip, imagining myself sipping mint juleps on the veranda in a frothy summer frock. Of course, I don't have a frothy, mint-julep-sipping frock, but my friend Susan let me borrow hers. Susan has a dress for everything.
"Another glass of wine?" Abdul asked.
"I believe I will."
"Ellie," David cut in. "Haven't you had enough?"
"Not yet."
Abdul Al Hamarani had appeared at exactly the right time, like a fairy-tale genie magically released from his bottle. He was buying film inside the rafting company's small office as Rachel and I stumbled inside after our ordeal.
"We should never have left the Greenbrier," I'd muttered, collapsing on a chair.
He turned around. Wearing khakis with sharp creases, a pressed shirt, and a safari vest, he looked like he'd stepped out of a J. Peterman catalog. "You are staying at the Greenbrier?"
I nodded.
"I am staying there as well." He was round-faced, with dark hair slicked back from a widow's peak and had he had wide, lively eyes. Pocketing his roll of film, he introduced himself and, after David joined us and settled up with the guide, insisted we drive back together in his rented Mercedes. David started to decline, but after our experience, I wasn't eager for the bumpy ride in the van we'd come in. We accepted Abdul's offer.
During the trip we learned he was a Saudi petrochemical sheik and distant relative of the royal family. He'd come in a few days in advance of an annual energy conference, one of those international global-policy making forums the Greenbrier is famous for hosting.
"I like to take pictures." He pointed to a canvas bag that had been slung on his shoulder. "Even your harshest summer is a relief from what I'm used to."
We continued getting acquainted over dinner.
"I beg your pardon, Ellie." He had a slight British accent. "But given your...your attitude, why did you embark on this rafting adventure to begin with?" He'd changed into a Savile Row suit for dinner. The softness around his belly said he'd dined at some of the best restaurants in the world.
"David suggested it." In truth, I'd been looking forward to a lazy day on the river, the hot sun beating down, the smell of laurel perfuming the air. I'd been warned it was too late for the delicate pink and white blossoms, but there might be some white rhododendron--a first cousin--in bloom. As the state flower of West Virginia, rhododendron grows in wild profusion at the side of the road, in ravines, and on mountains. Even the pat of butter on our plates was sculpted into the flower.
"Besides us, there were four teenagers. All of us first timers." The waiter set a fresh glass of wine down in front of me. "Everything was fine at the beginning. The first rapid was only a two. Nothing to worry about. But then we got to the Keeneys. They're the most powerful stretch of rapids on that part of the river."
I looked from Abdul to David, aware I was reworking and revising, turning the experience into a story. As if somehow that could blunt the terror I'd felt. But hey--I'm a filmmaker. I tell stories for a living.
"So we hit the Keeneys, which usually are a four, but because of the rains the other day, were closer to a five. Apparently, when we hit the first patch, the kids got spooked and forgot to stroke. Big mistake." I rolled my eyes and then clutched the arms of my chair. The room was starting to spin.
"Yeah," Rachel said. "Our guide kept telling us we had to keep stroking. All the way through the white water."
"If you stop, you lose control and balance, and you're at the mercy of the water." I took a sip of wine. "That's what happened."
Abdul patted his mustache and carefully trimmed goatee. A band of gold around his wrist glinted in the candlelight. "Yet you proceeded down the rest of the river without incident?"
David cleared his throat.
I looked away. "Actually, I told them they would either have to get a helicopter in to pick me up, or I would be hiking back. And since there weren't any helicopters handy--"
"And you're afraid to fly..." Rachel added.
I shot her a look. "You hiked back with me."
"I didn't want you to be alone." She swirled the Coke in her glass. The corners of her mouth twitched with silent laughter.
I settled back in my chair. The room was definitely starting to spin. I propped my elbows on the table. "One of the guides took us back. We climbed a fifty-foot cliff, then hiked five miles through the woods. That turned out to be the nicest part of the trip."
"Did you pass the abandoned railway and coal mines?" Abdul asked.
I raised an eyebrow. "How did you know?"
He thumbed his mustache. "I have been exploring the area myself."
"Of course. You said that." I looked at David and saw just the slightest shake of his head.
Abdul extracted a Gauloise from his pack and struck a match. Waving out the flame, he looked for an ashtray, but there was none. He casually dropped the spent match on the tablecloth. Our waiter, hovering nearby, promptly supplied an ashtray and whisked the match away, not even bothering to mention we were seated in the room's nonsmoking section.
Abdul didn't seem to notice, either, and launched into a discussion with David about the Russian oil market. As the director of foreign currency trading for a Philadelphia bank, David keeps current on issues involving global finance. After their initial friction, they'd seemed to warm to each other.
I stared at the spot where Abdul had dropped the match. A tiny smudge of gray marred the white damask tablecloth. I guess if you're related to the Saudi royal family, you can do things like that. After a moment I decided the smudge wasn't really jumping from one spot to another. I tried to focus on the men's conversation but heard only snatches of words, soft music, and the clink of silver on china.
I looked around. Ensconced in gilt-edged frames, a gentleman in a powdered wig and a lady in a low-cut gown gazed back at me. Did these stern representations of Southern society somehow sense my growing discomfort? After three days of rich food, soft accents, and scotch sprays, I was starting to feel overwhelmed. And that was before the river.
A curl of smoke rose from Abdul's cigarette, drifting into a haze above his head. I picked up a roll and buttered it, thinking how cruel it was to smash a tiny yellow flower with my butter knife. Who made those butterflowers anyway? Elves?
"So," I interrupted around a mouthful of bread. "You're here for an energy conference?"
Abdul looked over. "That's correct."
"Yeah? So what do you think? Have you decided what the government is gonna do for you next year?" I swallowed clumsily.
Irritation flashed in David's eyes. Rachel looked away.
"Don't look at me that way, David. Everybody knows the oil industry's in bed with the government. And--" I spread my hands. "What better place to be seduced? I mean, here we are in the lap of luxury in the poorest state in the country. And why is it so poor, you ask? Because all the coal that was so abundant here was stolen by certain interests while our government looked the other way."
"Ellie, that's enough."
"That's why those coal mines are abandoned, you know." I wagged a finger. "They were exploited by rich and powerful capitalishhts who didn't care what happened to the land after they raped it. And now they're shhtarting all over again with timber." I nodded in what I hoped was a sage manner. "Logging."
Abdul's mustache and goatee seemed to blur together.
"So you see, it's entirely fitting your meeting's at the Greenbrier," I went on. "Hey. I got an idea. Why don't we make a commercial? 'The Greenbrier--where the government goes to get grafted.'" I looked from Abdul to David. "Has a ring to it, don'cha think?"
Abdul inhaled deeply, the embers of his cigarette glowing orange. David avoided my eyes.
"You have passionate convictions, Ellie," Abdul said. "But you give me too much credit. I am just an observer, not a player in your politics." He picked up the menu. "Perhaps we should order our dinner."
***
After dinner Rachel headed to the bowling alley, where the kids at the resort hung out. David and I went to our suite: two large bedrooms and a lavishly decorated living room with a wraparound porch outside.
David loosened his tie and disappeared into the bedroom. I sprawled on the living room couch. He came back out, minus his jacket, and sat on the other end. I made no move toward him, which, even through my alcoholic haze, surprised me. Usually, I can't wait to get him alone. He gazed at me expectantly, his deep blue eyes wide. Soft lamplight glinted off his thick, white hair. Muscles bulged through his short-sleeved shirt. Though he was well into his fifties, he could hold his own with any Generation Xer. The silence between us grew. Then,
"Why do you always do that?" he said.
"What?" I said, after a beat. "Get drunk?"
He shook his head. "Why do you always make sure everyone knows you're an outsider?"
"I am."
"No more than Abdul."
"Sure. And I just happen to have twenty oil wells in my pocket, too." I sniffed. "My God, David, look at this place. The garbage alone could feed an entire West Virginia hamlet. And Abdul flicks his burnt match on the table like we're in some greasy spoon. I know some people are used to being waited on. But that just isn't me." I slouched. "Of course, nearly drowning today didn't help much."
David took a breath. "Ellie, I wanted us to enjoy our time together. Frightening you or Rachel was the last thing I ever intended to do."
That was probably as close as he would come to admitting that we _had_ brushed up against the void that afternoon. "But as far as belonging..." He sighed. "People spend their whole lives not knowing where they belong. Can't you just file this away as an experience?"
Maybe I was still too worked up. Or drunk. But his tone rankled. "Sorry. I forgot who I was dealing with. The guy who'll do anything to belong. Who needs acceptance from everyone."
He knelt in front of me. "Not from everyone," he said quietly. "Just you."
Something inside me came unhinged, and a weight in my chest I hadn't known was there began to melt.
"I'm sorry. That was way out of line."
"It's okay." He smiled and drew me into his arms.
Chapter Two
"Ellie?"
Abdul headed over to my table carrying a plate loaded with French toast, sausage, and grits. Draper's Cafe was done up in so many bright pinks and greens I felt like a prisoner inside the petals of a giant laurel. I folded my newspaper. "Good morning."
"Where's David?"
"Showering. He was working out."
Abdul's mound of food almost obscured the pattern of laurels bordering his plate. He smiled ruefully as he set it down. "I should follow his example. But I am not as disciplined."
I smiled, too, remembering our very undisciplined activities the night before. As if reading my mind, Abdul's grin widened. Mine faded. I felt like I'd been hit by a bus, and I was nursing a headache the size of Montana. "Abdul, I want to apologize for my behavior last night."
He sat down. "Think nothing of it."
"I was ugly."
"You were refreshing."
" _You_ are charming."
He pulled out a chair and sat down. Unfolding a pink linen napkin, he placed it on his lap, then reached for the syrup, drenching not only the French toast but the sausages and the grits in a sea of maple. My fruit cup looked Spartan in comparison.
He speared his sausage and swallowed it in two bites. If he was Muslim, he wasn't all that devout. "You are from Chicago, David tells me?"
"Born and bred." I braced myself, waiting for the inevitable comment about the Windy City, or "my kind of town," or other inanity non-Chicagoans feel compelled to express. As if we run around all day humming Sinatra, thrilled to live in a place with razzmatazz.
But all Abdul said was, "I am very impressed with your... your..."
"David."
"Yes. He has keen observations on the relationship between currencies and markets."
As much as I try, I can't summon up much enthusiasm for currency trading. Admittedly, I don't really understand it, and I keep wondering what I'm supposed to understand. As David explains it, it's a service that banks provide to their customers. The bank doesn't want to lose money, but they're not in it to make a killing. Except for the occasional scandal, which often turns out to be the result of poor judgment rather than deceit, currency trading just isn't very sexy. Which is fine by me. My ex-husband played the market. Badly. I have the debts to prove it.
But Abdul was clearly a man of wealth--and a member of the Saudi royal family. I should be polite. "I don't understand currency trading very well, but I imagine you need dollars because--well, why do you need dollars?"
A waitress in green and pink filled our coffee cups. He waited until she glided away. "You're a curious woman."
I shrugged.
He studied me closely, as if registering every detail of my appearance. I found it unnerving. I'm usually the observer.
"It is not that complex." He lowered his fork. "The price of oil is quoted in U.S. dollars, and most of my business is transacted that way. I use the proceeds to purchase currencies for my other investments."
"And what would those be?"
He hesitated. "I am always looking for new ideas and technologies to bring back to my people. For example, I have invested in a genetic engineering company which is experimenting with drought-resistant seeds. Also an Internet search engine that teaches children how to retrieve information more easily."
"Really?"
"It may be your David Linden and I will have more to discuss." He laughed, scraping up a mouthful of grits. "But enough business. You seem more--how do I say it--anchored this morning."
"Nicely put," I smiled. "The rafting...well, it isn't anything I plan to do again."
He laughed again and went back to his food. When he'd finished, he pulled out a copy of the _Journal_. "You don't mind?" He motioned to the paper.
I held up the _Chicago Tribune_ I'd bought earlier, and we settled back to read in companionable silence. I'd been surprised to find a Chicago paper in the mountains of West Virginia. But then, this was the Greenbrier. They probably had their own printing press in the back.
As I scanned the paper, a story on page nine caught my eye. A murder trial was about to get under way at criminal court downtown. The accused, a man named Johnnie Santoro, allegedly beat up and then shot his girlfriend at Calumet Park on the Southeast Side. He was pleading not guilty, but according to the article, there was a wealth of incriminating evidence. The last weeks of summer are usually the dog days in terms of hard news, so in the absence of anything more newsworthy, the case had been heavily covered by the media, the local stations promising all the legal maneuverings and high drama of the O.J. trial. I'd paid scant attention until now, figuring that whatever local TV wants me to watch is exactly what I should avoid.
Today, though, there was a grainy newspaper picture of Santoro in the paper. He was twenty-six, the article said, but he looked older. His eyes were hooded, and his hair was cut close to his skull. He was looking off camera, but his eyebrows were so overgrown and bushy they met over his nose, which gave him a simian look.
I stared at the picture and felt my skin grow clammy. I reached for a glass of water.
"Something is wrong, Ellie?" Abdul asked.
I gulped down a swig, then held up the paper. "This man who's on trial? He...looks familiar. I think I know him."
His eyebrows shot up.
"Good morning, troops." Strong hands squeezed my shoulders. I glanced up. David leaned over and kissed my cheek.
"It's the face," I said to Abdul. "I've seen it before."
David pulled out a chair. "What did I miss?"
I passed him the paper. "Look at this."
"What am I looking at?"
"The man in the picture. Who's on trial for murder."
David studied the article.
"I think I know him," I said. "But I don't know how."
I felt Abdul's eyes on me.
"Guy beats up his girlfriend and shoots her to death." David handed me back the paper. "What a nice person for you to know."
Chapter Three
Rachel was right, I am terrified of flying. I always have been, even before September 11. Going home, I put up a brave front, but by the time we landed at O'Hare, having bounced around one thunderstorm and flown through another, I was a quivering, quaking mass of Jell-o. And that was an improvement over the flight out.
I tripped over Rachel's bag as I came through the front door--she was already on the phone, her newest CD blasting. I lugged our bags in the house, for once not minding the stains in the carpet, the nicks on the wall, and all the other imperfections and flaws. A modest three-bedroom on the North Shore of Chicago, I managed to hang onto it after the divorce, and it looks like I'll be there forever; I can't afford to move. Tonight I was grateful.
I dumped our dirty laundry in the basement and went upstairs to my office. It used to be the guest room, but I appropriated it when Barry moved out. My computer, scanner, and printer fill most of the space, but I'd invested in an ergonomic chair last year, and I happily swiveled from side to side as I downloaded my e-mail. There was something reassuring about the clicks, bells, and blue bars that accompanied the unspooling of my messages. All was right in my little corner of cyberspace. It hadn't always been.
After I trashed the usual spam, only a few messages remained, none of them urgent, so I decided to unpack. As I was digging through the canvas bag that doubles as my briefcase and overnighter, I came across yesterday's _Trib_. It was folded to the page with Johnnie Santoro's picture. I studied it again and felt the same sense of familiarity.
I entered Santoro's name on a news database, and after a few seconds, more than a dozen articles popped up. I started scanning them. Santoro had been indicted for the murder of Mary Jo Bosanick, a young woman in her twenties. Mary Jo went to the Lakeside Inn, a tavern on the Southeast Side, to meet Santoro after a night-school class, but Santoro didn't show up until two hours later. A fierce argument erupted, and they both stormed out.
Mary Jo's body was found the next morning a few feet from Santoro's Chevy at the boat launch in Calumet Park. She'd been viciously beaten and shot twice in the head. Apparently, she tried to put up a fight; scrapings of Santoro's skin were found under her fingernails. The next day the cops arrested Santoro at the docks where he worked as a longshoreman.
I leaned back in my chair. There isn't much left of the dock life in Chicago--at least not on the same scale as Newark, Houston, or New Orleans. Back during the Fifties and Sixties, countless ships plied the Great Lakes, but traffic has since dried up. Competition from rail and trucks is one reason. The construction of larger, more efficient freighters is another. Weather also plays a part. Though they did dredge a terminal that operates year round, there's still only a nine-month season along the St. Lawrence Seaway linking Chicago to the Atlantic.
What little shipping remains, mostly steel and steel products, is centered on Calumet Harbor, not far from the park of the same name. On the rare day a ship does tie up, longshoremen queue up at waterfront warehouses for day jobs, like they've been doing for forty years. Most of the men are well past their prime, forced by meager pensions to take whatever work is available, but a few youngsters hang out there, too. That's where the cops found Santoro, stamping his feet in the morning chill, hoping for a few hours' work.
I looked through the window at the locust tree in my front yard. Its lattice of leaves, silvered by the moonlight, danced gently in the night breeze. Somewhere in the distance I heard the plaintive honk of a goose. Santoro might be a dock rat, but that didn't tell me how I knew him. I shut down the computer and went into my room.
Rachel had gone to bed, but I was still wired. I turned on the TV. The end of Ingmar Bergman's _Wild Strawberries_ was running on cable. Ostensibly about a professor who confronts the void of his life, the film opens with one of the best dream sequences ever filmed: boarded up windows, a clock with no hands, a hearse disgorging a coffin, the creepy, outstretched hand of a corpse.
Okay. Bergman is not what you'd call warm and fuzzy, but what can you expect from a Swede? His work even looks a little stagey now, but that's because every filmmaker in the world has copied his style. The way he works with light and shadow. The nuances of his camera moves. The way he imbues his characters with personality through one simple but perfect gesture. In the journeyman work I do for a living, I might set up a pretty shot or a smooth pan, but there's no emotional investment. No fusion of form and function. Even Bergman's outtakes are works of art.
As good as the film was, my eyelids began to droop. I forced them open a few times, but it was hopeless. I snapped off the tube and burrowed under the covers--and then sat bolt upright in the dark. The water. Nighttime. Outtakes. I knew where I'd run into Johnnie Santoro.
Chapter Four
For a month or so each fall, the mums take over Chicago. It's as if the Big Florist in the Sky has commanded, "Thou shalt have chrysanthemums and plant them everywhere." Huge planters of the red and yellow flowers, their spiky petals irrepressibly cheerful, flanked the door to Mac's studio as I headed inside.
When we started working together, MacArthur J. Kendall III had a tiny studio crammed with camera gear and editing equipment. Ten years later, his studio boasts two nonlinear editing suites, a soundstage, and one of the best editors in the galaxy.
Some people's bodies are made for the work they do. Michael Jordan. Martha Graham. Hank Chenowsky. With long, supple fingers, a lanky torso, and eyes that blink in sunlight like a mole's, Hank was destined to be either a concert pianist or a video editor. He chose editing, but to watch those fingers fly over the console is to watch a virtuoso perform.
He and I have spent more than a few late nights working, and he alone is responsible for the magic that makes our shows a cut above. I'm grateful to Mac, who appreciates what a talent he has and has managed to keep Hank happy. Though with Hank, happiness is more or less a permanent state of being. I've never seen him cranky.
I remember once asking him about his idea of heaven on earth.
"You first," he'd said.
"Okay." I squeezed my eyes shut. "A box of warm Krispy Kremes waiting for me...on a bed at the Four Seasons Hotel."
He cocked his head. "The Four Seasons?"
I opened one eye. "You ever been on one of their beds?"
"Uh, no."
"First of all, they're huge. Soft and firm at the same time. Perfect for sitting, and lying, and--well--they sell over two hundred beds a year, you know."
"And you know about beds at the Four Seasons because..."
"Um--uh--"
"Right." He cut in. "Okay. What flavor?"
"Pardon me?"
"The Krispy Kremes."
"Oh." I considered it. "Doesn't matter."
We exchanged nods.
"Your turn."
He bent over the keyboard to finish an edit. "Heaven on earth, huh?" He laced his fingers together and flexed them backward. "That's easy. Playing with Clapton."
"Where?"
"Shea Stadium."
"Instrument?"
"Bass, of course."
"Not piano?"
"That's my backup."
"What?"
"Playing with Count Basie at Carnegie Hall."
See what I mean about destiny?
But on this morning, garbled noise spewed out of the editing room door. It sounded like a flock of angry pigeons had taken up residence. I skirted the door and headed to Mac's office.
"Hey."
With his crewneck sweaters and khakis, Mac looks like an aging preppie, but it's dangerous to underestimate him. He's an excellent director and a shrewd judge of character. He looked up from a pile of paperwork. "What brings you here? A new client?"
I shook my head. "Things are slow." In my experience, industrial video production is an economic bellwether. When my work slows down, the economy isn't far behind. "I need the elements of the show we started for the water district."
"How come?"
"There's something I need to check."
He brightened. "They want to reedit?"
"You're slow, too."
He stood up. "It hasn't been this bad for years."
I followed him down the hall to a closed door, where he punched in a four-digit code on a wall panel. The door opened into a windowless room with gray walls and hundreds of gray cassettes on gray shelves.
"You'll be okay," I asked. "Won't you?"
"We'll survive." He started scanning the shelves. "You?"
"It's a little scary."
He tossed long brown hair back from his forehead. "I know that tune. Now...what were you looking for?"
"The Chicago water district."
"Our unfinished symphony."
"That's the one."
Last summer we'd started a video for the water district that showed how water travels from Lake Michigan to people's faucets. The journey begins at two intake cribs a few miles from shore. The cribs, anchored forty feet down on the lake bed, suck up enormous amounts of water, which is then piped through underwater tunnels to one of two treatment plants onshore. After the plants process and filter the water, it's piped through another series of underground channels to distribution centers at strategic points around the city as well as a hundred suburbs. These centers then pump it into homes. A simple concept; an engineering marvel.
Unfortunately, halfway through production, September 11 happened, and the water district cancelled the video. Given the situation, a how-it-works video on Chicago's water supply didn't seem prudent. Happily, they paid us for the time we'd already put in. Which is probably more than Schubert got.
"Found 'em." Mac motioned to a group of cassettes on the top shelf. "Which ones do you want?"
"The B roll we shot during the reenactment."
Mac climbed up a stepladder. "That _was_ one of your better ideas."
I smiled. We'd set out on the city tug from Navy Pier to scout one of the cribs. The Carter-Harrison crib is really two cylindrical structures joined together by a small suspension bridge. One cylinder has limestone and red-brick walls; the other has a white surface with pinkish stripes running down its sides. Except for a small tower rising through its center, it looks like a giant wedding cake. Although we wouldn't be shooting inside for security reasons, the Deaver crib, or candy striper, as boaters call it, is where the actual pumping takes place.
We'd disembarked at the limestone and brick cylinder, which houses living quarters for dozens of men. Before the pumps were automated, crews actually lived on the cribs to operate the machinery, and the facility, originally built around 1900, was equipped with bedrooms, a kitchen, and a dining area. Now, though, men bunk there only a few weeks each summer while they do maintenance and repair work. We'd planned to get some shots of them when the weather warmed up.
I remembered entering through a heavy iron door, half expecting to see a drawbridge slam shut behind us. But once inside, I was surprised. The cribs had been rehabbed, and the interior was all white walls and bright lights. At one end was a large, high-ceilinged kitchen and eating area; at the other, a series of bedrooms. A suit from the water district's PR department explained that during remodeling, some of the larger bedrooms had been divided up to accommodate more workmen.
As we passed one of the larger rooms, I saw a large wooden rolltop desk against a wall.
"What's that doing here?" I asked.
"Now, that's a great story." The PR guy glanced at the desk, then back at me. "But it's off the record."
I shrugged. I don't pretend to be a journalist; in no way does anything I do qualify as objective. In a pinch, you could call my shows infomercials, but--bottom line--if the client doesn't like it, it doesn't go in.
"It turns out that the cribs had quite a reputation. During Prohibition, this place was a speakeasy. And brothel."
I remember feeling my eyes widen. "No way!"
He laughed, clearly enjoying my astonishment. "You've heard of Big Bill Thompson?"
I nodded. Big Bill Thompson, aka William Hale Thompson, was one of Chicago's more avaricious mayors. A friend of gangsters, a taker of bribes, he'd amassed two million dollars in assets during the Twenties, almost twenty million by today's standards. However, he's remembered most not for his shady dealings--he had plenty of company there, anyway--but for his famous advice to his citizenry: "Vote early and often."
"Right," the PR guy said. "Well, this was his party pad. You should hear the stories about the booze and wild women. Capone used to come out here, too--one of his favorite hangouts, they say." He motioned toward the desk. "They even brought their own furniture."
"Are you kidding?"
He raised two fingers. "Scout's honor."
I looked around. "I guess they weren't just _voting_ early and often back then."
I'm still not sure how we did it, but we were so taken with the cribs' history that we persuaded the water district to let us stage a reenactment. We had to agree to shoot overnight so we wouldn't interfere with any work, but that was only a minor problem. We hired actors, dressed them in flapper costumes and sack suits, and through the genius of Mac's lighting, used smoky illumination and deep shadows to create a bordello atmosphere. The idea was to create a match dissolve, contrasting the bawdy revelries of the past to the techno-efficiency of today.
Now, Mac headed back to his office with the tapes and dropped one of them into his Beta player. I followed and settled into a chair. Color bars and tone appeared, followed by the drone of a motor. A dark blur filled the screen.
"That was a fun shoot," he said.
I stared at the screen. The night of the reenactment, Mac, the cameraman, and I set out from Diversey Harbor to shoot the trip out to the cribs. We were planning a sequence from the POV of the partygoers: shots of inky water, the gentle pitch of the boat, waves lapping the side. We'd gone as far south as Oak Street Beach when we started playing with the gain on the camera, trying to get the best exposure.
Shooting at night is tricky. Particularly on the lake and in the absence of artificial light. Not only do you lose detail, but if you're not careful, the image can turn grainy. You can use a night vision lens, but then your video might look green, like those shots of Scud missiles during the Gulf War. The solution is to try to include an existing light source, however dim, in the shot.
I watched as the shot on the screen panned from a red buoy back to shore, where streetlights rimmed a small park. The sound was on, and for a moment, we could hear the slap of waves, the murmur of our voices, the drone of traffic on Lake Shore Drive. But then a low, steady hum buzzed the sound track, punctuated with a crackle of static. Seconds later, a series of white lines streaked across the image, and erratic bursts of snow obliterated the picture.
I looked at Mac. "What the...What's that?"
Mac leaned forward, a frown tightening his face. "That's weird."
The tape kept rolling. More dropout and static zipped across the screen. "Mac, what's going on?"
He got up and stopped the tape. He rewound it. Then he punched Play. The damage was still there. "There's noise on the tape."
"I see that. How come?"
He kept studying it. "It looks like some kind of RF interference."
We exchanged puzzled expressions. Twenty years ago RF, or radio frequency interference, was a problem. Video equipment could pick up radio signals, which penetrated down to the camera heads, ruining the sound track or picture. Shooting at Sears Tower was a particular nightmare. Most of the local radio and TV station antennas sat on its roof, and you were as likely to pick up Phil Donahue or a Top 40 song on your track as the audio from your scene. Today, though, cameras are better shielded, and for the most part, the problem has disappeared.
"How can that be?" I asked.
"I don't know." Mac fingered the scar on the left side of his face, the result of a bad car accident as a teenager. "That's the first time I've seen it."
"There weren't any radio stations nearby, were there?"
He shook his head.
"A ship, maybe?"
"No way. Their frequencies are much lower. Anyway, it couldn't have been there when we first shot it."
"You know something? You're right. I would have seen it when I logged it in." I frowned. "In fact, I remember screening it before we set up for the match dissolve, and it was fine." I looked over. "So where have you been keeping this thing?"
"Ellie..." His scar started to turn red.
I raised my palm. "I'm sorry. Just kidding."
Mac always stores everything in his securely locked, temperature-controlled tape library. He fast-forwarded to the middle of the tape and punched Play.
"Look." I pointed to the monitor. "It's not so bad now."
Intermittent streaks still flashed the screen, but the snow was gone, and so was the low-pitched whine. We could plainly see a long shot of a park bench lit by a streetlamp. As the camera pushed in, what at first appeared to be a lump on the bench turned into a man curled up on it, his face toward the camera. As we zoomed in for a close-up, the man raised his head and looked into the camera. Thick, overgrown eyebrows cut across his forehead in an almost unbroken line, and there was a dazed expression in his eyes. He pushed himself up to a sitting position and tried to stand up, but as soon as he put weight on his legs, he collapsed back on the bench.
"Back up!" I said.
Mac rewound the tape. The man flew up, lay down awkwardly, and looked into the camera.
"Pause that!" I said.
Mac hit the remote.
"When does the log say we were there?"
He unfolded the piece of paper that had been inserted into the sleeve of the cassette. "Last July. The twenty-third."
"What time did we shoot?"
"Between midnight and one. Why? What's going on, Ellie?" I opened my bag and pulled out the copy of the _Trib_ from the other day.
Mac looked from the picture of Johnnie Santoro in the _Trib_ to the image paused on the screen and back again. "It's the same guy," he said softly.
"Check out the time he supposedly murdered that woman."
"July twenty-third," he said "Sometime between midnight and three." The lines on his face deepened. "But he was in the park. On that bench. I don't get it."
Our eyes met. "Neither do I."
Chapter Five
Hoisting my canvas bag over my shoulder, I locked the Volvo and walked to the elevators of the parking garage. The sharp, clean smell of gasoline tugged at my nostrils. I tried to smooth out my pants, but the two-hour trip downtown had welded permanent creases into the material.
Brashares and Associates had one of those LaSalle Street addresses that sounds much fancier than it is. Wedged between a real estate company and an accounting firm on the twenty-seventh floor, the office had an ordinary frosted glass door with black stenciling on it.
It was after five, but the lights were still on, and the door was unlocked. I pushed into a gloomy reception area with two chairs and a fern begging to be put out of its misery. A few feet away, a woman was bent over a keyboard. A copying machine in the corner spat out paper. Somewhere in the back, a phone was slammed down. The light on the woman's desk phone cut out.
"Gail, get in here," a churlish voice yelled.
The woman started, and then, as if embarrassed that I had witnessed her loss of composure, winced.
I smiled. "Ellie Foreman to see Chuck Brashares. I have an appointment."
"Gail, where are you, dammit?"
The woman at the desk returned a strained smile, picked up her phone, and buzzed someone. "Ellie Foreman's here."
I heard a drawer close and a chair squeak. A moment later a tall, slim man emerged from the back. Bald on top, he sported a sparse blond mustache. Ice blue eyes studied me from behind rimless glasses.
"Miss Foreman." He approached with feline, almost mincing steps, and we shook hands. "Thanks for coming down."
I figured him for a few years younger than me, but the tired expression on his face made him seem older. "Sorry to be late. The construction and the reverse commute are murder."
I cringed as soon as I said it, but he didn't seem to notice. He led me down a short hall.
His office had none of the upscale decor of my ex-husband's firm or the cheerful chaos of my father's old digs. A battered oak desk in the center of the room was piled high with folders. Two chairs sat in front, and a framed law degree from John Marshall Law School hung on the wall behind. Raised miniblinds gave onto a view of the building's air-conditioning shaft.
It seemed perfectly ordinary, if a tad shabby, until I stepped aside to let him close the door. Instead of the requisite family photos or framed landscapes, the wall behind it was covered with photographs of Brashares. There was a shot of him on skis wearing a red jacket and designer sunglasses. Another of him in a Sox uniform with a bat. Still another in running shorts with a number pinned to his shirt. There was even a shot of him with a helmet and an oar, white water swirling in the background. The shots were all eight by tens, and they were all precisely mounted in black matte frames, three to a row, three rows.
He watched me studying the wall. "I like to keep fit."
He seemed to expect some further response from me, maybe something along the lines of admiration.
"How'd you get the Sox uniform?" I asked.
"The Sox... I was high bidder for a day in the dugout at a charity auction," he said impatiently.
I smiled. Home run.
He gestured to the ski shot. "But that was Mount Snow. And that was the Boston Marathon. I run all over the world."
"Really."
He pointed to the shot of the rapids. "And that's the New River in West Virginia."
"I know that river."
His eyebrows shot up. "Do you raft?"
I stared him down. "Not anymore."
"Oh." He went back to his desk, pushed the files to one side, and sat down. "Well, let's go over what you said on the phone. You claim you have an alibi for my client?" He pulled out a yellow pad.
I sat down in the chair opposite him. "I'm a video producer, and I have tape of Johnnie Santoro in Olive Park at the same time he supposedly killed his girlfriend. I read you were representing him, and I figured you might want to see it."
"Olive Park? What--where is Olive Park?"
"It's a tiny enclosed area just north of Navy Pier. Near the water filtration plant. You can see it from Oak Street Beach."
"Santoro was at Olive Park?"
"Yes." I shifted. "But you already knew that."
His face was blank.
"You didn't?"
He pushed up the bridge of his glasses with his finger. "Why don't you tell me?"
"I was producing a video for the water district at the time. We were on our way out to the intake crib--"
"The intake crib?"
"The Carter-Harrison crib out on the lake."
He nodded.
"We wanted to get a few shots beforehand, so we took a boat out from Diversey Harbor. We were just south of Oak Street Beach when we started to try out different exposures. When you're shooting at night, well--" I shifted again. "Anyway, we took a few shots of the park, and Santoro was there. Passed out on a bench."
Brashares kept staring at me.
"There were two other people with me. I'm sure both of them would corroborate it." When he didn't answer, I crossed my legs. "You don't believe me."
"It's not that."
I waited.
He cleared his throat. "It-- it's just that Johnnie Santoro wasn't--how do I say it--all there that night. His brain was fried. Booze, dope, I don't know. Neither does he. But there's no question he was loaded. He can't remember what he did." He picked up a pencil. "Makes it hard to come up with a defense."
I recalled his dazed expression on the tape. How he struggled to get off the bench. "What are you going to do?"
"What is there to do? Try to work around it. And be grateful he remembers his name."
The phone on the desk trilled. He grabbed the receiver. "Yeah?"
If Santoro was as strung out as Brashares said that night, was he even capable of taking someone's life?
"I'll get back to you." Brashares replaced the phone. "Look, this is the first I've heard of any alibi. Why did you wait so long to come forward?"
I uncrossed my legs, surprised by the question. "I just made the connection. I saw his picture in the paper, and he looked familiar. I didn't realize I had video of him until the other day."
"Why didn't you go to the police?"
I looked over. I'd considered it, of course, but I read the papers. I know how Chicago cops can "lose" important evidence. Or "forget" to pass it through the system. But I'd just met Brashares. "The police are basically finished with their investigation," I answered carefully. "As I understand it, the ball's in your court."
"You know something about the legal system?"
"Both my father and my ex-husband are attorneys."
"Ah."
I got the feeling I'd climbed up a few points in his private opinion poll.
"Did you bring the tape with you?"
I dug into my canvas bag. "Do you have a VCR?"
"No. But you're prepared to leave it, I hope?"
I nodded and pulled out a copy of the tape, which Mac had made at the studio. "I also brought a copy of our video log. You'll see that it says we shot footage of the shoreline on the twenty-third around midnight."
Brashares took the tape and log and placed both at a precise angle to his pencil. For some reason, he didn't look as happy as a lawyer should who's just been handed a big break. But then, maybe he was just cautious. Or maybe he was already three steps ahead of me, formulating strategy to use in court. Or maybe he was just a lousy lawyer.
"There...there is a little damage on the tape." I explained about the RF. "You'll see some video dropout, and some snow from time to time. We just discovered it when we screened it the other day. But it wasn't there when we first shot it, and it didn't show up on any of the other tapes. I hope that won't be a problem."
He picked up the pencil. "I won't know until I take a look. But if it clearly shows Santoro in that park, we'll probably be okay."
"I hope so," I said. "I'd hate to think he was wrongly accused."
He frowned. "Up until now, I had no reason to believe he didn't do it."
"No?"
"They have a strong case. The car, the fingernail scrapings, his lack of an alibi. In fact, I'd almost persuaded him to cop a plea. He's taking a huge chance."
"Chance?"
"He's playing with a life sentence."
"Maybe he knows, in spite of his memory, that he really is innocent. They say even people with amnesia have gut feelings about these things."
"Sure. Him and all my other clients."
"You don't believe him?"
He shrugged. "He admits they fought. Says he might have taken a swing at her. But he claims she took his car and split. He never saw her again."
"So?"
"The other side has witnesses that saw two people driving into Calumet Park in his car."
"What does he say?"
"He can't remember."
"Which means the tape might be a real break."
"Maybe. But first I have to get it admitted."
"Why wouldn't it be? It's clearly Santoro. You'll see."
He leaned back. "Authenticity for one thing. Chain of custody for another. We have to prove both."
"Let's say you do. Then what?"
"Then, I'll do my best to see that he's acquitted."
"And then find whoever did kill the girl?"
He paused. "My job stops when I get him off. I'm not in the business of solving murders."
"But what if...what if someone framed him, and you get him off? What's to stop them from trying again?"
"You've just posed three hypotheticals, Miss Foreman. I can't deal with those. I deal with facts."
He got up and gazed at the wall of pictures, as if he was drawing inspiration from images of himself.
While he postured, I wondered how Santoro had become his client. Santoro didn't seem like the sporting type, and Brashares had probably never stepped foot on the docks. Then I recalled reading that Santoro's union card was up to date. Maybe the union had found him a lawyer.
He looked at me. "You're not planning to leave town in the near future, are you?"
"No. Why?"
"Because you'll probably have to testify."
Chapter Six
I called Rachel on the way home to see if she wanted me to pick up a pizza.
"No, that's okay. Katie and I are going to the mall."
"You're going where?"
"Her mom's on her way over."
"Whoa, girl. I don't remember giving you permission to go to the mall. Especially on a school night."
"Mom," she said, stretching the word into three syllables. "School just started."
"I'm aware of that. What about homework?"
"It's done."
"All of it?"
"Yes."
"What are you going to do there?"
"Mother, why are you always on my case?"
"Uh--how about I care about my daughter, and I want to know what she's up to?"
"Jeez, Mom. It's just the mall."
"I get it."
"Mother, it's my life. Stop invading my privacy."
I gripped the cell phone, prepared to launch into a discussion about study habits, responsibilities, and boundaries. "Rachel, let's get--"
"They're here, Mom," Rachel cut in. "Gotta go. Pick us up outside the food court at nine."
I checked my watch. It was barely seven. "Rachel, I didn't say you could go." I heard a distinct click, followed by silence. "Rachel?"
I drove another block with the cell pressed to my ear, then tossed it across the front seat. The Martians had landed, and they'd taken her brain. With any luck they'd send it back when she was twenty-five.
Dusk settled, cloaking everything in a mantle of purple as I wound through Skokie. The occasional shout of a child, the tinkle of music, and canned TV laughter spilled through the window. I turned onto Golf Road, feeling a twinge of regret at the loss of innocence, though whether it was Rachel's or my own, I wasn't sure.
Dad was watching the news when I unlocked his door. He lives in an assisted-living retirement home, although to hear him tell it, the only thing they assist with is the steady depletion of his savings. He glanced up from his leather wing chair, the one with gold tacking that had moved from the house with him. A plate with a half-eaten hamburger sat on the hassock. The smell of grilled onions hung in the air.
"Hi," I said, closing the door. "How ya doin'?"
He turned back to the tube. "That's the problem when you get old."
"What?"
"People come, people go. All day long. And everyone's got a key. It's a real invasion of privacy, you know?"
The joys of the sandwich generation. I slid the key back in my bag. "Sorry. I should have knocked."
He turned up his cheek for a kiss. A lamp on a nearby table threw a soft glow across his head, which was as smooth and shiny as a marble. But, at eighty-one, he's still alert and engaged. In fact, Susan says he reminds her of Ben Kingsley playing Ghandi.
I crossed to the window and opened it. "How's the new prescription?"
He'd been having problems with heart palpitations, and they'd changed his medication twice in two weeks. The first prescription fatigued him so much I was ready to take him to the ER until I tracked down his cardiologist, who was at a conference in Hawaii. He phoned in a new prescription and told me not to worry; we were only on the third of twelve possible drugs. If this one didn't work, he said cheerfully, there were still nine to go.
Fortunately, Dad did have more color tonight. "Any side effects?"
"Only if you call taking the boys to the cleaners today a side effect."
"Stud or draw?"
"What do you think?" He grinned. "You shoulda seen Marv's face after I bluffed the last hand. He thought he was drawing dead. He still hasn't figured out when I'm gonna do it."
It's hard to beat my father at five-card stud. I returned the grin, then gestured to his plate. "You eating enough?"
"Ellie, would you stop? I'll let you know when I'm about to die. Then you can worry."
"I'm not worried," I lied.
"I know," he chuckled. "So, what brings you here on a weekday night?"
I snapped off the TV and dropped a CD in his player. His face smoothed out as Sinatra started crooning. I felt a stab of envy. I remember intense discussions about pop music in my younger days. How it was an anesthetic, foisted upon us by the establishment to numb us to our suffering and political exploitation. Even now, I can't listen to a Motown riff without a twinge of guilt. But, as Frank's voice slid through the air, Dad snapped his fingers and closed his eyes, the tune clearly taking him back to happier times.
I waited until the song ended to tell him about Johnnie Santoro.
He was massaging his temples before I finished. "Ellie," he said, a rise in his voice. "What are you doing? Stay out of it."
"I can't. They may want me to testify at his trial."
"But you don't know that he's innocent."
"He was passed out on a bench near Navy Pier the night of the murder. Calumet Park's at least seven miles away."
"That means nothing. How do you know he didn't hitch a ride down there--or back up afterwards? I mean if he's really as forgetful as this lawyer says--"
"Dad, the guy was wiped out. He couldn't even stand up."
Dad pushed himself up. "Ellie. You have no idea who this man is, or who he associates with. The man was a longshoreman."
"So that means I shouldn't get involved?"
He flipped up his hands.
"That's odd, because I seem to remember someone else--someone close to me--who did the same thing."
Dad blinked. He'd grown up in Hyde Park but spent time in Lawndale, currying favor and running errands for a gang of hustlers in that thriving Jewish community. It was only for a few months before the war, but he still talks about it sixty years later.
"This isn't the same thing. This man could be a career criminal. The Mob runs the docks. And their unions."
"But I don't think he did it."
"So, who made you his savior?"
"Well now, that is the issue, isn't it? Where do you draw the line? When do you get involved, and when do you just step over the homeless man and pretend you didn't see him?"
He aimed a finger at me. "Ellie, this man is a potential killer, not a vagrant."
I folded my arms, and we glared at each other. Then he settled back in his chair, shaking his head. "I should know by now. Your mother was the same way--bringing home strays every Thanksgiving and Pesach. I never knew where she found them."
"Dad, if he's convicted, and I could have done something to help but was too scared or busy or wrapped up in myself, I'd carry that guilt forever. That tape could make a big difference."
"Maybe. Maybe not." He stopped talking and tapped two fingers against his chin. "You know," he said more softly, "there comes a time that you don't have to keep apologizing for thinking about yourself. You're allowed to live your own life. You're even allowed to enjoy it."
"I--I'm not that busy. I have time."
"Maybe you should spend it with your daughter or your boyfriend. Not get distracted on some crusade for a stranger. Deal with your own issues as they say..."
I looked away.
"How is Rachel?"
"She's fine."
"You sure?"
"Dad..."
"She called me this afternoon after school. She wanted to ride her bike down to visit."
"Rachel?" I was astonished. "My daughter wanted to voluntarily expend energy on some form of exercise?"
"She said she was bored."
The thing they don't tell you about the sandwich generation is that the two pieces of bread can gang up on the stuff in the middle. "What did you say?"
"I told her it was too far to ride all the way down to Skokie, and why didn't she go to the pool?"
The municipal pool, where Rachel hung out from dawn to dusk--at least last summer--was only a short bike ride from our house.
"What did she say?"
"' _Opa_ ,' she said--she sounded just like you do sometimes--'it's after Labor Day. The pool's closed. But even if it wasn't, swimming is for children.'" He got up, picked up his plate, and shuffled into the kitchen. I followed him in. "You know, it wouldn't hurt for her to have something to do after school." He dumped the remains of the burger in the trash and rinsed the plate in the sink. "Look. I'm not preaching. You've done a wonderful job. Considering. But she's thirteen. Sylvia said she still needs you, even if she doesn't think she does."
"Sylvia?"
I'm always surprised to find that an eighty-one-year-old man still blushes--all the way to the top of his head. "She just moved in."
"Uh-huh. And how old is Sylvia?"
"She's seventy-nine." He smiled. "But don't worry. She's pretty sure she can't get pregnant."
I giggled.
He smiled as he put the plate in the drainboard. "Sweetheart, I want you to stay out of this man's life. You have your own _tsuris_."
I noticed the determined set of his chin, and how much it resembled Rachel's. I felt like a piece of lunchmeat.
Chapter Seven
The phone chirped and the doorbell rang at the same time. I picked up the phone and opened the door.
"Fouad!" I smiled at the man standing outside. "What a nice surprise."
"This is Chuck Brashares."
"Sorry," I said into the phone. "Hold on, will you?"
I moved the phone away from my ear with one hand and shook Fouad's hand with the other. "It's so good to see you. How are you feeling?"
"And when I am sick, He restores me to health."
It's not unusual for Fouad Al Hamra, my friend and sometime gardener, to quote the Koran by way of greeting. He touched his fingers to his curly grizzled hair. He'd been shot a few months earlier but had recovered enough to resume work on a limited schedule.
I nodded and motioned to the phone. "I'll be out in a minute." I plugged the phone back in my ear. "Sorry, Mr. Brashares. You were saying?"
"I looked at the tape last night. Santoro is definitely on it."
I stifled an urge to say I told you so.
"I screened it several times, just to be sure. But I think we should proceed. I want you to testify. In fact, I've already spoken to the prosecution about it."
"So the quality of the tape isn't a problem?"
"Well, there is degradation, but it's not that bad when the camera's on him. You say you don't know how it was damaged?"
"No. It happened sometime after we shot it."
"Has the tape been stored in one place ever since?"
"It's been in a locked room at the studio. Only a couple of people have access to it."
His silence said he was satisfied. Then, "Well, it might not prove anything, but it should cast some doubt. I gave notice that I'll be calling you as an alibi witness. Expect a call from the other side. They will want to depose you before opening arguments."
I coughed. A deposition--at least the divorce kind--was not the sort of activity I looked forward to.
"They'll want to know where you got it, the circumstances of the shoot, where it's been since then. Things like that."
"I don't know. I--I didn't expect--"
He ignored my reaction. " There is one thing I should caution you about. Anytime a new witness shows up this close to the start of a trial, there's apt to be some skepticism on the other side."
"What do you mean?" I said, remembering Barry's lawyers a few years ago. "Are they going to be hostile?"
"Probably--er--cautious," he replied. "But don't worry. You'll handle it. In the meantime, I'll show Santoro the tape. Maybe it will jog his memory."
"Would it help if I met with him? Explain how we found him? He might remember more."
Another short silence. "I don't think that's a good idea. It could taint your testimony."
"But if he _could_ remember, wouldn't he make a better witness?"
"I'm not putting him on the stand."
"You're not?"
"He wouldn't make it past go. The prosecutor'll crucify him. Look. We're almost finished _voir dire_ , and the judge will probably grant the other side a motion to get up to speed on the tape. If the trial starts next Monday, and I think it will, it should only last a couple of days. We could get to you as early as Wednesday. But you and I should go over the questions before that." He paused. "By the way, I'm going to need the original of that videotape for the trial."
"You can't use a copy?"
"The judge will never allow it, given the interference. Best evidence rule."
"In that case, you'll need to rent a different player. We shot Beta SP."
"What's that?"
"A different format than VHS. More professional. Kind of like the difference between sixteen and thirty-five millimeter film."
"More expensive?"
"Sure."
"Well, it has to be done."
"Okay, but could you return it when the trial's over? I would hate for it to get lost."
"No problem."
I said I'd make a new master for Mac's files and messenger Brashares the original. In the meantime, we set up a time to meet so he could walk me through my testimony.
"Do you think he's got a chance?"
"I don't know. But we've got more than we had before. Thanks for coming forward."
"Chalk it up to civic duty."
After I hung up, I tried to figure out what bugged me about this guy. I couldn't put my finger on it. He wasn't incompetent. He _was_ doing the job, but I didn't have the sense he was committed to it. Then again, he was a defense lawyer. He couldn't be emotionally involved with every client. Still, I would have appreciated at least some comment about justice being served, or the truth coming out. I stood up. Maybe I was just reacting to his narcissism.
I threw on some sweats and joined Fouad outside. It was a brilliant, breezy day, the kind that triggers a yearning to be one with nature. Shading my eyes against the sun, I watched Foaud unload the spreader from his pickup. He'd lost weight, and the canvas pants he always wore when he worked hung low on his hips. Though he'd never had much excess flesh to begin with, now his dark eyes seemed enormous in his gaunt face.
My ex-husband considered lawn care a competitive sport. During the four years we were married, Barry spent thousands of dollars on landscapers, tools, and lawn care products in an effort to make our lawn the greenest, thickest patch of grass on the North Shore. At the beginning of April, even if snow still covered the ground, he'd demand that Fouad tell him precisely when fertilizer would be applied, the bushes trimmed, the weeding done. He suffered from an advanced case of "greenis" envy.
After we divorced, I didn't have enough money to keep Fouad on. For a few years the grass languished, weeds sprouted, and grubs feasted until the lawn looked like something out of the dust bowl. Fouad came back on a limited basis last spring, and we've made steady progress reclaiming the land.
"This will be the last time I fertilize before winter." He gazed ruefully at the grass, which bristled with weeds. "I'm sorry I wasn't able to come more often."
I bent over to pluck a blade of crabgrass, but a ladybug in speckled armor of black and orange was inching up its stem. Ladybugs are good.
I left it alone and straightened up. "Mother Nature will just have to understand."
Fouad smiled and poured a bag of what looked like orange sand into the spreader. "'Those who believe and do good deeds shall have gardens in which rivers flow.'"
Fouad has built a flourishing landscape service and a garden supply store, but he remains, at heart, a modest, spiritual soul. He rolled the spreader in a neat, straight row. Tiny bits of orange coated the green lawn. I followed him as he worked.
"Your visit to West Virginia went well?"
"Upsetting." I explained about the white-water rafting.
He stopped with his hand on the spreader handle. "You and Rachel were not hurt?"
"We were fine. Can't say I'll ever do it again, though."
"I understand."
I thought back to the hike through the woods, Draper's Cafe, Abdul's plate. Then I remembered who'd warned me I wouldn't see much laurel. "You were right about one thing."
"What's that?"
"The only laurel I saw was in pats of butter."
Chapter Eight
The fifth-floor courtroom at Twenty-sixth and California has high ceilings, marble walls, and polished mahogany railings around the witness stand. Unlike the cramped rooms on the lower floors, where a thick glass wall separates observers from participants and the ambiance is like a driver's license facility, this courtroom looks like a place where justice is meted out.
The trial started on Monday. As a witness, I wasn't allowed to attend, but a producer friend of mine at Channel Eleven knew the sketch artist for one of the other TV stations and told her to fill me in. The first witness was the police detective, who, through questioning by Assistant State's Attorney Kirk Ryan, confirmed the bullets that killed the victim came from a .38 revolver, although they never recovered the gun. Next was the medical examiner, who explained the victim's cause and manner of death. He had also recovered scrapings from the victim's fingernails, which DNA tests later proved to be consistent with Santoro's.
Ryan then led the victim's mother through a tearful testimony. Mary Jo was obedient, respectful, and ambitious, she said. Because her father was on long-term disability, the result of an accident at the steel mill, Mrs. Bosanick worked two minimum wage jobs. Mary Jo aspired to something better and was taking night classes, hoping to become a bookkeeper in a Loop office.
"But now my baby is gone. And our lives are destroyed," her mother sobbed. "By him." She pointed dramatically at Santoro.
Brashares didn't tear her apart during his cross. Instead, he worked around the edges, gently eliciting the fact that she and her husband had met Santoro several times and had even invited him over for dinner.
Next the prosecution placed both Mary Jo and Santoro at the Lakeside Inn the night she was murdered. The Lakeside was a gritty but quiet neighborhood bar not far from Calumet Park, the kind of place a single woman could occasionally drop in for a beer and not get hassled. The bartender testified that Mary Jo came in around ten, looking for Santoro. He knew Santoro was one of the dockworkers who only came in when they had cash in their pockets. On the night in question Santoro showed up around midnight. He'd obviously had a few, the bartender said, and when Mary Jo lit into him for being late, he lit back. Their argument became so loud the bartender told them to take it outside. Her body was found at the boat launch in Calumet Park a few hours later, the prosecutor reminded the jury. Lying next to Santoro's car.
But the star witness for the prosecution was Mary Jo's best friend, Rhonda Disapio. They'd gone to the same school, the same Catholic church, and Mary Jo had been maid of honor at Rhonda's wedding. A plump woman with bottle-blond hair, too much jewelry, and scarlet lipstick, Rhonda testified that Mary Jo had been complaining about Santoro's lack of money and ambition. Not only was he abusive, she said, but Mary Jo thought he was a loser. She was sorry she ever got involved. In fact, she was planning to break up with him the night she was killed.
Brashares immediately objected to her testimony as hearsay. The judge sustained it, but Brashares made a big show of asking for a mistrial. It was denied, but the judge instructed the jury to disregard the witness's comments.
Which was like telling them not to think about pink elephants.
Ryan concluded his questioning, and Brashares approached the stand. Again he chose not to attack Rhonda on cross. He did shake loose some inconsistencies, racking up points when she admitted she didn't know how Santoro and Mary Jo first met, nor did she know what they were fighting about on the night in question. As she stepped down from the stand, the sketch artist reported, she dabbed her eyes with a tissue.
By the time the prosecution rested on Tuesday, momentum was on their side. It was a circumstantial case, which, Brashares said, was the kind of case a jury loved to get. Drunk boyfriend follows angry girlfriend; girlfriend shakes him off; boyfriend flies into a rage and shoots her. It was easy to connect the dots.
Wednesday morning the room was packed with reporters, court-watchers, and gawkers. I was glad I'd worn my gray power suit. Especially after I met Brashares outside the courtroom.
"Who's on besides me?" I asked.
He frowned at me through his glasses. "A vice-president from the water district who'll talk about the hours Olive Park was open."
It turned out Olive Park, adjacent to the filtration plant, was owned by the water district. It had been open to the public until Nine-Eleven.
I nodded. "Good. Who else?"
"That's it." He smiled thinly.
I stared. "I'm it?"
"I couldn't find anyone else who saw Santoro. Maybe if you'd come forward earlier..." His voice trailed off.
"You couldn't get a continuance so you could keep looking?"
"The judge denied it."
"What about the night crew at the water treatment plant? Maybe someone saw Santoro walking around."
Brashares shook his head.
"Well, what about Mac? Or my cameraman?"
"They'll say the same thing as you. You called the shots, anyway."
"But Ryan'll crucify me." Prosecutor Kirk Ryan's conduct on cross had earned him the nickname the Hammer.
"Don't worry," Brashares said optimistically. "We have the tape."
I know enough about the legal system to know that when a lawyer tells me not to worry, that's precisely when I should.
The judge asked Brashares if he was ready. He nodded and replied in a clear voice, "May it please the court, we call Eleanor Foreman."
I tried to ignore the stir in the courtroom as I walked down the aisle, but all eyes were on me, including Santoro's. I stole a glance at him. He wasn't a big man, but he had broad, powerful shoulders. His buzz cut had grown out into a thick mat of dark hair, and he sat at the defense table, wearing a cheap brown suit.
As I mounted the step to the jury box, our eyes met. At first his were vacant, with a curious lack of focus. But then, in the next instant, an expression of hope flashed in them.
I swallowed.
"Miss Foreman, thank you for coming today," Brashares said after I'd been sworn in. "Tell us what you do for a living."
"I'm an industrial video producer." I answered succinctly, not volunteering anything extra, just as Brashares had instructed.
"And what does an industrial video producer do?"
I wanted to say, "Whatever it takes to get the damn show made." Instead I explained that while a producer's role depends on the director, the budget, and other circumstances, I generally did all the research, handled preproduction logistics, wrote the script, and supervised the location photography and postproduction.
Brashares nodded. "Let's turn to July twenty-third of last year, the night Mary Jo Bosanick died. Were you engaged in your profession that evening?"
"I was."
"What were you doing?"
"My crew and I were preparing to shoot a scene on the Harrison-Carter intake crib for the water district."
"Intake crib?"
I told him what it was and where it was located.
"And what were you photographing?"
I summarized the reenactment and what we had planned. I heard a few snickers when I got to the part about Big Bill and Capone.
Brashares waited until it was quiet. "Now. On the night in question, you didn't begin filming at the intake crib, did you?"
"No." I told him how we experimented with the camera gain and took a few shots near Olive Park before going out to the crib.
"Tell me, Miss Foreman. Was anyone in or around Olive Park that you see in court today?"
I pointed my finger at Santoro, the way Brashares had coached me. A murmur went up from the crowd.
"Let the record reflect that the witness identified my client, Johnnie Santoro. Now, Miss Foreman, what was he doing?"
"He was lying on a bench under a streetlight. He looked like he might have been asleep."
"Was he?"
"Not at first. He did try to get up. But he couldn't make it, and he collapsed back on the bench. He didn't move after that."
"How do you remember that? I mean how do you recall exactly where he was and what he was doing?"
"Because I recorded video of him doing it."
More murmurs went up from the crowd. Smiling faintly, Brashares paused to milk the moment.
"And when did you come to realize that the individual on the video was my client?"
"When I saw pictures of him on the news. I knew he looked familiar, but it took me a few days to realize where I'd seen him. When I figured it out, I immediately called you."
"Now." Brashares took a measured step in my direction. "It was your understanding, was it not, that my client wasn't moving all that well because he'd had a few drinks?"
"Objection," Ryan cut in.
Brashares blinked.
Ryan stood up. "Leading the witness. Plus, the witness has no knowledge of what condition the accused might be in. Anything she says is speculation."
"Your Honor, we intend to recall a witness who will talk to the number of drinks he had at the Lakeside," Brashares countered. "And Miss Foreman saw how he moved. Or failed to move. She can testify to what she saw."
The judge pursed his lips. "I'll allow it, but rephrase the question, counselor."
Brashares smiled. Ryan sat down, shaking his head.
"Now, Miss Foreman, what did you see Mr. Santoro do?"
I explained again what I had seen.
"As far as you know, did Mr. Santoro get up and leave the park?"
"Not while I was filming him."
"And what time was that?"
"Approximately one in the morning."
Ryan looked like he wanted to object, but then apparently decided not to.
"Now, Miss Foreman," Brashares continued. "You never completed the video for the water district, did you?"
"That's correct."
"Why not?"
I explained that it was canceled last September.
"However, even if it hadn't been canceled, you wouldn't have used any of the tape that my client appears on in your final product, isn't that right?"
"That's correct."
"Why not?"
"Those scenes were never meant to be part of the finished tape. They were outtakes. Shots we did to establish the right exposure."
"But since that time, you have since discovered something about those outtakes, haven't you?"
"That's right."
"Could you explain it to the court?"
"The tape with Mr. Santoro's image on it turns out to have been slightly damaged."
"Damaged how?"
"There appears to be some kind of interference on the tape."
"Radio interference?"
"Objection," Ryan shot up again. "She's not an expert on radio frequencies."
The judge looked at Brashares, then at me. "Sustained."
"Let me rephrase that," Brashares said smoothly. "Not being an electronics expert, perhaps you could explain the problem from a producer's perspective."
"Objection, Your Honor!" Ryan shouted again.
"Will both counsel please approach the bench?" The judge rose and stepped to the side of the bar.
While the lawyers and judge whispered, I looked around. Mary Jo's parents were sitting behind the prosecution's table. Next to them was Rhonda Disapio. Mary Jo's mother sat with her arms crossed, back straight. Her father stared at me with venom in his eyes. Only Disapio's face seemed to hold open the possibility I wasn't a lethal adversary.
I gazed at the row of people behind the defense table, wondering if any family members or friends of Santoro's had come to the trial, but from their detached expressions and body language, I surmised that wasn't likely.
Their side bar apparently now concluded, the two lawyers backed away from the bench.
"The objection is overruled," the judge said.
Brashares smiled at me. "Now, Miss Foreman, how did the problem manifest itself on the tape?"
I described what RF can do on a tape.
"And the RF was evident on the shots--excuse me, the outtakes--of my client."
"That's right." I was beginning to feel more comfortable. The questions were going the way Brashares said they would, and we were talking about subjects about which I had some knowledge.
Brashares moved to a separate table and picked up a videotape in a plastic sleeve. "Do you recognize this videotape?" He handed it to me.
"Yes. It's the original tape that I gave you."
"How do you know?"
I pointed to the label on the spine, which said Foreman Communications. "My label is on the edge of the cassette."
"Is this the tape that shows my client on the bench in Olive Park?"
"Yes."
"Does the tape fairly and accurately show how he appeared that day?"
"Yes."
"And to your knowledge, has that tape been tampered with or altered in any way, since it was recorded?"
"No."
Ryan scribbled furiously on his legal pad.
"Your Honor, I'd like to move this into evidence as defense exhibit number one," Brashares said. "With your permission, we will play it for the jury."
"Objection." Ryan again. "Chain of custody. Where was the tape from the day it was made until now?"
Brashares' eyes narrowed. "Counselor, I thought we worked that all out." He turned toward the judge. "Approach the bench, Your Honor."
The lawyers had another side bar with the judge, after which Brashares asked me a series of questions that elicited the fact that the tape had been in Mac's tape library since we shot it, and that the tape library was locked and accessible to only two or three people. Ryan seemed satisfied and sat down.
Brashares wheeled a cart with a video player and monitor to the front of the room. The jurors leaned forward, and the room quieted. Brashares inserted the cassette and pushed Play. The tape was cued to the scene of Santoro on the bench. We heard the buzz on the track, saw the streaks on the picture. The entire scene lasted less than a minute, after which Brashares hit Pause. There wasn't a sound in the courtroom. Brashares stepped toward the jury.
"Again, Miss Foreman, who is the man on the videotape?"
"It's Johnnie Santoro."
"And when was this shot?"
"July twenty-third of last year."
"Thank you, Miss Foreman." Brashares clicked his heels, turned around, and withdrew to the defense table. His face had a sheen, as if he'd just finished a five-mile run. He nodded to Ryan. "Your witness."
Chapter Nine
I took a sip of water. The mood in the courtroom lightened. A low buzz came from the observers, and people seemed to relax, except for the Bosanick family, who sat tight-lipped and silent.
But when Kirk Ryan rose, the murmuring stopped. People shifted in their seats. A woman in the second row licked her lips. The door at the back of the courtroom opened, and my father walked in. How had he gotten downtown? He nodded at me and sat down in back.
Ryan, a squat man with the confidence of someone much bigger, pushed a hand through wavy blond hair. Pasting a smile on his face, he ambled toward me as if we had all the time in the world.
"Good morning, Miss Foreman. Nice to see you again." He was referring to the deposition I'd had last week with his staff. Brashares had been right. They hadn't been hostile; in fact, everyone had been quite polite. I returned a weak smile.
"You're a documentary filmmaker, correct?"
"Not exactly."
"You're not?"
"I produce industrials--corporate sponsored videos."
"But you did produce _Celebrate Chicago_ for the city's millennium celebration, which subsequently ran on cable television."
"Yes. The City of Chicago sponsored that."
"So." He cupped his hands around an imaginary sphere. "Some of your products eventually do end up on television?"
I didn't know where he was going, but I had a feeling I wasn't going to like it. "Yes."
"And prior to being on your own, you worked at a television station producing news documentaries, correct?"
"Many years ago, yes."
"Even so, would you say you have an understanding of the news process?"
"Objection!" Brashares jumped up. "I don't know where this is leading, or how it's relevant to the proceedings."
"I'm laying foundation, Your Honor," Ryan replied quickly.
The judge rubbed his nose. "I'll allow it."
"So." Ryan turned back to me. "Miss Foreman, would you say you have an understanding of the news gathering process?"
"I suppose so."
"You watch the news regularly?"
"Local or national?"
He dipped his head, as if to acknowledge I'd scored a point. "Let's start with local."
"Not that often."
"Pardon me, but didn't you say you recognized Johnny Santoro from his picture on the news?"
"I saw it in the newspaper."
He ran his thumbs underneath the lapels of his suit. "So you do keep up with local news. Through the newspaper."
I nodded.
"Please respond audibly."
"Yes."
"And when was it that you recognized Johnnie Santoro's picture in the newspaper?"
"About two weeks ago."
"But the crime with which Santoro is charged occurred over a year ago. Are we to believe that you, a former TV news professional, haven't watched the news or picked up a newspaper in all that time?"
"Objection!" Brashares again. "The prosecution is assuming facts not in evidence."
"I'm getting to them right now," Ryan said.
"See that you do, Mr. Ryan," the judge said.
"Well, Miss Foreman? Have you not watched the news or read a paper in that time?"
I squeezed my hands together. "Of course I have."
"Then you know the Santoro case has been one of the major news stories of the past year, correct?"
I nodded.
"Please respond audibly."
"Yes."
"For someone who was once in the news business, someone who knows the value of timely information, someone whose shows are still broadcast on the airwaves, why did you wait so long to come forward with your..."--he made imaginary quotation marks in the air--"...discovery?"
"I didn't realize that Mr. Santoro was the man on the intake crib video until last week."
"But you read the newspaper, and you watch television. Tell me, how many hours of coverage do you think have been accorded to the Santoro case since his arrest?"
"I don't know."
"Would it be fair to say it's been in the news frequently?"
"I don't know." My stomach was churning.
"Yes or no."
"Yes."
"Once a month, perhaps? And now, with the trial, even more?"
"I wouldn't know."
"And in all that time, you haven't seen one photo or image of Mr. Santoro until last week?"
"That's right."
"And that one image just happened to spark your memory?"
"Yes."
"Isn't that convenient?"
"Objection!"
"Sustained."
Ryan turned toward the jury, making sure they saw the smirk on his face.
Several jurors exchanged meaningful glances. I caught a glimpse of my father, a defiant glare in his eyes. My cheeks burned. Compared to this, maybe white-water rafting wasn't so bad.
Ryan strutted back and forth in front of the jury box. "Now, Miss Foreman, you saw the defendant on a park bench July the twenty-third, is that right?"
"Yes."
"How much time did you spend taking his picture?"
"About ten minutes."
"And while you were there, you photographed other things besides the defendant, correct?"
"We were trying to find the right exposure."
"Yes. Now, you arrived in the vicinity at about what time?"
"About twelve or twelve-thirty."
"And you left at what time?"
"About one."
"And when you left, you motored directly out to the intake crib, correct?"
"Yes."
"Where you spent the next five or six hours, correct?"
"We wrapped about seven in the morning."
"However, after you left the vicinity of Olive Park, you really have no direct knowledge about what transpired, either at the park or onshore?"
"Objection!"
"Overruled. The witness may answer the question."
I looked at my shoes. "No."
Ryan faced the jury and smiled as if he had just revealed an important piece of information. "Now, Miss Foreman, let's talk a little bit about the damage to the tape for a moment. The alleged RF interference?"
I swallowed.
"What evidence do you have that the damage on the tape is indeed radio frequency interference?"
"I don't--I'm not sure I understand the question."
"Let me clarify. Have you taken the tape in for any kind of technical analysis?"
"No, but I didn't--"
"So you have no independent confirmation that RF interference really is the problem on the tape."
"My director agreed that's what it is. We've seen it before."
"But you didn't seek any kind of independent corroboration."
"We didn't need to. We knew what it was."
"Based on your experience."
"Yes. And that of my director."
"All right. Given that you knew what it was, you still never discovered where the problem originated, isn't that correct?"
"That's true."
"But it was serious enough that you wouldn't have been able to use this tape in the final product. If the project hadn't been canceled."
"That's correct."
"So, on this damaged tape, you know what the problem is, yet you can't adequately explain why it is there or where it's coming from. Is that right?"
"Yes."
"Very professional, Miss Foreman."
"Objection!" Brashares yelled.
"The jurors will disregard that last comment," the judge said.
"I apologize," Ryan smiled, baring his teeth. "Let's say we went back to Olive Park with a camera and tried to simulate the conditions that you found there. Would we be able to replicate the damage that we saw on your tape?"
The man was relentless. "I don't know."
"Why not?"
I hesitated. "RF interference can come from any number of different sources. And the tape didn't have any damage on it initially."
"How do you know that?"
"Because I screened it after we shot it, and it was fine."
Out of the corner of my eye, I could see Brashares stiffen, but Ryan's smile broadened, as if he knew he'd won big. "Let's see. The tape was fine after you screened it, but now, a year later, it shows significant damage. And you testified that it's been stored in a locked room at your director's studio for over a year, isn't that right?"
"Yes." I cringed. I knew what was coming.
"So, you don't know where the problem came from, and it's been a year since you looked at it. Yet you still maintain there's no possibility the tape has been tampered with." He didn't wait for my response but whipped around to face the jury. "Thank you, Miss Foreman. I have no further questions."
I sat on the stand for a moment, unsure who and where I was. Then I looked around the courtroom. A few faces looked back at me with sympathy, but most were curious, almost expectant, as if they were waiting for me to have a meltdown then and there. After all, I'd just been bushwhacked. Discredited. Hammered.
My father leaped to his feet and made his way to the door. In the space he'd occupied, I caught a glimpse of a man sitting behind him. Young, dark, somewhere in his twenties, he had crisp features with high cheekbones. Curly black chest hair poked through an open-necked shirt, and one arm was draped over the back of the bench. Even through my humiliation, I registered that he was sexy in a dark, Mediterranean kind of way.
I looked at him, hoping for a sympathetic nod or smile. He returned the look, but something on his face, a lilt of one brow perhaps, a narrowing of the other, gave me the feeling he could see through me and had decided there wasn't much there. A twinge of uneasiness passed over me. Averting my gaze, I stepped down from the box.
Chapter Ten
I testified on Wednesday, and the case went to the jury on Thursday. Ryan skewered me in closing arguments, implying I was the stupidest, most naive documentary filmmaker in the world. Why hadn't I come forward sooner? How did I know the tape hadn't been tampered with? Why couldn't I adequately explain the damage on the tape? Was I that technically incompetent? Either that, he said, or something else, something more sinister, was at work.
In either case, he declared scornfully, this was not an alibi. I might have seen Santoro at Olive Park, but what was to stop him from having traveled to Calumet Park either before or after? The tape was no more than a description of where Santoro ended up at a specific point in time. Indeed, when you added up the fingernail scrapings, the lovers' argument, and the fact that Mary Jo's body was found near his car, there was no way twelve intelligent jurors could possibly buy my story.
They didn't. On Friday they convicted Santoro.
The phone rang all afternoon--reporters, mostly, looking for a sound bite. Something that would sum up the conflict in ten seconds. Preferably at my expense. I decided I'd be damned if I'd give them one, and after a slew of calls during which my polite refusals to comment apparently weren't enough, I tried a new approach.
"Ellie Foreman?" a voice asked.
"Si?"
"Is this Ellie Foreman?"
"Si?" I stretched out the word.
"Uh--I'm looking for the video producer, Ellie Foreman. Is she there?"
"Meesus not home." I slammed down the phone before a fluent stream of Spanish could come back at me.
Small victories.
I was watching myself on TV when David unlocked the front door. I'd had no intention of turning on the tube, but, after polishing off half a bottle of wine, something drew me to the coverage--the same thing that draws gapers to an accident, perhaps. Or possibly a latent streak of masochism.
David took one look at me and went into the kitchen. The refrigerator door opened, a cabinet drawer closed. A minute later, he came into the family room carrying a plate of bagels, lox, cream cheese, and onions. He sat down on the couch.
"You haven't eaten today, have you?"
"I'm not hungry."
He spread some cheese on half a bagel, placed a thin slice of lox on it, and laid a strip of onion on top. The smell of the onion made my nose itch.
"You had a rough couple of days."
"It's a good life lesson. Never be a Good Samaritan."
He chewed slowly. "I suppose it won't help to say you did the right thing."
I gazed at the bagel and shook my head.
"What did your father say?"
"He said Brashares didn't do me any favors." I reached for the bagel. "Barry agreed with my father, by the way. He was almost compassionate when he came to get Rachel." I bit into the sandwich. "Well, as compassionate as an ex-husband can be."
David went into the kitchen. "What's his take?" he called over his shoulder.
"He says Brashares left enough holes in the case to drive a truck through."
"Like what?"
"Not objecting when he should have, for one thing. Not calling any other witnesses, for another. He said Ryan ought to be thanking his lucky stars his adversary was so incompetent. In fact, he was surprised Brashares didn't get a continuance--based on the tape and what I brought to the case. Admittedly, Barry is usually looking for a way to needle me, but he said the guy ought to be sued for malpractice."
David came back out with another bagel. "He would know."
"He also said Ryan did a masterful job. You know, limiting me to yes and no questions. Not letting me give any opinions." I finished the bagel. "But you know what bothers me the most?"
"What?"
"I think he's right."
David frowned.
"I've been thinking about it. Brashares did his job. But there was no feeling in it. No soul. I got the sense he didn't really care about Santoro. Or me."
"Can you blame him? Think of the scumbags he represents every day. He needs professional detachment."
"This was beyond professional detachment. And how can you do a good job for your client if you're not invested emotionally--at least a little bit?"
"Not everyone has the same passion, the same commitment as you, Ellie. You see an injustice, and your heart cries out to fix it. Most people don't bother. It's part of what makes you special."
I balled up a napkin and threw it at him. "Why is it you always know just the right thing to say?"
He tossed the napkin on the floor, moved over, and stroked the back of my neck. I settled back against the cushions, concentrating on the feel of his fingers. "That's good," I said thickly.
An hour later, I felt much better.
***
Before I fell asleep, I mentally played back the trial. I thought I was testifying for all the right reasons. Acting on principle. Serving justice. But now, lying in David's arms amid pillows, sheets, and blankets, I wasn't so sure. Was my concern the injustice that had been done--or the fact that my ego had been bruised?
The comforting weight of David's leg fell over mine. Maybe I should give it all up. Ratchet down a few notches. He'd never admit it, but David probably found me high maintenance. Wearying. I sometimes thought he'd be happier with a woman whose world view started and stopped with him. Someone who never questioned authority. Like the bimbo Robert Redford ended up with after he and Barbra Streisand broke up in _The Way We Were_.
I threw my arm above my head. David stirred, sleepily working the palm of his hand up my thigh. A shiver skimmed my nerves. Life with him would be easy. Pleasureable. I wouldn't have to work. I could dedicate myself to tennis. Join the garden club. And be bored--except in bed.
Chapter Eleven
There was a snap in the air as we came out of shul on _Rosh Hashanah_. My father rubbed his hands together. "I love fall days," he said cheerfully. "They always make me think of a fresh start. A new school term, new friends, a new suit for the High Holidays."
Rachel smoothed the skirt of her new outfit, a simple but elegant taupe knit from Nordstrom. With her blond curls, blue eyes, and pale skin, she looked like a princess. And much too grown up.
Dad draped an arm around David's shoulder and headed to the car. Though David was half a foot taller than Dad, he made the movement seem natural.
"What's for lunch?" David asked.
"You'll see," I smiled.
I'd made most of the meal before services, though perhaps "assembled" was a better word. Blintzes, bagels, and salad. The omelets I'd make once we got home. And, of course, apples and honey. Cooking has never been one of my core strengths. Don't get me wrong; I love to eat. Especially when someone else feeds me. But today was special.
"Oh, boy," Dad grinned. "We're in for a treat. Your mother's cooking."
Rachel rolled her eyes. "You must be feeling brave."
"Rah-chel," Dad said, using the Hebrew pronunciation. "It's a new year. Let's start out treating your mother right."
Rachel threw me an icy look.
I shot her the old arched-eyebrow in reply.
Her eyes narrowed, and she skipped over to Dad. "I'll bet we don't go to services tomorrow."
"Want that extra day off of school, huh?" I cracked.
My father glared at us. "Stop it. Both of you."
David cut in. "I wouldn't mind going again."
The way he looked at me made me suddenly ashamed of myself. "You know something? That's a good idea." I turned to Rachel. "We'll all go, okay?"
She shrugged.
To be perfectly honest, though, Rachel does have a point. I'm not as observant as I once was. My father says it's because I'm the product of a mixed marriage. My mother was raised Reform, about as assimilated as you can get. Her mother used to host an open house every Christmas Eve during which she wore a tiny Christmas tree on top of her head. My father, on the other hand, grew up in Hyde Park among a tightly knit group of observant German Jews. In fact, Mother used to joke that she was about as far as he could go and still marry a Jewish girl. Still, I suspected she was grateful that my father was there to teach me who I was and where I came from.
After services the next day, Barry took Rachel for the rest of the day. I dropped David at the airport for his flight back to Philly, took Dad home, then changed and headed to the mall. All the talk about new suits was inspiring. Once I got there, though, I lost my nerve. I usually need Susan's approval for a major purchase. I've brought home too many mistakes.
I window-shopped for a while, then wandered into a small, narrow gift shop with faux stucco on the walls. Merchandise was displayed on both sides of the aisle, and a blue-haired woman sat behind the register. She seemed to be the only employee in the store, but I was aware of one other shopper. I stopped in front of an end-aisle display of prettily packaged soaps, admiring the tiny butterflies, delicate flowers, and other designs painted on them. A sign declared that Soap Art was the latest thing. Guaranteed not to dissolve when wet. Maybe I'd get some for Rachel. A peace offering.
I kept browsing, admiring the wrapped baskets, ceramic pillboxes, and other _tsatskehs_ , then headed back to the soaps. The other shopper stood with her back to me, juggling two soaps in one hand. I was about to say, "Excuse me," so I could take some, when she slipped hers into her pocket.
I froze. After a moment she turned around--and froze, too, guilt and fear stamped on her face. I knew what I was supposed to do: demand she put the soap back, call the manager, shout for security. But I didn't. I was paralyzed, riveted to the floor.
We eyed each other warily, neither of us moving, until it must have dawned on her that I was either unable or unwilling to react. Then, something new edged into her eyes. Defiance, perhaps. Or triumph. She swept by me and exited the store.
I cowered in the aisle until the adrenaline drained out of my body. I picked out three soaps, took them to the counter, and paid. As the blue-haired woman handed me my bag, I felt an overarching guilt, as if I'd been the one to shoplift. It even crossed my mind to pay for the two the other woman took.
Instead, I left the store and trudged down the hall. I passed a colorful kiosk where a collection of nuts, sold by a woman who probably never shelled one in her life, gave off a pleasant, woodsy aroma. I moved on to the food court, bought a huge cookie with lots of chocolate chips, and wolfed it down. Heading toward the exit, I rationalized why I hadn't intervened. Since the trial, I was finished with trying to do the right thing. I'd been hammered enough. Let someone else pick up the ethical gauntlet. I brushed cookie crumbs off my shirt.
I hadn't gone very far when I heard footsteps behind me.
I stepped up my pace. So did the footsteps.
I slowed. They did, too.
At first I thought the shoplifter was behind me, but I couldn't figure out why. Was she planning to thank me? Explain why she did it? She didn't need to. I understood. I used to shoplift.
Shoplifting involves cunning. And _chutzpah_. I'd had both, once upon a time. I knew the rush, the high, the shame. And knowing that, I knew there was no way she was behind me. She wasn't ready to return the soap. Or even express remorse. She'd have to hit bottom first. I did.
I kept walking.
So did the person behind me.
It was a beautiful fall day; the mall wasn't crowded. So who was following me? The cashier from the gift shop? I didn't steal the soaps, but I didn't do anything to stop it. Maybe she'd noticed my tacit complicity and wanted to confront me.
No. That was just guilt talking. I couldn't take the moral high ground, but cowardice wasn't illegal. Besides, what clerk would leave the store unattended? I stopped and turned around.
Aside from a woman pushing a baby stroller, the hallway was empty.
I made a three sixty. No one. Turning back, I caught my image in a shop window. I scanned the reflection for any quick, unexpected movements. I did see a silhouette half in and half out of a doorway a few stores back. It wasn't the blue-haired woman, and it didn't look like the shoplifter. I waited. The figure turned away from me.
I started forward again. Within a few yards, the footsteps were back. I tightened my hold on my purse. Last year, my wallet was ripped off at a restaurant downtown. One man jammed the revolving door as I went through, while another squeezed into the same compartment as me. As I banged on the glass, yelling for help, the nearer man grabbed my wallet out of my purse. He took off when his buddy let go of the door. I wasn't hurt, but within an hour they'd racked up three grand on my Visa.
I ducked into a perfume boutique.
"Can I help you?" A saleswoman suddenly appeared at my side, suspicion flooding her face.
"No thanks. I'm just looking."
She planted herself in front of me.
I took my time inspecting a display case filled with perfume, amused at the irony of the situation. Then I exited the store, pretending I had nothing more important to do than spend the afternoon window-shopping. The clerk's sniff followed me out.
I passed more stores, anxious, now, to get back to the car and go home. I had just reached a bend where a walkway angles off the main corridor when a hand clapped me on the shoulder.
Chapter Twelve
I spun around and wrenched free. I grabbed my purse, swung it backward, and launched it at a blond head. Thanks to the bars of soap inside, it connected with a resounding thunk. My pursuer staggered into the walkway and collapsed on the floor.
"Please. Stop. Don't hurt me."
I stepped back, hugging my purse until the machine-gunning in my chest slowed down. The blond woman cringed against the wall. We were a few yards down the narrow hall that jutted off the main promenade.
"It's all right. I won't hit you again," I said.
When she tentatively looked up, I felt a jolt of recognition. It was Rhonda Disapio, Mary Jo Bosanick's best friend.
"You?" I cried.
Her face said she was at least as scared as me.
I leaned over and extended my hand. She hesitated, then took it. As she stood up, the musky scent of Tabu drifted over me. I hadn't smelled it since high school when girls sashayed down the halls, a heavy cloud of it trailing after them.
"You want to tail someone," I groused, "you ought to brush up on your technique. It sucks."
I looked around. Fluorescent lights flickered overhead, and the floor was institutional tile, not marble. Opposite us was a supply closet. A sign at the far end pointed to an employee washroom.
"You want to tell me why you've been following me?"
She blew air into her cheeks, as if she was wrestling with how to begin. "I--I'm really scared." Her voice was squeakier, more timid than I remembered. "I don't even think I should be here. But I don't know what else to do. I need your help."
"How did you find me?"
"I followed you. You're listed in the phone book."
I rubbed my chin. At the trial Rhonda had seemed smooth and self-assured. I recalled thinking her friend's murder was the most exciting event in her life and that she was reveling in her fifteen minutes. Now, as I took in her sloppy clothes, smeared lipstick, and earrings that didn't match her outfit, I could see she was stressed. Maybe I should feel some compassion. No, I reminded myself. She did sneak up on me, and I don't do surprise well.
"So talk."
She hoisted the strap of her purse up on her shoulder. A blue and white polka-dot scarf was knotted around the base of the bag. "When I testified at the trial, there are--well, things happened that didn't come out. I should have left town afterwards. But I couldn't." She shrugged helplessly. "I have a kid."
"What kind of things?"
She picked at the knot on her scarf.
"Rhonda, you found out where I lived. You followed me all the way up here. You stalked me through the mall. If you have something to say, now's the time."
"Yeah, okay. But please don't call the cops. At least, hear me out first."
"Call the cops?" I shifted uneasily. "Why would I do that?"
"Because of what I'm gonna say." She pressed her lips together. "The night Mary Jo got killed...I was with her."
"You were at Calumet Park?"
She nodded. "Mary Jo picked me up after she and Johnnie had that fight. She was driving his car."
"She took his car?" The fact that his car had been at Calumet Park was a key piece of evidence against Santoro.
"She had a set of keys. They were practically living together, you know."
"No, I didn't." No one did.
"She'd tell her parents she was staying with me," she said. "Anyway, after he'd belted her, she got really pissed, jumped into the car, and took off for my place. We picked up a bottle and went to the park."
I frowned. "I thought you said you had a kid."
She waved a hand. "It was after midnight. You know how kids sleep. My sister lives downstairs, anyway."
I bit back a reply.
"We drove over to the boat launch, see? We done it before. It's nice there late at night. Peaceful and all. You can really feel the lake."
"So it was you and Mary Jo those witnesses saw driving into the park."
She nodded. "So we're sitting on the rocks, getting kind of loaded, and Mary Jo's telling me she really did want to break it off with Johnnie. He was a fuck up; he wasn't gonna amount to nothin'. So we're talking and drinking and laughing, and then we see this boat come in--"
"A boat? At midnight?"
"It was summer. People fish all night long. Anyway, it's dark, and we can't see much, but it looks like there's two guys in the boat, and they're heading over to the launch. So, we start kiddin' around, like maybe we should hook up with those dudes--we might have more luck. Mary Jo even stands up, like she's gonna go over and start talking to 'em, you know? But I grabbed her and pulled her down. 'How do you know who these guys are, MJ?' I says. 'They could be creeps.'" Rhonda's voice wavered. "'Criminals, sex maniacs, drug dealers, you know?'"
"Go on," I said softly.
Rhonda ran her tongue around her lips, succeeding in smearing her lipstick more than it already was. "So MJ turns around and says--she says, 'What makes you think I don't already know about shit like that?'"
"Shit like what?" I asked.
"I asked her the same thing, but she shakes her head and says, 'Nothing...forget it.' But then she says, 'If there's any shit on that boat, they're hiding it pretty well. Look at all that crap.' So I look and I see the boat is filled with junk."
"Junk? What kind of junk?"
"I don't know, sort of logs, you know, like fireplace logs, but they were metal."
"Metal?"
"You could see them in the moonlight, but I didn't really take a close look 'cause I had to pee." She paused. "I should never have done that." Her voice cracked. "But I couldn't hold it." She dabbed at her eyes with her scarf.
I waited while she pulled herself together.
"There are these trees at the other end of the parking lot, and I went behind them. I must have been longer than I thought, because all of a sudden I hear voices. First MJ, then a man, then her again. Then she's saying 'Hey--stop it!' Then I hear someone running across the parking lot. And then she screams, 'Run, Rhonda, run!' And there's more steps. And then I hear the shots...and, and..." She covered her face with her hands.
"My God, Rhonda."
She dipped her head, as if she were answering a question. "Then they started across the grass. Coming right toward me. I could hear them talking."
"What were they saying?"
"I couldn't tell. It sounded like they might have been cussing. But they were whispering. Like they knew they had to stay quiet."
"Then what?"
"Thank God there's this hole in the fence behind the trees. With this red building behind it. A garage or shed or something. I was able to find it, and I squeezed through. Then I ran as fast as I could. I thought I was safe. But now..."
I saw the fear in her eyes.
"I think they're following me. They figured out who I am."
"From the trial."
She started to cry. "I didn't want to testify, but they made me."
"Rhonda, why didn't you go to the cops? This would have blown the case wide open."
"By the time I got it together, they'd already arrested Johnnie. I was afraid that if I went to the cops, the guys that killed MJ would come after me. Or my kid." She touched her fingers to a gold cross at her neck. "But now, they're coming anyway."
"Even more reason to go to the police. Or to Ryan."
A horrified look swept across her face. "I can't. He'd put me away for sure."
"At least you'd be safe," I said. "What do you think I'm going to be able to do?"
Her eyes flicked back to the head of the passageway, as if she feared whoever was following her might appear at any moment. "I saw you at the trial. I heard what Ryan said about you. You're one of those TV people."
"Not really."
"Yes you are. Like that blond on 'Inside Edition'? You know."
"Deborah Norville?"
Her face brightened. "Yeah. Her."
"Rhonda, I--"
She cut me off. "You know people. I bet you can fix it so they won't put me in jail. You know, make me one of those secret sources or something."
"You want me to interview you, is that it? Put you on TV--without revealing your identity--to tell the real story of Mary Jo Bosanick's murder? Is that what you have in mind?"
"Well, yeah. Maybe."
A flash of heat shot through me. "How about we put you on with an exclusive report? We'll call it a special investigation, hype it with a sexy headline: 'Confidential Source Comes Clean...Tape at Ten.'"
Her cheeks colored. "I know you don't think much of me. But you've gotta believe me. At first, I thought maybe Johnnie _had_ followed us down to the boat launch. But then I realized it couldn't have been him." Her eyes darted to the end of the corridor. "It was those guys. And now they're back."
"Why do you think they're following you?"
"Well, since the trial, I keep seeing the same car outside my place. One of them SUVs, you know? Dark. Like green or something. Then I saw it outside work--I work over at Hair Connection on Commercial. And then, yesterday, it parks outside my parents' house when we were there for dinner."
"Did you check the license plate?"
"I couldn't see it."
I was about to ask her if she knew the make when a series of noises suddenly exploded from the end of the corridor. Rhonda gasped. I spun around, clutching my purse like a club.
A group of teenage boys sprinted past the alcove, each of them trying to outdo the other with the loudest burps they could muster. When they saw us, one of them nudged his companion, and they erupted in wild, deep-throated laughs. I relaxed my grip, but when I looked back at Rhonda, her eyes were wild.
"Rhonda, you've got no choice. You have to go to the police."
"I told you. I can't."
I started to wonder why. Did she have a record of her own? Was she on probation? Or parole?
"Is there anything else you remember? Anything you saw or heard?"
She hesitated. "Like I said, they were whispering, mostly. But now that you mention it, I think I might have heard one of their names."
"Really?"
"I think one of them called the other Sammy. Yeah. Sammy."
"Sammy? Sammy what?"
"I don't know." She started in on the knot again.
"Anything else?"
"No."
She finally worked the knot on the scarf free, and it floated to the floor, a polka dot flag unfurled. As she bent down to retrieve it, there was a shuffling noise from the far end of the hallway, and a man in a beige uniform turned in, pushing a wheeled bucket ahead of him. He stopped, clearly surprised to see two shoppers in the alcove, but Rhonda was even more surprised. She let out a little scream. Then she lurched forward, snatched up her scarf, and sprinted past the man out into the mall.
Chapter Thirteen
By the time I got back to my car, the sun had surrendered to oily gray clouds, and a north wind was picking leaves off the trees.
As I drove down Skokie Boulevard, I tried to make sense of her story. Late night drinking. Mysterious men on boats. Casual references to drugs. If any of this had come out at the trial, I'd lay odds the outcome would have been different. At least Ryan might not have come after me with such enthusiasm. Although maybe that was just wishful thinking. For all I knew, Ryan might have found a way to dismiss Rhonda's story. He was the Hammer. He wouldn't have cared.
But Brashares would.
I left a message on his machine when I got home.
I was surprised when he called me back a few minutes later. I'd imagined him taking time off, running in a triathlon in some exotic location. But he said he'd been working nonstop. I filled him in.
"Disapio says she was there?" I heard a slight edge in his voice.
"Yes. She was too scared to come forward. She thinks she's in danger."
More silence.
I scowled into the phone. "I would think that gives you powerful ammunition for the appeal. I mean, doesn't that open up a whole new set of possibilities about Mary Jo's murder? Or at least cast reasonable doubt on the prosecution's case?"
"It might, but unfortunately, you can't raise new facts on appeal."
"But this--this could change everything."
"It is interesting. I'll admit that."
Lightning strafed the sky, and a crack of thunder rippled overhead. A sudden autumn storm sweeping in from the west.
"Let me see what I can do. I'll call you tomorrow."
I disconnected and held the receiver to my chest. Brashares seemed awfully casual about the information I'd given him. We might have been talking about the point spread on the Bears game. It wasn't what I'd expect from a lawyer whose client was facing a life sentence. But then, maybe he thought I was trying to tell him how to run his case.
Rain pelted the roof, and wind gusts whipped the windows. I poured a glass of wine, and thought about Rhonda Disapio. No question she'd boxed herself into a corner. She might well face serious consequences if she went to the authorities. But I didn't see any other solution.
I started to heat up a pot of water.
An hour later, the front door slammed, and footsteps pounded up the stairs. Rachel was home. I went upstairs and found her bent over her overnight bag, pulling out her clothes and flinging them on the floor. Rachel often comes home wired from the frenetic activity Barry puts her through; it takes a while to calm her down.
I kissed the top of her head. "Hi, sweetie. How was your visit?"
She whirled around. "Oh, hi, Mom." She went back to her bag and turned it upside down. A pair of gym shoes fell out.
"What's going on?"
"Can I buy some Steve Maddens?"
"Steve Maddens?"
"They're shoes, Mother. Cool ones. Everybody's got them."
"I didn't think you needed new shoes."
She picked up her gym shoes and tossed them into the wastebasket. "I do now." She balled up a T-shirt and pitched it on top of the shoes. "A Michael Stars shirt, too."
"A who?"
"A Michael Stars shirt. It's--oh, never mind. You'll never let me get one."
"I won't?"
"They cost a lot of money."
"How much?"
"About sixty dollars."
This was beyond wired. "That is a lot."
"See? I told them--" She clamped a hand over her mouth.
"You told who what?"
"Nothing."
"Rachel." I don't impose a lot of rules. But there is one: there shall be no gossip about the family--by the family--outside the family. You can attack, criticize, or scold in the bosom of the family, but never outside the home. It's probably a German thing I inherited from my father.
"Who did you see this weekend, besides your father?"
"No one."
Hmm. Daughter comes home, throws her clothes in the trash, and demands new ones. Yet claims she didn't meet anyone special over the weekend. Or tell them about her tightfisted mother. The rain drummed against the house like a bag of marbles.
I decided to try a different approach. I headed for the stairs. "You hungry? I'm making spaghetti and salad."
A puzzled look spread across my daughter's face. She shook her head.
"Well, I'll be downstairs." I started down.
Rachel was out of her room before I got to the bottom.
I smiled.
"Daddy's girlfriend was there."
I stopped smiling. I'd heard about the new woman in my ex-husband's life. Washboard abs, buns of steel. Barry was now working out with her. Or on her. Whatever. "Marlene, the aerobics queen?"
Rachel shot me a look.
"Okay." I raised my palms. "So she lifts weights too."
"Her daughter was there."
"I see."
"Her name is Carla."
"And how old is Carla?" I went into the kitchen.
Rachel followed me. "Sixteen."
I took out a knife and started chopping lettuce.
"She's got this really cool boyfriend. His name is Derek."
"And how old is Derek?"
"I don't know. But he drives."
I started chopping more briskly. I wasn't thrilled she was driving around with older teenagers. But Barry's a fairly responsible parent. They probably went out for ice cream. "Where'd you go?"
"Well, we heard there was this rave nearby, and--"
I spun around. "You went to a rave?"
Rachel immediately backpedaled. "We didn't go in. We just drove around the parking lot. And don't worry. I didn't do anything."
I clenched my fists so tight my nails bit into my palm. For a moment I thought I'd cut myself with the knife. "Rachel. You're only thirteen. You can't go to raves."
"I told you. We didn't go in. Everyone says I look older anyway."
I gazed at my daughter. Three inches taller than last year, she'd already lost that preteen, coltish look. Her body was starting to curve in all the right places. She could pass for sixteen. I forced myself to open my fists. _Stay calm, Ellie_.
"You're a beautiful girl, there's no question about that. But I don't care how old you look. You can't run around with sixteen-year-olds and go to raves."
"Why not?"
"Because you're thirteen. It's not appropriate. Or legal. Carla shouldn't be anywhere near them, either. I wonder if her mother knows? Maybe I should call--"
"Mom," she shrieked. "You can't!"
"If I hear anything more about raves, I will."
"I knew I shouldn't have told you." She fell into a sullen silence.
I turned back to the salad, but I'd lost my appetite.
***
Barry wasn't home when I called that night. Out with the aerobics queen, no doubt. I hoped they got drenched in the storm. An hour later, he still hadn't called back. I turned on the late news to make sure he hadn't been mugged, killed, or otherwise maimed and was using that as an excuse not to call.
The ten o'clock news is filled with let-it-bleed stories. Especially on weekends or slow news days, it's pretty much a litany of every accident, murder, and fire they can find within a fifty-mile radius.
I changed into a T-shirt and went into the bathroom to moisturize my face. Someone once told me I looked like Grace Slick, and I still consider it high praise, though both of us are now grayer, and, presumably, mellower. I was just finishing when the anchorman pulled on his serious face.
"A fatal accident on the Dan Ryan Expressway took the life of a twenty-four-year-old woman this evening. According to witnesses, the car veered out of control, skidded across the median, and hit an oncoming truck."
I jerked my head up and looked at the TV. Rain lashed the camera lens, blurring everything except for a swirl of blue and red lights. The picture cleared, and I saw a cop standing on the shoulder of the highway. Behind him was a car, the front end crushed and mangled. The camera panned over to two paramedics loading a gurney into the back of an ambulance. The body was covered by a plastic sheet but a corner flapped in the wind, revealing a piece of blue and white polka dot material.
Chapter Fourteen
It's usually around three in the morning that rational thought disappears, leaving dark conspiracies to hatch in its wake. The storm fell off to a soft rain, the sound of each drop distinct and perceptible, almost like the crackle of burning paper. I tossed and turned, my mind doubling back on itself.
A young woman covers up important information about the night of a murder, and a man who is probably innocent is convicted. Soon afterward the woman bares her soul to a video producer, telling her stories about boats and gunshots and fears that she's being followed. That night she dies in an automobile accident.
True, it happened at night, when drivers can be tired and less than careful. True, a storm made the roads slick. True, Rhonda Disapio might have been a rotten driver.
Still.
At six in the morning I ran across the grass for the paper. As if to apologize for last night, the sun was bright, and droplets of water sparkled like jewels on the grass. Mist rose from the ground, winding around the evergreens. The yard looked like an ancient fairyland. I took the newspaper in and brewed a pot of coffee, waiting for an elf or wood nymph to hop past the window.
I spread out the paper, dumped in a packet of sweetener, and sipped coffee from my "When the Going Gets Tough, the Tough Go Shopping" mug. Maybe it should say _shoplifting_. The steam from the coffee tickled my nose. Did cracking jokes about it mean I was cured?
The accident happened too late to make the morning edition, and TV didn't tell me anything I didn't already know. I thought about calling the state police, our version of the highway patrol. But they probably wouldn't disclose anything unless I had a compelling reason, and with my luck, I'd end up with a grumpy Broderick Crawford on the other end.
After dropping Rachel at school, I went upstairs, sniffing my coffee. Why is it the smell is always better than the taste? Don't get me wrong--if it tasted any better, I might have to get a new husband, clean the house in a shirtwaist dress, and greet him after work like a good little Maxwell Housewife.
In my office, I dug out my client list. The only other slow work period I recall was during the early Eighties, and I made an effort to hustle business. I culled through every corporate index at the library, wrote letters, sent demo reels. I even went on informational interviews--the kind where you know and they know there's no possibility of getting any work, but you go through the motions anyway.
I still think the only thing all that effort produced was the illusion that I was in charge. I had a plan. Kind of like the duck and cover drills the government made kids practice during the Cold War. About as effective, too. When the economy picked up, my work would, too, and it would come in the way it always has: word of mouth.
I started making calls anyway. I didn't expect anyone to call me back; mornings are a hassle for most people. I left messages, figuring I'd start to get callbacks that afternoon. I was rinsing my coffee cup in the sink when a thump sounded at the kitchen window.
Susan waved at me through the glass. "How about a walk?"
I grabbed my shoes and threw on a sweater.
Susan Siler and I are yin and yang. A tall, willowy redhead who always manages to look as if she's stepped out of _Vogue_ , she's a gourmet cook, has impeccable taste, and seems to glide through life without the bruises, blows, and jagged edges that perforate mine.
The cool, rain-washed air was overlaid with the tang of pine and woodsmoke. We skirted a couple of puddles left behind by the storm.
"Did you hear about Phyllis Hartford?" Susan asked.
"What?"
"George moved out last week. After twenty-seven years."
I didn't know Phyllis well, except for her baked goods. No holiday, school function, or community event ever took place without a plate of her pastries on hand. It was her knee-jerk response to life cycle events.
"She has no idea what she's going to do."
"She can make lemon squares."
Susan shot me a fierce look. "Watch it. I have the recipe."
We made our way to the bike path that cuts a swath through the forest preserve. The leaves were just starting to turn, and the trees were shot through with glints of red and yellow. A carpet of newly fallen leaves, still holding their colors, muffled our steps. I found myself treading more respectfully, trying not to disturb the balance of nature.
"Speaking of baking, Rachel had a meltdown last night." I told her about the clothes on the floor, the shoes in the trash, the demands for new ones.
Susan giggled.
"You think it's funny? I just bought her some fall things. Including a really nice suit."
"Hormones, Ellie. Get used to it. It only lasts another forty years."
"Yeah? Well, get this." I told Susan about her budding friendship with Carla and Derek. "She just turned thirteen, started eighth grade, and she's already talking about driving in cars with boys."
"So find something for her to do."
"She already takes piano lessons and plays field hockey. But hockey ends in October."
"What about one of those after-school programs? Justin took a great photography class last year."
"Do you know what it's like to sustain the interest of a thirteen-year-old girl whose brain has been corrupted by MTV?"
She flashed me her Mona Lisa smile. "I'm sure you'll find something."
I dodged a couple of bumblebees hovering on some goldenrod. Happily, they'd be gone soon. I don't like flying objects with stingers. As we rounded a corner, I told her how Rhonda Disapio had accosted me in the mall.
"Do you believe her?" Susan pushed her sleeves up to her elbows. "I mean, if she committed perjury on the stand..."
"I don't think she would have tracked me down and come all this way just to make it up." I hesitated. "But there's something else that kind of makes me believe her."
"What?"
"She died in a car accident last night."
Susan's eyes widened and then narrowed.
I explained what happened.
"It was a pretty bad storm," she said carefully. "The power's still out in some places."
"She kept saying she thought she was being followed."
It had to be at least sixty degrees outside, but Susan shivered. "So what are you going to do?"
"I thought of calling the state police to see if they consider it an accident--"
"Why wouldn't they?"
"I--I'm not sure. But even if they didn't, they wouldn't tell me. I'm not a relative or a friend-- I hardly know the woman. And since the trial, I doubt many people would believe much of what I said." I shrugged. "But I did talk to Brashares. You know, Santoro's lawyer."
"What did he say?"
"Not much. In fact, I don't think he's pursuing the appeal all that aggressively. For example--" I stopped short.
"What?"
I didn't answer.
"Ellie, what just happened?"
"I--I'm not sure. It's probably nothing."
"What is it?"
"I was just thinking that I told Brashares about my conversation with Rhonda as soon as I got home from the mall, and a few hours later, she was dead."
Susan slowed and arched her eyebrows. "Ellie..."
"You don't have to say it." I held up my palm. "I'm not jumping to any conclusions. In fact, I'm not even getting involved." I skipped a few steps ahead of her. "See? I'm fine. In fact, when I get home, I'm going to try to land some work."
We reached the end of the bike path and turned down Sunset Ridge. Ahead of us, a dark-colored SUV slowly turned the corner. I stopped and stared after it, shading my eyes with my hand.
"Now what?" Susan asked.
As it disappeared around the bend in the road, I felt my heart pumping. "Nothing." I couldn't say anything. Susan doesn't buy into conspiracies; she was just a baby when JFK died.
Chapter Fifteen
My mother always claimed I was a resilient child. I always bounced back, like one of those inflatable dummies. Though I prefer to model myself after the Black Knight in Monty Python, who kept challenging the king to battle even when his arms and legs were chopped off, by afternoon I convinced myself to move on with life. Put Santoro, Mary Jo Bosanick, and Rhonda Disapio behind me. Under the circumstances, I didn't see what I could do. Maybe Rhonda's death was just an accident. Maybe Santoro really did kill Mary Jo.
I called around to park districts and schools. Most of the popular after-school classes--acting, soccer, photography, computers--had been filled since July, but I did find two with space: Let's Learn Latin and Science Club. Neither would be high on Rachel's top ten, but I jotted them down.
I checked my machine. No callbacks yet. I eyed my Rolodex, wondering whether I'd have to cast a wider net. I wasn't looking forward to it; the leap from friendly voices to cold calls is a big one. I picked up the clothes from Rachel's floor and did a few loads of laundry.
The phone finally chirped around four. It was Karen Bishop, my longtime client from Midwest Mutual.
"Karen, how are you?"
"Good. Sorry I didn't get back to you sooner, Ellie. What's up?"
"Just checking in to see if there's anything I can help you out with. We haven't spoken in--"
I heard an exhalation of breath. "I had a feeling that's why you called."
"Excuse me?"
Karen and I have worked together for five years. A working mother herself, she's a no-bullshit person who's managed to survive, even flourish, in a corporate environment. Still, I wasn't prepared for what came next.
She hesitated. "Ellie, I can't use you. In fact, I don't think anyone will touch you with a ten-foot pole."
"What are you talking about?"
"It was your testimony at that trial. You attracted a lot of attention. People are a little leery of you right now." She paused. "You know how it is."
I gripped the phone and stared at a crack in the wall I hadn't noticed before. "No. Karen. How is it?"
"You know the mentality around here. People don't like anything that disrupts the status quo. That actually requires them to form an independent opinion. And you were kind of out there. Visible. Everyone saw you on the news--"
"Hold on. Am I being punished because I testified?"
"No, of course not. Even though Ryan did poke holes in your story."
"Does that mean I'm no longer capable of producing videos?"
"I didn't say that."
"What are you saying, Karen?"
She cleared her throat. "Frankly, Ellie, it's the issue of consent. You released video that technically didn't belong to you. At least that's what I'm hearing from our attorneys."
"Karen, the water district released it. They knew all about it."
"But you were the one who initiated it. The lawyers say you overstepped your boundaries. It's a bad precedent."
"But there was nothing proprietary on it."
"That may be, but the problem was you made the decision for them. No corporation wants their hand forced--especially by a third party. It wasn't your tape to begin with. No one's going to do anything about it, but they are saying it's indicative."
"Indicative of what?"
"Of--well, let's just say they've lost confidence in your professionalism."
I stiffened. "I can't believe this. What do you say?"
"Ellie, come on. What do you think?"
Through my shock and anger, I could tell this was hard for her, too. "Jesus, Karen. The man was accused of a murder he probably didn't commit. What was I supposed to do? Look the other way? Pretend it didn't happen?"
"I know, I know. But you know as well as I do, whether you actually did anything wrong doesn't matter. Appearance is everything. You weren't a team player."
"But I didn't do anything wrong."
Karen sighed. "Look Ellie, I don't have to tell you that people in corporations have blinders on when it comes to their own interests. Or perceived interests. They'll do whatever it takes to protect themselves--and their jobs. The bad news is as long as I depend on them for my paycheck, I have to toe the line. But there is some good news."
"Yeah? What?"
"It won't last forever. Memories are short. A few months from now, this will all blow over. Call me in the spring, and we'll talk. In the meantime, why don't you take some time off? After what you went through, I bet you could use it."
"Thanks."
I disconnected. Now I understood why no one was calling me back. The Chicago video community is small, and word travels fast. Particularly when it passes through corporate communications. And, to be honest, the water district hadn't been all that happy about releasing the tape in the first place.
But this was my livelihood, and spring was six months away. What if it didn't "blow over"? I could be blacklisted indefinitely. They might never let me back on the "team." Given that Barry's child support was, at best, erratic, how was I supposed to make ends meet?
I started pacing, a white hot anger skimming my nerves. Years ago, I would have been lauded as someone who, by virtue of suffering at the hands of the power structure, had become a person of value. But those days were gone, and I needed the corporate establishment--at least their largesse--to survive. Damn the suits. Damn Kirk Ryan. And damn Chuck Brashares.
It took six hours of self-pity, a hot bath, and two glasses of wine before I realized that Karen was right. No one had coerced me onto the stand. I'd come forward voluntarily. In a way, I had initiated the chain of events that destroyed my credibility. Karen was right about something else, too: _they_ didn't care if I ever worked again. _They_ had their interests to protect.
But I had mine.
I pulled back the sheets and climbed into bed. I'd gotten myself into this. I'd just have to get myself out.
Chapter Sixteen
You hear a lot about the North, South, and West Sides of Chicago, but no one talks much about the East Side, which was where I was going early Monday morning. Hugging Lake Michigan on its southeast side, the area includes working-class neighborhoods like South Chicago, South Deering, and Hegewisch.
A gassy odor filtered through the car as I got off the highway at 130th. If Chicago is the city of big shoulders, this is the meaty part. Farther east are streets with tiny bungalows, a bar on one corner, a church on the other, but 130th and Torrence is the industrial hub. Factories, warehouses, and cranes crowd together, abandoned rail cars line the streets, and smokestacks belch grit and God knows what else into the air.
I'd made a strategic choice. If the objective was to restore my credibility, I had a couple of options. I could try to verify Rhonda Disapio's story. The problem was, I wasn't sure how to go about it, short of setting up surveillance at the boat launch. Plus, if the boat men really did kill Mary Jo, I wasn't anxious to put myself on their turf. The other option was to ferret out Santoro's background, in an effort to prove he didn't kill Mary Jo. I already knew his haunts: the bar and the docks.
It wasn't a tough decision.
The Calumet River flows southwest from Lake Michigan to Calumet Harbor and eventually to the Mississippi River. Through yet another miracle of Chicago engineering, the harbor was dredged and transformed into a deep-water port so it could accommodate freighters from the St. Lawrence Seaway. Leading off the harbor are inlets that make the docks between them look like tines on a giant fork. It's at these docks that commodities are off-loaded. Years ago they were transferred to rail cars and shipped across the country. Now most of the cargo travels by truck.
I threaded my way around the Ford plant at Torrence and turned on 122nd. Turning again, I drove down a road that had been patched and repatched, and from the groan of my suspension, could stand yet another go-round. A mile down the road, a battered black and white sign said I had reached the Ceres Terminal. I swung into a lot studded with chunks of broken concrete and stopped behind a shabby brick building with a roof of corrugated metal. Two cars were parked at haphazard angles in front.
It was a cool October morning, and condensation coated the cars' windshields. Pulling on my Sox hat--I knew better than to wear a Cubs hat this far south--I wandered over to a group of longshoremen standing in front of a warehouse. Perched above them on a rusty steel scaffold was a fleshy, graying man with a clipboard. Most of the men looked old. Dressed in canvas coveralls and scuffed, steel-toed boots, several waved union cards in the air.
"Sorry, guys, that's all I need for today," the man with the clipboard said. "But I got a barge of steel coils coming in Friday. Be work for about a dozen of youse."
A collective grumble went up from the men, but it was surprisingly docile, as if they were used to disappointment. I shouldered my way through to the man with the clipboard, but he climbed down off the hiring stand and pretended not to see me. Pulling a tin out of his pocket, he opened it and pinched a wad of Red Man with his thumb and forefinger.
"Excuse me," I said as he packed it in his mouth. He squinted in my direction, one cheek plumped up like a chipmunk. "Do you know Johnnie Santoro?"
His eyebrows shot up, but he kept chewing.
"I know he used to work down here."
He spat out a clump of black goop, which landed a few inches from my left sneaker. "Haven't seen him in over a year. Don't expect to."
I stood my ground. "But you knew him, right?"
He looked me up and down. "You a cop?"
"No."
"Lawyer?"
"No."
"From the union?"
"No."
"Then I ain't got nuttin' to say."
He gave me his back and walked away. A few gulls swooped down in parallel arcs above his head, their bellies tinged with the morning sun. I considered groveling, beseeching him with the fact that my livelihood was at stake unless I could clear my name, but after glancing at the unemployed longshoremen still gathered by the warehouse, I reconsidered. I pulled the brim of my cap farther down and started back to the car. As I skirted a second warehouse with peeling paint on its sides, a flicker of movement caught my attention.
"Got a match?" A burly man with white hair, a bulbous red-veined nose, and skin the color of a dried apple drew a cigarette from behind his ear. The scent of booze clung to him, and there was a suspicious bulge in his pocket. I dug around in my purse and pulled out a frayed matchbook from the Italian Gardens, my favorite neighborhood restaurant.
He lit the cigarette with pudgy fingers and took in a deep drag. Then he blew it out so contentedly I was tempted to bum one, even though I haven't smoked in fifteen years. He grinned at me as if he knew what I was thinking, and slipped the matches into his pocket. "You're that dame I saw on TV." He studied me. "You stuck up for Johnnie. That took guts."
He could keep the matches. He knew Santoro. I tried to suppress my excitement. "It didn't seem to do much good."
"You never know." He brought the cigarette back up to his lips. "Why you come all the way down here? You ain't had enough?"
"I--I have some questions about him."
"Yeah." He spread his arms. "But how you know to come down here?"
"Oh." I'd misunderstood his question. "I called the union and asked where my best chance was to find some longshoremen. They said Ceres was the only place hiring today."
He nodded, then motioned for me to follow him to the edge of the dock. A barge was tied up a few yards away, its contents hidden under several tarps. Water lapped against the side of the barge. Across the inlet a freighter had tied up, and I heard shouts and saw men bustling to off-load materials. The smell of rotting fish was strong.
He flicked his ash into the water and took another drag, his belly ballooning in and out. "I'm Sweeney. What is it you wanna know?"
This was the best offer I'd had in weeks. "I'm not sure how to begin, but, well--tell me how you knew Santoro."
Sweeney inhaled. The tip of his cigarette glowed orange. "His daddy and me were buddies."
"Santoro's father is a longshoreman?"
"Was. He's passed on now. Died of cancer."
So did my mother. "Did he--Johnnie--come from a large family?"
He took another drag. "Not so big. Four kids, I think. Three girls and Johnnie."
"Do they live around here?"
"Not far." He flicked his cigarette off the dock. It landed in the water with a tiny hiss. "What is it you want, lady?"
I sucked in a breath. "Mr. Sweeney, I don't think Johnnie Santoro killed his girlfriend. But the jury didn't believe me, and unless I can prove it, I may not ever work again. I'm trying to find any information, any evidence, anything that would help prove he didn't do it. I figured I'd start by coming down here."
He stared at me, sizing me up for another long moment. Then, "In that case, I dunno if I'm gonna be much help."
"Why?"
"I--well, let's just say I didn't much like Johnnie."
A small boat chugged down the waterway. The barge rocked gently in its wake.
"Why not?"
"Johnnie was one of those guys who always wanted something for nothin'. You know what I mean? Thought just because his father worked down here, he was--entitled." He folded his arms.
"Did he work regularly as a longshoreman?"
Sweeney scoffed. "Not much. And when he did, he was always struttin' around like he owned the place. Mouthing off, too."
"About what?"
"His friends. His deals. How he was gonna score big. Bullshit like that."
"Deals? Was Santoro dealing?"
"Don't know." He looked off onto the water.
I waited.
He coughed hard, a smoker's hack, and took out another cigarette. "But seems to me, a couple months before he got busted, I can remember him sayin' he wouldn't have to be doing this much longer."
"Doing what?"
"You know. Scroungin' work down here."
"Why not?"
"Said he was working a big deal."
"But you never asked about the details."
He looked at me under hooded eyes. "Ain't none of my business, now, was it?"
"Did he ever mention a guy named Sammy?"
He dug out the Italian Garden matches, frowning. "Not so's I remember."
He lit another cigarette, waved out the match, and let it drop to the cracked concrete.
I cleared my throat, phrasing my next question carefully. "Did Johnnie have a union card?"
"Oh yeah, his daddy made sure of that. That was part of the problem. Charlie couldn't say no to Johnnie."
I paused. "Well, given the way things are down here, you think he might have been mixed up with the wrong people? People who didn't like the way he behaved, and--"
"You mean like the people what still control who gets hired and how much of our pension they're gonna rip off, even though there ain't no work? Those kind of people who you mean?"
I nodded.
He hesitated. "I couldn't say. All I can say is Charlie and I weren't never mixed up with that crowd. Those guys'll bleed you dry." He sniffed. "Of course, twenty years ago, it didn't matter. There was plenty of work. You could still make it. But now...it ain't never been this bad. A boat don't tie up but maybe once a week. No way you can live on that."
He stole a glance at me, then unexpectedly grinned, baring a set of yellow, stained teeth. "Now, I ain't gonna deny that Charlie and I mighta helped something fall off the back of the boat once or twice. Like the time a bunch of Corvette engines came in on a freighter. Some of 'em ended up in cars all up and down the South Shore. I heard the FBI took to casing the McDonald's over at Seventy-ninth and Phillips, making all them high school kids lift up their hoods so's they could check out what was inside their Chevys." His belly shook with quiet laughter. "But those days are gone. There ain't nothing left to steal. I mean, who'd want a load of steel coils?"
"So, it's not likely Johnnie was--"
"Like I told you. I keep my head down."
"I understand." I looked out over the waterway. The sun was sewing the surface of the water with tiny bursts of light. "Tell me something, Mr. Sweeney. Has anybody else come down here asking questions about Santoro?"
"Like who?"
"Cops, investigators, lawyers. Anyone."
"Not so's I'd notice. But this ain't the kind of place people come if they don't have to."
"Yeah. Well, thanks. You've been a big help."
He straightened up. "Like I said, Charlie was my friend."
I headed toward the car. Just before I rounded the corner, I turned around. Sweeney was gazing out over the water, as if the docks had stolen his soul, but it wasn't worth the effort to get it back.
Chapter Seventeen
As I drove north on the Bishop Ford, a giant pair of red lips on a white billboard reminded me of Rhonda Disapio. How her mouth squeezed into a tight, crimson ball against her pale skin. How her lipstick was smeared the day we met. It occurred to me that she'd come north to see me on the same highway. Except she never finished the return trip.
I tightened my seat belt. The fact was, I probably hadn't done myself any good at the harbor. Johnnie Santoro sounded like a punk who was mixed up with the wrong people. Not someone I'd care to help. I wondered whether I'd have gotten involved with him in the first place if I'd known.
The Dan Ryan runs from 95th Street to the Loop. As I approached 95th, I slowed. Calumet Park, the lake, and the boat launch were only a few miles away. I could try to check out Rhonda's story. I'd never have incontrovertible proof--Mary Jo had died over a year ago--but at least I could see if Rhonda's version of events was feasible. I wouldn't even have to get out of the car. I could just drive around.
I made the turn.
East of the expressway the neighborhood is largely black until you hit the lake, where it turns Hispanic. The streets off 95th are narrow and lined with row-houses and bungalows, but they're clean and neat, as if they're struggling to stay respectable.
They do parks well in Chicago, and Cal Park was no exception. The 200-acre stretch of land is a tranquil haven, with graceful curves, wide promenades, and lots of trees. I passed a few kids on bikes--they had to be playing hooky--and two women pushing strollers. I rolled down the window. Sun-warmed air swept through the car.
I swung into a parking lot at the northeast end of the park. Directly in front of me the lake curved around a wide bend. To my left were a few trees, but not enough to obstruct the view. I cut the engine and watched a few gulls march past the car, their tiny heads bobbing back and forth. The sun fired the trees with splashes of copper, red, and gold. A soft breeze skittered the leaves. I climbed out.
In front of me stretched four narrow wooden piers supported by pilings. Between them were lanes of water wide enough for a boat to navigate. Asphalt backed up to the water's edge, and two men were dragging their boat from the water to a trailer hitched to the back of their van. A metal breakwater angled around the north end of the boatyard, sheltering it from the worst of the lake's excesses.
To my right, an expanse of rocks hugged the shore, bordering a path wide enough for joggers and bicyclists. I walked over, trying to imagine the scene as Rhonda had described it. I hunkered down on the rocks, pretending it was late at night and I had a bottle. I gazed at the pilings, imagining a shadowy boat as it coasted into the launch, hearing the drone of its motor as it slowed. I tried to feel the spark of interest that must have run through Mary Jo and Rhonda when they realized two men were aboard. Though the launch was far enough away that you couldn't make out faces--I could barely see the men securing their boat to their van now--I could picture the two women giggling, daring each other to make the first move.
I retraced my steps to the parking lot. A grassy, leaf-strewn area, now stubbly and brown, lay in front of it. To my left, running the length of the lot, was a chain link fence. Behind it was a one-story red building, a Park District facility of some sort, I thought.
Rhonda had said she'd escaped through a hole in the fence. I walked over and started down its length, jiggling and shaking it as I went. Halfway down, something went slack. I stopped and shook it again. The bottom of the fence had come loose. I leaned over and lifted a section of fencing. Was this where Rhonda slipped through and escaped?
I replaced the loose section and straightened up. Along the fence line was an accumulation of litter, pushed up against the links where grass had grown through. Discarded coffee cups, beer cans, fast-food wrappers, even a few swatches of material that might have been shirts at one time. A few yards farther down, up against the fence, something glittered. I explored it with my foot. A silver charm bracelet was tangled up in a comb. Bending over, I extricated it from the comb. It had a small silver heart. Probably belonged to some little girl who cried for days when she discovered it was lost.
I studied the bracelet, looked both ways, then dropped it in my bag. I still struggle to resist things that don't belong to me, but this time I was rescuing something from oblivion, not shoplifting. That was different, wasn't it?
As I headed to the boat launch, the two men with the van, having attached their boat to their trailer, were pulling away. More debris had collected where the breakwater met the shore. Soda bottles. A dented gas can. Shards of glass. I watched all of it disappear under a wave and reappear moments later. Did everyone expect someone else to clean up after them?
I was about to walk out onto one of the piers when I heard a noise behind me.
I spun around. A couple of kids on bicycles were closing in on my Volvo. Though it's over ten years old, it's not a beater, and I planned on getting another couple of years out of it. I threw my bag over my shoulder and started over. As I did, one of the kids parked his bike and planted himself on the driver's side. Stretching out his arm, he strolled from the front to the back of the car, running his hand along the side. The other kid watched, laughing.
"Hey!" I sprinted toward them. "Stop messing with my car!"
The kid who'd been laughing turned around, his grin fading. The boy who'd been at the car ran to his bike and jumped on. They both pedaled furiously in the opposite direction.
"Hey you! Stop it right there!" I yelled.
But I was no match for young male bikers at warp speed. By the time I got to the car, they had turned the corner and were out of sight. Breathless and damp, I stopped at the spot where they'd been. A long, wavy scratch extended from the front end to the rear bumper.
Chapter Eighteen
When I got home I called Mark Lefferts, an old friend from high school who owns a body shop in Glenview. We dated for about a month during senior year, one of those relationships that burned hot and furious and then turned to ashes when he decided he liked Angie Sawyer more than me. Angie, a cute blonde and a cheerleader, reportedly had a fondness for the backseats of cars. No wonder he made a career of automotives.
He could fix the scratch for about twelve hundred dollars, he said. Once I started breathing again, I said I'd learn to like it and hung up. So much for old boyfriends. Although he did have great weed, I recalled. Back in the days when grass cost thirty dollars an ounce, he had sinsemilla before anyone else.
I wandered into the kitchen. Thinking about grass reminded me of Mary Jo's statement to Rhonda at the boat launch. Something like "What makes you think I don't know about dealing?" At the time, I'd thought it was a strange comment. Out of context. But now, I wondered. Is that what this was all about?
I started wiping the counter with a sponge. Maybe the men in the boat were bringing in drugs from Canada through the Great Lakes. Hell, if the Calumet River was involved, they could have come up the Mississippi. Was it possible Mary Jo wasn't at the boat launch by chance? What if she was there to intercept the shipment for Santoro? Sweeney hadn't denied Santoro was into dealing; in fact, when I asked him point blank, he'd kept his mouth shut. And Mary Jo was Santoro's girlfriend.
But then, why would she have brought Rhonda Disapio with her? Unless Rhonda was involved, too. No, that didn't seem right. Maybe Santoro had ordered Mary Jo to intercept the stash, but she refused. Maybe that's what they were fighting about at the bar. Maybe she didn't want anything to do with dealing and was trying to make a break for it in his car.
I wiped the burners on the stove. Or was it the other way around? Maybe she was trying to get _more_ heavily involved. Freeze Santoro out. Nobody ever said Mary Jo was an angel--except her mother. Maybe Mary Jo made off with Santoro's car, left him stranded, and proceeded to the boat launch herself. But then, after she got there, the deal fell apart. Maybe the men didn't know her. Or didn't buy her story. Or thought she was a cop. They panicked. She ran. They killed her.
Either way, Santoro would have been caught in the middle. He might not have been at Calumet Park, he might not be guilty of murder, but you wouldn't call him an innocent.
I rinsed the sponge in the sink. There was only one problem with my theory. There hadn't been any talk of drug dealing at the trial. Not a hint. And while I realize it's probably not a great idea to admit to one crime when you're on trial for another, I doubted if Brashares had even entertained the possibility. Which was too bad. If he could have established that Mary Jo was acting as Santoro's go-between, it might have buttressed the tape.
I squeezed water out of the sponge. I was spinning, formulating theories without proof. Even so, Brashares should know. I called and got his machine.
"Hi. It's Ellie Foreman. Something's come up that I thought I should run by you. It's about Santoro's background and what those men might have been doing when Mary Jo was killed. It might give the tape more credibility. Then again, it could be nothing. But I thought I should at least mention it."
As I hung up, sun streamed through the window, and the reds, oranges, and yellows of the leaves put on a show. All this talk about scams and drugs and murders was making me feel dirty. I went up to take a shower.
***
That weekend Dad and I barbecued on the tiny deck off my kitchen. Figuring it might be the last one for the season, I bought thick steaks and tried not to think how many arteries I was plugging.
Dad still fires up the coals better than any man I know. And he does it without any props except lighter fluid. Within minutes, he had flames licking the side of the grill. When the coals were edged with a thin border of white, I brought out the meat.
"You recovered from your experience in court?" He poked the steaks with the tongs.
I sank into a deck chair. "You know, I think you were right. In a way, I'm kind of sorry I ever got involved."
He slid the meat onto the grill. "Didn't the woman who testified just die in a car accident?"
"How did you know?"
"Ellie, I may be old and slow, but most of my cylinders are still firing. It was on TV."
"Rhonda Disapio was Mary Jo Bosanick's friend," I said. "What you don't know is that she came to see me on the day she died."
He looked up. "Why?"
"She had a pretty strange story." I told Dad about my encounter with Rhonda.
"What did she expect you to do?"
"Put her on TV so she could stay out of jail. She thought I worked for the news. Ryan did make a big point of that, remember."
"Why didn't she go to the police?"
"She said she was too scared."
"I'm not one to speak ill of the dead, but no one would ever accuse her of being the sharpest knife in the drawer."
"Could be. But I'm starting to wonder whether it all revolved around drugs."
"Drugs?"
I sketched out my suspicions but didn't tell him how I arrived at them. He wouldn't approve of my field trip.
"So," Dad said when I finished. "Santoro might not be the innocent you thought he was?"
"Maybe not."
He picked up his scotch. The ice cubes clinked against the glass. To his credit, he didn't come out with an I told you so.
"That might also explain why Brashares was so strange," I said.
"Santoro's lawyer?"
I nodded. "I kept thinking he was just going through the motions. Doing the minimum required but nothing more."
"You think he knew Santoro was dirty?"
"It's possible. Maybe Brashares didn't want to expend all that energy on a loser. Isn't that the way defense lawyers think?"
"If they do, they ought to stop being defense lawyers."
Through the kitchen window I caught glimpses of David and Rachel washing lettuce for the salad.
I turned back to my father. "I called Brashares to let him know. But he hasn't called me back."
Dad flipped over the meat, then eased himself into a chair. Sandburg had it wrong. It's age, not fog, that creeps in on "little cat feet."
"Ellie, why are you still calling this lawyer? The trial's over."
I shrugged.
"Ellie..."
"Okay." I sighed. "Since the trial, no one will hire me. I can't even get any callbacks. Karen Bishop, my client at Midwest Mutual, says it's because of the tape. Apparently, I forced its release, and people, especially corporate people, don't like that. I've lost a lot of credibility. I was trying to do some damage control."
"That's _meshuga_. Leave it alone," he said wearily.
"Dad, I have to work."
A peal of giggles came from inside. David and Rachel were playing catch with a cucumber, pretending it was a football.
"Where is it written that you have to support yourself forever?"
"Don't go there, Dad."
It was my dependency--or what Barry claimed was my dependency--that triggered our problems when we were married. I only worked when I felt like it, he complained, while he was expected to bring home the regular paycheck. But he was an attorney in a full-service firm, billing two thousand hours a year. He never really understood the nature of freelancing. It never comes in at a steady pace. You can write four proposals for every project you get. Go to appointments, lunches, and meetings that ultimately produce nothing. When I wasn't actually producing a video, he was quick to call me a princess. Or worse.
I didn't intend to revisit the pattern with David. But that was a conversation for another time. I picked up the tongs and checked the meat. "You know, there's another possibility. About Brashares."
"What's that?"
"He might be on somebody's payroll himself."
"Whose?"
"No one liked Santoro very much. Sweeney said--uh--I mean I heard he had a big mouth. Maybe some--someone with influence told Brashares not to try all that hard to get him off. Maybe they were happy to see Santoro take the fall."
"Now you think he was framed?" Dad's voice hardened.
I didn't answer.
"Now I know you're _meshuga_."
"Hold on. Suppose the business down at the boat launch did have something to do with drugs. We all know that where drugs are involved, organized crime isn't far away."
"You don't think you're stringing a few inferences together into a huge assumption?" Dad's eyes narrowed. "Ellie. You began this conversation saying I was right. That you were wrong to get involved. Sounds to me like you're getting in deeper."
"This isn't getting involved. It's just talking. I thought, given your experience, you might have some perspective."
"My experience?"
"Skull. Lawndale. Before the war."
My father snorted. "Sweetheart, that was over sixty years ago. And Skull was no mobster."
"That's not what you implied."
"Skull was a...a street thug. With pretensions. Anyway, you're talking about a different world. A different time. Life wasn't as...as coarse. There were standards."
"A shark is a shark. No matter when they attack."
"You think so?" He got up to inspect the steaks. "Lemme tell you a story. When I went into practice for myself, a couple of guys came to me with one of those offers. You know. The ones you're not supposed to refuse." He faced me. "They wanted to help me build my practice. Said they could steer a lot of work my way.
"I knew what they were asking me. And I thought about it. It was tempting. You were a baby, and I was supporting your _oma_ and your _opa_." He jabbed the steaks with the tongs. "But after a week or so, I called them back and said, 'Thanks. But no thanks. I'm going down a different path'. They understood. In fact, they said, 'If you ever change your mind, come talk to us.'"
"That really happened?"
"What? You think I made it up? My point is there were boundaries back then. Limits. You could say no, and the Outfit would leave you alone. Not anymore." He waved the tongs. "Today they'd find a way to finagle it so I'd have to work for them. Threats. Extortion. Blackmail. There's no respect anymore. I mean, you're talking about the same scum who ripped off scrap metal from the World Trade Center."
"But Dad, in a way you're just confirming my suspicions. Maybe Santoro was mixed up with these goons. Maybe he pissed them off. Maybe--"
"Ellie, I love you dearly, but you're as headstrong as your mother. You can't live with ambiguity. So you latch onto some crazy idea and try to convince everyone it's true. Even if it isn't."
"At least I come by it honestly," I grumbled.
He waved a hand. "Let's say you're right, and he was involved with some scumbags. What are you gonna do about it? You got no idea who they are. They might not even be wise guys. Today you got your Russians, your Eastern Europeans, your Asians--"
"Tongs."
He looked toward the grill. "They're here."
"I meant--never mind."
"I'll tell you what the problem is." He brandished the tongs. "There's no respect for life anymore. The sanctity of life. Nobody gives a damn. Take these young suicide bombers. You know, the ones who kill themselves for the glory of Allah. How were these children raised? They're nothing more than cannon fodder. What kind of people are their parents? It's a _shonda_."
I watched him spear the steaks and take them off the grill. "You know why they're doing it. It's their jihad."
"Don't you believe it. They're doing it because some crazy Arab seduces these poor _shlubs_ by convincing them they'll be heroes." He shook the tongs in the air. "You know, if I had a nickel for all the fools in the world, I'd be a millionaire. And something else..."
I realized that was all I was going to get out of Dad tonight. But, then, he was allowed; age confers a license to rant.
***
After dinner, David, Dad, and I sat in the family room, trying to ignore the pounding bass that vibrated down from Rachel's room.
"I spoke to Abdul earlier," David said. "He said to send his regards. He hopes you're all right."
"Abdul?" I asked.
He smiled shyly. "He asked me to help him finance the purchase of a chemical plant in Indiana."
"Smooth. I guess the rafting trip turned out to be profitable. At least for you."
"You helped. He's very fond of you."
My father beamed. "You make a good team."
David went on. "I told him about the trial and what's been going on."
I shot him a warning look. I didn't want Dad to get curious again. I shouldn't have worried.
"Wait a minute. Did I hear you right? Abdul?" The lines on Dad's forehead deepened.
"We met at the Greenbrier," I said. "He's a relative of the Saudi royal family. He owns oil wells."
Dad looked over at David, then at me. "You couldn't find a Jewish sheik?"
David and I traded smiles. I got up to kiss my father, thinking how lucky I was to be surrounded by the people I love, when the phone trilled. I ran into the kitchen to get it.
"Ellie?" It was Susan.
"What's up?"
"You'd better turn on Channel Nine."
I ran into the family room and punched on the nine o'clock news.
"According to police," the anchorman was saying, "the body of attorney Chuck Brashares was found in his Loop office earlier tonight. Police say Brashares was shot in the head approximately three days ago."
Chapter Nineteen
A political scandal in the governor's office pushed Brashares's murder off the front page, but the story on page three was chilling enough. His office had been broken into while he was working late. The cops found evidence of a struggle, there were bruises on his face, and it looked like his arm was broken. The office had been trashed, the safe cleaned out. Police speculated robbery was the motive.
His death cured me of any further involvement. I had no reason to think it was connected to Santoro's case, but three people were dead: Mary Jo, Rhonda, and now Brashares. That was enough. I forced myself back to the business of living, cleaning closets, washing the car--the key scratch gave the Volvo a tired dignity, I decided--and taking long bike rides.
David didn't come in the next weekend, and Barry didn't take Rachel. Friday night she came into the family room with a smile, a bowl of sudsy warm water, and a cigar box full of nail polish. After soaking my hands, she proceeded to file my nails, tighten my cuticles, and apply not one but three coats of polish. The result was a purple base, green tips, and a thin orange stripe separating the two. My nails were a vision.
Afterward, we made popcorn and watched a video. The film, a techno-thriller with big stars, great locations, but cardboard characters, was way too predictable, and I was dozing off when two beams of light suddenly poured through the window. Startled, I leapt up and ran to the window. A dark-colored SUV was pulling up to the curb.
A tiny ice crystal formed in the pit of my stomach. Hadn't Rhonda Disapio been followed by a dark SUV? And hadn't I seen one when Susan and I went for a walk? I wondered whether to double lock the door.
But in that split second between thought and action, Rachel sprinted over and threw it open.
"Rachel--what are you--"
She ran out the doorway and down the driveway. A car window slid down, and she stuck her head in. I raced after her, my heart thudding, but there were no shots. No screams. No nothing. Rachel turned around, her eyes beaming.
"It's Carla and Derek. They want me to go out with them. Can I, Mom? Please?"
I sagged against the locust tree. "Whose car is that?"
Rachel looked at me, then back at the car. "Derek's parents."
I nodded, pressing my lips together. There were probably about five thousand dark-colored SUVs on the North Shore.
Rachel's face lit up. "Thanks, Mom. You're way cool. I'll be home in a couple of hours."
"Hold on." She'd misunderstood my nod. "You're not going anywhere."
"But you just--"
"That wasn't permission to go out." I started back to the house. "Rachel, it's after ten. You can't go out this late."
"But Mother--"
"We've been through this before. No driving around. No late dates. Anyway, you have that Science Club project to finish."
After a pitched battle a few weeks ago, during which I'd maintained it wasn't a yes or no option, she'd decided it was less humiliating to be a techno-geek than a linguist.
"But it's the weekend."
I glared at her.
She threw me a hateful look. "Daddy thinks you're neurotic, you know."
"You don't live with your father."
"Maybe I should."
"With behavior like this, I'm open to negotiation." I looked at the car. Two forms were in the front seat, their heads close together. "You just tell them you're not available tonight."
Rachel didn't move.
"If you won't, I will."
Her bottom lip curled, the way it does when she's about to scream or cry or shout. "You don't want me to have any friends."
"Rachel..."
"You don't want me to be popular. You want me to be a freak like you."
I pointed to the car. "Go."
She trudged back to the car and stuck her face in the window. A moment later, the car backed out of the driveway. As it sped off around the corner, she ran inside, tears streaming down her face, and raced up the stairs. The sound of the door slamming reverberated through the house.
Chapter Twenty
Monday I got a call from Great Lakes Oil, which, until their merger with a British multinational, was one of the country's largest oil companies. An aide to Assistant Vice-President Dale Reedy asked if I was interested in bidding on a potential training video about the process used to extract oil from shale. Reedy would be out of town for a week or two but wanted to meet as soon after that as possible.
I tried not to accept too effusively. I've always felt somewhat proprietary toward Great Lakes; it was _our_ gas station when I was a kid. My mother used to collect the glasses they gave out as premiums. I remember one sunny day riding my bike down to the corner to fill up my tires with air. How I told the manager we only needed one more glass to make up a set of eight. How he slipped me a free one, which I presented to my mother with a flourish.
The blue and white signs that used to dot the country have for the most part disappeared, but the Great Lakes skyscraper still towers over the Loop, and every time I pass it, I think of those glasses. Indeed, if it were possible for me to feel warm and fuzzy about any corporation, it would probably be Great Lakes.
I said I'd be delighted to meet Dale Reedy, and we set up a date. I hummed as I got off the phone. At least one corporation was willing to deal with me. And Great Lakes was a first-tier company. This could be major bucks. Things were looking up.
***
"Where are we going?" I asked as David and I headed downtown Friday night.
"It's a surprise." He pulled into the left lane, weaving between cars.
"Pretty sure of yourself, for someone who didn't know how to find the lake six months ago."
"I had a good teacher. Plus..." he said, gunning the engine, "it isn't my car."
I fastened my seat belt, but I didn't need to. Chicago takes its cultural cues from the West Coast, and thanks to the snarl of traffic well past rush hour, we'd apparently absorbed their worst nightmares, too. There was no rational reason for the tie-up: no Cubs game, no accident, no construction. Nevertheless, we crawled down the highway for the better part of an hour. By the time we pulled up at the Four Seasons Hotel, I felt as wilted as yesterday's salad.
The doorman opened the door, his uniform festooned with more ribbons and buttons than a veteran on Decoration Day. David took my arm and guided me inside. He'd asked me to put on my black slacks, a white linen blouse, and the dangly silver earrings he bought me that make me feel dressed up. I laced my arm through his. He used to stay at The Ritz-Carlton, but after we met he switched--he was an equal opportunity hotel guest--and we'd spent our first nights together here. Long, languid nights lost in the touch and taste of passion. Thoughts of the most perfect bed in the world danced in my brain. Could Krispy Kremes be far behind?
I grinned. "Is this the surprise?"
"Well, sort of."
"Sort of?"
He hesitated. "Abdul is in town, and he asked us to have dinner with him."
"Abdul?"
"I couldn't get out of it, and he really wanted us both to come."
My smile faded. I'd only met the man once. As a client of David's, especially a new client, he merited a modicum of courtesy, but he wouldn't be my first choice for a dinner companion. I was about to say so when David pulled a room card out of his pocket.
" _This_ is the surprise," he said. "After dinner."
A warm feeling started to radiate through me. "What about Rachel?"
He checked his watch. "Katie's mother ought to be picking her up right about...now."
"Okay." I kissed him. "I forgive you."
Our heels clicked across the marble floor of the lobby. We strolled past a polished mahogany table with a huge flower arrangement on top. Behind us was an oak hutch filled with elegant china, to one side a banister with ornate scrollwork. A silk carpet lay beneath our feet.
I stopped to smell the flowers, a mixture of giant sunflowers, calla lilies, and smaller blossoms I couldn't identify, though they looked like tiny orchids. Their sweet, delicate fragrance tickled my throat. David leaned over, picked one of the small blooms, and placed it behind my ear. Looking up, I caught our reflections in a gilt-edged mirror. Soft lighting bathed us in a warm, golden glow. The pale, cream-colored flower made a stark contrast to my black curls.
I touched my fingers to the flower. It was just a tiny flower; people take them all the time. But as I stroked its soft, velvety petals, it occurred to me that, innocent as it was, David would never have done something like that six months ago.
We crossed to the elevator. He clearly had loosened up since we met. That was a good thing, wasn't it? Then why did I feel so uneasy? The walls of the elevator felt like they were closing in.
"Are you okay?" David asked.
I looked over. I knew what was bothering me. It wasn't the flower.
I still can't accept it when good things happen to me. Especially when it comes to relationships. If it seems too good to be true, it probably is. I make sure of it.
"What are you doing?"
I crushed it on the floor. "You've come a long way, haven't you? I've even got you stealing now."
He gazed at me, then, without speaking, picked up the flower and deposited it in the ashtray. The elevator slowed and stopped on the forty-fifth floor. As the doors opened, a young couple wrapped in each other's arms hurriedly broke apart and edged past us into the car, giggly and gay. We got out. The doors closed with a whoosh, but not before the man reached for the woman.
David faced me. He had to be furious. He was probably going to tell me how hypocritical I was. Attacking him for one of my own failings. But I couldn't tell him the truth.
Instead, he caught my chin in his hand and brushed his fingers across my cheek. "Ellie, it was only a goddamn flower. I'd give you a whole garden if I could."
I wasn't expecting that. Most men would have retaliated. But David wasn't like most men. He was utterly without guile. I sagged against the wall. Maybe I was wrong. This was no big deal. Maybe I'd been overreacting. Even a teensy bit maudlin. This was supposed to be a pleasant evening, not fraught with tension. I straightened up and pasted on a smile, determined to be a charming dinner companion. David smiled back and knocked on 4520.
"Good evening, Ellie." Abdul opened the door and planted kisses on my cheeks. He was wearing a loose-fitting dark blue silk shirt and white linen pants. He gave off a heavy scent of cologne.
"Abdul. How lovely to see you again. When did you get in?"
"David and I took the same flight out." He smiled. "When he told me what a trying period you've been through, I insisted you join me for dinner."
I glanced over at David. "It was a wonderful idea."
He ushered us into his suite, which was furnished with settees, thick carpets, and Louis XVI chairs in rich patterns of red, gold, and blue. In the center of the room was a table set for three with crystal glasses and elegant china. Heavy drapes framed a picture window with a view of the Hancock and beyond that the lake. A breeze swept off the water, sharpening the edges of buildings and making the lights twinkle. The soft, dark blanket of water was pierced by an occasional flash from a boat or buoy. If the view from our room was half as beautiful, it would be heaven.
Abdul took a bottle of wine from a silver cooler and filled one of the glasses. "Try this."
I sipped. "Excellent."
He showed me the label. "It's Joseph Heitz. One of your Californias."
He put it back and picked up a crystal plate layered with triangular toast points. A small bowl in the middle held black caviar. I took one, scooping up a dab of caviar, scallions, and chopped egg. Abdul smeared his with a thick coating and bit into it.
The meal started with grilled shrimp marinated in a coriander lime sauce and progressed to rack of lamb with a caramelized shallot and thyme crust. Each course was served by two unassuming waiters, who whisked silver-domed covers off the plates. I reminded myself to tell Susan about it.
Abdul regaled us with stories about the small village in which he grew up, and despite his occasional lapse in manners that I attributed to the difference in cultures, I felt myself warming to him. The wine and the food did their job, too, and by the time the waiters served us sorbet topped with lavender blossoms, I almost believed my crisis over the flower was just a blip. An aberration. I'd been jumpy since Brashares' death. That's all it was.
"What brings you to Chicago?" I asked.
"I am looking at a small chemical company in Indiana. Great Lakes Oil has put it up for sale. David is helping me finance it."
I sat up straighter. "Great Lakes Oil?"
He nodded. "Since their merger, they're looking to spin off their smaller operations."
"What a coincidence."
Abdul angled his head. "Why is that?"
"I just got a call from them. Inviting me to bid on a video. An assistant vice-president wants to produce a video on shale oil. The industry flirted with it thirty years ago. But I guess with the price of oil what it is, they're resurrecting all their toys."
"Indeed." He smiled.
I felt myself color. I'd forgotten to whom I was talking.
He rose and went to a small table with a silver humidor on top. Bending over, he opened it and extracted two cigars. "What is the executive's name, out of curiosity?"
"Dale Reedy."
He hesitated, then pulled a clip out of his pocket and snipped off the end of one cigar. "I don't know the name." He lit it with a silver lighter, then handed David the other.
Surprise flickered through me; I'd never seen David smoke. "Why are you looking to buy an American chemical company? Why not build your own in--in Saudi Arabia?"
Abdul puffed on the cigar. "That is our ultimate plan," he said. "But Great Lakes produces an additive that extends the storage life of gasoline. It seems to work well in dry, hot climates." He blew out a stream of smoke. "We want to bring it to Saudi Arabia. As you may know, the money from petrodollars does not stretch as far as it once did. We have only one job for every two men. And if men don't work in our part of the world..." He waved his cigar, not needing to explain what could happen to a generation of young Saudi men with too much time and not enough money.
David cut in. "Did you get the correlations I faxed yesterday?"
Abdul turned to David. "You have anticipated me. As usual. However..." He handed his cigar clip over to David, who fiddled around with the tip. "It may be that our closing date is more fluid than I previously thought. Is there a way we can incorporate that into the hedging strategy?"
"Of course. Just remember, the more flexibility we incorporate, the more expensive the hedge."
Abdul touched the flame of his lighter to David's cigar.
"When you get a moment, e-mail me the parameters, and I'll work up some new strategies."
"I am fortunate to have you on my side."
David smiled.
I rose and moved to the window, queasy from the smoke. The side of the Hancock, its windows lit in random patterns, looked like a giant Tetris board. I grabbed the metal base of the window and pulled. To my surprise, the window opened, and a strong gust of air rushed in, peppered with the blasts from car horns, shouts, and squealing brakes. Startled, I sprang back.
David scrambled up. "Are you all right?"
"I'm sorry." I shook my head in embarrassment. "I--I didn't expect the window to open." I should have known. At the Four Seasons, everything works, including the windows.
"No." Abdul extinguished his cigar. "It's my fault. I did not ask if the smoke would be a problem."
Gusts of air whistled through the room, scattering a sheaf of papers on a small table nearby. I reached up and shut the window, then bent over to pick up the papers. "No. It's my fault. I should have said something."
"Here. Let me." Abdul crossed the room and bent down, too. Our heads bumped. He laughed nervously.
I patted my head. He retreated into the other room with the papers. A latch snapped open and shut. He came back out and motioned me back to the table.
"Now, tell me about this trial." He poured me a fresh glass of wine. "You must have been disappointed at the jury's decision."
I took the glass. "You could say that."
"After David told me about it, I read some of the stories online. I must admit I became curious about one thing."
"What's that?"
"The reports said something about RF interference. That it was raised during your cross-examination. What is this RF?"
"Ryan made mincemeat out of me on that," I sighed. "Radio interference. It affected our equipment and damaged the tape."
"And you never discovered the source?"
"We didn't even know it was there until just before the trial."
"Why did your lawyer not make that clear?"
"Well, first of all, he wasn't _my_ lawyer. But to answer the question--" I hesitated. "As a matter of fact, that is a good question. I don't know."
"This is the same lawyer who lost his life."
"You _have_ been keeping up." I paused. "The police say he was the victim of a botched robbery."
"What do you say?"
My gaze slipped from him to David. "I say...well, frankly, I'd rather not have to think about him, or Santoro, or Mary Jo Bosanick again."
Abdul scratched his goatee. "Then it is good that it is over."
***
Our room wasn't as plush as Abdul's, but we weren't there for the decor. I padded over to the bed. My feet sank into deep pile carpeting. I perched on the edge of the mattress and bounced up and down. Perfect.
David smoothed a hand down my hair. I faced him, letting him trace the line of my jaw with his finger. Suddenly, we were full of each other. Hair, skin, smell. His arms wound around me, his mouth settled on mine. I fell back and pulled him on top. Our clothes came off, and our bodies took over.
Afterward, we lay beside each other in the dark. The light from the window threw spiky shadows across the room. David ran his hand down my leg.
"I'm sorry about tonight. But when Abdul called, he wouldn't take no for an answer."
I reached across and took his hand, ran it up my side, and covered it with my own. "It worked out just fine."
"He likes you a lot, you know."
I giggled. "Then I guess you better watch out."
"Why?"
"They're allowed to have more than one wife, aren't they?"
"He makes a move, he'll be one dead sheik."
"Proprietary, aren't we?"
He leaned over to kiss me, then buried his face in my neck. "I'm glad the trial is over," he murmured.
David's father died before he was born, and he'd lost his mother at seven. He'd gone into foster care after that, bouncing from one home to another. Some were good. Some weren't. He didn't talk about it much. But he didn't need to. I knew what he wanted. Stability. Security. Routine. For him it was more than a want. Or a need. It was a prerequisite--the defining quality of his existence.
Some time later, when his even, quiet breathing told me he was asleep, I crept out of bed. Our room faced west, and I stared out the window. Lights twinkled, marking the streets in a repetitive series of grids that stretched to the horizon. It was hard to get lost in this city. You always knew where you were. David liked it that way. I wasn't so sure.
Chapter Twenty-one
"I thought we'd go out for dinner," I said when I picked Rachel up from Science Club. "I have to go to the studio later."
There was no response.
"Want to go to that salad place?"
"With you?"
"Uh--yeah."
She rolled her eyes.
"I take it that's a no."
She leaned forward and snapped on the radio. A loud bass hammered the console, and an angry voice yelled about white sluts and guns. Mercifully, Rachel changed the station, but another rapster, sounding very much like the first, flooded through the speakers. Looking over, I was about to suggest she turn it off when it occurred to me that she'd reprogrammed the buttons in the Volvo. I keep classic rock and NPR at my fingertips; I don't do rap.
What's more, Rachel knew she wasn't supposed to play with the radio without special dispensation. I caught her watching me out of the corner of her eye.
That's when I got it. She'd changed them deliberately and was waiting for my reaction.
I had to make an instant choice, one of those small, perhaps insignificant parental decisions that, nonetheless, fills me with panic. Should I remind her of the rules and reinforce my role as a disciplinarian, which would escalate the conflict between us? Or should I let it slide, thus giving her a degree of power she hadn't yet earned? What was the right choice?
I mulled it over. It was a minor incident. Neither of us would remember it five years from now. But isn't that what parenting is? An aggregate of unimportant decisions that mold a child into an adult? What if I made the wrong choice? Would she resent me for the rest of her life? Would she turn into an ax murderer? I waited for divine inspiration.
"Okay," I said when it didn't come. "How about Italian?" Better to have _shalom bayit_ --peace in the house--at least for today.
She slouched lower in her seat. Her eyes slid to the radio, then narrowed into horizontal slits like they do when she's happy. "Cool."
***
I sank onto the couch in Hank's editing suite, wrapping my jacket around me to ward off the chill. He'd agreed to stay overtime to help me edit a new demo reel for Great Lakes Oil. Styles in video production change, and I wanted to include some clips with an MTV look: quick cuts, strobed action, hot music.
While Hank set up the decks, I studied his collection of frogs, a cheerful jumble of amphibians given to him by clients, including a frog wearing a beret, a toad in a turban, and my contribution, a frog holding a menorah.
He swiveled around, and saw what I was looking at. "Got a new one coming."
"What's that?"
"A frog with chopsticks. Guy's bringing it from Shanghai."
"It was probably made in Japan."
Shrugging, he turned back to the Avid and loaded a CD into the drive.
"Is that my old reel?"
"Yup. I backed it up."
"You're so smart. We'll be out of here in no time."
"It doesn't matter." He sighed. "Not much else is happening."
How could I resist with an opening like that? "And what does that mean, kimosabe?"
"Kimosabe?" He got up and headed down the hall. "You are a dinosaur."
I followed him to the tape library, where all Mac's shows are stored. "I'm donating my bones to the Field Museum."
He grunted as he punched in the code on the wall panel. "What shows do you want to add?"
"How about the most recent one we did for Midwest Mutual--you remember--the one for Claims? And the promo for the Jewish Broadcasting Network. And maybe the opening of Atlantic Wireless."
"No Marian Iverson?"
I shot him a look.
"Hey, we got paid."
"I thought we all agreed the price was too high."
Back in the editing room, he hunched over the keyboard. He set up the Avid for digitizing, then hit the Record button. As video played through the monitor, his shoulders sagged.
"Okay, Hank. What's wrong?"
For a moment, I thought he wasn't going to answer. Then, "There's this girl..."
The light from the monitor cast a pale glow across his face. For some reason, I'd never associated him with a woman before. Not that I thought he was gay. But with his slender build, ponytail, and magic fingers, he seemed almost androgynous. A sprite, too ethereal for the messy emotions the rest of us get mired in. But now, watching him fidget, it occurred to me how blinding the myopia of self-absorption can be.
"Tell me about her."
"She's a musician. Alto sax. I met her at the White Hen. She was buying cereal and milk." He smiled wistfully. "At two in the morning."
"What's her name?"
"Sandy. Sandy Tooley." It rolled off his tongue. "We got together a few times. She was really nice, you know?" His eyes were faraway and unfocused. I knew that look. It's the one that says, _I can still taste her skin, her lips, her body_. "I thought she really liked me. I mean, she acted as if--" He broke off.
"It's okay," I said softly.
He swallowed hard. "Everything was great for a couple of weeks. Better than great. Then I called her the other day--night--when I got off--to tell her I was on my way over. Except she said not to come. She said she had things to do. I wasn't--well--real happy about it. I really wanted to see her, you know?"
"So you went over there anyway."
He didn't answer.
I shaded my eyes. "And when you got there, she was with another man."
"How did you know?"
"I'm sorry."
"She said it was her old boyfriend, and that she would call me later." He took a shaky breath. "That was Monday, Ellie. I haven't heard from her."
Today was Wednesday.
"Maybe they were just talking."
"For three days?"
An hour later we'd finished digitizing the new pieces and cut in the excerpts. We were just winding up when the phone rang. Hank grabbed it. Though I only had a view of his back, I could tell it was Sandy. His spine straightened. His voice grew silky and eager. He ran a hand through his hair.
I ducked out of the room and wandered into Mac's office. It was a comfortable room with two floor-to-ceiling windows that spilled pools of yellow across the dark expanse of lawn. The studio was tucked away on an industrial block in Northbrook. At night, without the bustle from nearby businesses, it was quiet and isolated.
Hank's muffled voice drifted through the air. "He was? You're sure?" I heard a relieved exhalation. Then, in that eager, breathy voice, "Yes. About an hour." A pause. "Me, too." Then, "Don't get dressed." The receiver was replaced with a click.
I strolled back into the editing room. Hank was beaming, his smile so contagious I had to return it.
"She was out of town."
"Get out of here, Hank. We can finish tomorrow."
His smile broadened.
"Out." I pointed to the door.
"Tell you what. Lemme finish this edit, and I'll dub it in the morning."
"Better yet, if you set up the machines, I'll run the dubs myself."
"You don't have to."
"It's okay. I can lock up."
"Well..." Indecision and desire warred on his face. "Mac--"
"Don't worry. I'm sure he trusts me to lock a door."
Desire won. Hank finished the last edit and added black to the tail of the piece. Then he went into a side room to set up the dubs. After checking to see that the VHS machines were in sync, he started them rolling. "Thanks, Ellie. This means a lot."
"Go away before I change my mind."
He grabbed his backpack and bolted. I heard him race down the hall and out the door. Young love.
Seating myself in his chair, I swiveled in front of the bank of monitors. We'd added three new excerpts and deleted three others. As the signals changed from digital impulses to magnetic signals and then to images and sounds, I marveled at the magic of technology.
The reel was less than eight minutes. When it was over, I checked the dubs to make sure an image had indeed been recorded, then rewound and ejected them from the decks. The silence was sudden and deep. Hank had said not to shut down the Avid, so I gathered my bag and the shows we'd pulled.
As I walked back into the tape library, I mentally indexed my clients of the past few years. Midwest Mutual; Seagrave's Food Service; Van Allen, the paper company; Brisco Chemicals. I'd produced shows for them all. The corporate handmaiden.
It hadn't started out that way. I'd graduated college with dreams of becoming the American Lina Wertmuller who also produced substantive documentaries on the side. Seamlessly segueing from the arts to politics in a highly versatile and acclaimed career. Instead, I got married.
I was restacking the Midwest Mutual show on the shelf, thinking how time really does mellow us all, when the door to the library slammed shut. I stood where I was, uncomprehending. Then I realized it had to be Hank. He must have forgotten something.
"Hank?"
I thought I heard footsteps on the other side of the door. "What'd you forget, lover boy?"
No answer. I went to the door, intending to give him _shtick_ about Sandy and how she'd be dressed to the nines if he didn't get over there soon.
I twisted the knob. It didn't move. I tried again. Nothing. "Hank, are you there? The door's locked."
Silence. I heard a squeak. "Hank. Stop screwing around."
I listened again and thought I heard a quiet rustling on the other side. Like paper being shuffled. Then a sharp, pungent smell. Familiar. Almost tangy. I banged my fists against the door.
"Hank. Come on. Something's wrong. Open up."
No one responded. I kept banging until my fist was sore. I pressed my ear against the door. I felt a sensation of warmth. Strange. I hadn't expended that much energy. I leaned my palms against the door. More warmth. I looked down. At the bottom of the door, orange light flickered.
My brain connected. The smell. Like a parking garage! Gasoline!
I broke out in a sweat. Fire! And I was trapped. "Help!" I screamed. "Anyone. Fire! Open up!"
I beat on the door with my palms until they stung. When nothing happened, I threw myself against it, hoping to smash the lock. Pain radiated through my shoulder, but the door held.
The room seemed to have heated up ten degrees. "Help! Please!" I looked wildly around. Wasn't there supposed to be a fire extinguisher in every room? Not here. No windows. No pictures. Not even a nail in the wall. But when I scanned the ceiling, a wave of relief surged through me. A sprinkler. Of course. Water would gush down and extinguish the fire. All I had to do was wait.
I started pacing. I should call the fire department. I automatically looked for my bag, then realized I'd dropped it--and my cell--on the other side of the door. Damn! Meanwhile, crackles replaced the rustles on the other side of the door. The doorknob was too hot to touch. Wisps of black smoke seeped under the door. Didn't I read that most fire fatalities came from smoke inhalation, not flames? I covered my mouth with my hand. Why weren't the damned sprinklers working? Mac would never let fire prevention slip below code, would he? Should I stuff something under the crack in the door?
Another smell, like burning tires, wormed itself into my nose and throat. I tried to remember what I knew about fire. Never open a door if it was hot to the touch; a new source of oxygen would fan the flames. No problem. It was so hot I couldn't open it.
Now thick curls of smoke were rising on my side of the door. The heat pressed against my skin. I was starting to sweat. Where was the water? The only way out was through the door. I might have to break it down to open it. But if I did, I might create a back draft. What should I do? I couldn't wait much longer.
I started to case the library, trying not to feel desperate. But aside from the tapes, the shelves, and the stepladder, which was too heavy to lift, there was nothing in the room. No windows. No furniture. Not even a trash can. I sucked down hot air.
The shelves. They were the do-it-yourself kind that could be disassembled and put together in multiple configurations. Studying them, I got an idea. When they got going, the sprinklers would help douse the fire. If I could somehow use a shelf to break through the door when the water started, I might make it out.
But that required the sprinklers to kick in. I looked up at the ceiling. Sweat dripped down the back of my neck. What was taking them so damn long? The ones in the hall, at least, should have been on by now. My heart sank. Mac probably hadn't updated the system since he first moved in. And that was ten years ago. It was possible they weren't going to work.
Smoke billowed under the doorjamb and started to rise, saturating my clothes and hair. Heat blanketed the room like a shroud. I struggled to take a breath. If the sprinkler didn't start soon, it wouldn't matter. I dropped to the floor to find some breathable air. My stomach leapt to my chest. Flames licked the bottom of the door.
I got up and lunged at the closest shelf. As the tapes on it clattered to the floor, I banged on the underside to dislodge it. But the metal teeth gripped the slots in the frame. Nothing moved. The smoke thickened and moved lower. I coughed. Sweat poured off my forehead. I kept pounding the underside of the shelf.
Finally I was able to maneuver one of the teeth out of its slot. I kept banging; another one popped out. Grabbing the free end, I twisted and jerked. The shelf came free.
It was a bulky, awkward piece of metal, about a yard long, a foot wide, an inch thick. I looked up. Smoke was dimming my eyesight, but the sprinklers were still dry. I was running out of time. I stepped back, holding the shelf like a battering ram. I swung it back to gain momentum, then smashed it into the door. The door shook. Something cracked, but it held. I backed up, clutching the shelf, but a spasm of coughing stopped me. There was too much smoke. The shelf slipped from my hands.
I dropped to the floor and crawled to the other side of the room. But the air over there was just as smoky. I felt woozy. I forced myself to start naming the fifty states. I couldn't give up.
When the water finally streamed down, its force stung my skin and startled me awake. I was lying on the floor dazed and sleepy. The spray drenched me and seemed to dissolve the wall of smoke. I mouthed a prayer of thanks.
Slowly I got to my feet. I picked up the shelf one more time and rammed it into the door. This time the veneer splintered, and a jagged hole appeared. I tore at it with my hands, breaking off slivers of wood. Finally, the hole I'd made was large enough to thrust my arm through. I stripped off my jacket and wrapped a sleeve around my hand. Then I reached through to unlock the door from the other side. Grabbing the shelf, I flung myself into the hall.
Flames danced along the floor and walls, but no fireball engulfed me. The sprinklers were doing their job. Using the shelf as a shield, I staggered through rising steam toward Mac's office. I could make out the dim shape of the windows. I stumbled over to one, drew back the shelf, and rammed it as hard as I could. Glass shattered. An alarm sounded. Using the shelf, I broke off shards of glass that still clung to the frame and crawled through the window.
Chapter Twenty-two
I was still gulping down air when the fire department arrived. My throat felt gritty, I was dizzy, and I was bleeding in two places on my legs. After checking my vitals, the paramedics insisted on taking me to the ER, but I refused. I did let them lead me to the ambulance, where they gave me a wet towel and a bottle of water, and bandaged my cuts. I wiped off some of the soot that covered me and slung my jacket over my shoulders. By the time Mac arrived, the fire been reduced to a residue of sodden debris.
"A shelf?" After being briefed by the battalion chief, Mac came over and grabbed my shoulders. "You broke out of the library with a shelf?"
"Someone locked me in."
"Where was Hank?"
"He left."
"Are you okay?"
I thought about making a crack about being toasted on a stick like a marshmallow, but when I looked at Mac, I changed my mind. Usually a consummate prep, he was wearing wrinkled khakis and a stained T-shirt. That stiff-upper-lip Wasp thing he does had vanished, an expression of fear and relief in its place. I nodded.
"Christ, Ellie. You could have been killed."
I started to shrug, but the movement turned into a shudder, and the shudder into a sob. The tears started, and I sagged against Mac. He held me until it passed.
***
I washed my hair three times, but it still smelled like smoke the next morning. Mac called to tell me the police had picked up Hank and held him for six hours at the station. They let him go around five.
"They can't think he had anything to do--"
"Not anymore." Mac's voice was grim. I got the feeling it hadn't been a fun time. "His girlfriend waited for him. They went back to her place."
Two points for Sandy.
"They _are_ treating it as an arson, aren't they?"
"They won't confirm it, but I overheard the firemen talking about burn patterns and accelerant."
"Have you been back over?"
"The hall's totally gone. So is the Avid. Hank's editing room is in bad shape, too, and the tape library is ruined." He sighed. "And then there's my window."
"Oh, God. I'm so sorry, Mac."
"Yeah, well, I've been meaning to reorganize. But the camera gear is okay. And the other editing room is okay. Once we clean up the smoke damage, we'll be back in business."
Always the optimist. "No ideas who did it?"
"Not that they're telling me."
I cleared my throat. "Mac..." I stopped. Rachel was standing at the kitchen door. "I'll call you later."
I sat her down and told her an abbreviated version of what had happened. She blanched, then jumped up and threw her arms around me. "I want to stay home from school. With you."
"I love you too, sweetie." I hugged her close. "But you can't get out of it that easily."
Somehow I forgot to call my father.
Village Detective Dan O'Malley showed up around nine. With shaggy red hair and freckled skin, he looks almost like a kid, except for his moustache and his height. He's at least six four, and he fills any room he enters. But I'd dealt with him before, and we'd achieved a grudging respect for each other--an accomplishment, considering my attitude toward law enforcement and his toward nosy women. I poured coffee, aware that he was looking me over. I imagined him opening with "A fine kettle of fish we're in now, Ollie."
He sipped his coffee. "How you feeling this morning?" His voice was surprisingly soft for a man of his bulk.
"Like a slab of ribs at an all-you-can-eat barbecue."
"You seem to have a talent for attracting trouble."
"I guess you could look at it that way."
"Why? How do _you_ look at it?"
"The same way I did last night when your officers questioned me. I think it has something to do with Johnnie Santoro."
"The man whose trial you testified at."
I nodded. "His lawyer was killed a few days ago."
"So I hear."
I leaned against the counter. I was certain that the fire was linked to Santoro, Mary Jo, and Calumet Park. First Rhonda Disapio dies in an "accident." Then Brashares in a robbery gone bad. Now someone was trying to turn me into a crispy critter.
The problem was I couldn't prove it. I couldn't provide _any_ evidence. And with nothing to back up my suspicions, the cops last night didn't take me seriously. But, then, why should they? I'd been put in my place at the trial. Hammered by a rising star in state law enforcement.
As if reading my mind, O'Malley looked over. "If there's something you want to tell me, now would be a good time."
I hesitated, then ran him through the events since the trial, including what I'd learned from Rhonda and Sweeney. "Bottom line: I think Santoro was working a deal, and Mary Jo was his mule or his courier or something."
"Drugs?"
I nodded. "It fell apart, they panicked, and Mary Jo was killed."
"They?"
"Before she died, Rhonda Disapio told me two guys showed up at the boat launch at Calumet Park. She said they killed Mary Jo."
"Why didn't she say that at trial?"
"She was scared. They tried to come after her, after they got Mary Jo, but she got away. She didn't want to take any chances."
"I don't know." O'Malley shook his head. "Sounds weak."
"Not if they were mixed up with the Mob."
"Who?"
"The guys at Calumet Park. Santoro, too. He might even have ended up taking the fall for them."
O'Malley brushed a finger across his mustache. "You have any proof?"
"It depends on your definition." I told him that Santoro was a longshoreman but wasn't well liked. And that he'd told Sweeney before the murder that he was onto something big.
"Like I said, do you have any proof?"
"Well, Rhonda Disapio did die in that 'accident.'"
"After she told you about the men at Calumet Park."
"And a few days later, Brashares was killed."
"And you think it's all connected."
"Brashares could have known the men who killed Mary Jo. Maybe they pressured him to make sure Santoro took the fall. But maybe he had second thoughts. Maybe he threatened to blow it wide open and they had to shut him up."
"Got it all figured out, huh?"
"Just coming up with possibilities."
"And now you think the Mob's behind this alleged arson. That there's some kind of conspiracy--I don't know--to silence you."
"It is possible, isn't it?"
"But why? Why would they be coming after you?"
I bit my lip. "Because I figured it out?" I said.
He shook his head. "Ellie, how would they know? It's not like you've been broadcasting it on the news."
He had a point.
"Tell me," he said. "What evidence can you provide that would help me find out?"
I didn't answer.
He tapped a finger on his cup. "Aside from this Santoro business, is there anyone else you can think of--besides the Mafia--who'd want to do you harm?"
I wouldn't meet his eyes. "Not at the moment."
"I see."
The most I could get out of him was a promise to call the detectives assigned to Brashares' murder.
A young investigator from the fire department showed up after O'Malley left. He ran through what I gathered was a required checklist. He asked where I'd been when the fire first appeared; what I saw, heard, and smelled. He asked about the color of the smoke and flames, and whether I heard an explosion. He pulled out a sketch of the studio's floor plan and asked me to retrace my steps from the time we finished the dubs until I crawled through the window. He left a few minutes later, a satisfied look on his face.
I'm glad someone was satisfied. I felt like I'd spent a hundred dollars at the grocery store and come home with nothing.
***
When Rachel and I got back from school that afternoon, Fouad was tramping across the lawn, waving a leaf blower. He turned it off when he saw us.
"I heard about the fire on the radio." He looked worried.
"News travels fast." I skirted the piles of leaves he'd collected.
"You are not hurt?"
I shook my head.
"That is good." His eyes fastened on something behind me.
I turned to see Rachel with a worried expression of her own. "Aren't you coming in, Mom?" She pulled on the straps of her backpack.
"I want to talk to Fouad for a minute. You could start practicing the piano."
"You'll just be a minute, right?"
"You bet." I brushed a curl off her forehead. "You can watch me through the window." She nodded and went inside.
"What is going on, Ellie?"
I turned around. "I think someone is trying to kill me."
Fouad moved here from Syria over thirty years ago, knowing his appearance, accent, and customs would always mark him as an outsider. That he would never be treated with the back-slapping heartiness white America reserves for itself. Yet this outsider had risked his life for me. There weren't many people I trusted more.
His eyes narrowed. "Who?"
"I don't know. I don't know anything--except that it began with Santoro."
I took him through the chronology. When I finished, he took the leaf blower off his shoulder. He doesn't dwell on it, but Fouad knows about the dark, evil underbelly of human nature.
"Why do you think it's the Mafia?"
"Because whoever is behind this doesn't want something exposed, and they're using a lot of resources to make sure it isn't. I don't know many other organizations with that kind of clout."
We walked back to his pickup where he put down the leaf blower. "But why are they after you?"
"I--I'm not entirely sure. I did meet with Rhonda Disapio before she died. She was the one who told me about the two men. She thought she was being followed. Maybe they saw us together."
He pulled out a rake from the truck. "But this has been the only incident directed against you? Since the trial?"
I thought about the SUV I'd seen when Susan and I took a walk. You couldn't really call that an "incident." I wasn't even sure it was significant. "There was nothing," I said, "until Brashares died."
"And he died--they broke into his office and attacked him."
"Tossed the place and cleaned out his safe."
Fouad was quiet as he raked the separate mounds of leaves together into one large pile. Then he looked up. "Perhaps there was something in his office that connected them to you."
"In his office?" I kicked a few leaves and watched them swirl in the air before settling. I hardly knew Brashares. I'd only been in his office once. In fact, since the trial, we'd only talked once or twice. Most of our communication was on answering machines. Playing phone tag.
The phone.
I looked up.
"What?" Fouad asked.
"I left a message on Brashares' machine."
Fouad's jaw tightened.
"I said something about Santoro and the men at Calumet Park. I hugged my chest. "Do you think that's it? I mean, if they were following Rhonda, they already suspected I knew something. And then, when they heard the message..."
"What are you saying?"
"I'm saying that the people who broke into Brashares' office might have listened to the messages on his answering machine. And heard the one where I mentioned the 'men at Calumet Park.' That could be the link." The temperature was in the fifties, but my palms were sweaty. "Oh God. Me and my big mouth."
Fouad tried to comfort me. "The Koran says, 'Allah does not impose upon any soul a duty but to the extent of its ability.' You were only doing what you thought you should."
"Even so, it backfired." I chewed on a finger. "Fouad, what do I do? The police don't believe any of this."
"Then you must convince them."
The plink of piano chords floated through the window. "How? I don't have any evidence."
He smiled. "You will find it; I am certain of that."
I wasn't quite sure how to take that, but coming from Fouad, it had to be a compliment. He bundled the leaves into a canvas tarp, tied the ends, and carried it to the back of his truck.
I followed him. "Oh. I almost forgot. I met someone from your part of the world the other day."
He looked over.
"A new client of David's. A Saudi oil sheik. He says he's related to the royal family."
"What's his name?"
"Abdul Al Hamarani. He's trying to buy a plant from Great Lakes Oil."
"There are thousands of royals in Saudi Arabia," he said. I must have looked crestfallen because he added, "I have a friend from Riyadh. I'll ask about him when I see him at prayers."
I went into the kitchen to think about dinner.
Rachel called from the living room, "Next week is the end of Science Club, you know."
"Already?" Where had the time gone?
"Well, the first session. They're having Parents' Day on Friday. Are you coming?"
I missed a lot when Rachel was young. Swimming lessons. Soccer games. Her violin recital. I remember thinking they couldn't possibly be as important as my work. After the divorce, my priorities changed. Now I try not to miss anything.
I went into the living room. "Of course I'm coming. Why? What's up?"
"It's a surprise." She grinned. "But you'll like it."
I swatted her on her rear end. "Tease."
***
O'Malley got back to me that night. "I called down to Area Three and talked to the dicks handling Brashares' case."
"And?"
"They're sticking with the program."
"A botched robbery?"
"They say he was in the wrong place at the wrong time."
"Convenient, isn't it?"
"Ellie." O'Malley cleared his throat. "I know you had problems last summer. But lightning doesn't strike twice. Unless you can give me something, there's nothing I can do. Christ, I wouldn't know where to start anyway. Your story covers almost every friggin' police jurisdiction in Cook County."
"It's not a story."
"Well, it isn't a case." He paused. "Look, you know how it works. Give me something I can work with. Otherwise, all I got is a suspicious fire. Which could have been set by anyone."
I thought about the message I'd left on Brashares' machine. That wasn't evidence, either. At best, it was conjecture. But it was obvious O'Malley wasn't eager to take me on.
I kept my mouth shut.
Chapter Twenty-three
I'd always considered our village a quiet place where nothing much happens, until I discovered some history no one talks about much. Apparently, there used to be a bar behind the train station. It was a popular watering hole, especially on Fridays, when the owner took it upon himself to cash his customers' paychecks. Except for one Friday morning around three, when four masked men robbed the place at gunpoint and escaped with fifty thousand dollars. The community was shocked. Shocked. What sort of people would keep that much cash on the premises? It came out later that the owner was running a "finance and loan" business on the side.
The bar is now gone, but the owner's family isn't. Specifically, Joey DePalma, aka the Surgeon, and his brothers. They were part of the old Grand Avenue crew but moved to the suburbs in the Sixties. His brothers didn't stay; their bodies were found in a Wisconsin field a couple of years later. DePalma made a precipitous retirement after that.
I once asked O'Malley why they called him the Surgeon. He said DePalma was known for his skill with a knife. But that was a long time ago, he added. DePalma led a quiet life now, enjoying his grandchildren and garden. And the careful scrutiny of village cops.
The next morning found me driving down a residential street a mile from my house. Some of the homes, products of remodeling, were upscale two-story structures, but most were modest splits and ranches. Midway down the block was a brown brick ranch with a cedar shake roof and a well-tended lawn. I was surprised how well the house blended in; I'd expected something showier.
I climbed out of the Volvo and made my way to the front porch. The screen door had one of those fancy _D_ s in the middle. I was about to ring the bell when I stopped. What was I doing? You don't just drop in on a mobster for tea. I started back to the car.
"Can I help you?"
I spun around. Coming around the side of the house was a man pushing a wheelbarrow. He appeared to be in his seventies and had a big belly that spilled over baggy pants, but his shirt revealed brawny arms and shoulders. He wore thick black glasses, and his skin looked as if he'd had a bad case of acne as a youth. Oddly enough, that made him seem more approachable.
I pasted on what I hoped was a sincere smile. "I was just--I was just admiring your garden."
He flicked his eyes toward his flower beds, which, since the frost last week, consisted of dead marigolds, withered salvia, and a few scraggly petunias.
"I mean, over the summer," I stammered. "It must have been gorgeous."
He looked me up and down, then picked up the wheelbarrow handles. "If you're selling something, we're not interested."
"I'm not selling anything," I said. "My name is Ellie Foreman. I live in the neighborhood."
He paused, then straightened up and motioned toward the house. "My wife Lenora handles the charity donations."
I turned around. A soft, round woman was watching us from the door. She wore beige stretch pants and a long, flowery tunic, and her hair was tinted a brassy red. She was wearing glasses, too. Oversized, with blue frames.
"I'm not here for money, sir." I took a breath. "The truth is--I need your help."
He gave me another once-over. "You say you live around here?"
"A few blocks away."
After a long moment, he beckoned me to follow him and went inside the house. As he brushed by his wife, he said, "Go into the kitchen, Lenora."
She disappeared without a word.
I followed him in. To the left was the narrow hallway Lenora had just passed through, to my right a sunken living room. The carpeting was beige, the furniture, too. A crucifix hung over the fireplace. End tables were crowded with photographs of small children and young parents, most with sunny smiles on their faces. But the hall we were standing in was gloomy, and the open door hadn't filtered out that musty smell that clings to old people's homes.
"What's the problem, miss?"
"I think someone may be trying to kill me. But I'm not sure who it is or why they're doing it." I felt the trepidation in my voice. "I--I'm afraid, and I want it to stop. I don't know who to turn to."
His eyebrows arranged themselves into an annoyed expression. "I'm a retired senior citizen, living on a pension. Whaddaya think I can do?"
I swallowed. "I think it might have something to do with my testimony at Johnnie Santoro's trial."
His expression didn't change.
"Somebody thinks I know something. But I don't know who it is or what I'm supposed to know. I'm a single mother. I have a daughter." I looked over at the photographs. "I'm all she has. Sir," I added.
DePalma looked me over for what seemed like a long time, though it probably was only a few seconds. Then, "You have a problem, get in touch with my lawyer. William Casey. At Brickman, Casey, and Scott. He'll help you."
"Mr. DePalma, with all due respect, your lawyer can't help me, and I think you know it."
"Young lady, like I said, I'm just a retiree on a pension. I can't help you." He took a step forward. "And now, you're gonna have to leave."
My stomach twisted. "Please, Mr. DePalma. I was almost killed in a fire the other day. It was arson. But the police don't know who's behind it. And it doesn't seem like they want to find out."
He paused. "Where was the fire?"
"In Northbrook. At a video studio."
He pulled out a clean, white handkerchief from his pocket. I took it as a hopeful sign.
"I thought perhaps you'd consider looking into it, and... well maybe..." My voice trailed off.
He held the handkerchief to his face, blew his nose, and tucked it back in his pocket. Then he placed his hand on my arm. The back of his hand was covered with dark hair, and his fingers were thick and stubby. I could see that hand wielding a knife.
"Ms. Foreman, it's time for you to go."
The Surgeon guided me out. The door closed quietly behind me.
***
David was en route to London and wasn't reachable until evening. Because of the time difference, I woke him up. When I told him about the fire, the drowsiness in his voice vanished.
"My God, Ellie. I'll fly back tomorrow."
"Don't. I'm fine."
"Are you sure?"
My throat was scratchy, and I still thought I smelled smoke everywhere, but he didn't have to know that. "I'm sure."
Silence. Then, "What about Rachel?"
"She's fine. Katie's sleeping over. It's Halloween."
When I was young, Halloween was my favorite holiday. Not anymore. Grisly costumes and nasty pranks have stripped the holiday of all its charm. I can't understand people who spend hundreds of dollars to celebrate the macabre.
Happily, Rachel was too old for trick-or-treating, but one of her friends was having a party the following night, and the girls were trying on every article of clothing in Rachel's closet in an effort to pull together some costumes.
"...isn't good, Ellie."
I realized I hadn't been listening. "I'm sorry, David. What was that?"
"I said this isn't good."
"I know. But, at least no one was hurt, and--"
"No," David cut in. "Not that."
I stared at the broom closet. The door was half open. Maybe I should close it all the way. "What?"
He paused. "I'm concerned you might be in danger. I want to be with you."
"I don't need your protection." I bit my tongue. "I'm sorry. I didn't mean that."
"Maybe you did."
"David--"
"Look, I know you can take care of yourself. But when you care for someone, at least when I do, I want to make sure they're safe."
"Please don't take this the wrong way, but don't you think you might be projecting a wee bit?"
"What if I am?" His voice rose. "I'll admit I'm not a big risk taker." He paused. "The biggest risk I ever took was falling in love with you."
I swallowed.
"I didn't say that to make you feel guilty. I--I guess--I just wish none of this had happened."
"Do you think I should have kept my mouth shut? Even though I didn't--and still don't--believe Santoro killed Mary Jo?"
I heard his sigh over seven thousand miles. "No, of course not. But there isn't a day that passes that I'm not afraid for you."
I cleared my throat. This probably wasn't the time to tell him about DePalma.
"Ellie, I don't want this to sound like an ultimatum. But I'd like to suggest we both do some thinking."
I gripped the phone. "About what?"
"About us and how we can make this work. We're such different people."
"I thought that's why you were attracted to me. You know, action woman meets pensive man."
I heard a strangled sound on his end. I closed the door to the broom closet.
"Why don't we both do some thinking?" he said after a pause. "I'll call you next week."
"David?"
"What?" I heard the rise of his breath.
I bit back a reply. "Nothing. I--I'll talk to you later."
I hung up and started to load the dishwasher, but as I transferred a plate to the machine, it fell to the floor and shattered.
"Dammit." I kicked the cabinet under the sink. "Shit." My toes throbbed.
By the time I swept up the pieces, it was dark outside, a thick, heavy darkness in which objects lose definition. I put the broom away. What was I doing? Why was I picking fights with David? He was right about one thing. The bubble wrap of suburban life was no guarantee of safety. But his response was to avoid risk. Insulate himself with routine. Mine was to rush the front lines wielding a saber, refusing to cave in to fear.
It was a problem.
Chapter Twenty-four
Mac called with good news on Monday. The insurance company would pay for most of the cleanup, plus the equipment replacement. In fact, Mac had decided to upgrade to a better Avid. Faster chips, better processor, and the ability to author to DVD.
"So everything in the tape room is gone?" I asked.
"I'm sorry, Ellie. I know a lot of your shows were in there."
"No. I'm the one who should apologize."
"Why?"
"I may have been the target."
"Yeah. The police asked me about that."
"What did you tell them?"
He didn't say anything, but I heard an entire conversation in his silence.
"Mac?"
"Look, Ellie," he said. "I have a lot of rebuilding to do. I have a family to support. If someone was sending us a message, I heard it loud and clear. I don't want to get involved."
"So you're--"
"I don't know who was responsible for the fire. Or why. And I don't want to. I just want it all to go away."
"So I'm out there on my own."
"You don't have to be."
I changed the subject. "How's Hank? I've tried calling, but we keep trading messages. Has he recovered?"
Mac laughed. "Let me put it this way. If this is how he reacts to stress, I'll have to pile more on."
Sandy must be something.
I hung up and looked outside. The sky was that heartbreaking shade of blue you only see in autumn. I called Susan to walk, but she works part-time in an art gallery and had already left the house. I threw on sneakers and sweats, feeling resentful toward everyone who had a place to go and a job to do.
I stretched and jogged over to Voltz Road. Twisting and turning through the forest preserve, Voltz has no sidewalks, just narrow shoulders of gravel. Huge trees shield the estates on either side. But the canopy of leaves now looked ragged, and leaves crunched under my feet.
Two years ago Rachel and I were driving down Voltz when we saw a fawn lying in the middle of the road. As we drew closer, it started to make small jerky motions, and I saw blood seeping out from under it. The creature's legs and back were broken, and it couldn't roll over, much less stand up.
We stopped and called the cops. Then I carefully picked it up and carried it to the side of road. As I lowered it into a ditch, the fawn's large, obsidian eyes locked onto mine, and I was sure I saw an awareness in them that this was not the right order of things. That something very fundamental in its young universe had irrevocably shifted.
When the cop showed up, he inspected the fawn and said, "You know what I have to do."
"No," Rachel cried, clutching my arm.
I pulled her to me and spoke over her head. "Could you--would you wait until we leave?"
I led my sobbing daughter to the car and drove away. A shot rang out. Neither of us looked back.
Now, as I jogged past the ditch, caught up in the ache of the memory, I barely registered the black limo that eased past me. But limos weren't unusual in this neighborhood. Neither were SUVs, and I didn't make any connections when a dark green one passed behind the limo, followed, seconds later, by a gray sedan.
A moment later, I stopped. An animal, about fifteen inches long, was blocking my path. Clumps of straw-colored fur stuck out from scaly, pink skin. The creature had beady eyes, tiny floppy ears, and something that could have been a tail. I stared, not quite sure what I was looking at: a skinned rat, an albino raccoon, some mutant forest being. Then it trotted toward me on squat little legs.
A dog.
"Spike." A male voice cut through the air. "Get over here."
The animal hesitated, as if it were weighing whether to obey the command, but before it could decide, a man stepped out from the bushes. He was short, wearing a fancy sport shirt, slacks, and Italian loafers. Gray streaks ran through dark hair. He was holding a white leather leash studded with colored rhinestones that flashed in the sun. Dark glasses covered his eyes.
As we faced off, a fierce squeal came from the dog. I took a startled step back. The dog lifted his leg against a bush to mark his territory. When he was finished, his ears pricked up, and he barked again, though it wasn't clear at what. The man bent over and scooped up the creature with one hand. He flipped up his glasses with the other.
"Sorry." Cool eyes gazed at me. "Spike ain't having a good day."
"What's wrong with him?"
The man shrugged. "The vet don't know. Cushing's disease, he says. Maybe hypothyroid. Who knows? The little shit's on six different medicines. Costs me a fuckin' fortune."
"Oh." I started to resume my run. The man blocked my path.
"Hold on there, Miss Foreman."
I froze.
I briefly considered making a break for it. I might be able to put some distance between us--he was weighed down with the dog. But when a beefy man who looked like a linebacker suddenly materialized at the next corner, I changed my mind.
"The car's around the corner." Hitching his thumb, the man with the dog smiled. He had crooked but very white teeth. "How 'bout we go for a ride."
A cloud moved across the sun, not darkening as much as leaching the color out of things. I wiped damp hands on my sweats. The goon came forward and, clamping a firm hand on my arm, led me around the corner. He opened the door to the limo, unfolded the jump seat, and directed me in. Spike and the man climbed in the back.
We drove past the bridge at the west end of Voltz and headed north on Waukegan Road. Spike curled up on a blue blanket that lay on the backseat. At close range, I could see the dog's skin was flaking off. Large patches settled on the blanket, and his body gave off a sour smell.
"It started about ten months ago." The man ran his hand down Spike's back. "I thought it was cancer, but they keep tellin' me it ain't. Turns out Maltese get these weird diseases."
We passed the Park District pool Rachel lived at most of last summer.
"Just drive around for a while, Vinny."
"Sure thing." Vinny drove a few miles per hour under the limit.
"Who are you?" I asked.
He ignored the question. "You got a dog?"
Rachel had pestered me for one, but I didn't give in. I saw _Old Yeller_. I know what happens when you let a dog into your life. I shook my head.
"Maybe you should. Keep you from bothering nice old men."
DePalma.
We passed a church with a billboard advertising their Friday night fish fry and bingo. The man picked up Spike, letting the dog lick his cheek. "You can't go up to people and ask them things like you done. These are quality people. They need peace and quiet."
"I didn't know what else to do."
"Someone torched a place you were in, huh?"
DePalma had briefed him. Or else he already knew.
Spike settled down in the man's lap, sinking his head on his paws. "Why don't you tell me about it."
I let out a nervous giggle. The fact that I was in the back of a limo, expected to spill my guts to a wise guy was pretty far out, even for me. "Do you know Johnnie Santoro?"
"The one who offed his girlfriend at Calumet Park."
My pulse sped up. He knew. "I was just wondering. Was he--"
"How 'bout you lemme ask the questions."
We crossed into Lake County where Waukegan Road slows. They say it's because of construction, but it's been that way for years. The cost overruns had to be lining a few pockets quite nicely by now. We inched forward, surrounded by cars, delivery trucks, and a yellow school bus filled with children.
"You had that videotape of Santoro. You testified at his trial."
"Yes."
"What makes you think I know him?"
I took a breath and launched into an explanation. I told him what I'd heard about Santoro's background, the men at the park, and Rhonda Disapio and Brashares' deaths. The fire. But as the words spilled out, the series of events I'd strung together sounded flimsier and more elusive in the retelling than they had in my mind. Not as conspiratorial. Possibly even coincidental. I felt foolish, and I could tell from my companion's expression, which changed from guarded to puzzled to exasperated, that he agreed.
"That ain't a lot to go on," he said.
I looked out the rear window. The dark-colored SUV was in back of us this time. I stiffened. "Are we being followed?"
"Of course." He waved a hand. "The most elite crime-fighting organization in the world checks in every day."
"The FBI?"
"You got it." He twisted around and saluted through the window. "They got these new mikes can pick up anything they point 'em at." The SUV dropped back and switched lanes. A few car lengths back was the gray sedan. It made for an odd procession. He twisted around. "Vinny, you can head back now."
"Yeah, boss."
We turned off Waukegan and started east. As if sensing the change in direction, Spike raised his head and sniffed the air.
"Listen." The man paused. "There ain't nothing there. This Santoro--he ain't connected. He ain't a friend. He ain't even a friend of a friend."
I shifted uncomfortably. "But I thought--"
"You thought wrong."
"I know I don't have what you'd call hard evidence. And I probably haven't done a very good job explaining it. But three people are dead, and I almost died in a fire. Someone's doing something."
His mouth tightened, as if he was losing patience with me. "Look, lady, I don't know who or what's causing your problems. I don't know who torched the place. And you know something? You're probably better off not knowing." He allowed his words to sink in. "I'll tell you what I think. I think you should get out of town for a while. Go on vacation. A nice, long one. It'll do wonders. You'll have a whole new attitude."
We turned south on Skokie Highway. Spike yawned and licked his hairless paws. The man picked him up and nuzzled his neck, oblivious to the flakes of skin that drifted down onto his pants. As we veered off on Sunset Ridge, I squinted against the glare of the sun. When we reached Voltz, we were a block closer to my house than where he'd picked me up.
"Vinny, let her out," he said. The car slowed. "By the way, this conversation never took place."
"How could it?" I said. "I don't know who you are."
He nodded. "Good way to keep it." He motioned for me to open the door. I slid over and climbed out. He leaned to the side to close it, but I kept my hand on the handle.
"I hope it works out for Spike."
He gave me a curt nod.
***
I trudged back to the house. The sidewalk was flecked with chips that glinted like diamonds. DePalma's pal could be telling the truth; then again, he might not. But then, why come all this way to tell me the Mob wasn't involved with Santoro? I kicked a stone on the path. Dad was right. I was building a house of cards with my theories about drug scams and double-dealing. I'd made a fool of myself. That wasn't anything new, of course, but the fact remained that nothing made sense. If the Mob wasn't involved, if there was no drug deal, why was Mary Jo Bosanick dead? And Rhonda? And Brashares?
I picked up the stone and rolled it between my palms. I supposed I could visit Santoro in jail and ask him point blank, but I shuddered at the thought of going down to Cook County alone. What was I going to say to him? Are you or have you ever been involved in dealing? If he was as big a jerk as Sweeney said, my odds of getting a straight answer were low. And if they framed him as well as I thought they did, he was just as much a victim as me. Or Mary Jo. Or Rhonda. No, _genug iz genug_. Enough is enough.
I looked both ways, then hurled the stone as far as I could. There's never an open declaration that a conspiracy exists. They unfold gradually and subtly, through events that, if they're connected at all, are often considered coincidental. Odd, perhaps, but not necessarily malevolent. It's only in the revelation and retelling that one sees the intent, the planning, the duplicity.
It's like a tree falling in the forest, I thought. If no one reveals it, is there still a conspiracy? Maybe Rhonda did die in a tragic accident, Brashares in a robbery. Maybe the fire was some type of insurance-related arson.
And maybe pigs can fly.
I called David at his hotel that night. It was about four A.M. London time, but he didn't pick up. I turned out the light and stared at the wall in the dark.
Chapter Twenty-five
The chalk squeaked as the handsome young man wrote on the blackboard, adding the words "baby monitor" to a list that included
garage door openers
alarm systems
cell phones
walkie-talkies
satellites
GPS systems
television
"Very good."
I crowded into the back row with the other parents, mothers mostly, all of us feeling the awkwardness of being in close quarters with strangers. The teacher turned around. Thick eyebrows and a pronounced chin made him look a little like George Clooney, and when he smiled, I realized why I hadn't heard any complaints about Science Club.
"You can see--just from this list--how many things depend on radio waves. Everyday items as well as the exotic." He picked up a chart and propped it against the blackboard. Across the top of the chart was a band of colors, identified by initials, such as VLF, LF, and EHF. Underneath were terms such as AM Broadcast, FM Broadcast, and Radar. At the bottom were numbers: 10kHz, 1MHz, 100 MHz, and 10 GHz.
"These are some of the frequencies of radio waves, and that's what your children have been studying. Radio waves are just one part of the electromagnetic spectrum, which also includes visible light, infrared, X-rays, gamma rays, and other forms of electromagnetic energy."
I tried not to let my eyes glaze over. One of my biggest regrets is not having a better grounding in science. I thought I understood the concepts, but whenever I tried to apply them, I usually got it wrong. Apparently, speed isn't the same thing as velocity, and acceleration doesn't always mean speeding up. I had the poor grades to prove it.
It might have been my teachers. In high school, my science teacher was an Indian woman who wore beautiful saris but whose accent was so thick I could barely understand her. And in college, the TA taught us how to handicap racing forms but not much about physics.
Rachel, though, seemed to have lucked out.
"So. Bearing in mind that we were studying radio..." the teacher continued, "...we decided to build one."
"I didn't know that," one of the mothers whispered to another.
The second woman mouthed back, "Me neither."
"Don't worry," he added hastily. "I asked them to keep it a surprise. In case it didn't work."
The kids giggled, and the adults traded amused glances. "I'm Brian Matson, by the way. But this is really your kids' class. They're anxious to show you what they did."
Several kids rose from their desks. A young boy introduced us to transistors, capacitors, and inductors. Another explained oscillating sine waves. Rachel got up to explain what a diode was and how it worked in a receiver. As I watched, I felt pride that this knowledgeable, confident young woman was my daughter.
The kids turned on a small receiver sitting on one of the desks and left the room. A few seconds later, we heard a click and some static from speakers on both sides of the blackboard.
"Afternoon folks, and welcome to WSCS. That's WSCS, the Science Club Sensation. I'm Paul in the P.M., and I'm here to play your favorite tunes."
We whistled, cheered, and clapped enthusiastically. The faces of the kids still in the classroom lit up. Two songs pulsed through the speakers: "Somebody to Love," which, knowing my hero worship of Grace Slick, had to be Rachel's doing, and something by U2. A girl handed out apple juice and cookies. When the songs were over, Paul signed off, and the kids returned to the classroom.
"That was an ultra-low power AM radio station," Brian said. "Small. But real." We clapped again. "Thank you. It's been a great session. Your kids were terrific."
The kids broke out in cheers of their own. I heard snippets of "Way cool," "Best class I've ever had," "Awesome dude."
Brian waved an embarrassed hand. "Since we still have some time left, I thought you might want to take a look at other types of radio systems. I'm a ham radio operator, and I brought in some of my gear. I'd be happy to answer any questions."
I munched on a cookie as we milled around, thinking back to my first job in college, tearing copy at an all-news radio station. It wasn't glamorous, the pay was rotten, but I got hooked. Not just on the rush of breaking news, but also the thrill of shaping, in some tiny, insignificant way, the presentation of history.
The following semester, I signed up for a course on Edward R. Murrow, where I listened to his rooftop broadcasts from London during the blitz. By the time I saw his documentary on McCarthy, I knew what I wanted to do with my life, and I changed my major to film.
But that didn't mean I knew anything about the technology of broadcasting. Happily, Rachel now did. Maybe she would become a techno-geek. Maybe we would go into business together: she the technical expert, me the content provider. Foreman and Daughter. It had a ring.
"What's that?" I pointed to an object about the size of a cell phone that was plugged into a small black box with knobs on the front. A computer sat beside it.
Brian smiled. "That's called a handie-talkie."
"Is that like a walkie-talkie?"
"Well, it is both a transmitter and receiver. Like a walkie-talkie. But this is much more sophisticated. It's part of a packet radio setup."
"A what?"
"An amateur radio system. You know, ham radio. But this is digital. Packet combines radio and computers. The little box you're looking at can transmit and receive both voice and data."
"No way. That little thing?"
"Well, you need an antenna and a computer," he said. "And there needs to be an unobstructed view--a direct line of sight--between the transmitter and receiver. You also need this piece of equipment." He ran a hand over the black box with knobs on it. Now that I was closer, I could see a needle on a dial swinging back and forth.
"That looks like a VU meter," I said. "You know. It measures audio levels when you're working with video."
"This is a TNC. A terminal node controller."
"What does it do?"
"It's kind of the brains of the packet system. It's the interface between the computer and the handie-talkie. The chips inside have all the functions the equipment needs."
"Are you saying that if you have this, and a computer, and that handie-talkie, you can run a radio station?"
"Absolutely. Like I said, you also need an antenna, but you can put one of those almost anywhere these days."
"So, tell me something. If we already have the computer, how much would the rest of this cost?"
He laughed. "It's not as bad as you think. You could probably get a basic setup for a few hundred." He glanced over at Rachel. "Are we thinking Christmas?"
"Possibly Chanukah."
He was about to reply, when one of the other mothers buttonholed him and asked whether her son ought to apply to MIT.
***
Back home, I turned on the news and boiled water for pasta. The trial of a suspected terrorist had begun on the East Coast, and, once again, they were replaying the video of the Twin Towers. I never thought I'd become inured to those images, but mindless repetition by the media had almost stripped away the horror. I snapped off the tube and turned on the radio. As Smoky warbled about the tracks of his tears, I tried to pretend I was fixing dinner for William Hurt and Kevin Kline.
Outside the fading light turned everything into shades of granite, but a bunch of kids were still playing outdoors, determined to keep dusk at bay. One of them booted a ball past my house. Two boys scrambled after it, but it rolled under a gray car that was parked a few houses away. I was watching them retrieve it when I noticed two figures inside the car. A twinge went through me. How long had they been there?
As if hearing my thoughts, the men opened their doors and climbed out. The one who'd been in the driver's seat had steel gray hair and a mustache. He wore a suit jacket, the material tight across the back. With his thick neck and barrel chest, he could have been an ex-boxer. The other man was lean and younger. He had on jeans, a blue T-shirt, and a billed cap.
The older man bent down under the car, got the ball, and tossed it back to one of the kids. The boy caught it and stared up at the man. The man smiled and gave the kid a thumbs-up. The kid ducked his head and ran back to his game.
The older man cut across the grass to my house while the man in the billed cap strolled up the driveway. The doorbell rang.
I opened the door cautiously.
"Miss Foreman? Jerry Coates, FBI. We'd like a few minutes of your time."
"Why?"
"We'd like to talk to you."
"Could I see some identification?"
The older man showed me a black leather billfold engraved with a gold shield on one side. When he flipped it open, a grainy color photo identified him as Special Agent Jerome Coates. The letters FBI were stamped across the photo.
The second man held up his ID. "Special Agent Nick LeJeune."
His hair was shorter in the picture, and he was wearing a suit, but it was the same guy. I studied him, noticing the crow's feet around his eyes, the light stubble on his chin. He tugged the brim of his cap, which was emblazoned with "Different Drummer Fishing Charter" in white letters.
I led the way into the family room and sat primly on the sofa. Coates sat on a chair. LeJeune settled on the other end of the couch.
Coates began. "Your legal name is Goldman, isn't it?"
"That's my ex-husband's name. And my daughter's. I changed mine back to Foreman when we divorced."
"And you've been living here for ten years?"
"That's right. Last August."
"And your daughter is thirteen?"
"Yes."
Coates took out a memo pad and made a note. LeJeune ran his hands over the nubby beige material of the couch.
"So, you want to tell me why you were riding around in Dominick Morelli's limo the other day?" Coates asked.
My jaw went slack. "That was Dom Morelli?"
Dominick Morelli was one of the leading figures in the Chicago Outfit, reputed to be involved in gambling, juice loans, labor racketeering, and most recently an aggressive--but thus far unsuccessful--effort to open a casino in the suburbs.
"He never told me his name."
Coates' expression said he didn't believe me.
I flashed back to the man who'd stroked Spike so lovingly. "He didn't identify himself, and I didn't think I should press it, you know?"
LeJeune covered his mouth with his hand. Was he stifling a smile?
Coates' frown deepened. "You always go for limo rides with strangers?"
"He wasn't a stranger. I mean, it was obvious he was someone. He knew my name. But I didn't think it was my place--"
"So you did know him."
If they were following him as closely as Morelli claimed and had that mike he was talking about, they already knew everything we said. I leaned back against the couch. "How come you're not in the SUV today?"
The two men exchanged glances.
"I mean, it's definitely more North Shore than a gray--a gray--"
"Plymouth," LeJeune cut in. I thought I heard a slight lilt in his voice. Southern. But soft.
"Right. Plymouth. Hey, how long have you been following me anyway?" Coates looked confused. "I wish you'd identified yourselves sooner. I was really scared. Rhonda Disapio was, too. In fact--"
"Mrs. Goldman--"
"Miss Foreman."
"Miss Foreman." He scowled. "What the hell are you talking about?"
I looked from one to the other. Then it clicked. "You were following _Morelli_ , not me."
They exchanged another glance.
My stomach pitched. "In the Plymouth."
Coates nodded. LeJeune was eyeing me.
"Then who was in the SUV?"
"Why don't you let me ask the questions, Miss Foreman?" Coates said.
That was the second time in a week that a man had said that to me.
I studied the men more closely. They didn't seem like partners. Or even particularly close. There didn't seem to be much awareness of each other's rhythms, none of the shared patterns couples acquire when they've worked together over time.
LeJeune seemed more attuned to his environment, his gaze processing the posters on the walls, the crowded bookshelves, my mother's silver bowl. I hoped he didn't notice the tarnish. His eyes swept over a glossy news magazine on the coffee table where a woman was mirrored in endless reflections in a story on cloning. When he realized I was watching him, he looked up. The green in his eyes was flecked with black.
"How's about I do some summin' up?" His accent was definitely Southern. Ss that sounded like zs, a bit garbled, as if he was talking around a marble in his mouth. "You were out jogging. Morelli picked you up. You went for a ride. Yes?"
I nodded.
"And you know who Mr. Morelli is."
"I do now."
Coates interrupted, a vein on his forehead starting to pulse. "What kind of business did you have with Dominick Morelli?"
"I didn't have any business with him."
He thrust a finger into his shirt collar. I frowned. Was it possible the Feds didn't know who was in the SUV? Maybe that's what they were trying to find out. If so, I should lighten up. The answer would benefit us all.
"I think he was doing a favor for Joey DePalma," I said.
"Joey DePalma?" Coates' voice spiked. "Now you're gonna tell me you know--excuse me--you don't know the Surgeon, too?"
LeJeune took off his cap and ran his hand through his hair. Sandy. Threaded with silver.
"I went to Joey DePalma's house a few days ago."
Coates folded his arms. "Why?"
"I needed some answers."
"And you needed answers because..."
"You know."
"Know what?"
His blank face sent another twinge through me. "You don't?"
"Look, Miss Foreman," Coates said, "it's been a long day. Let's not give each other a hard time."
They didn't know. "I thought someone was trying to kill me."
LeJeune laid his arm across the back of the couch. Coates leaned forward. "You want to run that by me again?"
I told them about the tape and the trial and what had happened since. Rhonda Disapio's version of the night in the park. The men on a boat. A man named Sammy. How she died. The fire. Brashares. I sounded more convincing than I had with Morelli, but whether that was because I was more composed or the danger seemed more real, I wasn't sure.
LeJeune spoke up when I finished. "You don't think Disapio's death was an accident?"
"The timing was suspicious. She died a few hours after we talked at the mall."
"Who else knew about your meeting with her?"
"Brashares."
"The lawyer who died."
I nodded.
He looked into space, his eyes clouding. "Why do you think his death has anything to do with you?"
"I'm not sure it does--anymore." I told them my theory about Santoro being framed and how I thought I had stumbled into the middle of it. As a result of my conversation with Morelli, though, I'd seen the error of my ways. "I might have been overreacting," I admitted.
"Why is that?"
"I couldn't get any new clients for a while, and I was worried. But that seems to have worked itself out."
"Oh?"
"An executive at Great Lakes Oil wants to talk to me about a project."
"Great Lakes Oil?"
I nodded.
"But the genesis of all this--this worry--was that tape? The one you played at the trial?"
"Yes. Unfortunately, he was convicted anyway."
"Why do you think that happened?"
"I guess Ryan was a better lawyer. And there was that slight technical problem on the tape."
"Technical problem?"
"Some kind of RF interference." I shrugged. "Twenty years ago it was pretty common with video equipment."
LeJeune raised the brim of his hat. "You still have the tape?"
I nodded.
"I'd like to see it."
I got the tape from my office, dropped it into the VCR, and hit Play. When the breakup streaked across the screen, LeJeune scratched his chin. When it was over, he looked at me. "You mind if I take this?"
"Go ahead." I ejected the tape from the player. Coates took it, turned it over in his hand, then handed it to LeJeune.
LeJeune asked more questions about the shoot. What the weather was like. The lake traffic. Whether anyone radioed in to shore from the crib. I answered as best I could. Then he dipped his head and looked at Coates. As Coates nodded, a sound from the hall distracted me.
"Mom, when are we eating? I'm starved."
I turned around. Rachel stood at the entrance to the family room.
LeJeune moved to the door. "We won't take up any more of your time."
"I appreciate it." As I walked them out, I asked, "You'll let me know what you find on the tape?"
He dug in his pocket and pulled out a card. "If you don't hear from me in a week, give me a call."
I glanced at the card. "LeJeune--that's a French name, isn't it?"
"Acadian."
"You're from Louisiana?"
He grinned, his eyes crinkling up in the corners. "Yes, ma'am. Lafourche Parish."
"Where is that?"
"In the southeast part of the state. Between Thibodaux and Raceland." My silence apparently prompted him to add, "About fifty miles west of New Orleans. On the bayou."
"Cajun country?"
"Yes, ma'am. Most beautiful country you'll ever see."
Cajun. That was the accent. I opened the door.
Coates aimed a stern finger at me. "Be careful who you go joy riding with. You never know when you're gonna hit a speed trap."
I slid my eyes to LeJeune. He pulled on his hat and winked.
Chapter Twenty-six
A blanket of fog wrapped itself around the upper reaches of the Great Lakes Oil building, obscuring the view. In the heart of the Loop, the building is the second highest in Chicago, and on a clear day its gleaming white façade stretches eleven hundred feet into the heavens, an eighty-three-floor testament to fossil fuels and capitalism. Today, though, peering out at a curtain of gray from the sixty-eighth floor, I couldn't see the horizon or any landmarks. I felt an eerie sense of disorientation, like the passengers in that old _Twilight Zone_ episode, when their plane disappears into a time warp.
I sat in the reception area and thumbed through last month's _Training and Development_. It was a boring read, but after encounters with wise guys and FBI agents, I was happy to be bored. I was back on familiar turf. The rules of engagement in the corporate world are predictable. I'd spent years learning them.
Two goals are paramount: profitability and accountability. Augment the first; avoid the second. The effort you apply to each depends on the state of the economy, last quarter's results, and your pecking order in the organization. Of course, appended to all this is one crucial corollary: everyone spins.
In fact, corporate politics, though often subtle, can be more insidious than the public kind. The media doesn't troll its inner sanctums; as a result, knives can be twisted more frequently. And if an "awkward" story does leak to the press, the company can always rationalize it in the name of shareholder value.
For me, the fascinating part is figuring out individual agendas. It isn't hard. People tend to confide in third parties, perceiving--correctly, in my case--that I don't have a stake in the outcome. As an outsider, though, I'm also the first to be blamed when something goes wrong, so, I end up measuring everything I say, too.
I was led down a corridor with impersonal art on the walls and thick beige carpeting. Assistant Vice-President Dale Reedy's office was big, but it wasn't a corner office. Still, he was clearly senior enough to have an aide call and set up an appointment. And usher me down to meet him.
I hate to admit it, but I was taken aback to see a woman rise and walk around the desk to greet me. She was about five two, with short, glossy blond hair, pale skin, and a pug nose. She wore a severe navy suit with no blouse. She looked somewhere in her thirties.
"Delighted to meet you, Ellie," she said in a clipped British accent. "I've heard good things about you." She smiled and extended her hand. Her nails had been chewed to the quick.
"Thank you, Ms. Reedy."
"It's Dale." Stale tobacco smoke clung to the beige carpets and drapes. She waved me over to a table in the corner where copies of today's _Wall Street Journal_ , the _New York Times_ , and both Chicago papers lay. I pulled out a wicker chair and stumbled over a pair of shoes. I bent down to pick them up.
"Sorry," she laughed. "My running shoes. Just pitch them in the corner, would you?"
Another fitness geek. "You're a jogger?"
"I am. No time for a health club. I run along the lakefront."
Okay. I could handle that. I sat on the chair and snuck a glance at the headlines. She settled in the opposite chair and pulled out a pack of Royals.
"Slow news day." She struck a match.
I looked up. She was smiling, but her brown eyes had a hard-bitten look, as if life had let her down somewhere along the way.
"Wait until summer." I stared at her cigarette.
"For the record, I still do an eight-minute mile. If I quit, I could probably do a bloody triathlon." She touched the flame to her cigarette, then exhaled a stream of smoke. "You're better looking in person than you are on the telly."
My stomach turned over. She knew about the trial. I'd never get the gig.
"Bloody hell." She jumped up, the wicker on the seat of her chair crackling. "I forgot my notes."
Her desk was cluttered with papers, books, a desk phone, and a cell. Underneath the mess was a sheet of brown blotting paper inside an old-fashioned leather blotter. Behind the desk was a credenza with two shelves. A framed photograph of two boys sat on the top shelf. With dark hair and dark eyes, both were in the universal soccer pose: one knee on the ground, a ball in their hands. Curious. She was fair; they were dark. Rachel and I, in reverse.
She moved back to the table.
"So you joined Great Lakes after the merger?"
She nodded. "They needed some shaking up. Especially here in T&D. Early retirement, attrition, deadweight." Which judging from her no-nonsense style, she must have promptly jettisoned.
"How long have you been stateside?"
"About eighteen months."
I gestured toward the photo of the boys. "Long enough for them to join a soccer team, I see."
"They're back in England." Her face was impassive. "I didn't know how long I would be here. So they stayed." She stubbed her cigarette out in a large ceramic ashtray. No mention of a husband. "Now Ellie. What do you know about shale oil?"
I dug out a file from my bag. I'd done a little research last night. "It's a fossil fuel that's extracted from shale by heating it to very high temperatures. But the process isn't widespread, mostly because of the cost. There are also environmental issues. Greenpeace raised a fuss about greenhouse emissions in Australia--so much that the company developing the shale down there ultimately pulled out."
"Quite right."
"Here in the U.S., the federal government owns a lot of the shale reserves, but they're leasing or selling them off bit by bit. And because environmental controls are more restrictive here, there hasn't been the same outcry. At least not yet. The entire process was looked at during the first energy crisis. But because of the cost, nothing much was done." I closed the file.
"Impressive." She leaned back. "Tell me. Do you like to ski?"
I raised an eyebrow.
"Great Lakes has some shale reserves in Colorado that we're starting to develop, and we want to produce a training video about them. Well--" She tipped her head. "Part training. Part PR. We want to take a leadership role within the industry. Position ourselves on the leading edge of an emerging--or re-emerging--technology, I should say. Twenty-first-century vision. That kind of thing."
"Why now?"
"The costs are more manageable." She laughed. "And who knows? It might even become profitable."
"I read that Great Lakes had an opportunity to bid on the Australian project but declined. Why?"
She extracted another cigarette. "We wanted to start fresh. Without any baggage." She stole a look at me. "You can understand that."
I leaned my elbows on the table. "Why me?" _Given my baggage_ , I almost added.
She took her time lighting her cigarette. "You come highly recommended."
"By whom?"
"Midwest Mutual. The Mayor's Office. Brisco Chemicals." She blew out another stream of smoke. "And as for the others, well, bugger them, I say."
I sat up straighter. I liked Dale Reedy.
We spent a few more minutes discussing the audience, the timetable, the budget, and possible elements. I imagined myself on the slopes of Aspen, gracefully crisscrossing a hill. A nice fantasy, considering I've only skied twice.
"So when can you get me a proposal?"
I was about to answer when a knock interrupted, and an older man appeared at the door. Gray hair, stylishly cut. Nice suit. Cuff links. "Dale, I wanted to make sure you got those RFPs--oh, sorry. I didn't mean to interrupt."
_Sure you didn 't._
Reedy rose and introduced me to her boss, the vice-president of training and development. He clasped my hand.
"You look familiar." He tipped his head to the side, then aimed a finger at me. "Aren't you the woman who was on TV a few weeks ago? That trial business."
I felt myself color. "Guilty."
He studied me, then looked back at Reedy. Her face was curiously blank. "I see. Nice to meet you." His voice reeked with false politesse. "Dale, come see me when you're done." His lips tightened.
She nodded and watched him go. My spirits sank. Dale dabbed at her hair. Had she noticed? "Sorry. Where were we?"
"I--I brought a demo reel for you to screen. It's not my most current version, unfortunately, but I can supply references."
"I should think the reel would be sufficient." She fixed me with a serious look. "Ellie. May I ask you something?"
"Of course."
"It's about that trial."
I'd been wondering whether she was going to bring it up. I braced myself. "Go ahead."
"Do you still think he's innocent? I mean, now that it's over?"
I wasn't sure how to answer. If I went into any detail, I might scare her off. She'd think I was unreliable, too flaky to work with. But if I didn't say anything, she might think I was holding out on her, something you never do with a client.
"Yes," I said slowly. "I still think he's innocent. And if I'd been smarter, or more persuasive, maybe the jury would have agreed."
"But everyone else was so certain."
"I know."
"I thought they had quite a bit of evidence."
"I suppose so. But nothing's happened since to change my mind. In fact--" I stopped. "Never mind. It doesn't matter."
She cocked her head.
I shrugged. "His lawyer is dead, and I don't see anyone jumping into the ring to take his appeal. Although the court will have to appoint someone eventually."
She tapped her pen on a pad of paper, shutting me out. "Of course."
"To be honest, I've been trying to put it all behind me." I looked out the window. Most of the fog had burned off, leaving wispy clouds scuttling across a blue sky. She followed my gaze. I looked back at her.
"But you still think about it."
"A little," I admitted. "Especially when I'm driving down Lake Shore Drive. You know, the cribs are only a few miles from here. And Olive Park is even closer." I waved a hand. "You could probably see them from your window."
"I doubt it," she said crisply. "I have a southern exposure."
The room felt suddenly chillier.
"Oh. Um, by the way, is there someone--some resource person I could call while I'm working on the proposal?"
"Resource person?"
"I'm sure to have some questions that I don't need to bother you with. Background on Great Lakes Oil. And shale development."
"Let me give you our librarian's name. I'll tell her to expect a call."
Back at her desk, she pulled out a flat board that looked like a drawer and ran her hand down a page that was taped to it. She scribbled a number on her pad, tore off the sheet, and handed it to me.
"So then, why don't we set up a meeting for next week?" She picked up her PDA and pushed a few buttons. "How is Monday, the fourteenth? I'd like to get moving."
"Sounds fine." I stood up.
"Ellie, it's been a pleasure. I look forward to working with you."
"Same here. I'll be in touch."
I felt her eyes on my back as I left.
***
A young stud in the parking lot brought my car around, his head bobbing to the beat from my radio--some rap tune encouraging him to kill The Man. After shelling out twenty bucks for a measly two hours' parking, I could relate. I peeled out of the garage, my tires squealing impressively.
Traffic slowed on the expressway. Sandwiched between a truck and a van, I took out my cell and called Dad. A woman answered.
"Hello?" The voice was throaty but sweet.
"I'm sorry. I must have the wrong number."
"This is Sylvia Weiner."
"Oh, hello, Sylvia. This is Ellie Foreman. How are you?"
"I'm fine, dear. Just fine. And what can I do for you?"
"Uh--is my father there?"
"Your father? Who are you trying to reach, dear?"
I hesitated. "Jake. Jake Foreman."
"I'm sorry. I don't know anyone named Jake. You must have the wrong number."
I heard a slight commotion, followed by the swish of the phone being transferred.
"Ellie?"
"Dad? Is that you? Is everything okay?"
"Sure, sure," he said. "Everything's fine."
"So that's Sylvia?"
"That's Sylvia," he replied. "A hell of a girl." I heard giggles in the background. His voice dropped to a whisper. "She doesn't remember so good."
"Is it--"
"I think so." He answered. "Just starting."
I sighed. "I'm sorry."
"Hey. Nothing's forever. That's why you gotta enjoy every day."
"In that case, I'll let you go."
"No. I'm glad you called. _Nu_?"
"I just wanted to remind you about the Eskin Bar Mitzvah this weekend."
"What time?"
"Service at nine. Kiddush and lunch afterwards."
"Long day."
"Your friends."
The Eskins and my parents played bridge together for years. Their son, Danny, was the same age as I, and our parents had hoped we'd find each other. For a while, I thought it might happen. In Sunday school, he used to borrow _zedakah_ money from me. A high honor when you're five. But after he borrowed a twenty on our one date in high school, I decided he could bestow the honor elsewhere. He became an accountant and got married, but we kept in touch in that almost-family kind of way. I went to their wedding; they came to mine. They weren't invited to Rachel's Bat Mitzvah, but we'd kept it small.
"The Torah service starts at ten," I said. "I'll pick you up at nine-thirty."
"Sounds good."
"Say hi to Sylvia for me."
"Who?"
Such a joker.
Chapter Twenty-seven
I was going over my notes from my meeting with Dale Reedy the next day, thinking how much I admired her for blazing a trail in the corporate world at the expense of a family life, when David called.
"Hi. How's the jet lag?" I tried for a cheerful tone, but it sounded artificial.
"I'm okay. I got back Sunday." Today was Tuesday.
We chatted about unimportant things, both of us tiptoeing around the edges, as if confronting what was really on our minds would bruise us, scrape our skin raw. He sounded pleased when I told him about the Great Lakes proposal. He said Abdul was still working on his deal. When we ran out of prattle, I took a breath.
"I called you in London. The night after we spoke. There was no answer in your room." He didn't say anything. "Were you with someone?"
"Are you accusing me of something?"
"I--I was upset."
He was silent a moment. Then, "Ellie. I wasn't with anyone. I knew it was you. I didn't feel like talking."
"But we've got to."
"Why? We're not going to resolve anything over the phone."
"Then how do we? Resolve things?"
"I don't know." His words reverberated over seven hundred miles of fiber optics. "Have you done any thinking?"
"Yes. But I don't know if you'll like it." I glanced down at my notes on shale oil. The words looked garbled and meaningless. "You told me--just after the trial--how I'm always trying to right what I see as injustices. Maybe I do. But I try to be careful. I don't look for danger. Occasionally, though, events do spiral out of control, like they did last summer."
"What about now?"
"I don't know that I can change the way I approach life. Or that I want to." I paused. "You know, sometimes I get the feeling you want to put me in a glass jar where I'll be safe. I know it's motivated by love, but that's not what I need. What I need is your support. It doesn't help when you tell me how I'm going out on a limb or making a fool of myself. I do enough of that for myself."
"So now I haven't given you enough support?"
"David, you're the best thing in my life. It's just I can't crawl into a cocoon with you and hide from the world."
"Is that your impression of me?"
"Well..." I paused again. "You are pretty quick to tell me when I'm venturing too far afield."
He didn't answer.
"I--I know it's because you care. And I know you don't want to cramp my style. It's just--" I stopped. "Sometimes I think I'm a bad influence on you. That I'm forcing you into situations and circumstances you'd never find yourself in were it not for me."
"You don't trust me very much, do you?"
I winced. He was veering far too close to the truth.
"What do you mean? Of course I--"
"No, you don't. Listen to me. Whether I steal a flower at a hotel and put it in your hair isn't your responsibility. I'm an adult. I make those decisions myself. By the same token, if I choose to share your life, unpredictable as it may be, it's because I want to. But I can't just let you put yourself--and Rachel--in jeopardy, if it can be helped. And you can't expect me to."
"You're sorry I ever testified, aren't you?"
"That's not the issue."
A streak of anger shot up my spine. "It was easier to react. You should know that given the same circumstances, I'd probably do it again."
"I understand," he said tiredly. "That's where this conversation began. Look..." He paused. "I hope you understand what I'm about to say. I think we should take a breather."
My body went still. _This is how it starts_ , I thought. _With stillness. No movement. Just words._ "A breather?"
"I think we both need to decide--before we get any deeper--whether this is something we want to work out." His voice was shadowed with pain. "That can't happen when we're seeing each other. We get--distracted."
I had an image of us in bed, his body against mine. I pushed it away. "How long of a breather?"
"I don't know."
More silence.
"What do I tell Rachel?" My voice was small.
"That I love her. You, too."
The stillness dissolved. My throat got thick. "Then why?"
His voice filled up. "You know why. Don't make me say it again."
It was useless to try to change his mind, and he cut the connection before I could. I stared at the phone, thinking about the flower I'd torn out of my hair at the Four Seasons. If I'd been trying to sabotage the relationship, I'd been more successful than I imagined.
***
I stayed up late burying myself in research so I wouldn't have to think about David. I found out more than I'd ever wanted to know about shale oil from Googling, but there wasn't much on Great Lakes Oil's web site. I'd have to call the librarian tomorrow. I rummaged around for the sheet of paper Dale had ripped off her pad.
The light must have hit at an odd angle, because as I held up the sheet, I noticed a residual imprint of numbers near the top of the page. She must have jotted them down on a sheet of paper she'd torn off before. She writes with a heavy hand, I thought, because the numbers weren't hard to discern. The first three were three-one-two, the area code for downtown Chicago. Then seven digits. And four more. A phone number and an extension. I squinted at them. Something about the extension was familiar: four, five, two, zero.
I stared at them for a while, then typed them into my notes. Maybe they were one of those numbers I call all the time and just don't realize--like tech support at my ISP. Or somebody's fax number, which somehow had burrowed into my memory. I couldn't quite grasp it. I balled up the paper and pitched it into the trash.
I rolled my shoulders, then shut down for the night. I checked on Rachel. She'd kicked the covers off and was curled on the edge of her bed, a stuffed tiger in her arms. I covered her with the sheet. But it was a cold night, and her window was cracked. I added a quilt.
I padded into my bathroom and peered into the mirror. Where would I be in twenty years? Was I destined to spend the rest of my life alone? Rachel would be living her own life. Would I become one of those bitter old women who wait all week for a call from their children and then complain about everything when the call came?
Enough. The best thing I could do now was end this day. I climbed into bed. It wasn't the Four Seasons, but it was soft and warm. I pulled the covers over my head, felt myself getting drowsy, falling free.
I roused with a start and threw off the covers. Racing into my office, I snatched the crumpled paper out of the trash and smoothed it out. Then I picked up the phone and dialed the numbers. It answered on the second ring.
"Good evening. Four Seasons Hotel. How may I help you?"
"I--I'm sorry. I must have the wrong number."
I hung up and stared at the phone. The number on Dale's sheet of paper had been the Four Seasons hotel. And four-five-two-zero was Suite 4520.
Abdul's suite.
Chapter Twenty-eight
"How come David's not here?" Rachel asked the morning of the Bar Mitzvah. Sipping a glass of orange juice, she alternately kicked her heels against the legs of the chair and pointed her toes. _People_ magazine was open to a picture of Tom Cruise and some Hollywood babe. "Is he still in Europe?"
I looked up from the newspaper. "No, honey."
"Is he sick?"
"No." I folded the paper and propped it on the table.
"You're fighting again, aren't you?"
A tic of irritation passed through me. "We're not fighting." Rachel's frame of reference as far as relationships were concerned was rigid: people either fought, or everything was okay. There was no gray, no middle ground. But, then, with Barry and me as role models, what did I expect? "We both have some thinking to do."
"About what?"
I flicked the newspaper so I could read below the fold. "If it were any of your business, I'd tell you."
She wrinkled her nose.
I stood up and straightened the cropped silk jacket Susan and I had found in the Lord & Taylor outlet. Smoothing out my black silk pants, I said, "Let's get moving. We have to pick up _Opa_."
As I drove down to Skokie, I wondered why Dale Reedy had Abdul's number. He had said he was working on a deal with Great Lakes Oil, but Training and Development is a long way from Acquisitions. Maybe he wanted to find out how to train his people to manufacture the additive he'd been talking about.
***
After the service, during which Sean Eskin, Danny's son, recited a dogged _aftarah_ and an even more dogged _drush_ , the audience decamped to a hotel for lunch. Dad, Rachel, and I piled into the car, speculating on how lavish it would be.
"Danny's an accountant. He knows the value of a dollar," I said. "I'll bet he chintzed on the food."
"I don't know," Dad said. "The kid's an only child."
"You think it's gonna be _Goodbye Columbus_?"
He shrugged.
"Care to put down a slight wager?" I grinned.
He grinned back. "You really want to gamble...with me?"
"Five bucks says it's a tightfisted affair."
"You're on."
***
Our first clue came outside the ballroom, where two hundred table assignments were alphabetically laid out on a table. Instead of a number, guests were assigned to a "team." Dad and I got the Bears, Rachel the Blackhawks. On each side of the table were life-sized blowups of Sean posing in different sports uniforms. In one he was wearing a Sox uniform with a bat slung across his shoulders. In the other, he was shooting a hoop in a Bulls uniform.
Dad clapped an arm around my shoulders and held out his palm with the other. "Like taking candy from a baby."
Groaning, I pushed through the door to the ballroom, which had been transformed into a sports arena. Stadium lights blinded us with their glare; a set of real bleachers hugged the walls. A regulation hoop was set up at one end of the room; a ball-pitching machine occupied the other. Over a dozen kids were lined up waiting to take a swing.
Silver and blue balloons covered almost every surface, including the ceiling, and a quilted Thermos bottle with Sean's name engraved on it sat on each plate. More blowups of Sean in a Bears, Cubs, Blackhawks, and Fire uniform were strategically placed around the room.
But the highlight of the decor--if you could call it that--was Cubs pitcher Rusty Steiger. Live. Dressed in his uniform, he was signing autographs over at the ball machine. Dad tapped me on the shoulder. I dug out a five from my wallet and handed it over. He palmed it cheerfully.
Once we were seated, the room went dark, and that twinkly, twangy music they use to introduce the Bulls spilled out. The DJ, in a creditable imitation of announcer Ray Clay, shouted, "And now, your host...the incomparable, the one, the only...Sean Eskin!"
A spotlight was thrown up. A moment later, Sean, one hand in his mother's and the other in his dad's, skipped into the room. At the DJ's exhortation, the crowd applauded wildly. All three Eskins looked slightly embarrassed but gamely raised their arms in a salute.
The lights snapped on again, and chatter filled the room. Before digging into my fruit cocktail, I waved my spoon. "Play ball!"
By the end of the main course, which consisted of baked chicken dressed up in some kind of sauce with wild rice and something that resembled green beans, I felt like asking Dad for my five bucks back. But before I could, Sean's parents rose to thank the rabbi, the _chazzan_ , the tutor who'd worked with Sean on his Hebrew, and everyone else in the universe. Then Sean's grandmother, my parents' old friend, made her way onto the parquet dance floor. She was wearing a Chanel suit. Blue and silver. Not a hair out of place.
"Sean," she said in a quavering voice, "I only wish your _zaideh_ Leon was here to see you today." Sean's grandfather had passed away six years earlier, around the same time as my mother.
"If he did, he'd have another heart attack," Dad whispered.
The grandmother went on to _kvell_ about the wonderful job her grandson had done, then proceeded to name all her siblings and those of her late husband. I looked at my watch.
During dessert, the DJ led the kids, Rachel among them, around the room in a conga line. After snaking past all twenty tables, it ended up on the dance floor where a limbo pole suddenly appeared. When it was her turn, Rachel slid gracefully under the pole. The DJ threw one of those neon necklaces around her neck. Blushing, she straightened up and tried to pretend she wasn't having a good time.
Dad's eyes sparkled as he watched her. "She's beautiful, Ellie."
She was wearing a satiny white blouse that barely skimmed her waist, a short gray skirt, and heels.
"She's going to be fighting them off with sticks."
"Tell me about it," I said.
"You'll live through it." Chuckling, he squeezed my hand with both of his. "So, you're okay now, sweetheart?"
I nodded. I wasn't going to tell him about my visits with DePalma, Morelli, or the FBI, but I did tell him about Dale Reedy. "Looks like I might be getting some work after all. I met with a woman from Great Lakes Oil."
"You see? All that worry for nothing. You should listen to your daughter. What does she say? Take a chill pill."
I smiled. "Yeah. But something odd did happen. Do you remember us talking about David's new client? The petrochemical sheik from Saudi Arabia?"
Dad released my hand and rubbed his nose.
"He's buying one of Great Lakes Oil's chemical plants in Indiana. But apparently, he called my client. Dale Reedy."
"So?"
"She's in Training and Development, not Acquisitions."
"Like I said, so?"
"So, when we had dinner together a few weeks ago, he said he didn't know her. In fact, at the time, we both thought she was a he."
Dad's eyes slid toward the dance floor where the Eskin family was gathering. "I'm still waiting for the punch line."
"Dad, why would he call her? I can understand him talking to the lawyers. Or the Acquisitions people. But Training and Development?"
"How do you know this happened?"
"I found _his_ number on _her_ pad of paper."
"Maybe some question came up about training people at the plant."
"But he specifically said he didn't know her. And he knew I was going to be meeting her. Don't you find that coincidental?"
My father fixed me with one of his stares: the one that says _Back off and stop making trouble_. Then he got up from the table and walked over to Rachel, who was scooping up the last of her ice cream. He bowed and held out his hand. A minute later, they were dancing.
Dad still did an excellent foxtrot, and Rachel followed beautifully. As he waltzed her around the room, people at some of the tables pointed at the elderly gentleman with the young girl. When the music ended, he dipped Rachel with a flourish. She bent back almost prone and pointed her toes like a pro. I heard a smattering of applause.
It was after four when we got back into the car. The afternoon light was fading, but I felt disoriented, like you do when you come out of a movie in the middle of the day. As I turned out of the lot, Dad fidgeted in the front seat.
"What's wrong?"
"Something's poking me in the back."
I pulled to the side of the road. He eased himself off the seat and shoved his hand into the space between the seat cushion and back.
"Something's stuck in here."
"Hold on." I started to open my door so I could walk around.
"No. I got it." He shook his head and pulled out a piece of silver jewelry. The bracelet from Calumet Park. "What is this?"
I looked over, surprised. "That's strange. How did that get there?"
"What is this, a bracelet?"
"I found it a few weeks ago. I thought it was in my bag."
Dad looked puzzled. "You should keep it in your jewelry box."
"I guess I should." I was about to shove it into my pocket when something made me check the rearview mirror. Rachel, her eyes down, kept winding a strand of hair around her fingers. She didn't look up.
Mystery solved.
I retrieved my canvas bag from the floor under Dad's feet and dropped the bracelet into it. She and I would have to have a chat about privacy. Subhead two, paragraph six of the Boundaries discussion. But that was later.
Once we were on our way again, Dad looked over and grinned. "I have a confession to make."
I raised an eyebrow.
"I'll tell you the truth. I never wanted you to hook up with Danny. It was your mother's idea."
"It wasn't you? How come?" I expected him to launch into his we-are-German-Jews-and-the-Eskins-aren't routine, but he surprised me.
"Danny wasn't what you'd call an Einstein. And it's obvious the kid's not much better."
I smiled.
"By the way, where's David?"
My smiled faded. "He's in Philadelphia."
My father cocked his head, as if to ask why.
I shook my head.
Rachel dangled her arms over the front seat. "Mom says they're--"
My father placed his hand on her arm. "Rachel. _Iz genug_."
That's enough.
Chapter Twenty-nine
A chilly breeze swayed branches against the darkening sky, but after the brittle noise of the party, the feel of the wind was soothing. As we pulled into the driveway, I noticed a silver sports car parked at the curb. I didn't think too much about it until Rachel squealed.
"Mother, look. I can't believe it. A Spyder. Right in front of our house."
"A Spyder?" I shivered.
"The car, Mother. It's only one of the tightest cars ever."
"Tight?"
"Cool, Mom. Tight is cool."
I looked, but I'm not much of a car person. Or a teenage linguist.
"Look, Mom, a guy's getting out." She craned her neck and let out a wolf whistle. I giggled, but the giggle died in my throat when Nick LeJeune slid out of the driver's seat. I threw the Volvo in park.
Rachel looked over. "You know this guy?"
"You do too. He was here last week."
I climbed out of the car and adjusted my jacket. LeJeune leaned against his car. He was wearing jeans and a black leather jacket, and his Different Drummer hat was pulled low on his forehead. I walked over, aware that under the brim, his eyes were on me. "What brings you back this way, Agent LeJeune?"
"Is this your daughter?" He looked past me.
I turned around. Rachel was eyeing him curiously. "Rachel, this is Nick LeJeune."
He flipped up his hat. "You're almost as pretty as your mama." He held out his hand.
I could see her color, even in the dim light.
"You didn't answer my question," I said.
"I told you I'd get back to you."
"On a weekend?"
"The Bureau never sleeps, _ch er_."
_' Cher'?_ I paused. "You're lucky you caught us."
"I agree."
I made a show of looking him up and down. "So, what can I do for you?"
"I was hoping you'd go for a ride with me."
Rachel sucked in a breath.
***
Twenty minutes later, I squeezed into the Spyder next to Rachel. I'd changed into jeans, work boots, and a heavy jacket. LeJeune made a loop around the village, cruising the main streets well over the speed limit. Rachel pumped him with car-talk questions, and he seemed genuinely pleased to reply. I kept my eyes peeled to the road, only half listening to their chatter. Our village is known up and down the North Shore for the cops who lurk on side streets just waiting to snare speeders.
Thankfully, there were none today, and as we dropped her off at Katie's, Rachel was full of smiles. She agreed to be home by eleven and to call my cell if she needed a ride. Then she raced inside to tell Katie and everybody else she'd ever known about her adventure. LeJeune turned onto the Edens and headed downtown.
"You made a friend," I said.
"She's a cute kid."
"The best."
"Knows her stuff about cars."
"Rachel?"
"She knew the model number, the horsepower, the torque. Even knew the manual transmission doesn't have a clutch."
I looked down. Sure enough, there was no third pedal on the floor.
"It's controlled by computer now."
I ran a hand through my hair. I should have listened more carefully. Where did she learn that stuff? Her father? Or someone else--like some kid who'd just gotten his license? Should I worry?
"You want to know who taught her, don't you?" He read my mind. "And what she had to fork over in return."
My hand dropped to my lap. The guy was pretty sure of himself. Probably thought he was God's gift to the world of crime fighting. Women, too.
He grew quiet as he threaded through traffic. The Spyder sat lower on the road than I liked, but it _was_ well balanced, and LeJeune was a good driver.
"I got it this spring," he said. "A twenty-year reward to myself."
It occurred to me that a sports car was something a single man would buy. Unless the man had money to burn. Which FBI agents didn't. Or wouldn't flaunt if they did. "You've been an agent for twenty years?"
"That's right."
"But you said you were from Cajun country."
"I've been here since eighty-two."
He was quiet again. The Spyder slipped through traffic with ease, the air was crisp, and the lights on the highway sparkled. I felt lighter than I had in weeks. I didn't even care where we were going.
At Fullerton, he turned east and then south when he reached Lincoln Avenue. The street hasn't changed much, despite the yuppie invasion that followed the urban pioneers. More fake gas lamps and wrought iron, maybe, but a lot of the same restaurants and clubs. The area used to be a mecca for blues joints; some are still there. But the front end of the Chevy that once jutted out of a brick wall twenty feet above the street was gone, and so was the blues club underneath it. In its place was a Thai restaurant with a uninspiring façade.
"I miss it, too," he said, following my gaze.
We parked in a lot just off Lincoln. Summer is Chicago's best season, but people don't hibernate until January, and despite the cold, the sidewalks were crowded. As we rounded the corner, we heard the wail of a saxophone. The guy who stations himself on the Michigan Avenue bridge Monday through Friday was moonlighting here tonight. LeJeune threw a bill into his case.
"Where are we going?" I zipped up my jacket.
"I thought we'd have a drink...listen to some music."
"A drink. Music?"
"Unless you've got other plans..."
Before I could answer, he opened the door to Blues Alley, and we walked into a large room. Muddy Waters spilled out of the jukebox. Twenty tables surrounded a stage, half of them filled. The blades of a ceiling fan circled lethargically, not doing much to disperse a thick cloud of smoke.
I sat at a table while LeJeune went to the bar, returning with a draft and glass of wine. I wondered how he knew what I drank.
"Okay," I said. "What's going on? Why did we come all this way?"
"You like blues?"
"Yeah, but--"
"Well..."
A woman in jeans and a tight green sweater squeezed by us, her attention so focused on LeJeune that when she brushed the edge of the table, a few drops of beer spilled out of his glass. He pretended not to notice. He tapped a fist on the table to the music.
When the riff ended, he looked over. "You've got balls, _ch er_. You know that?"
"Excuse me?"
"Going to DePalma's--that took guts. And the way you handled Morelli. You don't let people give you any shit."
An official commendation from the Bureau? Was this why I was here? "I told you. I was desperate."
He smiled. "Thoreau says, 'It is characteristic of wisdom not to do desperate things.' But I'd make an exception in your case." The jukebox went quiet. "So, Ellie Foreman. How did you get involved with video?"
The guy jumped around as unpredictably as a fly. He was either very cunning or totally incompetent.
Feeling off balance, I curled my fingers around my wineglass. "My mother took me to see _Old Yeller_ when I was eight, and I cried my eyes out. Then I saw it again with my best friend and realized I was being jerked around. I decided to figure out how they did it."
He laughed. "I was right. You're a Valkyrie."
A literate FBI agent. Wasn't that an oxymoron? "Thoreau says it's better to be the jerkor than the jerkee," I said.
"Take no prisoners."
"War is hell."
" _C 'est vrai, ma petite_."
"Speaking of being jerked around, what's with the _ch er_ and _petite_ stuff?"
His grin deepened. "That's the way we talk to our women back home."
"Except you're not home, and I'm not your woman."
He looked away. Smoke from a nearby table drifted over. LeJeune rose and left the table. For an instant, I started to second-guess myself. Had I been too harsh? Too abrasive? Was he ticked off? Maybe I should make nice. Well, at least courteous. He came back a minute later with another round of drinks.
"So," I smiled. "How'd you get to be an agent?"
He leaned back. "I wanted to catch the bad guys."
"Which ones?"
"Oilmen, for starters." He sipped his beer. "My daddy tried a _vacherie_ , but he lost his shirt."
" _Vacherie_?"
"Cattle ranch," he said. "He didn't make it, so he took a job with the oil company. Had over twenty years in when he lost his leg. They fired him. A year short of retirement. Never gave him another penny."
I winced.
"It's an old story--at least in my part of the world. Hell, even the Kingfish couldn't bring them to their knees."
"Kingfish," I said. "As in Huey Long, Kingfish?"
He nodded. "Before he became governor, he sued an oil company. Trying to get workmen's comp for men like my father. He lost, but he kept on fighting for the little guy. Problem was, the corporate interests didn't like that too much, so the same oil company tried to impeach him ten years later. They lost, too."
"Nice story." I shifted. "But the FBI wouldn't be at the top of my list of crusaders against corporate greed."
"You would know."
I once wrote for an underground newspaper. I read my three Ms: Marcuse, Marx, and Mao. I tried hard to be a revolutionary. Unfortunately, it didn't take. I was told I was too _bourgeois_. That the most I could aspire to was running a safe house. "You've been checking up on me."
He didn't answer.
"Then you should know I don't do that kind of thing anymore."
"That's okay. Chasing down wise guys isn't what I do, either."
"What is it you do?"
"Take pretty ladies out for drinks."
Who was this guy? First he comes to my house and asks about mafiosos. Now he's flirting like I'm some Friday night special. I tilted my head, wondering whether I had enough cash for a cab home but hoping I wouldn't have to use it. Despite his pretense, if that's what it was, I was enjoying his company. I changed the subject. "Was your mother Acadian, too?"
"Italian. My daddy met her in New Orleans."
"Are they still down there?"
"My daddy is. My mother passed about five years ago. Cancer."
"Mine, too." And her death burned a hole in my heart that would never heal.
I finished my wine. "Sit and Cry" jangled out of the jukebox. Buddy Guy.
People jostled our table as they passed. Though it was barely seven, the place was filling up. LeJeune got the check.
"Let's get something to eat." He stood up and took the check to the bar, ignoring the interested look of the female bartender. I allowed myself just the tiniest gloat.
Chapter Thirty
We drove north and east to Diversey Harbor. A veil of navy blue edged the western sky, pierced by an orange glow from Chicago streetlights. LeJeune drove around the inlet and stopped at the end of the Diversey boat launch, a ramp of concrete that slopes down to the water. Hundreds of boats anchor here during the summer, but now only a group of skinny pilings stood sentinel. He cut the engine and leaned an arm across the back of my seat. "There's a settling quality about water, you know?"
"Settling?" Traffic whined on Lake Shore Drive.
"It gives you what you need."
Black waves slapped against the pilings. "How do you figure that?"
"My daddy used to take me fishing on the bayou in our _bateau_. Sometimes it was so quiet you could hear the heartbeat of a hummingbird. Sometimes it was meaner than a wasp with two stingers. But every day, it gave me something to remember." He looked out. "You might not like what it's offering, but it's there for the taking."
A sudden gust of wind rocked the Spyder. I thought back to the rapids of the New River in West Virginia. No metaphysical discussion could ever convince me water would meet my spiritual needs.
"Water doesn't give up its secrets. Even when it tears your heart out."
"Sounds like Cajun folklore. With a little voodoo thrown in."
He grinned, keyed the engine, and made a graceful one eighty. Soon we were heading down Clark Street. He parked on Arlington and guided me to Federico's, a restaurant with red-checked tablecloths, soft music, and garlic-scented air. As we walked in, the host gave me a once-over and led us to a table in the back. LeJeune took off his leather jacket and draped it over his chair. He was wearing a white button-down shirt that made him look like a young collegiate.
A waiter appeared. "They are fresh today, Signor Nick. And large."
Without asking me, LeJeune ordered a bucket of steamed mussels and more drinks. He settled back in his chair, looking very much at home.
"You like mussels?"
Enough already. I blew out a breath. "Look, it's nice of you to buy me drinks and dinner, but I think you ought to tell me why we're here. I got that it's not just social, but I don't take too well to...to subterfuge. If you want something, ask."
"You're right. It's time." He looked over and smiled. "But I want you to know I have been enjoying myself. I don't meet many women with looks and brains."
I started to say something, but he cut in. "We looked at the tape you gave us." His voice was low; I strained to hear him over the music.
The waiter came with our drinks. Wine for me, Molson's for him. LeJeune waited until he left.
"I need you to answer some questions."
I nodded.
"You went out to the crib the same night you took those shots of Santoro, right?"
"That's right."
"What did you shoot out there?"
"We shot a reenactment of what might have gone on during the Twenties. We hired actors, dressed them up, staged a speakeasy kind of scene."
"That was last summer, right?"
"Mid-July."
He looked off into space for a moment. "Did you screen the tape after you recorded it?"
"Of course. We had to log in the time code."
"Time code?"
I explained that the time code is a series of numbers that pop up on the bottom of the screen and allow you to select precise frames for editing.
"Does time code correlate to real time?"
I shook my head. "It's an arbitrary clock that's set at the beginning of the shoot."
He looked disappointed.
"Why?"
He took a sip of his Molson's. "Doesn't matter. So you looked at the tape, and it was fine."
"Right."
"But then, a year later, it turns out to be damaged."
"Right."
"Did the damage show up on more than one tape?"
"Not that I know of."
"You checked the others?"
"Mac, my director, did. It was only on the one tape. Why?"
He didn't answer. We were moving on his timetable, not mine. "So...going back to last summer, you screened the tape, and then you took it back out there a couple days later?"
"We needed it to set up the match dissolve."
"The what?"
I explained that a match dissolve was a special effect in which you dissolved between two shots made from the same location and angle, but at different times.
"Kind of like time lapse?"
"Exactly. But just one shot."
He pulled out a sheet of paper and a pen. "I need you to walk me through everything you did with that tape the day you went back out there. Draw me a diagram. Show me exactly where you were."
"Are you crazy? I can't remember that. It was over a year ago."
"You have to."
I stuck out my chin, about to tell him what he could do with the crib, the tape, his paper, and his pen, but the look on his face stopped me. The intense, engaged man across from me was nothing like the smart aleck who'd been spouting Cajun _shtick_ an hour ago. I took the pen.
"What's the first thing you did when you got down to the harbor?"
"Well, we boarded the _Versulis_ --that's the tug that took us out to the crib from Navy Pier."
"How many people were on the tug?"
"Come on. You can't expect me to--"
"Try. Please."
I searched my memory. It had been a cool day, I recalled, shrouded in fog, but the chop on the lake whipsawed the tug, making a steady shot impossible. I remember asking Mac if he thought it would be stable enough to shoot. The cameraman was there. And the soundman. And the PR guy from the water district. And the crew on the tug. "I think there were five of us. And three of them." I ran it through my mind again. "Eight, all told."
"Good. See, you can remember."
"Maybe a little," I said grudgingly.
"What happened when you docked on the crib?"
"We unloaded the gear--"
"Where?" He motioned to the paper.
I sketched a rough diagram of the crib's surface, pointing out where we'd docked and unloaded the equipment. "We shot some exteriors near the entrance. Then we went inside to tape."
"Draw me a sketch."
I turned over the paper and sketched out the living quarters, the kitchen, the eating area, the large half-filled tank that sat to one side.
"Where was the tape from the reenactment at this point?"
"It was in my bag."
"Your bag?"
I leaned over and felt for my bag before I realized I didn't have it with me. "I usually carry a canvas bag on location. I keep a stopwatch in it, a penknife, gaffer's tape, sometimes a mike."
"You carry it over your shoulder?"
I nodded, wondering why that was important.
"Okay. Then what?"
"We set up in one of the bedrooms." I pointed it out on the sketch. "The same room where we did the party scene. Then we took the reenactment tape and played it back through the camera so we could duplicate the same shot."
"What was the shot?"
"Actually, there were two. We did a shot of the rolltop desk, moving in and out--I'm sorry, zooming close and then moving wider on it. Then we also did an establishing shot of the bedroom. They were both on the reenactment tape. We figured we'd decide later which worked better."
"Now, tell me something. Where on that reenactment reel was the shot that you used for the match--whatever you call it?"
"Dissolve. But I don't understand the question."
He repeated it.
"You mean where on the cassette did it physically lie?" When he nodded, I answered. "Pretty much toward the tail. The end. We'd already recorded a good deal before we got to it."
"And what did you do with the reenactment tape afterward?"
"After what?"
"After you used it for the dissolve."
"I put it back in my bag."
"Good." He took another pull on his Molson's. "Then what?"
"Then, nothing." I was growing impatient. "Come on, Nick. What's this all about?"
A determined look passed across his face. He shook his head.
"Yeah, well." I pushed my wineglass away, feeling cranky. "Maybe if I knew why I'm supposed to remember, I could be more helpful."
He studied me, as if weighing how much to say. "I'll tell you as much as I can when we're done. Okay?"
"This better be good." I studied the wall behind him, where a still life of a bowl of fruit and bottle of wine hung. " _Ch er_."
He gave me a little smile. "Then what?"
"I think we went outside to grab some exteriors. Yes, that's right." It was coming back now. The fog had burned off, but it was still cool and cloudy. The lighting would be flat but even. "That must have been when we went up to the suspension bridge."
"Show me."
I penciled in the bridge that connects the two structures on the crib. "A guy was up there painting, or coating it with rust remover or something. We thought it would be a cool angle, so we went up."
"Did you take your bag up with you?"
"I--I don't remember."
His jaw tightened. "Try."
I struggled with the memory. The bridge was narrow and not very long. About thirty feet. Because of that, Mac didn't come with us. It was just me and the cameraman. "I'm not sure--it was pretty cramped up there."
He shifted. "Okay. Let me ask this. Where on the bridge did you position the camera?"
That I did remember. "The cameraman was about halfway across so he could pan across from the lake to the guy on the bridge." I placed an _X_ where he had set up.
"And where were you?"
I looked at the sketch, then at LeJeune. "I would have to have been behind him--out of camera range. Near the candystriper."
"The pink and white structure?"
I nodded.
"Show me."
I put another _X_ at the end of the bridge.
He angled the drawing toward him. "How much tape did you shoot up there?"
"Not much. A total of maybe two or three minutes."
"Okay. What did you do after the scenes on the bridge?"
_What did we do?_ I closed my eyes. I remember standing up, leaning over the railing of the bridge, looking down, waving to Mac.
My eyes flew open. I got it. "Before we came down, we shot down onto the surface of the crib. Four or five overhead shots." I paused. "And then we went back down to the surface of the crib and shot more workmen."
"That's good." His eyes glittered. "And then?"
"We came back down." I remembered the whistle of the wind, the sound of the gulls, the dense, gray light.
The light.
"I forgot! We lugged a light kit up with us, in case we needed some fill on the bridge. In the end, though, we managed to squeak by with available light. When we were done, the cameraman picked up the camera, and I picked up the light kit and my bag, and we--" I stopped. "My bag!" I concentrated on the memory, testing its veracity. Yes. "My bag _was_ there, between the light kit and the wall. Wedged against the candy striper."
A grin broke across his face. "Show me."
We leaned over the diagram, and I drew it. Then he folded the sheet of paper and put it in his pocket.
Our mussels arrived in a black bucket with a cloud of steam.
"Dig in," he said.
I pried apart the shells, extricated the meat, and, after dipping them in butter sauce, let them slide down my throat. The waiter was right. They were fresh and large, and the smooth, hearty scent of garlic infused each bite.
We were quiet as we ate.
When there was nothing left but broth, LeJeune pushed the bucket to the side and tore open one of the wipes the waiter had brought to the table. "I like a lady who's not afraid to eat."
I wondered whether I could sop up the broth with a piece of bread. If I'd been with David, I wouldn't have hesitated. As LeJeune handed me the other wipe, an image of a restaurant in Philadelphia sprang into mind. David, Rachel, and me last summer. Newspapers on the table, wooden mallets, a pitcher of soda, a mountain of hot, spicy hard-shells. All of us sucking juice out of the tiny orange legs, laughing when it dribbled down our chins. A sharp pang stabbed me.
LeJeune didn't seem to notice. "Just one more thing." He tossed the wipe into the bucket. "When you were done with the reenactment tape, did your cameraman rewind it before he took it out of the camera?"
"Come on. How would I remember that?"
He kept his mouth shut and his eyes on me.
"Lemme think," I sighed. "If he's in the middle of a setup or scene, he usually dumps the cassette and loads a new one right away so he can pick up the shot."
"Is that what he did?"
"I'm sorry. I wasn't watching." I searched his face. "Your turn now. Why is that important?"
He didn't answer.
"Come on, bayou boy. Why is the location of the cassette or whether it was rewound important?"
He spoke in that hushed voice again. "We went out to the intake crib a few days ago. We took a look around. Listened with some equipment. We were out there for hours, but we didn't pick up any radio signals."
"Radio signals?"
"Your tape, Ellie. The RF."
An uneasy tingle ran through me.
"Our analysts say there's a possibility that the degradation on your tape was the result of a powerful blast," he said quietly. "Not a recurring signal. Just one. From a very close distance."
I thought about the grilling he'd just put me through. "Are you saying," I said slowly, "that the tape could have been damaged out on the crib?"
"It's possible."
"Where?"
He folded his hands on the table. "That's what I'm trying to find out."
"Why does it matter? What's so important about this signal?"
He shrugged and looked away. If he knew the answer, he wasn't about to tell me.
"Who are you, Nick LeJeune?"
He raised his eyebrows. "I told you."
"Bullshit. You show up at my house with an agent who tracks mobsters. But you're not on any mob squad. You tell me stories about RF signals from the intake crib. Why do you need to know about the tape? What made you seek me out? At least tell me that much."
"You're right." He cleared his throat. "Okay. Coates comes into the men's room down at the Bureau a while back. I'm there doing my business, and he's shaking his head, telling me he's gotta follow up on some ditsy broad who's riding around with mafiosi on the North Shore. So we're having a laugh about it, and then he says, 'It's the same broad who testified in that trial about the intake cribs.'" He steepled his hands. "I didn't have much else going on, so I came with him."
"Yeah? What do you do at the Bureau--monitor the Coast Guard?"
"Me?" He hesitated a fraction too long. "I do odd jobs."
"Odd jobs."
"That's right. Hey." He unfolded his hands and waved one in the air. "You want some pasta? The stuffed shells here put Maggiano's to shame."
***
Bands of rain streaked across the windshield on the ride home. The Spyder's headlights barely pierced the gloom. LeJeune switched on the defroster while I wiped the inside of the windshield. We were behind an SUV, a Ford Explorer. As LeJeune swerved around it, it occurred to me I hadn't seen any dark SUVs recently. Since the day I'd been driving around with Morelli, in fact.
I looked over. "You remember that Keystone cop routine you and Morelli and that SUV went through the other day?"
"Yeah."
"Did you ever find out who was in it?"
He was quiet for a minute. "Someone muddied the plates up. We couldn't get a read."
"So you don't know who they were?"
He shook his head.
"Or if they were following Morelli?"
He shrugged.
"Rhonda Disapio thought she was being followed by an SUV. And she died."
"But you're alive. And how many SUVs have been following you recently?"
"None," I admitted. "But--"
"So maybe it was a coincidence."
"And maybe it wasn't."
"You said you were through with conspiracies."
"Then why are you pumping me about the cribs?"
He didn't answer, and we drove the rest of the way in silence. As we pulled up to my house, he kept the engine running. I took that as a sign to get out.
"Thanks for dinner." I opened the door.
He leaned across and tipped up my chin with his finger. "You know, in this light, _ch er_, you're a dead ringer for Vivian Leigh." He paused, as if waiting for my reaction.
"Does this usually work for you? I mean that Cajun shit just keeps oozing right out."
He grinned, not at all disconcerted. "Just like mud on the bayou."
Chapter Thirty-one
I got into my PJs, trying to make sense of the evening.
The tape with the shots of Johnnie Santoro might have been damaged on the cribs.
By a nonrecurring radio signal. Okay. Fine. But why was the FBI so interested?
There had to be a logical reason. I checked the clock. Almost eleven. Rachel would be home soon. She knew something about radio. I could ask her.
I turned on _Saturday Night Live_. A skit about homeopathic medicine was followed by a heavy metal band screaming their way through a song about war. I suffered through it with gritted teeth, then checked the clock again. Eleven twenty.
Rachel was late.
I went to the window. Rain sluiced down over the glass, and the disk of the streetlight at our end of the block was rimmed in fog. Beyond it stretched a dark expanse of space. I went into the kitchen and poured a glass of wine. When I came back out, it was eleven twenty-two.
No Rachel.
I picked up the phone and dialed.
Katie's mother answered.
"Patsy, Hi. It's Ellie Foreman."
"Oh, hi, Ellie. How are you?"
"Should I come get Rachel now? She didn't call for a ride, but it's past her curfew."
I heard a slight hesitation. "Umm, Rachel's not here."
"Oh, did Frank give her a ride home?" Frank was Patsy's husband.
"Ellie, Rachel left a long time ago."
"What?"
"Her cousin picked her up around nine."
I gripped the phone. "Her cousin?"
"She said you knew all about it."
I kept my mouth shut.
"My God, Ellie--you didn't? Oh no. What can I do?"
"Did you see this 'cousin'?"
"No. I think they honked--the rain and all--and she went running out." She sounded more upset than me. "Tell you what. Let me wake up Katie and ask her. She'll probably know."
"I think that would be a good idea, if you don't--" A pair of headlights threw beams of opaque light through the window. A dark SUV was pulling into the driveway. I tensed. The side door opened, and Rachel emerged, jacket over her head. The SUV backed down the driveway.
"You know what?" I said into the phone. "She just pulled up. Thanks, Patsy."
I was at the door when Rachel jabbed it with her key. I planted my hands on my hips and waited. It seemed to take her a while to open it. I thought I heard her humming.
When the door finally opened, she took her time withdrawing her key.
I cleared my throat.
She looked up. Her mouth split into a wide, lazy grin. "Hiya, Moommm."
"Rachel, where have you been? And who was that?" I motioned toward the driveway.
She didn't answer but slowly walked into the kitchen, deliberately putting one foot in front of the other. Her shirt was bunched up above her waist, and her hair was tangled.
"Rachel?" I followed her in.
She stumbled over her feet, arms flailing. I caught her before she fell. She tried to push me away, but she missed. Her fingers clumsily flicked across my arm.
"Rachel?"
She looked at me with flat, glassy eyes.
"Rachel, sit down." She ignored me. I tightened my hold on her arm.
She flopped down at the table and propped up her head with her hand.
"You've been drinking."
"No, I haven't."
_Stay calm, Ellie. You can handle this_. "Who were you with?"
She shook her head in a slow, exaggerated way.
"I know you weren't at Katie's. I talked to Mrs. Shearson."
She didn't say anything.
"Where were you, Rachel?"
She slouched lower in the chair.
"Rachel, were you with Carla and Derek?"
She lifted a finger to her lips. "Shhh. Can't tell." Her shoulder dropped and she slumped to one side. Bleary eyes slid from my face to the table, then rolled back in her head and closed. "Rachel, do you need to throw up?"
She opened her eyes and gave me a lopsided smile. "Nooope. I'm fine." Then she threw up on the table.
***
I called early the next morning. "Barry, it's me."
"How ya doing, Ellie?" He sounded chipper.
"I've been better."
"What's wrong?"
"Your daughter came home drunk last night and puked on the kitchen table."
He went quiet. Then, "Is she all right?"
"She's still passed out." I stood up. "Barry, I've been meaning to talk to you about this. Did you know she's been hanging around with Carla and Derek?"
"Marlene's daughter?"
"Yes." I restrained myself from adding her "aerobics queen" handle. "Apparently, she was with them. I wondered if you'd ask Marlene if that's true. And then, maybe we can all sit down and--"
"Hold on, Ellie. What makes you think she was with Carla?"
"Well, for one, she admitted it."
"You asked her?"
"Of course." I started to pace around my office, the familiar irritation of dealing with Barry surfacing.
"Well, what did you expect her to say?"
"The truth."
"The truth." He snorted. "She'd say anything to get you off her back. What evidence do you have?"
"Evidence? What evidence? This isn't a court--"
"Where did she go? Who drove? Who slipped her an ID? Come on, Ellie. Instead of running off at the mouth, show me the evidence."
"Running off at the mouth?" I clenched my jaw. "Barry, lose the attack mode, will you? Rachel's in trouble. We have to deal with it responsibly."
"Attack? You accuse my girlfriend's daughter of corrupting Rachel, and you don't have a shred of evidence to back it up. You tell me who's attacking."
I squeezed my eyes shut and counted to five. No way would I make it to ten. "Barry, I insist that Rachel and Carla stay away from each other. If not, I'll have to take action. In the meantime, you might want to have a little talk with Marlene, just to make sure she knows where her daughter is when she's flexing her glutes or abs or whatever else she's doing all day."
A hostile silence followed. Then, "Ellie, you're the one who doesn't know where your daughter is or who she's with. If you can't handle Rachel, maybe we should reconsider her living arrangements."
A swell of fury shot through me. I slammed down the phone.
I didn't want to admit it, but Barry was right. I _didn 't_ know where Rachel had been. She'd lied to me. Katie's mother, too. But up until now, it never crossed my mind not to trust her. I always thought we had an honest relationship. But maybe an honest relationship with a teenager was an oxymoron. I'd been no angel myself, I recalled. If Rachel was dissembling at thirteen, what would she be doing in a few years?
I sank into a chair. When Rachel was young, I joked about raising children according to the school of bribes and threats. But that was when parenting consisted mostly of cuddling, potty training, and making sure they ate a tablespoon of vegetables a day. Now that she was becoming a person, I felt ill equipped and clumsy. What was the right tack? Cajole or insist? Negotiate or demand?
I gazed at the picture of her and David on my desk. I'd taken it at the Botanic Gardens last summer. Why wasn't he here with me? He might not know what to do either, but at least we could muddle through it together.
***
By evening Rachel was a palette of miserable colors: her eyes were yellow, her nose red, and there was a greenish cast to her skin. I brought ginger ale and aspirin and tucked her back in bed. Her bedside lamp threw an arc of light on a clutter of stuffed animals in the corner.
"Now I know what a soccer ball feels like," she moaned.
"Sounds about right."
"I'll never do that again."
"You heard her," I said to a poster of four young men with black T-shirts and tattoos, who were glowering at me from the wall.
"Why do people get drunk?"
I didn't answer.
"It wasn't even fun."
"You want to tell me about it?"
She sighed. "Derek got this stuff. Sloe gin. We drove to the park and drank it."
Sloe gin fizzes. Tasted like punch but kicked like tequila. The kind of drink you give to underaged kids. I remember worshiping the porcelain goddess myself years ago, thanks to sloe gin. I also remember why my date kept pouring them down my throat. "Rachel, did anything happen--I mean--after you were drinking?" I imagined Derek fondling my baby. Or worse.
"Carla and Derek started to make out, but then she threw up, and we came home."
"What about you? Did Derek try anything, well... inappropriate?" If she answered yes, I'd claw his eyes out.
"Mom, I was in the backseat. He was driving."
I relaxed.
"I'm sorry, Mommy."
"Shhh." I stroked her forehead. Maybe it had taught her a lesson. Maybe she wouldn't be so eager the next time. "Try to get some sleep."
"Will you read to me?'
We hadn't done that in years. "What did you have in mind?"
"Harry Potter? I never finished number four."
I grabbed _Goblet of Fire_ from her bookcase and started in. Harry and Malfoy had set to each other using their wands as weapons, and Hermione's teeth were growing at an alarming rate. Snape cut off the scuffle in the nick of time, but no one seemed happy it was over. When I looked up at Rachel, she was asleep. I finished the chapter myself.
Chapter Thirty-two
Rachel and I discussed the consequences of her behavior at breakfast. She was grounded until Thanksgiving: no play dates, mall trips, or evenings out. Especially on weekends. She would write a letter of apology to Mrs. Shearson for lying, and she'd volunteer at the soup kitchen twice a week.
"What about Science Club?" she asked. "We're starting chemistry. Can I still go?"
I thought about it. "Yes, that's permissible."
She beamed and polished off her cereal. I'd pictured her spending her teenage years in a dusty basement, fooling around with transistors, crystals, and diodes. But hey, if it meant she wouldn't be cruising around raves or getting drunk in the park, I'd throw in a chemistry set.
After I dropped her at school, I put on my black Garfield and Marx suit and headed downtown. Traffic clogged the expressway, the sky was bleak with hard-edged clouds, and a bitter wind tossed the last of the leaves around. November had arrived. We'd be lucky to see any sun before April.
Ninety minutes later, I was ushered into Dale Reedy's office. She was wearing a gray suit, almost a replica of her navy, but either the color was bad or she hadn't slept well. She looked more chalky than a British complexion would suggest.
"How are you, Ellie?" She came around the desk and shook my hand.
"Fine. And you?"
She led me over to the table. "I'm anxious to see the proposal."
No chitchat today. I dug out my copy, wondering if something was troubling her but knowing I couldn't ask.
"I think it turned out well," I said. "But I do have a couple of questions. I noted them with asterisks."
She took the papers. "I'll just start going through these, shall I?"
As she read, I looked around. She'd moved the pictures of her boys to a different shelf, and the daily newspapers were stacked in a chair. I didn't see her running shoes. Had she suspended jogging for the season?
"This is good." She pointed to a paragraph on the second page. "Especially the bit about the leadership role we're taking."
You spoon-fed that to me, I wanted to reply. "Thanks."
She flipped through the document. "Yes. I think this will do. I'll have to show it to Tribble, of course, and take some time to--"
"Tribble?"
She yanked a finger toward the door. "My boss."
I remembered the older, gray-haired man who'd been so cool, even disapproving. My expression must have telegraphed my concern, because she added, "Oh, don't worry about him. He lets me do whatever I want. He's usually ripped by afternoon, anyway."
We exchanged knowing looks, and she turned to the last page. "Let's go over the budget, shall we?"
I cleared my throat. "I wasn't sure whether you wanted me to take a crew from Chicago or pick up one out there. Either way would work." I leaned forward. "And I wasn't sure how much you wanted to invest in postproduction."
"Postproduction?"
"Editing and special effects. And duplication. But the special effects are the prime consideration. If we--"
"Duplication?" She tapped a pencil. "What are we duplicating?"
"We'll need to know--at some point--how many copies you're going to want of the finished show. We use a duplication service, and the price breaks according to how many dubs you order. I listed them here." I pointed to a line item at the bottom of the page.
"Oh, I see." She gave me a rueful smile. "But tell me. How many copies do you make of the unfinished tapes?"
"Unfinished?"
"The tapes that you shoot in the field."
"You mean the originals we shoot on location?"
She nodded. "If that's what you call them."
I shrugged. "We normally don't make any. Most clients aren't interested in the elements. They just want the finished product."
She frowned. "But didn't I hear about a copy of a tape you made on the telly?"
I tried not to react. "That was the tape for the water district. That...that was a special situation."
"Did you make copies of everything for them?"
"No. In their case, we returned the originals. At least most of them." It was my turn to frown. Where was she going with this? "But the tapes are your property. If you think you're going to need copies, I'll be glad to make them."
"Well, now. I'm not sure."
"Most of our clients are happy to let us keep them in storage. That way they know they're safe, and they won't get damaged." I flinched as soon as I said it, but Dale already had arched her eyebrows.
I tried to backpedal. "Uh...the water district was an abnormality. An aberration."
"Indeed." Her eyebrows smoothed out. "But you returned all your originals to them?"
"Not all of them. They only wanted the ones that showed the operation of the filtration plant and the cribs. For security reasons."
"Of course. But then--" she paused "--how were you able to use that tape at the trial?"
A muffled trill cut through the air. "Blast. I thought I turned that off." She slid open her drawer, picked up a cell phone, and looked at the LED. Then she flipped off the switch. "Sorry. Where were we?"
"It's all right. As a matter of fact, the tape that we played at the trial was one of the ones I didn't have to return. It was a reenactment. We used actors and dressed them up in costumes. The only thing we shot were interiors. We probably could have done the shoot in a hotel room, and no one would have known the difference."
She smiled. "But you ended up making copies of it, anyway. For the trial."
"That's right." I was puzzled by her questions, but she was the client.
"So, how many--" Her office phone rang. "It never ends." She looked back at me. "I'll let my secretary get it. Give her something to do besides read the _Enquirer_."
But the phone kept ringing. She glanced impatiently at the phone, then held up the proposal and tapped the edges of the papers on the table. Finally, the ringing stopped. She laid the papers flat and clasped her hands together.
"Now..."
I was about to continue when there was a knock at the door.
"Yes?"
The door cracked, and a woman with dark hair and darker skin poked her head in. "I'm sorry, Ms. Reedy, but a Mr. Sam says he needs to talk to you now."
Dale's jaw tightened. "Tell him I'll call him back."
"He was very persistent."
"Lavinia." Dale's voice was icy. "I told you--I'll call him--"
The woman tensed, her face saying she didn't want to be held accountable if Dale refused the call.
Dale got it. "Oh, never mind. I'll take the bloody call."
Lavinia withdrew, looking relieved. Dale stomped to her desk. "Twit. Doesn't even know how to screen calls."
I shifted in my seat.
She picked up the phone. "Yes?" A pause. "No." Another pause. "That's correct."
I stood up and wandered over to her window, trying to give her some privacy. The cloud cover had lifted to a high overcast. Her window was larger than most office windows, and if I pressed close to the glass I could see both east and west, despite the southern exposure. To the west was the heart of the Loop, a patchwork of irregularly shaped buildings. I could even see the Eisenhower Expressway, which runs west from the junction of the Kennedy and Dan Ryan. Millions of commuters use one of those three highways every day.
"Listen to me." Dale's voice grew more agitated. "I will handle it. Don't worry. I'll call you back."
I looked the other way. The traffic on Lake Shore Drive was a shifting pattern of dots, and the lake, gunmetal today, looked deserted and cold. Leaning my forehead against the glass, I could just make out the intake cribs in the distance. If I craned my neck farther left, maybe I could see Navy Pier.
As I looked, I noticed something running the length of the window at the edge of the glass. At first, I thought it was a crack. I reached up a finger to touch it, but it felt bumpy, not smooth, like you'd expect with an embedded break. I ran a finger down its length. A delicate wire, with a clear insulation, was taped to the window. It was barely more substantial than a thread, not something you'd see if you weren't looking for it...
I took a step back and followed its path with my eyes. Down to the floor, across the baseboards, around the corner, behind Dale's desk. I looked up. Dale was watching me, her phone in her hand, but when she caught me looking at her, she flicked her eyes away and returned the phone to its base.
She didn't say anything.
Neither did I.
***
Rachel's Science Club teacher had said antennas were flexible. You could put them anywhere. But why would Dale Reedy have an antenna in her window, I wondered on the way home. Did she have a radio setup there? Was there some connection between the oil company and the cribs?
There must be another explanation. Maybe Dale had a shortwave radio. Or a ham setup. Maybe she used it to keep in touch with her boys back in England. Families did things like that when they were separated by distance, didn't they?
But then, where was the rest of the equipment? And why didn't she say anything about it? Because it was clear from her expression when I found it that I wasn't supposed to. In fact, her behavior during our entire meeting was strange. Our discussion about the video was perfunctory. The enthusiasm she'd mustered during our first meeting was gone. The only thing she'd been interested in was the duplication of the tapes.
I thought back to our conversation. How she'd seemed to have trouble grasping the difference between originals and edited shows; how she kept coming back to the water district tape. Wanting to know how many copies we'd made. Whether I'd returned the original. My stomach tightened. She was pumping me about the tape we'd shot at the cribs! The tape with the RF on it.
The familiar landmarks on Ontario Street took on a sinister aura as I headed west. Buildings were darker, more hulking, cars and trucks more aggressive. Pedestrians wore menacing leers. What was so frigging important about that tape? First LeJeune. Now Dale Reedy.
But Dale had a wire on her window. And a direct line of sight to the cribs.
***
When I got home, I dug out LeJeune's card and punched in his number at the Bureau. His voicemail picked up. I left a message, telling him I needed to talk to him about my meeting at Great Lakes Oil. I was still uneasy two hours later, and on the way home from school, I quizzed Rachel. "Sweetheart, remember those radios your Science Club teacher brought in for Parents' Day?"
"Sure."
"What are they used for?"
"Which ones?"
"Wasn't there something called a packet?"
She nodded. "Packet's awesome."
"Why?"
"Once you hook up a computer to it, you can do just about anything. Transmit voice, data, send signals to make things happen."
"Yeah?"
She twisted around. "We told you all that at Parents' Day. Weren't you listening?"
"I was, and you did a great job."
She nodded, as if the compliment was her due. We pulled into the garage.
"But tell me something, Rach. Can you send just one signal with radio if you want? You know, just one blast at a time?"
"Of course." She pointed to the garage door opener. "That's what that thing does."
"Gotcha. But you'd still need an antenna with a line of sight between the two points, right? Even with only one signal?"
"Uh, duh."
I went upstairs to change. But as I hung up my suit, safe in the confines of my home, I started to second-guess myself. I could have misinterpreted the situation. What if the wire wasn't Dale's? Perhaps it had been left by the former occupant of her office. She hadn't been in the country that long. Maybe her predecessor had an affinity for shortwave or ham radio, and when Dale inherited the office, she never got around to removing it.
For all I knew, moreover, Dale's behavior today could have been job related. God knows she was in a high-stress environment. Maybe she was in political trouble. It had been known to happen. Zealous female outstrips boss. But if said boss is a member of the old boys' network, guess who gets the shaft?
I put on jeans and a turtleneck and went outside to rake leaves. Fouad hadn't been around for a while, and a thick layer of them covered the grass. They were wet and heavy and speckled with black rot. It felt like moving rocks. I cleared a section of lawn then bagged the debris and dragged the bag into the garage. I'd been working less than half an hour, but I'd worked up a sweat, and my hands tingled. I went back inside. I'd mulch the bulbs later.
Back in the kitchen, Rachel threw open the fridge. Grabbing a can of pop, she snapped off the top and swilled down half the can in one gulp. Then she let out a long, resonant burp.
"Lovely." A wave of cold air drifted over me. I closed the refrigerator door.
"Ummm." She took another swig. "By the way," she said on her way out, "he called while you were outside."
"Nick?"
She shot me a curious glance. "No. David."
"Oh."
She stomped up the stairs.
***
I called David back after dinner, but he didn't pick up. I left a message, then channel surfed for a while. The late news was full of the terrorist's trial. Acting as his own attorney, he was raging about the injustice of the American legal system. I turned off the TV.
After checking my e-mail, I started to clean up my desk. I'm fairly casual about housework; with a teenage daughter, you have to be. The only exception is when I feel life slipping out of control. Then I charge through the house like an army of cleaning ladies, straightening, dusting, and scrubbing, as if the imposition of physical order might magically extend to my mind.
I pitched scraps of paper, rubber bands, and candy wrappers into the trash. Then I took everything off the desk and wiped the surface. As I was moving a couple of paperbacks, I noticed a corner of yellow paper inside one. I slid it out of the book. It was the sheet of paper from Dale Reedy's legal pad. With the imprint of the Four Seasons' phone number. And Abdul's suite.
Was there a connection between them? It was possible. Except that a few weeks ago he said he'd never heard of Dale Reedy. Didn't even know she was a woman. An uneasy sensation hummed my skin.
Chapter Thirty-three
When Dad has to walk any distance, he uses a cane that once belonged to his grandfather. Made of dark, polished oak, it has a knobbed silver handle that resembles a crown. It's a work of art, with delicate engraved motifs and carvings. He was rubbing it as we parked outside Irv's clothing store for men.
"We'll just run in, find a wool overcoat, and come out," he said impatiently.
"Okay." I got out and took his arm. "But you might want to consider a down jacket."
"Why would I want to do that?"
"Dad, it's a new century. They have different materials. They're really comfy. And warm."
"What's wrong with a nice, double-breasted camel's hair?"
"Not a thing. I'm just saying you could try something new."
He sniffed as we pushed through the door. Irv's is one of those no-nonsense places that sells menswear at a discount.
"So, how's Sylvia?"
He thumped his cane on the parquet floor. "Lovely lady, that Sylvia. Makes a mean batch of chicken soup."
"We've moved up to soup, have we?"
"She made Shabbos dinner last week. Brisket just like Barney Teitelman's mother used to make with lots of onions and gravy."
I smiled. "Anything else going on you want to tell me?"
"If there is, you won't hear it from me."
We moved down aisles filled with men's clothing. To me all those suits, jackets and slacks are drab--too many pinstripes, grays, and browns--but I soldiered on. A salesman hovered at a discreet distance.
Coats were in the back. I thumbed through a rack and held out a dark green down coat with a zippered lining and hood. "How about something like this?"
He looked over from the rack he was browsing. "What am I, an explorer in the tundra?"
He turned around and pulled out a long brown wool with a reddish orange fleck weave. "How about this?"
"It looks like it was made in the Forties."
"Exactly." He took out the hanger, slipped on the coat, and moved to a full-length mirror."So what's going on with David?"
I cleared my throat. "Thank you for rescuing me the other day in the car."
He gazed at me in the glass and buttoned the coat. "You're having _tsuris_?"
"We--we have some things to work out."
"You should work them out quickly. You never know how much time you're gonna have."
"Dad, don't be maudlin."
"Just being realistic." He pirouetted in the mirror, then unbuttoned the coat and shrugged out of it. "Okay, let's see that Alaskan snowsuit."
I held out the down coat. He tried it on, checked himself out in the mirror, and arched his eyebrows. "Is that why you look like somebody shot your best friend?"
"That's part of it."
"What's the other part?"
I told him about Rachel's bout with sloe gin.
At first Dad looked concerned. Then his face smoothed out to a knowing look, and, by the time I finished, he was chuckling. "Sloe gin, huh? Reminds me of the time I was fifteen. Barney and I found a bottle of hooch behind the bar at Teitelman's. Figured it was left over from Prohibition. So we drank it. Boy, were we sorry."
"But Dad, she's only thirteen. Two years is a big difference."
"If she had anywhere near as bad a hangover as I did, she learned an important lesson."
"Are you're saying I shouldn't worry?"
"Tell me something. What were you doing at thirteen, Eleanor?"
I opened my mouth. Nothing came out.
"I rest my case." He waved a hand. "Don't worry. Rachel's a smart cookie. She'll be all right."
"Maybe," I sighed. "But the worst part is that Barry and I can't talk about it rationally. It was his girlfriend's daughter Rachel was with. I've been thinking I might call the woman. You know, discuss it mother to mother."
His answer came fast. "Don't do that."
"Why not?"
"If your husband thinks--"
"My ex-husband."
"Him, too," he shot back. "If he thinks you're sneaking around behind his back, he'll make your life miserable."
I didn't want to admit it, but he was right.
"Promise me, Ellie..."
"All right. You win."
Nodding, he shrugged out of the down coat. "And now, just so we're even, so do you."
I cocked my head.
He patted the down coat, a twinkle in his eye. "Let's get out of here before I change my mind."
On the way home, Dad tapped his cane on the floor of the car, humming tunelessly. I smiled. I should start thinking about Thanksgiving. I'd told Dad to invite Sylvia, as well as his buddies Marv and Frank. Rachel would be with us too, and she was planning to invite a classmate who'd just moved here from China. I needed to round up a turkey. Buy sweet potatoes, green beans. I'd make up a Jell-o mold, of course, and pecan pie. And the apple and chestnut stuffing recipe Susan found in a gourmet magazine. We'd probably have way too much food, but we could take the leftovers to the soup kitchen.
I was mentally preparing my grocery list when it occurred to me I didn't know whether to count on David. A pang went through me. We'd only been together a few months, and our relationship was already fraying. Was I too reckless for him? Or was he too cautious, unable to loosen up? Or was all of this just an excuse to ignore my own demons? I chewed my lip. Analyzing the situation wouldn't help if he stayed in Philadelphia while I was here. Why couldn't things go back the way they were? Why couldn't we rewind the past few weeks?
Rewinding my life made me think of the tape, Dale Reedy, and the wire on her window. I looked over at Dad. He might use a cane to get around, but his mind was still sharp. I'd been reluctant to get him involved: last summer he'd ended up in the hospital because of me. But David and I were hardly talking, and LeJeune was who knows where. Mac and Susan didn't want to get involved, and I didn't want to burden Fouad. I didn't have many options. I needed to talk it through with someone.
I edged out of my lane to pass a Mercedes. "Dad, I need your advice."
He looked over, still rubbing the knob of the cane.
"I was wrong about something. You remember the lawyer who was killed? Brashares?"
"Santoro's lawyer?"
"Right. Remember how I thought the mob might be involved?"
His sigh sounded like escaping steam. "Ellie, I thought that was over and done with."
"I thought it was, too. But a few things have come up. And I can't--well, I'm starting to worry." I paused. "It started again at Mac's studio. I was working late there one night when a fire broke out, and--"
"You were in a fire?"
"I wasn't hurt," I added hastily. "At the time, I thought it might be connected to my testimony at the trial."
"What are you talking about?"
"I guess I need to tell you the whole thing."
I explained what I learned about Santoro, how that led me to DePalma and Morelli, how the FBI suddenly took an interest in the tape. "They're trying to identify the source of the RF on the videotape. They think it's somewhere on the intake cribs."
He squinted. I had his full attention.
"But now I'm not sure who's doing what or why." I told him about Dale Reedy and the wire on her window.
Dad put a hand on his cane and the other on the handle. "You say the fire department hasn't solved this arson?"
"They don't have any suspects."
"But somebody set that fire."
I nodded.
"And you thought it was the Mafia coming after you--because of something you were supposed to know. That Brashares and the Disapio girl probably also knew."
I nodded again.
"But now not only the FBI but this oil executive is asking you questions about the same tape. The tape you showed at the Santoro trial."
I considered telling him about Abdul and his possible connection to Dale Reedy but decided not to. I wasn't sure how--or even if--they were connected, and the fact that Abdul was in touch with David would just give Dad another reason to worry. "That's about it."
I exited the Edens on Old Orchard Road and drove east. Dad looked straight ahead, a frown on his face. The only sound in the car was his cane tapping.
He seemed to become aware of something slowly. "Maybe you've been looking at it the wrong way."
"What do you mean?"
"Maybe it wasn't you they were after. Maybe it was the tape."
"The tape?"
"It sounds like some people don't want that tape to exist."
"The woman at Great Lakes Oil?"
"Among others."
"Because of the RF."
"Which the FBI is trying to analyze." He looked over. "Tell me. How many copies of that tape did you make?"
"That's what Dale Reedy wanted to know."
"What did you tell her?"
"Actually, not much. We were interrupted by a phone call." I thought back. "And then I saw the wire on her window."
He rubbed his chin. "So, how many are there?"
"Let's see. I made two copies before I testified. One of which I took with me to Brashares' office the first time. Then there was the original Beta that we played at the trial. There was also the master dub that we made for the files--in case we never got the original back. That's the one that was destroyed in the fire." I stopped at a light. "Brashares may have copied the copy for the prosecution, but then again, he was so cheap he might have just lent them the original."
"If the prosecution wanted a copy, they would have paid for it."
"Okay. So, I'm not sure what Brashares did."
"Too bad you can't ask him." He cleared his throat. "So, as far as you know, we're talking about four tapes."
"Yes."
He laid the cane down and ticked them off on his fingers. "You gave Brashares the original and one copy."
I nodded.
"And there was another one in the studio that was burned."
"Right."
"What about the fourth?" He squeezed his pinkie.
I didn't answer. It had been in my bag until I gave it to the Feds. But nobody, except Dad now, knew I didn't have it.
Chapter Thirty-four
There were two calls on my machine when I got home. The first was a terse message from LeJeune. He'd be out of town for a few days but would be in touch when he got back. That was it. No mention about getting my call. Nothing about looking into Dale Reedy or Great Lakes Oil.
The second was from Abdul, who was back in Chicago. "I am sorry I did not reach you," he said on the tape. "Please call me back." He reeled off the number of the Four Seasons.
As if I didn't know.
I deleted the message.
A bone-chilling rain mixed with sleet pounded the area that night. The weather people congratulated themselves on accurately predicting the first snow of the season. Never mind that they'd predicted the same thing a few days ago, and nothing materialized. It's as if they can't wait to proclaim that winter has, in fact, arrived in Chicago. It must be written into their contracts. I turned up the heat and threw extra quilts on the beds.
Dad wanted me to let Dale Reedy know that I had given the fourth tape to the FBI. I wasn't so sure. Given her odd behavior the other day, that seemed like the wrong kind of signal to send. If she thought I was onto something she didn't want known, telling her that I had surrendered the tape wasn't going to convince her I was suddenly not a threat.
But that left me not knowing what to do--or whom to trust. I punched in LeJeune's number again. I knew he wouldn't be there, but maybe he'd call me back. "Hey, Nick. It's Ellie. I know you're out of town, but I really need to talk to you... Give a call, okay?"
As I hung up, I heard a grunt from the hall. Rachel stood in the doorway, hands on her hips. "You're dumping David, aren't you?"
"What?"
"You're dumping David for Nick."
"Are you crazy? Of course not."
"I don't believe you. You're lying."
"Rachel, what's gotten into you?"
"You know something? Daddy was right." Angry red patches flared on her cheeks.
"What are you talking about?"
"He said you're too dysfunctional for a normal relationship. He said you'd probably run through a lot of men."
I stared at her, slack jawed. "He said what?"
She didn't answer.
"Rachel, there's nothing between us. You'll have to trust me on that. And, as for your father--"
"I saw how he looked at you the night he came over. He asked me a lot of questions, too."
"Rachel, he's an FBI agent. That's his job."
"Questions about David and Daddy?"
"Young lady, I don't know what you're getting at, but I don't like it one bit. I think--"
Her face was turning purple. "You get after me for drinking, for breaking the rules. But you're the real hypocrite. You dump one guy, then go out with another. I wonder who it'll be tomorrow? You know something? I want to move in with Dad. At least he and Marlene are stable."
She stomped out of the room.
Chapter Thirty-five
Angry gray clouds scudded across the sky as I pulled into the lot at the supermarket the next day. They matched my mood. I grabbed a cart and headed inside. Rachel's outburst had been unnerving. Not just because of her emotional swings, which I knew were the result of hormones kicking in. Or even her anger, which was understandable--she'd seen me with David, and then, a short time later, with Nick. She could be legitimately confused.
What _was_ making me crazy was Barry. I thought, after years of hostility, we'd reached a plateau where we could interact with civility if not warmth. But he had blindsided me again, spinning half-truths behind my back. In the past, I could usually work around him. Stop--or at least deflect his blows--before he did any damage. But this time I'd played into his hands. David was gone, LeJeune had appeared. I was his best accessory.
I snatched two bags of chocolate chips off the shelf. I tore one open and shoved a handful in my mouth. As the chocolate slid down my throat, I wasn't sure whom to blame: Barry or myself.
***
Hank Chenowsky lives in a three-flat in Wrigleyville, not far from the ballpark. It was an older building, and as I climbed to the second floor, a musty smell sifted through the walls. Hank opened the door, a surprised look on his face. I wondered why; I'd called him from the grocery store. He was taking the day off; the editing room wasn't quite ready. I got my answer when I sniffed the air.
I swore off grass years ago, choosing alcohol instead. It was a Hobson's choice. I was all for "better living through chemistry," but I knew weed could lead to lung cancer. Some studies linked it to brain damage. Alcohol could trigger heart attacks and brain damage. Since brain damage was a given, I went with liquor, figuring a heart attack would kill me quicker than cancer. Oh. And booze is legal.
Hank's eyes were bloodshot, his pupils dilated. "Oh, man. You did say you were coming down. Sorry."
I looked around. "Where's Sandy?"
"Giving a music lesson."
"Too bad. I was hoping I could meet her."
"Me, too." He smiled beatifically. "She's awesome."
At least somebody's love life was good. I followed him back to the kitchen, feeling envious. His apartment had hardwood floors, high ceilings, and a back porch off the kitchen. My first apartment in Old Town had a similar layout. A memory of winter weekends with Barry flashed through my mind. Both of us stripping off boots, Levi's, turtlenecks, and sweaters, desperate to get our hands on each other, even though we'd just gotten dressed. Passion and sex are easy when you're young.
Hank opened the fridge and scratched his head. "You want something? Juice? Tea?"
"I'll settle for diet soda."
He whirled around, a look of horror suffusing his face. "Ellie, do you know how bad that shit is for you?"
Considering his present state of consciousness, I bit my lip.
"You should purify your system, you know? Cleanse all the additives polluting your body. Your body is your temple, man." He sniffed with the zeal of a convert. "Sandy won't bring anything into the house that isn't organic." He rummaged in the fridge and pulled out a pitcher of something dark and murky. "Here. Try this oolong. It's organic. It flushes out toxins." He poured a glass.
I took a sip. Bitter and sharp. I had a sudden craving for a Big Mac. "I feel better already."
Brightening, he poured one for himself, and we went into the living room. A framed eight-by-ten photo rested on a table. Hank with a young woman. Almost as tall as Hank, she had long, frizzy red hair and wore granny glasses. Her skin was so pale it was almost translucent. Their arms were wrapped around each other, and they both wore loopy smiles. I saw the lake in the background.
"Hey, this is the first time you've ever been here," he said, as if the thought had just occurred to him.
"That's right, Hank."
He nodded his head. "Cool, man."
I settled back on the couch. Hank has a big-screen TV with every conceivable accessory attached to it: DVD, video deck, satellite receiver. He even has a connection to his computer in case he needs to see something he's downloaded on a really big screen.
"So why are you here?"
"Well, like I said, I was hoping I could meet Sandy." I pointed to the picture.
He flashed me the same loopy smile. "She's working. Teaching."
"A music lesson."
"How'd you know?"
I set the glass down on the table. "So how have things been going since the fire?"
"We're getting there. Another few weeks, we'll be finished."
"Still no word on who might have done it?"
"No. Mac says the case is still open, but since the insurance came through, I don't think he cares too much."
I nodded. Next to the picture of Hank and Sandy was a frog in a red and white striped shirt, steering a gondola.
"Hank, do you remember the RF on that tape from the cribs?"
He rolled his eyes. "Oh, man. Not again."
"Well, a few questions came up recently, and you know so much more about that kind of thing than I do."
"I don't know. I kinda wanta forget about that."
"Just a couple of questions. Please."
He flipped up his palm. "Let's have it."
"Thanks." I set down my tea. "Okay. Let's say you have interference on a tape, and you find out that rather than being continuous, it might have been just one single, powerful burst. What does that tell you?"
He squinted, and rubbed his chin with his fingers. "I give up. What?"
"Seriously, Hank. The tape is being analyzed--" I didn't say by whom-- "and they're not sure the interference came through the camera."
"That's weird."
"Not if the tape was sitting next to a source that was transmitting radio waves."
"Is that what they're saying?"
"They're not saying anything. I'm asking."
He rubbed his chin again. "Man, I don't know. Anything I say would just be a guess."
"Guessing counts."
"Well, when you're talking about one burst, no matter where it's coming from, you might be looking at some kind of data transmission."
"Data?"
"Voice transmission is continuous. More or less steady. Depending on the conversation, of course. But when you transmit data, it comes in a binary burst. Kind of like..." He paused and then expelled a loud noise, part belch, part word. "BRAAAP."
I suppressed a giggle. "So the signal might have been one of those--er, BRAAAPs?"
"Yeah. BRAAAP." It sounded like an imitation of a sick frog. "BRAAAP. BRAAAP." He grinned like a kid who's discovered a new way to annoy his mother.
"That's pretty much what Rachel said, too. Well, not in as many words." I shifted. "So it could be a data transmission. Theoretically."
"Sure," he nodded. "You have enough power, you can put an RF signal on anything that's magnetic."
"Power? How much power are you talking about?"
"Man, I don't know. I'm a video guy, Ellie, not an engineer. Enough to trigger the signal." He tossed his long hair, then gathered it as if he was making a ponytail. "Where was it?"
"The transmitter?"
"Right."
"I don't know. But is there any way to tell whether a signal is transmitting voice or data from the pattern of RF on video?"
"What do you mean?"
"Well, hypothetically, could there be streaks on the tape if the signal were voice, but snow if it were data...something like that?"
"Sorry, Charlie."
"Why not?"
He squinted at me. "You ever take any science courses?"
"As few as possible."
"It shows. Listen. You're dealing with the electromagnetic spectrum. It's all the same shit. The only thing that changes is the frequency. The wavelength."
"Which means?"
"In your case, it means that just because you see it doesn't mean you can tell what's causing it."
I sighed. "Okay. I got it."
"Really?"
"Well, maybe."
He grinned. I stayed a few more minutes, thinking Sandy might show up, but when she didn't, Hank walked me to the door. As I took the steps down, I turned around.
"Hey, thanks for the tea."
"BRAAAP." He saluted.
***
Angry whitecaps roiled the lake as I took the Drive north. Between the afternoon rush, which seems to start around three these days, and an early dusk, it would take over an hour to get home. I was heading west on Peterson when I noticed the SUV following me. At first, I tried to put it out of my mind. If I ignored it, it didn't exist. But three minutes later, when it was still there, I checked the rearview mirror for plates.
There weren't any.
At least in front. I pulled over to let it pass so I could spot the ones on its rear. But as I slowed, it did, too. A ripple of unease ran through me. Finally, it turned off onto a side street.
Susan showed up after school, looking chic in black wool pants and a royal blue sweater. I've never seen her with a hair out of place, a stain on her shirt, a snag in her pantyhose. I don't know how she does it. She's just as busy as me--maybe busier. I brewed coffee, feeling grungy in my sweats.
We took our mugs into the family room. A rerun of _Nova_ was on TV. It was a show about sharks and the divers who photographed them off an island near Costa Rica. There were lots of dreamy underwater sequences where hammerheads and manta rays peacefully coexist. I wondered what kind of video equipment the divers were using and how they could shoot film and breathe at the same time.
Susan settled into a chair. "I have a good one for you."
I flipped off the tube. "Shoot."
Susan has her fingers on the pulse of village life, a situation for which I'm exceedingly grateful. Without her, I'd be bereft of the giggles and snide comments a good gossip supplies.
"You know Carol Bailey, right? Two small kids, really involved in IAS?"
I nodded. The Infant Aid Society luncheon is an annual September tradition on the North Shore. Over five hundred women, in elegant fall finery, gather inside a huge tent on a palatial Winnetka estate for lunch and a fashion show. The proceeds help provide day care for disadvantaged mothers struggling to get their lives in order. Having gone to the luncheon once or twice, I feel nothing but admiration for the hostess who sacrifices her lawn to a thousand shoes and metal stakes every year.
"Which one is Carol?"
"She's on the board. Always talking up the Society and the vital services they're delivering."
A hazy image floated into my mind. "Tall, thin, blond, I-hate-you-cause-you're-gorgeous looks?"
"That's the one." Susan paused, a twinkle in her eye. "Well, Carol was arrested last week."
"What?"
She dropped her voice to a whisper. "Child endangerment."
"No."
"She left her kids in the car to go in for a manicure, and when she came out, two police officers were waiting for her. She had to beg them not to call DCFS."
"My God. What happened?"
"Her husband eventually showed up." Susan tore open a packet of sweetener and dumped the whole thing in her mug. "I guess they worked it out. But still. There's this new state law, you know. Twelve thousand dollars if you leave your kids alone in the car."
"You think she paid it?"
She sipped her coffee. "Probably not. Family connections, you know."
"I know." I sipped my coffee. "People like that make me mad."
"People with connections?"
"No. People who are hypocritical about themselves." I waved a hand. "Like people who drive to an Earth Day rally in their SUVs."
"Or give money to MADD and then drive drunk?"
"Or get ticked off when a dog poops in _their_ yard, but won't use a pooper scooper on _others_ '."
We both laughed. She raised her mug. "This is good."
"It's vanilla."
There was a clatter from the kitchen. I turned to see Rachel righting a cereal bowl she'd somehow upended on the counter. I watched as she got milk out of the fridge, poured it into the bowl, and grabbed a spoon from the drawer, all the while conspicuously avoiding my eyes.
I turned back to Susan, whose eyebrow was arched so high it could have been in St. Louis. "All is not happy in paradise, I see."
I shrugged.
"What happened?"
I told her about Rachel's tantrum.
When I finished, Susan fixed me with a penetrating look.
I braced. "Okay. Let's hear it. You're not happy with me, either."
"The issue isn't whether I'm happy, Ellie. It's whether you are."
"Susan, you need to understand something. David was the one who said we needed to take a break. Not me."
"Why?"
"You know what's been happening since I testified. Things around here haven't been what you could call normal."
"Oh, I don't know. Maybe getting trapped in fires is part of your self-improvement program."
"David can't handle it."
"Can you blame him?"
I fumed. "I know he's concerned, but if it were up to him, I'd live in a perfect little room with perfect furnishings--you know, like that room Keir Dullea ended up in in _2001_."
Susan put her mug down. "Ellie, you're probably my closest friend. You could rob a bank, overthrow the government, and I would still love you. But sometimes I wonder if you know what you're doing."
"Susan--"
"No, let me finish. You have this wonderful man who adores you and your daughter. There's nothing he wants more than to be with you for the rest of your life. So, what do you do? Dredge up some lame philosophical excuse why it's not working out, push him away, and then go running around with an FBI agent, who--" she made imaginary quotation marks in the air "--you're suddenly 'helping' on an important case."
"Susan, I told Rachel, and I'm telling you. There's nothing there. It's a totally professional relationship."
"Okay."
"Anyway, that has nothing to do with David."
"Except for the fact that he's not around, and this guy is." She peered at me over her coffee cup. "Oh yes. And the fact that David loves you."
I frowned. I thought of the weekend at the Greenbrier. The surprise at the Four Seasons. The way he took care of me after the trial. The plans he was always making. "But he's always doing nice things."
"Always doing nice things, huh? As in 'I love you and I want you to be happy' nice things."
I didn't answer.
Susan flipped up her palm. "Hmm...Let's see. Here we have a generous man, who wants a loving, intimate relationship." She flipped up the other. "And here we have an FBI agent who gallops in like the Lone Ranger, and will probably gallop right out after whatever 'case' you're working on is over. But, of course, he'd be glad to give you a ride on Silver first." She alternated raising her hands, as if weighing the scales of justice. "Gee, I wonder which is the better deal?"
The way I was feeling, a ride on Silver might not be a bad idea. Fast. Fun. No strings attached. But I couldn't say that. "Susan, you can't really believe I'd break up with David for an FBI agent who thinks he's God's gift to the world at large. He's out of town anyway. I haven't talked to him in days."
She nodded toward the kitchen. "I'm not the one you have to convince."
I caught a glimpse of Rachel, pretending to do her homework. She had to be listening to every word. "But I will admit to one thing."
"What's that?"
"He's about the only one who's taking me seriously."
Susan picked up her coffee cup. "Ellie, do you think you might have a few issues with intimacy? Maybe you should consider seeing someone."
Chapter Thirty-six
Maybe Susan was right, I thought, as I pulled into the gas station the next morning. Maybe I was incapable of sustaining an intimate relationship. I'd never thought of myself as any more or less dysfunctional than the rest of society, but given my problems with Barry, David, and Rachel--even a near brush with Susan--perhaps I should reevaluate.
I wrestled the hose into the tank, imagining a _dybbuk_ inside gleefully laughing at me, though whether it was because of my mood or the fact that gas prices were bleeding me dry, I wasn't sure. While I waited, I decided to clean out the back of the car. It beat watching dollars and cents zoom up at lightning speed. Or dwelling on my shortcomings.
I started with my canvas bag, which was wedged underneath the front seat on the floor. I took it over to a large metal trash container. I set it down on the concrete island and felt around the bag. Two objects seemed to be stuck together. I pulled them out. The silver bracelet from Calumet Park was tangled up around my stopwatch.
As I started picking at the bracelet to unravel it, I thought about the VHS copy of the tape I'd given to LeJeune. If Dad was right, and someone _was_ after the tape, they'd been going to extraordinary lengths to get it. Break-ins, arson, and--assuming Brashares' death was part of it--even murder. But that didn't explain why Rhonda Disapio was dead. Or Mary Jo Bosanick. They had nothing to do with the tape. Mary Jo Bosanick never knew it existed.
I studied the bracelet. I was willing to concede that my theory about drug dealing and gangsters was far-fetched. Even harebrained. But how likely was it that the two women's deaths were random acts of violence? Two girls, best friends, party at Calumet Park on a summer night. Two men motor into the boat launch. One woman dies, the other makes a narrow escape. A year later, she dies, too. Meanwhile, the men disappear. No one knows or believes they exist. Except me. And the only thing I knew was that one called the other Sammy.
Someone at the next pump whistled. I jumped back, nearly losing my balance. I looked at the pump; it was still. I went inside to pay, the bracelet and stopwatch in my hand. I set them down on the counter and dug out a twenty.
The young man behind the counter looked at his digital readout. "It's twenty-two fifty, ma'am."
Damn. I try to keep gas at twenty bucks a pop, purely on principle. Never mind that I make more trips to the gas station; those are the little ways we fool ourselves. As I fished out a few more dollars, the guy behind the counter eyed the bracelet.
"Looks like the one I bought my girlfriend."
I looked up. "The bracelet?"
He was wearing a striped uniform shirt with his name, Sam, embroidered in red on the pocket. He pointed. "The heart thing. I got the same one for her."
I picked at some grime on the charm. "I hope your girlfriend's was in better shape than this."
"It was." He grinned as he handed me back my change, and I headed out to the car. I was two steps away when I froze.
I'd found the bracelet in Calumet Park, where one of the men called the other Sammy.
Someone named "Mr. Sam" had called Dale Reedy the day I was with her.
I climbed in the car, threw the bracelet on the seat, and started the engine. Sammy was one of the guys in the boat. Coming into the boat launch the night Mary Jo was killed. According to Rhonda, it was Sammy and his cohort who killed her.
As I swung out of the gas station, my mind started to race. What--exactly--had Rhonda said? A hot, humid night. Mary Jo and Santoro had fought. Mary Jo took Rhonda to the park to drink it off. While they were there, two men came into the boat launch, their boat loaded with gear.
I'd assumed they were running drugs, partly because of Santoro's background--but also because of the comment Mary Jo made: "What makes you think I don't know about dealing?"
I had been wrong. But if it wasn't drugs, what was it? Why would two strangers kill a woman they didn't know--and then her friend as well? I circled the village park, deserted and bleak in the November chill. Frigid water collected in troughs and depressions around the field.
People kill for many reasons, but one of the biggest is fear. Fear that they'll be killed first. But Mary Jo and Rhonda weren't threatening.
Fear of being caught is another. Rhonda thought the men were just fishing, but were they? Or were they doing something else? Something they didn't want revealed. Something with such high stakes--at least for them--that killing two young women was their only option.
A lone figure struck out across the park. His jacket was pulled close and shoulders hunched against the cold.
What was it? What were they hiding? Something on the boat? The boat was carrying some cargo. Rhonda had said something about it. But what? "A lot of shit" were her words, I recalled.
I turned the corner and headed back to Willow Road. As I passed the drycleaners and hardware store, the sun made a brief appearance, glinting off the Volvo's hood.
Glinting. Something glinting in the moonlight. That was it.
Metal. Logs. Metal fireplace logs.
I frowned. Something that looked like metal fireplace logs. What was Rhonda trying to describe? I squinted through the windshield.
A metal container of some kind.
Sure.
One of those metal trash containers with a foot pedal to open it up. Maybe they held a stash of drugs.
Or maybe something else.
A fire extinguisher? No. Most fire extinguishers are red; they wouldn't necessarily glint in the moonlight. And someone would have to be pretty hard up to kill over a fire extinguisher.
Think, Ellie.
The men were coming in off the lake. Late at night. With metal containers. What if those containers had something to do with the water? Maybe they held water. Or you used them in water.
An image of divers filming hammerhead sharks sprang into my mind.
A tank. An oxygen tank.
Scuba diving equipment.
Is that what Rhonda saw? A boat filled with diving equipment?
Why would someone be diving in the middle of the night in Lake Michigan? And why wouldn't they want anyone to know about it?
I tried to piece it together. A man named Sammy was at Cal Park a year ago. The night Mary Jo Bosanick died. Possibly ferrying scuba diving equipment.
Dale Reedy got a call from a man named Mr. Sam. She had a wire taped to her window. She'd been quizzing me about a tape I'd shot at the intake crib. And she had a line of sight to the crib.
The sun disappeared behind a cloud.
***
Rachel's door was locked, and when I knocked, there was no answer. I went into my office and went online. The computer chimed I had mail. The return path read Greatlakesoil.com. I clicked on the message.
_I 'm frightfully sorry, Ellie, but we're going to have to cancel the project. The economy has been bumpier than we anticipated, and we simply can not justify further discretionary expenses at this time. I know how much work you put into the proposal, and I would be happy to compensate you for your time to date. I hope there will be an opportunity for us to do business together in the future._
Short. Concise. Definitive. And obviously code. Something had happened.
It could have been her boss, Tribble. When he'd come into her office during our first meeting, I was sure he knew who I was. He would never have forbidden her to hire me; the corporate world doesn't work that way. But, drunk or not, he might have asked pointed questions about my credibility, reliability, perhaps even my talent. Dale might have started out defending me, but faced with his volley of questions, she would have realized something was off, and, over time, she might have concluded it wasn't a battle worth fighting.
Then again, maybe it wasn't Tribble. Maybe it had something to do with the wire, the tape, and a man named Sammy.
I stood up and started to pace. There was no way I could find out, and I could only think of one person who could. Where was LeJeune?
Chapter Thirty-seven
Rachel spent Saturday camped in her room with the door locked. An occasional exclamation through the door was the only proof she was alive. I made her favorite pasta and tomato sauce for dinner as a peace offering. I wasn't convinced it was up to me to make peace, but Jewish American guilt goes a long way toward accepting responsibility for sins you haven't committed. But she waited until the sauce had turned cold and the noodles rubbery before sneaking down for a plate.
Around nine, I made a run to the video store. I'd get something we could watch together. Maybe we'd even start talking. I grabbed an Adam Sandler tape and a comedy with Cameron Diaz and was back on my block in less than thirty minutes. I was passing my neighbors' house when I slammed on the brakes.
A dark-colored SUV was pulling away from the house. Two figures were in the front. I tried to make out the license plates as it sped away, but it was too dark. I swerved into the driveway and raced into the house.
"Rachel?" I yelled. "Rach, where are you?"
Silence.
I ran upstairs and checked all the rooms. No Rachel. I checked the closets. No one. My pulse throbbed in my ears. I ran downstairs. The basement was empty. I raced back up and opened the front door. It was a frigid night, and an icy wind stung my skin.
Where was she? Maybe she'd left a note. If she did, it would be in the kitchen. I ran in. Nothing. I checked the clock. Almost nine-thirty. She knew her curfew was eleven. Had she gone out deliberately? Maybe I was wrong not to give her a cell phone or a pager. Lots of parents did these days, but I'd considered it excessive. A badge of conspicuous consumption.
I picked up the phone and called Barry. The phone rang four times, after which his machine kicked on. Another weekend in Door County with Marlene, no doubt. But no Rachel. I called Katie's house. No answer there, either. I thought about calling Susan, but I knew her machine would pick up; she and Doug are always out on Saturday night.
I huddled on the couch struggling to keep panic at bay. A plane flew low overhead, triggering an instant of fear. But it passed safely, its thunder shaking the walls.
A nightmarish conspiracy unspooled in my mind. What if the men in the SUV knew the exact moment I'd be alone and vulnerable and deliberately chose that moment to strip me of the only thing in my life that held any meaning?
I flashed back to the New River in West Virginia, when I failed to rescue my daughter. Was it happening all over again? They had taken Rachel, and they were going to do something unspeakably horrible to her if I didn't--but what? What was I supposed to do? Give them the tape? Tell them what I knew? What did they want?
I gazed around the room. The walls, the bookshelves, the furniture all looked solid, almost comforting in their ordinariness. Still, an overwhelming sense of futility washed over me. I sank back on the couch. If she wasn't home by eleven fifteen, I'd call the police.
At midnight I was about to pick up the phone when a sweep of light tore through the window. I raced to the hall, my heart pumping, and before I could really think about it, grabbed my father's Colt .45. Checking to see that it was loaded, I released the safety. I hoped I still knew how to chamber a round.
I flattened myself against the door. My mouth felt like it was filled with cotton. I waited--for a window to shatter, a knob to turn, a door to fly open. When the bell rang, I sidestepped to the glass panel inset on the door. Under the porch light, looking hollow-eyed and slightly green, was a cop.
I sagged against the wall. I should have realized anyone ringing the doorbell in the middle of the night wasn't out for a nefarious purpose. I slipped the Colt back in the cabinet.
"Good evening, Ms. Foreman." The officer was one of the cops who'd questioned me after the fire at Mac's studio. "I came by to tell you your daughter's okay."
My breath caught. "What do you mean, okay? Where is she?"
"She's--she's at the station."
"At the police station? Why?"
"Ma'am--er, we brought her in on an unlawful possession charge."
Over on the expressway, a truck rumbled by. Its echo reverberated through the trees. I stared at his badge, uncomprehending. He could have been speaking Chinese.
"What?"
"I was there when they brought her in. Detective O'Malley sent me over as a courtesy."
"Rachel's been arrested? What for?"
"For unlawful possession of a weapon, ma'am."
My jaw dropped. "A weapon?"
"A firearm."
I gasped for breath. "A gun?"
He nodded. "She's in custody now."
"In custody?" When he didn't answer, I added, "What's going to happen to her?"
"Well, ma'am, that kind of depends on her. And you. She's with the youth officer now."
"Oh my God." My hand flew to my mouth.
"Don't worry. She's fine--a little shaken up is all. But you need to get down there."
***
The village police station, a modern brick building, sits in the middle of an upscale residential neighborhood. Set back from the road, it could pass for a school or a community center, except for the phalanx of cruisers in the parking lot. After parking the car, I ran past a flagpole to the front entrance. White boulders, bloodless in the weak moonlight, lined the walkway.
The lobby resembled a modest office complex with tiled floor, white walls, and fake plants. Doors led off both sides. Near one of the doors was a pass-through window, behind which lay several desks and an array of communications equipment. The combination of fluorescent lights and crackles from the radio was unworldly.
I announced myself to the dispatcher, an older, heavyset man with thin bands of white hair stretched across a pink scalp. Then I sat in a black molded plastic chair. I felt like I'd stepped through the looking glass.
"Ms. Foreman?"
I looked up. An attractive blonde stood in front of me. Her name tag read Officer Georgia Davis, but she was dressed in a pair of tailored black slacks, black boots, and ivory sweater. Her shoulder-length blond hair was curled in a perfect flip, and her eyes were large pools of brown.
She flashed me a hesitant smile. "I'm the youth officer. I'm handling your daughter's case."
A wave of embarrassment washed over me.
"Why don't you come with me?" Again, a tentative smile.
She waited for me to gather my bag, and we pushed through one of the doors.
"Is she okay?"
We went down a long hall and rounded a corner. She pointed to a door. "She's been waiting for you."
As I opened the door, I clamped down on my tongue. I was in a small, windowless room, about eight by ten. The walls were cinderblock, and a built-in bench stretched along one wall. Two vertical steel bars, the kind you see in wheelchair-accessible bathrooms, were attached to the walls. A pair of handcuffs-- _handcuffs_ --dangled from one of them. Rachel was curled in a ball at one end of the bench.
At the sound of the door opening, she looked up. Her skin was waxy white, her expression one of abject fear. When she realized it was me, her eyes widened like they used to when I'd come home from an out-of-town trip. She propelled herself into my arms.
"I'm sorry, Mommy." Tears streamed down her face. "I'm so sorry."
I hugged her tight. "It's okay, baby. Mommy's here."
There was a knock on the door, and Officer Davis came in. Rachel pressed herself more tightly against me. Spying a box of tissues at the other end of the bench, I gently disengaged from Rachel and handed her the box. She cringed, but I brushed a hand across her hair, trying to telegraph that it was okay. Davis leaned against the back of the door, and read from a clipboard with papers attached.
"At about eleven o'clock Officers Randall and Brewster stopped a black SUV Lexus speeding south on Waukegan Road near Dundee. At first the driver attempted to outrun the officers, but eventually he pulled over. They apprehended Derek Harrington, Carla Sager, and Rachel Goldman. When they began questioning the youths, they observed a thirty-eight caliber handgun on the floor of the passenger side of the vehicle. A revolver." Davis looked up. "It was loaded."
"A loaded revolver?"
Davis held up a hand. "Your daughter was in the backseat at the time we apprehended them. We have no reason to believe she handled the gun at any time. Is that right, Rachel?"
Rachel sniffled into her tissue.
Davis went on. "The officers who took them into custody ran a check on the car and found it was registered to Robert and Alexa Harrington of Glencoe. No one in the vehicle had a Firearms Identification--"
"Where are the other kids?" I cut in.
"In our other interview rooms. The Harringtons are on their way down, but we haven't been able to reach Mrs. Sager or Mr. Goldman."
"That's because they're--"
"I understand. Rachel told me they're--away. We were able to contact an aunt. She's coming down."
I felt oddly relieved I wouldn't have to explain how Rachel's father happened to be with Carla's mother. "So, what happens now?"
Officer Davis looked at Rachel, then at me. "Well, Ms. Foreman, Rachel and I have talked. Why don't you tell your mother what happened?" Davis gave her a slight nod.
Rachel ran her tongue around her lips. "Well, Derek and Carla rented _Natural Born Killers_ --you know with Woody Harrelson and Juliette--"
I nodded. Oliver Stone's New Age _Bonnie and Clyde_. Two young serial killers shooting their way across the country and loving it.
"They picked me up and we went back to Carla's to watch it. I thought it was creepy, but they thought it was awesome." She looked down. "Then we went back to Derek's, and he found the gun."
"Where?"
She shrugged. "It was his father's, I think."
"What the hell did you think you were doing?"
Rachel fell silent.
"Rachel?"
"I didn't know."
"Were you high?"
"No. No one was doing anything."
I cut my eyes to Davis. Had they been stoned, maybe I might have understood. But they were cold sober. Davis raised her eyebrows back at me as if to say, _This is the way it is these days._
"Derek found some ammo in his garage, so we started to ride around. Derek started saying all these things, like first we'd shoot out some windows, and then--" She shuddered. "We'd get you, and--"
"Me?"
She wouldn't meet my eyes. I looked at Davis, who gave me a brief shake of her head. A fresh torrent of tears started down Rachel's face.
"I didn't mean it, Mom. I was just really mad. I would never have--" She hunched over again, bowing her head in her hands.
I put my arms around her. "It's okay, Rachel." I whispered. "I know."
Davis cleared her throat. "It's clear to me that Rachel feels very remorseful. We've been talking about choices, and she realizes she's made some unwise ones. Particularly in the area of friends. We've also talked about things she can do the next time she feels angry and upset. And she's promised me that we'll talk again after Thanksgiving--if that's okay with you."
"That's fine."
When Rachel looked up, I flashed her a smile. So did Davis. The anguish on her face started to recede.
"What happens now?" I asked.
"She can go home."
Rachel's face brightened. "I can?"
Davis nodded. "Get some rest. You've had a rough night."
Rachel stood up and moved to the door. Then she turned around and came back to Davis. Standing on her toes, she kissed her on the cheek. Davis colored.
"Thank you," I said.
"She's a good kid." She put an arm around Rachel's shoulder. "I've explained this to Rachel, so I should tell you." She extracted some papers from her clipboard. "Your court date is in five weeks. She doesn't have to appear; her lawyer can be there for her. I'll be there, but I'll recommend that the charges be dropped."
Officer Davis and I shook hands.
As we left the station, I noticed a couple, about my age, in the black plastic chairs. The woman was sobbing into a handkerchief, and the man's arm was around her shoulders. Derek's parents, probably. I pushed through the door without stopping.
Chapter Thirty-eight
Rachel and I slept late the next morning. Then we did a huge Thanksgiving shopping at the supermarket. She was subdued; we chatted about inconsequential things. I wanted to fully process last night before we talked. For now, I was grateful that she wasn't hurt. And that the SUV turned out to be Derek's.
When we got back, Fouad was on the front lawn removing the last of my annuals. It was hard to believe the wiry twigs and stems he was pulling out were once petunias and impatiens. He helped us unload the groceries. Rachel put them away.
Back outside, he started in on the prairie grass flanking my driveway. The weak November sun made the dry stems seem luminous. Against an empty, lavender sky, the effect was pure Georgia O'Keefe.
Fouad's red and black lumber jacket was open at the neck. Wiry black chest hairs spilled out above his T-shirt as he slashed through the grass.
"They're saying it might snow tonight," he said.
I drew in a breath. Sometimes there's a tangy, metallic scent that precedes snow, but I didn't smell it.
"Where have you been?" I asked. "I haven't seen you in a while."
"My son had some problems. We went to Duke to work them out."
Ahmed was a stellar premed student. I couldn't imagine what kind of problems he'd have. I asked.
"Someone set fire to his dorm room."
I swallowed.
Fouad didn't look up. "Fortunately, the damage was minor."
"Fouad, I'm so sorry. What did you do?"
He shrugged. "We spoke to the dean, his advisor, and the dormitory monitor. They were full of apologies." He kept hacking through the grasses.
I waited for him to say more.
He didn't.
I shook my head. "I don't know how you do it, Fouad. You've been here thirty years. How do you deal with it--with such--equanimity?"
He was quiet. Then he got up, looked at me, and moved to the other side of the driveway. "I want to tell you a story. It was told to me by your friend David."
"My David?"
He nodded and started in on the grasses on that side. "Last summer, the night we went looking for you, we were here...." He motioned to the house. "We were worried. We did not know where you were. So we agreed to wait for a few minutes."
I remembered the night.
"While we waited, we talked. He told me about a young girl in Germany. During the Thirties. About the same age as your Rachel is now. Maybe a year or two older.
"She grew up in Freiburg. Her father was a tailor. Not a wealthy man, but he managed. She had a brother and a sister. She went to school, had friends. She had a happy childhood." He threw a glance over his shoulder. "But then, she was told she could not go to school anymore. Her friends were no longer permitted to play with her. Her father was forbidden from working. The family was restricted as to where they could go. They were forced to wear a sign on their clothing. They endured cruel taunts from neighbors who a few months before had been their friends. One day she was forced to watch her father strip down to his underwear in the middle of the street while others--their former friends and neighbors--gathered around to jeer."
He put down the scythe and looked up. "Whenever I think it is bad here, I remember the story about David's mother. And I thank Allah I am where I am." He scooped up the ends of the prairie grass and stuffed them into a plastic bag. "You understand?"
I nodded.
"There are many Muslims who share my view. Despite what you hear on the television." He stood up, the bag in one hand, the scythe in the other. "I take care of the landscaping at the mosque over in Northbrook, you know. I hear the young men, the students, talking before and after prayers. Most of them love this country. They are grateful to be here." We walked back toward his truck. "They do everything they can to fit in. They dress American, they eat American, they even Americanize their names. _Fariq_ becomes Frank, _Samir_ becomes Sammy, _Rayann_ becomes Ray--"
I stopped. "What did you say?"
He turned around. "I said they do everything they--"
"No. The name. Sammy. It's short for what?"
" _Samir_. S-A-M-I-R. It means entertaining companion." He tilted his head. "Why do you ask?"
I shook my head. "I--I--it's probably nothing. I mean, there are probably lots of Samirs in Chicago, right?"
"Yes. It's a very common name. Surname, too."
"A surname, too?"
"Sam is." He went on. "If the family name is _Sam_ , a young man might call himself Sammy. There is also _Sami_ , which means high, lofty, or elevated. Or _Samman_ , which means grocer."
I followed him to his pickup. "Is there any way to determine which name someone who calls himself Sammy might be using?"
He shook his head. "It would be like someone who calls himself Al. Is he Albert, Alfred, or Alphonse?" He lay the scythe in the bed of the truck. "By the way...speaking of names...I spoke to my friend the other day. The one from Riyadh."
It took me a few seconds to focus. "Riyadh?"
"Your friend from the royal family."
A jolt of uneasiness shot through me. "Abdul."
Fouad nodded. "My friend said there is a database on the Saudi royal family on the Internet."
Of course. Why hadn't I thought of that?
"It is not an official site, you understand. Just a private effort to keep track and organize some of them. Over two thousand names are listed. It is not complete by any means; there are over five thousand royals. I searched for the name you gave me."
"You didn't have to."
He held up his palm. "No, do not thank me. I found nothing."
I stiffened.
"But you see, most Muslim family records are based on the progeny of the mother, not the father."
"Like Jews."
He nodded. "If a mother is not listed, it could mean that no one, including the Sauds themselves, is sure who was doing what with whom. There also is the issue of polygamy. It is sometimes difficult to identify the children when there are four or more wives in the family. And there is also the desire for privacy, particularly where females are concerned. Sometimes information comes to light only when the mother's obituary is published."
"What are you saying, Fouad? Is Abdul a phony?"
He answered carefully. "I can only tell you that the name does not come up on any of the branches of the family that are publicly known."
***
I couldn't sleep that night. I'd gone online and found--or didn't find--the same thing as Fouad. Did that mean Abdul was an imposter? Posing as a Saudi royal while David helped him purchase a chemical company? And if he wasn't, why would he lie about it? What was he trying to hide? Even if it turned out he was a member of the royal family, he was still spending a lot of time and money in Chicago. And harboring some kind of connection to Dale Reedy.
And now there was a man named Sam who called Dale Reedy. Plus a man named Sammy from Calumet Park, possibly ferrying diving equipment, who may have killed Mary Jo. And Sam and Sammy were common Arabic names.
I bunched up the pillow in front of me. I'd read how terrorist operations were sometimes financed by wealthy, seemingly legitimate Arabs, who, in reality, were covertly supporting a terrorist cell or two. Was Abdul one of them? He conducted business at the highest levels of society: the Greenbrier, international currency markets. He seemed to have money to burn. And I had reason to doubt his veracity.
Was Sammy part of his cell? He wasn't enrolled in flight school--he took diving lessons instead--but what difference did that make, if the goal was some terrorist action?
I rolled over and turned on the light. I couldn't accuse Abdul of terrorism; that would be the worst kind of racism. There was always the chance that Abdul _was_ a member of the royal family. Maybe his mother, an exotic princess or daughter of an emir, had slipped through the family records.
Still, there seemed to be some connection between the RF damage on a videotape, a British woman at Great Lakes Oil, the intake cribs, and possibly an Arab. Something just beyond my grasp.
I threw the covers off. I felt as if I were in the maw of some mysterious creature, unable to figure out what it was. My theories might make sense if I knew its genus, its habitat, its routine. But it was keeping itself hidden. Unapproachable.
I stared at the phone, feeling frustrated and helpless and alone.
Chapter Thirty-nine
David answered on the second ring. "Linden here."
"Hello, David."
"Ellie." His voice was unreadable. "How are you?"
I felt like skipping the ritualized dialogue that begins most conversations, but David probably needed the words--and the time. "Okay. You?"
"I'm good," he said. "It's pretty late."
I checked the clock. Almost midnight. "Sorry."
"So, what's up?"
Pass the ball. State your case.
"I--I need to ask you a couple of questions. About Abdul."
"Abdul?" Disappointment colored his voice.
I felt like I should apologize and irritated that I felt that way. "It's important."
He sighed. "What's the question?"
"How did he become your client?"
"What kind of a question is that? You know how it happened. You were there."
"He met you at the Greenbrier, was impressed by your credentials, and decided to hire you?"
"Well...basically, yes."
"Well then, why did you take him on?"
"What is this, the third degree?"
"I'm sorry. I guess what I'm asking is why you didn't delegate him to one of your staff."
"He needed someone with expertise in foreign exchange. I'm the head of the department. Ellie, what's this about? Have you been bothering him? Because if--"
My irritation grew. "Actually, it's the other way around. He called me a day or so ago. He was here."
"I told you before, he likes you."
"David." I hesitated. "I think he's a phony."
There was silence. Then, "What the hell are you talking about?"
"David, I don't think he's related to the Saudi royal family-- Fouad checked. And now I'm starting to wonder if he's really a businessman."
"Ellie. Stop. Don't go any further. Abdul and I are working on a major purchase. I don't need any interference. Especially from you."
"Okay. Answer this. Where are you in the deal, David? Has Abdul put up any money yet?"
"That's none of your damn business."
"David. I would never insinuate myself into your business dealings without a good reason, just like I know you wouldn't insinuate yourself into my work."
Silence.
"He hasn't put up a dime, has he?"
"We're not at that stage yet. You heard him at dinner a few weeks ago. We need to rethink a few things."
"David, he's stalling you."
"Why would he do that?" His voice sounded less certain.
"Because he isn't who he says he is. Because he's using you as a cover for what he's really up to."
"And what would that be?"
"I'm not sure."
I heard a long exhalation. "Ellie--"
"David, listen to me. I can't explain it. But the timing is suspicious. You meet him, and barely a week or so later, he's your client."
"It happens."
"And he just happens to want to buy a plant near Chicago?"
"Ellie..."
"Which forces him to come out here on a regular basis."
"So what?"
"How do you know he doesn't have some ulterior motive? That he's playing you?"
"What possible reason would he have to do that?"
I should have told him my suspicions. Explained everything that was happening. But I was afraid he wouldn't believe me. And given our problems, he might even think I was sabotaging him. I couldn't risk it.
"Remember how he played host at the Four Seasons? He was smooth, wasn't he? It made me wonder if he was a pro."
"A pro? At what?"
"Pumping me about the trial and the tape and the RF."
It takes David a long time to trust someone, but once he does, he's enduringly loyal. It would never occur to him to question a friend's motives.
"Ellie, what are you accusing him of?"
"Didn't you ever wonder how he knew about the abandoned coal mines at the Greenbrier?"
"What are you talking about now?"
"David. Remember the rafting? When I hiked back through the woods with Rachel? Abdul knew we would be passing the old coal mines in the area. Tell me something. How does a Saudi sheik know that?"
"Maybe he took a walk. Maybe he had been there before. What's your point?"
"Something isn't right. It hasn't been since I met him. I get the feeling he already knows the answers to the questions he asks. And now, I've discovered a connection between him and a woman at Great Lakes Oil who--"
"I would hope so."
"David, this woman doesn't have anything to do with his acquisition. She's in Training and Development. I did a video proposal for her, and I found out they've been in touch with each other. Then she abruptly cancels my video. It's all getting very bizarre."
He cut me off. "Ellie, I know things are not going well right now. I know you've got some problems. But this is off the charts. You can't make my client out to be some kind of scam artist. I won't permit it."
A flicker of anger pulsed through me. "Fine. Just remember it was our trip to the Greenbrier that started everything."
"What do you mean by 'everything'?"
"Everything you can't seem to handle." Damn. It slipped out.
"I see. We're back to me now. Ellie, I don't have to justify myself to you. Who I do business with is not your concern. If you want to pick a fight with me, you're going to have to do it another way." He cleared his throat. "Look, it's late. I need to get some shut-eye. I think this conversation is just about over."
"No, wait," I cried miserably. "I'm sorry. I didn't mean it."
"Then what do you mean?"
As an instrument of communication, the telephone has its limitations. "I--I don't know. Things are just--very strange. And I miss you."
He was quiet. Then, "That's not what I hear."
The bitterness in his voice jarred me. "Who have you been talking to?"
"You're not my only friend in Chicago."
"Rachel. You've been talking to Rachel. I can't believe it. You and she--"
"Ellie--"
"It is her, isn't it?" He didn't answer. "Tell me."
"No." His voice was soft but emphatic.
"Damn you, David. Someone is spying on me, and you won't tell me who it is? How dare you? I have enough trouble with that right now."
"Ellie, what--"
I couldn't take any more. "You know something? You're right. This conversation is over."
The phone hit the base unit with a thud.
Chapter Forty
After dropping Rachel off at school on Monday, I came home and went online. A quick thirty minutes on the net surfaced more than a dozen places in Chicago that offered scuba diving lessons. An equal number were scattered around the suburbs.
I sighed and started down the list. Many had already closed for the season, and their machines told me to leave a message. I frowned. That wouldn't do me any good. On the ninth listing I reached a human, but he didn't want to reveal anything about his customers and seemed annoyed I had the _chutzpah_ to ask. Another man accused me of engaging in industrial espionage.
I took a break and reassessed my methodology. Clearly, a different approach was required. I thought about it. Ten minutes later, I hung up the phone in triumph. It had worked; the person I called actually checked their customer database but didn't find any Sammys or Samirs. Still, I was buoyed by my progress. I refined my technique on the next call. Again, they checked their records, but no luck.
Finally, on the twentieth call, I reached a friendly female voice.
"Diving Unlimited."
"Hi," I said cheerfully, sliding right into character. "My name is Grace Barnett Wing. I work in the personnel department at Walgreen's."
"Yes?"
"I'm checking up on the application of a young man who says he took diving lessons from you." I heard the soft click of keys in the background. "We like to verify our applicants' extracurricular activities, as well as their professional ones. Would there be someone I could talk to about that?"
"Extracurricular activities? You're kidding."
"I wish I was. You can't be too careful these days, you know what I mean?"
"I guess."
"I know it's an imposition. But I really would appreciate the help."
"What's the name?"
"Well, you're not going to believe this, but I did a really stupid thing." I paused. "I spilled coffee all over his application, and it's kind of hard to read. And I know my boss'll kill me if I screw up. The guy's being considered for a management position."
She hesitated. "Well, what do you think the name is?"
"His first name could be Sammy. But then again, that could be his last name."
She was quiet for a moment. "Miss--what did you say your name was?"
"Grace Wing. Barnett Wing." Forgive me, Grace.
"Well, Miss Wing, if all you've got is one name, I don't know how I can help."
I lowered my voice to a stage whisper. "Well, I'm not supposed to say this--I'm sure you can understand--but he--um--he's definitely--well, we're pretty sure he's Arabic. You know, from the Middle East."
I heard an intake of breath.
"And, well, I was just wondering if you could check your customers for the Ss to see if, well, you know, you'd find any names like--"
"You say you're from Walgreen's?"
"Yes. The corporate office. I know I was careless, but--"
"Do you know when he signed up?"
"I'm sorry, but I don't."
"Sammy, you say?"
"Yes. But it could be Sam." I considered whether to tell her it was a common Arabic surname. No. Not good.
"Well, let me check last names first." More clicks. Silence. I held my breath. "No Sammy."
"What about Sam?"
"Nothing. We have a Samson, and a Samos, but nothing that looks Arabic."
I crossed my fingers. "Can you run down first names?"
"I don't know. That might be kind of tricky. I could try to pull up every Sammy or Sam we have on our database, but how would I know if it's the right person?"
"Maybe his last name will pop up, and, like I said--well, it might be obvious."
She sighed. "I suppose it's worth a try."
I heard the clicks of her fingers on a keyboard. Outside the sound of a truck roared down the block.
"Well, now, this is interesting."
My pulse picked up.
"When did you say he was here?"
"I didn't."
"I have a Samir Hanjour. He enrolled a year ago last spring."
"Really?"
"Yes. It looks like he took some lessons but never completed the course."
"I'm surprised. He seems like the type who would follow through."
"People stop for all sorts of reasons, you know. Sometimes their ears can't take it, you know. The pressure. Other times, they move, or their jobs change. It's not so unusual."
"No, I suppose not." I hesitated. "Tell me, is he the only diving student you had with the name of Samir, or Samman, or Sami?"
"Hold on." A few moments passed. "Yes. That's it."
"Then that's got to be the young man I'm looking for. The address I have looks like he lives in--well, I can't tell." I cleared my throat. "The coffee."
"We have him living in Orland Park."
"That's it. Yes. Where in Orland Park?"
She reeled off an address. I copied it down. "Do you want the phone number?"
"Sure."
She gave me a number with a 773 area code. I wrote it down.
"Oh, hold on. You know what? There's a "w" by the number I just gave you. I think I might have given you his work number by mistake. Do you want the home number instead?"
"Sure."
She repeated another number with a 630 area code.
"You've been wonderful. You probably just saved my job. I can't thank you enough. What's your name?"
"Mary. Mary Rhodes."
"Thank you, Mary. I'll be sure to note how helpful you were in our files."
"My pleasure."
As soon as I disconnected, I tried the home number, but it was out of service, and there was no forwarding number. Then I punched in the work number. After five rings, a man's voice picked up.
"Yeah?" Gruff. Breathing hard. I'd pulled him away from something.
"I'm trying to reach Samir Hanjour. Is he there?"
"Who?"
I repeated the name.
"There ain't no one by that name here."
"Oh, dear. Maybe I have the wrong information. I thought he worked there."
"Well, maybe he did, but he don't no more. I never heard of 'im."
"I'm sorry to have disturbed you. This--this is--Walgreen's, isn't it?"
"Walgreen's? Lady, you got the maintenance room at People's Edison."
People's Edison? The huge utility that provides most of Chicago's power? "Oh. I'm terribly sorry. I must have the wrong number."
I carefully put the phone back on the base. I picked it up a second later and called People's Edison's corporate headquarters and asked to be connected to personnel. A moment later an officious voice told me there was no way she could release any information about PE employees unless I had clearance from her department head. I thanked her and hung up.
I stood up and started to pace. An Arabic man named Sammy took scuba diving lessons last year. Apparently, he also worked at People's Edison. Or did when he started the scuba lessons. I wondered if he drove an SUV.
***
I attacked the lump of dough with a rolling pin like a tiny steamroller. The dough bulged, cracked, and finally surrendered to a higher force. Once it was uniformly thin and even, I transferred it to a nine-inch pie plate, trimmed off the extra, and fluted the edges. I rotated the plate and smiled. Martha Stewart had nothing on me. I was starting in on the filling when the phone rang.
"Hello?"
Silence.
"Hello?"
A click. I hung up and wiped a floury hand across my brow. A wrong number. That's all it was.
I finished the filling and put it in the fridge. Then I rummaged in the cupboard for onions. As long as I was feeling domestic, I should get a head start on the stuffing.
Damn. I was all out. But it was barely one o'clock. I threw on a coat and grabbed my keys.
I noticed the SUV on the way home from the store. A hundred yards behind. It was still there when I turned onto Happ Road. Two figures were inside. Men.
Fear skittered around in me. I pressed down on the gas and sped past my block, praying that the cops who hide at the side of the road were there. But they must have been taking the day off. The SUV accelerated and kept pace.
My fear spread.
I got to the end of Happ, careened around Sunset Ridge and onto Voltz. I checked the rearview mirror. Nothing. But Voltz twists and turns and cuts off your sight line. At Lee, I turned right and raced toward Shermer.
I needed to find someplace safe. Someplace no one could get to me. The mall? No. Too big. Too isolated. Too many empty corridors. The library? It was close by, and it was my sanctuary as a child. But it had been remodeled recently; there were lots of small study rooms and cubicles. I needed a place where everything was out in the open. Where there were people.
I was still deciding when the SUV reappeared in the mirror. Closer now. Shortening their rope. My heart hammered in my chest. I flew across Shermer, then Dundee, and sped back to the grocery store. I tore into the parking lot, threw the car in park, and sprinted through the door.
My breath was ragged, and I was trembling. Positioning myself so I had a clear view of the front window, I walked up to one of the checkers, a woman I've known for years. I hugged my arms across my chest.
"What'd you forget this time?" she smiled, then took a closer look. "Hey. Are you all right?"
"Couldn't be better." I tried to take a long, cleansing breath. "How's the handicap?"
"My handicap?"
"Yes," I panted. She was a golfer.
"Good," she said uncertainly, as if she had no idea where I was coming from but was too polite to say so. "I shaved off another stroke this summer."
I looked out the window. The SUV had pulled into the lot and was inching down the lane where the Volvo was parked. I jumped back from the window and said a prayer. The SUV slowed, stopped, and then slowly pulled away.
"That's great, Debbie." I blew out a breath. "Just great. Golf sure is a great sport."
***
I wandered through the supermarket aisles, thinking I'd hide out there until it was time to pick up Rachel. I was stunned to find heart of palm was over three dollars a can; a tiny jar of caviar was only six. I wandered over to the candy aisle. More my style anyway, but even here, the prices were up to nearly a dollar a bar.
As I scanned the array of brightly colored packages, a familiar itchy feeling rose in my throat, and it dawned on me that a grocery store was not a good place for me to be right now. I felt alone. Defenseless. Out of control. It would be easy to find myself with a case of sticky fingers. I forced myself to walk to the coffee bar at the front of the store, where I bought a latte and made myself sit to drink it.
Once I had Rachel, I drove down to Skokie, taking Hibbard and Illinois instead of the expressway. Every few yards I checked the rearview mirror; no one was tailing us.
"Where are we going?" Rachel asked as we wound through the quiet streets.
"To Dad's."
"Is _Opa_ okay?"
"He's fine. I--I just want to check up on him. "
"Oh." Rachel seemed abnormally quiet, and I wondered whether I'd subconsciously projected my fear onto her. I needed to be more careful. As we turned onto Hunter, we passed a yard already crowded with Santas, candy canes, and a large sleigh filled with packages.
"Look." I waved. "It's not even Thanksgiving, and they'll probably leave them up until February."
Rachel didn't say anything.
"If we can live through this," I cracked, "we can live through anything."
Rachel recoiled as though I'd struck her.
"Christmas, honey. The decorations."
She burst into tears. "I don't want to go to _Opa '_s."
"Rachel, what are you talking about?"
She sobbed. "He's going to yell at me. And so will you."
"Oh." Now I knew. I pulled to the side of the road. "Honey, that's not it."
Her sobs grew louder. I drew her into my arms. She threw her arms around my neck and buried her face in my shoulder.
"I thought--I thought I was going to jail, Mommy." She wailed.
"Shh." I brushed my fingers across the curls framing her forehead. When she was little I always thought she looked like one of those angels with golden halos. "It's over now, honey."
A few minutes passed. Her sobs began to hiccup. "They--were--so--mean."
"Officer Davis was mean?"
"Not--her." She sniffled. "She was--okay."
I thought she was okay, too. Better than okay.
"The others. The ones who arrested me." She took a shuddering breath. "They told me if I got into trouble again, I'd go to juvenile detention. They treated me like I was--like--I--was a--a--" She started to tear up again.
"A criminal?"
She nodded, her eyes glassy and wet. "When we got to the station--they took our fingerprints--and then they put me in that cell--and--they handcuffed me to the wall."
I winced. I remembered the time I was arrested for shoplifting. How frightened I was. How ashamed. How alone. I hugged her tighter.
"Then they asked me all these questions. But in a really mean way. They kept saying they knew someone at school was dealing, and I had to tell them who it was. And then--" She stopped short, a horrified look on her face. "Mother, are they going to tell the school what happened?"
I pushed an unruly curl behind her ear. "No. The school doesn't know anything about it."
"What about _Opa_?"
"I haven't told him."
"Mommy...please...don't."
I looked over. "I won't. Unless you say something first."
"Never." She shook her head and sniffed. "Never." She looked up. I saw the determined tilt of her chin. "I never want to see Carla again. Even if I have to make all new friends."
I forced a smile. "How about we talk about it over the weekend? I don't want you to forget, but I don't want it to ruin Thanksgiving. We'll figure out how to keep our noses clean after Thursday."
"Our noses?"
"Ours," I said, silently thanking God hers was on just fine and that she seemed to have survived her ordeal with only minor damage. "Yours and mine. I'd like to spend more time with you."
She nodded and wiped her eyes with her hands. For the first time in days, the hint of a smile cracked her face. "Mom?"
"Yeah?"
"Do you think I could set up a chemistry lab in the basement?"
Chapter Forty-one
"My favorite girls!" Dad swung open his door. "What a surprise."
"We just happened to be in the neighborhood..."
Dad squinted as we trooped inside. He knew I was lying. "Are you okay?"
"We're fine," I said hastily, exchanging a glance with Rachel. "We--er--wanted to have dinner with you."
He looked at me, then Rachel. "Chinese?"
Rachel nodded eagerly, and Dad went into the kitchen to hunt for the take-out menu. Rachel took off her coat and plopped down on the couch.
I prowled around the apartment. With only two rooms and a kitchen, it didn't take long to make a circuit.
"Sit down, Ellie," Dad said as I passed the kitchen. "You're making me nervous."
I sat at the dining room table. Dad brought in the menu, and after a group consultation, called in an order of egg rolls, sweet and sour chicken, and lo mein with Cantonese noodles. "Can't fill up too much," he winked at Rachel. "Not with turkey day coming up."
My cell phone trilled. I jumped up and dug it out of my purse. "Hello?"
There was no response. "Hello?" Silence. "Damn it." I looked around. "No one's there."
Rachel and Dad watched me with curious expressions. I looked back at the cell, hoping a "missed call" display might pop up along with the number. Nothing. I shoved the cell back in my pocket.
"How about a game of chess?" Dad asked.
"Cool." Rachel went to the cabinet, pulled out his chess set, and proceeded to set it up on the table.
"I'll skip this round," I said.
Dad nudged Rachel. "A comedian, your mother."
Rachel giggled.
I went to the window. It was close to five, but the skies, swollen with thick gray clouds, were more luminous than usual. A snowstorm was coming. For real, this time. I looked back at Rachel and Dad, engrossed in their opening moves. I ducked into the bedroom.
"Ellie, your _schpilkes_ are driving me crazy."
I came back out. I was driving myself crazy. "Why don't I go pick up the food?"
"Awesome," Rachel said. "I'm starving."
Dad stared at me through his glasses. "We could have it delivered."
I felt around in my bag for my cell phone. "I need some air. It's okay."
"You sure?"
I nodded. I headed to my car, trying to be aware of everything in front of and behind me. Five painted rocks bordering the lawn. Four cracks in the sidewalk. Two streetlights angling in on the lot. I started to count how many cars were there, but lost count when I dug out my keys.
As I fitted the key in the lock, I felt a sudden presence loom over me. Closing in fast. I didn't have time to get in the car. What should I do? My key! I'd rake the car key across his face. When I sensed he was almost on top of me, I threw my hand in the air and whipped around.
LeJeune caught my wrist.
I staggered back. "Jesus Christ!"
"I wouldn't go that far, _ch er_."
He was wearing a dark, bulky parka, and his "Different Drummer" hat was pulled low on his face. But his eyes smiled down at me.
"Damn you!" I waved my keys. "You almost lost your smooth Cajun skin!"
He loosened his grip on my wrist. "You do have a way with words."
I shook off his hold. How dare he act as if he was just casually dropping by? As if nothing was wrong, the past week never happened?
"How did you find me?"
"The Bureau has its ways, _ch er_."
I didn't know whether to curse him or just walk away. I started to open the car door, but now that he was back, the fear, the not knowing, the sense that things were closing in on me--it all suddenly seemed to be too much. My composure snapped.
"Oh God, Nick." My voice trembled. "I've been so alone. And scared. I'm being followed. And I don't know who or--" Burying my face in his coat, I started to cry.
He waited patiently, his arms around me, until I calmed down. When there was only a sob or two left, he tilted my chin up with one hand and brushed away my tears with the other. He leaned over, and the next thing I knew, his mouth was on mine. Doing things I hadn't felt in a long time.
***
As we drove to the Chinese restaurant in the Volvo, I wondered what had just happened between us. But he didn't say anything, just looked through the windshield with a half-smile on his face. Maybe it wasn't that important to him. Just the cost of doing business. It was probably in the FBI handbook: kiss hysterical woman, calm her down, then get what you need.
Whatever it was, we'd have to sort it out later. There were more important issues at hand. I told him the SUV was following me again. I also told him how I'd shaken it--for the moment. He nodded but didn't ask any questions. I wondered why.
"What am I going to do?" My voice sounded shrill as we parked and headed inside. "I can't go back home tonight. It's too dangerous."
"I know."
"You know?" I looked over. "Damn you again. If you know I'm in danger, where the hell have you been for the past week? Didn't you get my messages?"
"I got them."
"Then why you didn't call me back? I might have been--Rachel and I might have been--"
As we reached the door to the restaurant, he cut me off. "I was out of the country. I couldn't talk on an unsecured line."
We pushed through the door. Basically a carry-out, the restaurant was small, with a high counter that stretched across two thirds of the room. Three small tables sat in front. The sound of splattering oil drifted out from the kitchen, and the scent of Asian spices permeated the air.
"When did you get back?"
"This morning."
Now that we were in the light of the restaurant, I saw the stubble on his face and the dark pouches under his eyes. When he realized I was checking him out, he dipped his head. I checked the bag of food on the counter. The name Forman was scribbled on the receipt. They always forget the e.
I gestured to the bag. "You want something?"
"Just coffee."
I nodded at the proprietor, who filled a plastic cup and handed it to LeJeune. As he took it, his movements seemed jerkier, less fluid than usual. A subtle tension seemed to have come over him.
After paying, we headed back to Dad's.
"Nick, I need to tell you what's been going on."
He sipped his coffee. "This is good. They still don't brew coffee right in London."
"London? You were in London?" I stopped at a red light. Dale Reedy was from England. I thought about the length of time he'd been gone, what had happened before and since. When the light changed, I said, "You're on an antiterrorism squad, aren't you?"
He looked at me for a long moment. Then he nodded.
"Why didn't you tell me?"
"I couldn't." He leaned his arm across the back of the seat. "A few months ago we received credible information from Saudi intelligence about a planned terrorist attack in the Midwest. Something specifically involving water. It was confirmed by the Mossad. And British intelligence. They said it would go down after the verdict."
"What verdict?"
"The guy who's on trial now. If he's convicted--"
"Which, in all likelihood, he will be...."
He shrugged. "Yeah. Well, if he is, there's supposed to be a nasty surprise afterwards."
"In Chicago?"
He nodded.
"Why here?"
"Why not? Chicago's been relatively unscathed so far. It is the Second City. And we have reason to suspect there's a sleeper cell here."
"This is all connected to the RF on my tape, isn't it?"
"Yes."
I parked behind Dad's apartment and switched off the engine. I sat very still. "How did you find out? That it was connected to me?"
"We didn't. Not at first. But you testified at the Santoro trial about RF interference. Out on the water. In the Midwest." He shrugged. "It just seemed like something we ought to check out. Especially after we got word you were taking rides with Outfit guys."
"What did you find out in London?"
He motioned to the bag of food. "Why don't you drop that off?"
I hesitated. Dad had never met LeJeune, and, aside from his car, Rachel wasn't too fond of him. But now that he was back, I didn't want to let him out of my sight. I wrestled with what to do. "Do you want to come up?"
Nick must have sensed my indecision. "Why don't I wait here."
Relieved, I opened the door and took the bag up to Dad's. I told him I wouldn't be staying.
"Why?"
"It's kind of hard to explain."
"Try."
I told him how LeJeune found me in the parking lot. "We--we have some things to work out."
Dad fixed doubtful eyes on me.
"Business," I added hastily.
Dad took the bag of food. "Be careful."
I gave him a hug. "I will. I'll be back."
LeJeune was on his when I got back to the car. I climbed in. When he was done, he stretched out his arm on the seat and nuzzled my neck
"Come here, _ch er_." His voice was hoarse.
I looked over. I saw in his eyes that he wanted me, and I realized, with a jolt, that I wanted him to. He moved close and traced a finger down the side of my face. My stomach fluttered.
"I've been meaning to do this for a while."
I tried to steady myself. "Before we get--distracted..." I pushed his hand away. "...there are things we need to discuss."
He curled his fingers around mine. "Like what?"
"Like Sammy. I think I know who he is."
"Sammy?"
A few droplets of sleet spattered the windshield, fat and heavy. I switched on the wipers. "The guy who was at Calumet Park the night Mary Jo Bosanick was killed? The one who came in on the boat? I think there's a link between him and the intake crib."
LeJeune cocked his head.
"His name is Samir Hanjour. He lived in Orland Park. He was enrolled in a scuba diving course at Diving Unlimited. But he dropped out."
He sat straight, fully alert now. "How'd you find that out?"
I told him about the calls I'd made. "There's more. There's a woman involved. A British woman who works for Great Lakes Oil."
The lines on his forehead drew together. "What do you know about her?"
I told him about Dale Reedy and the wire on her window. I'd barely finished when he punched in a number on his cell and repeated what I said.
"Yes. Could be. Get a team out there ASAP and get back to me." He paused. "Have them meet at the police marina." He listened. "Call him." Another pause. "And we got a possible ID on the cousin. Samir Hanjour."
"Cousin?"
He held up a finger and repeated the address in Orland Park. Then he snapped off the phone. "You've been a busy woman."
I shot him a look. "Are you patronizing me?"
"No, I assure you, I'm not."
"Then, what is this about a cousin? And why do I get the idea you already know about Dale Reedy?"
"We didn't know about the antenna."
"But you do know about her."
"That was one of the reasons I was in London."
"But you couldn't tell me."
He didn't answer.
I crossed my arms. "Let me see if I get this. You've known about her--since when?"
"We got the first intelligence in May."
"You knew she might be part of some terrorist action, and you let me deal with her anyway?"
"We hadn't confirmed it, and by the time we did, she'd already made contact with you."
Snow started falling in earnest. I turned the wipers on high. "So you let me go ahead and risk my life doing business with her?"
His eyes flashed in the dim light. "Tell me something. Would you have broken off with her if we'd asked?"
He had a point.
"You tipped us off to her, anyway."
"Me? How?"
"When we first came to your house, you said something about a call from an executive from Great Lakes Oil. Look, Ellie. We didn't know anything concrete. It was only when she surfaced a second time--through you--that we started to piece it together."
"How did she surface the first time?"
He shook his head. He wasn't going to tell me.
I tried another tack. "So you went to London because of Dale Reedy?"
"That's right. Turns out the woman's got quite a track record. Fringe human rights movements. A real left-winger. Came to the attention of Scotland Yard when a bomb went off in Grosvenor Square fifteen years ago--she was indirectly involved. But then, she suddenly turns herself around. Changes her name. Gets married. Takes a straight job. Starts working her way up the corporate ladder."
"She wouldn't be the first. Look at Jerry Rubin."
"Jerry Rubin wasn't married to a Saudi Arabian expatriate."
"What?"
"Dale Reedy, aka Darlene Eaton, is married to a suspected terrorist by the name of Dani Aziz. British intelligence has been looking for him for years. But he's slippery; he stays underground. Always travels. Meanwhile, she's here. And their kids live with her folks."
I remembered the photo in Dale's office. Two boys in soccer uniforms. Cute. Dark hair. She'd never mentioned a husband. I'd assumed she was single.
I felt stupid. "Are you saying her husband, this Dani, is Samir?"
He shook his head. "Pakistani agents saw Aziz in Peshawar last month. But he has a cousin. And no one has seen him for over a year."
I paused, trying to assimilate the information, but I kept coming back to a question. "Why are you telling me all this now? What's changed?"
"You've been followed, right? By someone in an SUV?"
At my nod, he pulled out a piece of paper and unfolded it. "Turn on the dome light, _ch er_, and take a look at this."
I stared at the scan of a photo. It was a lousy quality. Grainy. High contrast. Probably a copy of a copy of a passport picture. My stomach lurched anyway. The dark eyes. Mediterranean features. The cold expression. "I know him," I whispered.
LeJeune's eyes burned into me. "Is he the one in the SUV?"
I shook my head. "He was at Santoro's trial the day I testified."
"You're sure?"
I remembered how he looked at me as if I were some inanimate object. A piece of garbage to be disposed of. I shivered. "He was sitting in the row behind my father."
LeJeune reached for his cell phone.
I reached across to stop him. "Wait. I'm not finished. I think there's a link between Dale Reedy and the financier of the thing. Whatever the thing is."
LeJeune reached for his coffee instead. "Financier?"
"Abdul Al Hamarani. He tells people he's related to the Saudi royal family. Spreads money around as thick as butter. Stays in fancy hotels. His cover is that he's buying a chemical plant from Great Lakes Oil."
"Abdul, eh?"
"Abdul Al Hamarani. He's a client of my--of a man I know. He and Dale Reedy have been in contact." I explained how I found his number on her pad of paper.
Before he had a chance to reply, his cell phone trilled.
Chapter Forty-two
LeJeune stared through the windshield, his answers short, deferential. A superior giving him orders. When he was done, he twisted around.
"Our men found an antenna on the crib. I have to go. You'll have to--"
I grabbed his arm. "I can't--you can't leave. Not again. Not with Samir--"
"But I can't--" He checked his watch. "Shit. There's no time. I need to borrow your car."
"Only if I'm in it."
"But I can't--"
"Nick..."
He looked through the window, then at me. "Okay. Let's go."
Relief and fear swept through me simultaneously. An odd duality of emotions. "What's going on?"
"I don't know." His face was grim.
I called Dad as we pulled away and told him where I was going. He didn't say much. Then, "Rachel can stay here. I'll wait up."
"I love you, Dad."
A mixture of snow and sleet fell as we slogged through traffic. The streets were slick, but rush hour was at its peak. I wiped the inside of the windshield with my sleeve. LeJeune kept up a fast tap on the floor.
Over an hour later, we parked downtown near the police marina. One of their boats took us out, but the ride out was nothing like I remembered. A bitter wind raked the lake's surface, turning my face numb in minutes. My stomach pitched with the waves, and for the first time I could understand how the _Edmund Fitzgerald_ got into trouble. By the time we stepped onto the Carter-Harrison intake crib, almost two hours had passed.
The crib was swarming with men, most of them in FBI jackets. A complement of Chicago police officers was there, too, and a few others, I guessed, from the water department. Arc lights were strung up, and the snowflakes caught in their glare looked iridescent. A boat, which might have been Coast Guard or possibly military, was anchored a few yards away with tanks and scuba diving equipment on its deck. Funny. Rhonda Disapio was right. From this distance, they did look like logs. Metal fireplace logs.
I peered into the lake, watching snowflakes dissolve and disappear into murky black water. What was going on? Was something hidden in its depths?
LeJeune joined a group of men at the limestone and brick structure. Some of them glanced my way. Feeling self-conscious, I studied a bronze fish that sat like a gargoyle on top of the limestone wall. Flakes of snow blew into my face.
Two men up on the suspension bridge pointed at something. I squinted, trying to see. It was a set of double windows near the top of the pink and white structure.
LeJeune came over. "They turned off the pumps."
"Why?"
"So we could send divers down in the candy striper."
"What for?"
"So they can find whatever the antenna's attached to."
"Where was it--the antenna?"
He pointed up to the bridge. "It was attached to the wall. Just above those windows. Next to the suspension bridge."
The suspension bridge. "I left the damaged cassette on the bridge. Right next to the candy striper," I said slowly.
"Right," LeJeune said.
A swell of noise on the other side of the crib distracted us. A couple of men gestured. LeJeune went over to listen. Then he got on his cell. He came back over, his face unreadable. "The divers found something."
I tensed.
"We're gonna bring in some help. You're gonna have to clear out."
I started to object, but he cut me off. "Go home. I'll call you later."
I shook my head.
He looked over at the men, then at me. I sensed him come to a decision. "Okay. There's a white van parked over at DuSable Harbor. No one will be in it. Wait for me there."
I nodded. "Who does it belong to?"
"A friend."
"There's no way you can come?"
He shook his head. "Not yet."
"Are--are you going to be okay?"
He brushed a hand across my cheek "You can count on it, _ch er_."
Twenty minutes later a marine police boat docked at the crib, and half a dozen men in bulky dark hazmat suits and spacesuit helmets disembarked. Seven of us, including the cops and the men from the water department, took their places, and we motored back to shore. We crowded into the semienclosed cabin behind the pilot's chair to keep warm. Nobody talked. As we approached shore, the Great Lakes Oil building loomed over the cityscape, its pale walls a mosaic of reflected light.
We docked at the police marina, and a cop walked me over to DuSable Harbor. A white van was parked on the semicircular drive. Four stubby antennas protruded through a metal plate on its roof. The plate looked like a stop sign laid horizontally. Two more antennas stuck out from other spots on the roof.
"What is this?" I asked the cop who'd walked me over.
"Don't ask me. Some kind of radio gear, I think."
"You sure I can go inside?"
The cop motioned to the police boat that had ferried us ashore. It was just backing out of the marina. "The guy who it belongs to just hopped a ride out."
There was no answer when I tapped, so I slid the door open. A beam of light spilled out of a tiny desk lamp clamped above the driver's seat, but most of the van was in shadows. There were no seats in the back, and the space was crammed with equipment. I saw VU meters on a few pieces. Speakers hung on both sides of the wall.
The only other light was a greenish hue from a laptop on the floor of the van. I crawled over and saw a green bull's-eye, almost chartreuse, with brighter green circles inside it. A bright green splotch in the center looked like one of those TV radar maps of a storm, except here a dotted radius ran from the center of the splotch to the circle's circumference. Numbers and words, including Display Source, Sector, and Decay Rate, appeared around and on the circle. I had no clue what they meant.
The interior of the van gave off a slightly stale odor, but compared to the crib, it was warm and dry. I hunkered down behind the front seat. The window was streaked with sleet, but I thought I saw a large boat move slowly past, its dark shape massed against the darker black of the lake. A metal chain clanked in the distance. Despite the tension, or maybe because of it, my eyes felt heavy. I yawned.
The next thing I knew, the van door was opening, and a blast of cold air rushed in. I startled awake to see LeJeune.
"Getting your beauty sleep, _ch er_?"
"What--what's happening?"
He climbed in and brushed his lips across mine. His jacket smelled fishy, but his lips were soft. I closed my eyes and kissed him back.
When we broke apart, I was breathless.
He grinned. "For a welcome like that, I'd go back and do it all over again."
Before I could answer, the door slid open again, and someone else climbed in. A man crawled past me, settled himself in the front, and turned the tiny desk lamp to high. I blinked in the harsh light. The man was in his twenties, I thought. He was wearing a blue warm-up suit with a white stripe down the side, but a thickness around his middle implied the clothes were just for show. A headband around his forehead held back a mane of curly, dark hair.
"I'm Clarence." He dipped his head. "A friend of Drummer's."
"Drummer?"
He pointed to the words _Different Drummer Fishing Charter_ on LeJeune's hat. Now that I was thinking about it, I'd never seen him without it.
"Are you with the FBI?"
"Sometimes."
I leaned up against the side of the van. "Why is it I can't ever get a straight answer from any of you guys?"
Clarence cleared his throat and looked at LeJeune. Then he crawled over to his laptop.
"You will this time," LeJeune said. "I want to tell you what we found." He took a breath. "It was a watertight, hermetically sealed box. The size of a suitcase--maybe thirty-six by twenty-four by eight. When we opened it up, we found two compartments. One contained radio equipment: a small transmitter, a receiver, and built-in power source. The other held--" His face was impassive. "An explosive device."
"A bomb?" I clamped a hand over my mouth.
He nodded. "Don't worry--it's been disabled by now." He flicked his eyes over to Clarence. "But--" He faltered. "--it was nuclear."
I bit down on my hand to keep from crying out. I'd heard about suitcase nukes. Small nuclear bombs. Both the Soviets and the U.S. made them, but some had gone missing when the Soviet Union collapsed. Experts feared they'd ended up in the hands of terrorists.
"Was it--was it--one of the Russians'?"
"We don't think so." He shifted. "Let me rephrase that. It's unclear if any Soviet nuclear tactical weapons would even work after twenty years. They need regular maintenance and upkeep, which, given what's going on in that part of the world, isn't happening. But someone may have gotten one to use as a prototype. Or maybe they built one from scratch."
"That's possible?"
"Given enough money, there are plenty of disaffected Pakistanis, former Soviet nuclear scientists, even Iraqis, who would do it in a heartbeat."
"I thought the technology was way beyond--well, too sophisticated for terrorists."
"The hardest part is getting weapons-grade uranium. We've heard rumors it's been coming out of Turkey." He waved a hand. "Who knows? Assuming you can get your hands on some, you can cut corners, and--well--it can be done."
I felt sluggish and heavy, as if I was trying to tread water but was sinking into its depths. I wondered if I was in shock. "How small?"
"Excuse me?"
"You said it was a small device. How small?"
"It's just a guess at this point, but probably less than a kiloton. One fifteenth of what they used at Hiroshima."
"But powerful enough to take out a couple of city blocks," Clarence said.
"Or the water supply of Chicago," LeJeune said.
"That's what they were doing? Sabotaging the water supply?"
Clarence and LeJeune exchanged another glance.
"What? What is it? Why are you looking at each other like that?"
"Because that's the good news," LeJeune said. "If it had detonated, the radiation would have made parts of downtown Chicago uninhabitable." He paused. "For at least a century or two. And, if the wind was blowing the other way, the lake would be poisoned for about that long."
I opened and closed my mouth like a fish, half expecting him to break into a grin and tell me this was all a joke. A prank he and his Bureau buddies were playing. His expression was hard as granite.
"That's just for starters," he went on. "A blast like that, if it had gone off in the Loop, would incinerate anyone within a one-block radius. A quarter mile away, over 250,000 people would die within a day from radiation sickness. A half mile away, you still have thousands dead. Within five to ten miles, the environment would be irreversibly poisoned."
"Did you know there's only one fucking hospital in the entire country that knows how to deal with radiation sickness?" Clarence said. "And that's in Tennessee, for Christ's sake."
"There's something you can take to ward it off, can't you?" I asked.
"Iodine tablets," Clarence said. "But they only work when you know it's coming in advance. And even if you knew, how are you gonna get enough to everyone in Chicago?"
"But they didn't target the Loop," I said. "They sank it on the crib."
"I guess we can be thankful for that," LeJeune said. "Although obliterating the water system is plenty serious. Humans can't survive without water more than three days. What happens when all the bottled water's gone?"
I pressed my lips together.
"Order would break down. You'd have looting. Panic. Chaos. Hospitals would be overwhelmed. And don't forget downtown Chicago's evacuated. Abandoned. No commerce. No transportation. Nothing. For decades to come." He shook his head. " _Ch er_, you're looking at something that would make September eleventh look like a birthday party."
I covered my face with my hands. A tenuous silence settled over the van, broken only by the whine of the laptop.
LeJeune gently pried my hands away from my face. "But that's not going to happen, Ellie. None of it."
I looked up.
"You know why?" He tipped up my chin with his hand. "Because you came forward at that trial."
"The RF," I said softly. "On my tape."
"That was our break." He motioned to Clarence. "Tell her."
"It wasn't just a simple transmitter and receiver. It was a sophisticated packet radio setup. We found gauges that indicate they were monitoring the internal environment of the box and reporting all that data back. Temperature, humidity, pressure, battery strength. Other stuff, too. "
"That was all transmitted back to the Great Lakes Oil building?"
"Yeah. But who knows where it went from there? That's the beauty of it, see. The scientists monitoring the box--or the guys with their finger on the button--could be anywhere. Chicago, the Middle East, Asia. All you need is a computer and a modem."
"But we shot out at the cribs over a year ago. Are you telling me the suitcase has been underwater since then?"
"Looks that way," Clarence said.
"So it was planted before September eleventh."
LeJeune nodded.
"How did they get it there?"
"Probably brought it in through a port. In a steel container. Then barged it up the Mississippi."
I felt my eyes grow round. What if Santoro had offloaded it? What kind of irony would that be?
"Where is Dale Reedy?"
"We're looking for her," he said. "She won't get far. We have a team on their way over to Great Lakes."
I rocked forward and hugged my knees. "I don't get it. How could no one have found the antenna on the crib before now?"
"There weren't many people out there, even in summer. And remember, you can't find something you're not looking for. They used thin, flexible conduit. Against a surface, it's almost invisible." His hands sketched out the path in the air. "They ran it from the pit of the candy striper, up the wall, and out the set of windows above the suspension bridge. The antenna itself was less than six inches long."
"But you went out there to look around."
Clarence answered. "We used the van for a couple of hours, then took a field strength meter out on a boat, but we weren't out there long enough. Looks like the transponder woke itself up every six or eight hours to transmit or receive a signal. No way we would have caught it."
"But we did," I said, "because we were shooting out there over ten hours."
Clarence aimed a finger at me. "Exactly."
"How did they power it?" I asked, thinking back to my conversation with Hank. "What kind of battery lasts almost two years?"
"A fuel cell battery," Clarence replied. "They use 'em on the space shuttle. They're just starting to show up commercially. They convert small amounts of fuel into electrical energy. Make a power source that lasts for years. Somebody built one into the suitcase."
"How much you want to bet Samir studied electrical and computer engineering at DePaul or IIT?" LeJeune said.
"In between his scuba diving lessons," I said.
"It was his job." LeJeune shrugged.
"But how were they able to sink it on the crib without anyone seeing?"
"Before September eleventh, security on the cribs was a joke. Kids used to swim out there, smoke weed, dive off the side at night. And in winter, there were weeks when no one was out there at all." He smiled thinly. "Hey. You bring everything out on a boat late at night, break into or dive down in the candy striper, hook up the cable and the antenna, then sink the box. No big deal."
I rocked back on my haunches. "They accounted for every contingency," I said bitterly.
"Except one. They never expected your videotape would end up near their antenna."
I shook my head slowly. "It was luck. Blind, stupid luck."
LeJeune smiled. "My daddy always says luck is 'Labor Under Correct Knowledge.'"
Chapter Forty-three
I didn't want to stay in the van. I wanted to go home to Rachel and Dad. I was just about to ask Clarence to drop me off at my car when LeJeune's cell buzzed.
He picked up. "Yeah. Got it. Okay." He turned to us. "The bomb squad finished disabling the device. They replaced it with pipe, and they're taking it to the lab."
"Thank God." I slumped against the side of the van.
LeJeune pocketed his cell. Clarence started to fiddle with a plastic box about the size of a paperback book.
"What's that?" I asked.
"A display unit."
"What does it display?"
"If I tell you, I'll have to kill you." He glanced at me. "Just a joke," he said with an edgy laugh. "It's part of the Doppler direction finder. It helps detect the direction that a radio signal is coming from."
"Is that metal stop sign on top of the van part of it?"
"Drummer said you were smart." He nodded. "The Doppler is mostly used by amateur radio buffs--you wouldn't believe the games they play with it--but it does come in handy in situations like this."
"But I thought you said it didn't work the last time."
"That's because we weren't listening long enough, and we didn't know the frequency. But now..."
"You have the frequency?"
"It was on the transmitter. They were using an out-of-the-way ham radio band. In the 220 megahertz band." He looked at me over his laptop. "Which was smart."
"Why?"
"Less chance of being picked up by people like me." He went back to his toy.
"So what are you doing now? I thought it was all over."
He looked up. "It is."
"Then why are you setting up more equipment?"
"Uh--uh..." His voice trailed off, and he threw a glance to LeJeune.
LeJeune's jaw tightened.
It occurred to me that since his last call, LeJeune was preoccupied and distant. I was willing to chalk it up to his reluctance, maybe his inability to express emotion, but now I wasn't sure. The danger was over. Why wasn't he more relieved? Where was the cocky FBI shtick? I reviewed what he said after the call. They took out the bomb, he'd said, and replaced it with pipe.
"They replaced the explosive with pipe," I said slowly. "Why did they do that, Nick?"
Clarence moved to the front of the van and turned over the engine. We pulled away from the harbor.
"Why did they replace it with pipe?"
After a long pause, LeJeune answered. "So we could drop it back in the crib."
Why were they doing that? They should have removed the bomb and disconnected the radio. Stripped everything down. Shipped it to the NSA or CIA or whoever did the kind of analysis they needed. I felt a bite of anxiety.
"Why?"
Clarence made a wide turn and headed west.
LeJeune seemed to be choosing his words with care. "We have a line on Reedy. But we still want to flush out Samir."
"Samir? He's probably on his way back to Saudi Arabia or Yemen."
"Not necessarily. He might not know we've disabled the device. But even if he does, he might stick around."
"What do you mean?"
"Ego. It was his thing. He wants to see it through."
"So why not let Dale Reedy lead you to him after you pick her up?"
"There's no guarantee she'll cop to anything when we find her. Remember, the asshole from the World Trade Center--the one who's on trial now--still isn't talking." He went quiet.
I didn't like the feel of it.
"Ellie," he said slowly, "we need you to help us out."
I sat very still. Snow thudded against the windshield, splattering into tiny craters on the glass.
"Samir thinks you know what's going on. We want him to keep thinking that."
"You want him to think the bomb is still there?"
"We want to keep the signals going. So we can flush him out. And you're the best way to make that happen." He leaned forward, his voice perversely soft and sweet and full of Cajun lilt. "We want you to go back out there. Pretend you're doing another video for the water district. A sequel."
My jaw went slack. "That's crazy. No one goes out there this time of year. He'll know it's a setup."
"Not if the water district announces they've decided to finish the video they started last year. And that they've rehired you to produce it."
"Who's going to believe them?"
"We only need to convince one person."
I sat in stunned silence. He must have taken it as acquiescence, because he leaned in closer.
"But, even if he doesn't believe it, he can't afford to let it happen. He can't risk any more attention focused on the crib."
"You want to use me as bait."
He didn't answer.
I scuttled away from him. "You want to use me as bait," I repeated.
He acted as if he hadn't heard me. "We're pretty sure we didn't miss any cycles, and unless he's got a spotter out there, which is almost impossible, given that the crib is a few miles offshore, he's not gonna know we intercepted the bomb.
"On the other hand, since we exchanged one material for another, the data values that are transmitted back to them may change."
My body itched with anger.
"With lead pipe in there instead of a demolition charge, the internal environment--the pressure, the temperature--will be different. That's going to confuse them. They'll be anxious. They'll want to know what's going on. Obviously, they can't go out and check it themselves. So the fact that you are going out there will make them crazy."
"Great. Why don't I just paint a bull's-eye on my back?"
"Ellie." He faced me. "You won't be in danger. We'll be waiting for them. Agents. SWAT teams. Coast Guard. Chicago marine police. We'll be with you every step of the way. If Samir or his people get within fifty yards of you, we'll pick them off. I won't let anything happen to you."
I glared at him. "Is there some reason I should believe you?"
He pushed up the brim of his hat. "Is this the same woman with the finely tuned sense of justice? The one who wanted to clear her reputation?"
"That finely tuned sense of justice is tempered by an equally fine-tuned sense of survival."
An edge crept into his voice. "In that case, you might want to think about your daughter. Or your father. You sure as hell won't be much use to them dead."
"You bastard," I hissed.
He grabbed my shoulders. "Listen to me. Who was on the bridge next to the antenna? Who had interference on their tape? Who saw Reedy's antenna? Hanjour's been trying to get those tapes for weeks. Now he's obviously coming after you. Damn it, Ellie. It's only a matter of time. He's panicking, and panic makes people dangerous."
He let that sink in.
My fingers prodded my forehead. He was right; I had no choice.
He leaned back. "Here's the deal. We want you to go back out there. Friday morning, day after Thanksgiving. Just before dawn. To scout the location or do whatever it is you do." He went on. "The thing is, we want you to make sure Dale Reedy knows what you're doing. E-mail her. Leave her a voice mail. Tell her it's okay the project got canned. That you got something else. And be sure to tell her what it is." He looked over. "You know how to do that."
"I thought she was gone."
"We're confident the message will get to the right ears."
I swallowed. "Then what?"
He explained that the _James J. Versulis_ would be waiting for me at the pier. I was to board and proceed out to the crib.
"What if he tries something before that?"
"We'll have a sniper team in the parking lot. And a SWAT team on the docks. We'll have men on the tug, too, and a team on the crib."
"And if they try before Friday?"
"We're posting men at your house. Starting tonight. Twenty-four seven."
"No. I have family coming on Thursday. And guests. What if they--"
"You'll be all right. So will your family. I guarantee it." A muscle in his jaw pulsed. "Believe me, _ch er_, there's only one person you should be scared of."
"Who's that?"
"Me."
"Why is that?"
"Because I always finish what I start."
Chapter Forty-four
Sleet dribbled sideways, then up. Layers of clouds raced across the sky. Clarence inched through the snow and sleet, trying to maintain traction. I stared out the window, catching glimpses of our surroundings.
I shifted uncomfortably. It might be hours before I got back to Dad's. But maybe that was good. I could just see us on Thursday. The table groaning with food. Dad about to carve the bird. The guests coming up to the house, skirting a gray Plymouth with two sullen men inside.
"Don't worry, folks," I'd warble cheerfully. "I'm the target of an FBI sting. An Arab terrorist is after me, he could strike at any time. But the FBI says not to worry, they'll protect me. You, too. Happy Thanksgiving."
We'd been driving ten minutes before I realized we were headed in the opposite direction from the marina where my car was parked. When Clarence turned onto the Eisenhower Expressway, I twisted around.
"Where are we going?"
"To my place," LeJeune said after a beat.
"Excuse me?"
"I live in Oak Park--it's close. The roads are a mess, and all of us need some shut-eye. Clarence will run you back to your car in the morning."
"I want to go home." Back to Rachel and Dad. To the people I love.
"Ellie, it's after one in the morning. You can't drive in this mess. We'll get you home by seven."
"No. Turn the van around."
" _Ch er_, it doesn't make--"
"BRRAAAP!" A loud noise blared out from the van's speakers. It sounded like a cross between a foghorn and a wounded goose.
I looked at LeJeune. "What was that?"
LeJeune scowled. "Clarence, what the fuck was that?"
A series of high-pitched tones beeped, like a microwave oven just ending a cycle. Clarence slowed and pulled over. He threw the van into park and crawled back to his laptop.
The same green bull's-eye was on the screen, but there were more lines, and it looked like new numbers and words had appeared. Clarence studied the monitor, then pressed his lips together. His expression was grim.
"What is it?" Nick's voice was tight.
Clarence pressed a key, bringing up a screen with columns of three-digit numbers. He tapped another key, and a different column came up. He sucked in a breath.
"Well?"
"It's another signal."
LeJeune shook his head. "It can't be."
"It is. On the same frequency as the crib."
"But we're miles away from the crib."
"That may be, but I've got a signal registering three on my S-meter."
They exchanged looks.
"What does that mean?" I asked quietly.
Neither of them answered.
"What are you saying, Clarence?" I asked, my voice louder.
He turned around. "Another signal, tuned to the same frequency as the device on the crib, was just transmitted--from somewhere around here."
"But we're heading west on the Eisenhower. Away from the lake."
"I know."
Another signal. "Does that mean there's another..." I hugged my knees, trying not to panic. My throat suddenly felt full of dirt. We were on the Eisenhower. Going away from the crib.
The Eisenhower.
Something about that picked at my brain. Something I should know.
The snow swirled up, then down, then circled in a vortex of its own design. I couldn't see farther than a few feet. The Eisenhower. The complicated cloverleaf design connecting the Eisenhower to the Dan Ryan and the Kennedy. Could that be--
"Oh God!" I turned to look at LeJeune and Clarence. "Dale Reedy has a line of sight from her window to the Eisenhower! I saw it from her window."
There was dead silence. Then everything happened at once.
"Holy shit!" Horror spread across Clarence's face. "Fuck it, Drummer! You gotta stop the team at Great Lakes! Now!"
LeJeune dug out his cell phone and started punching in numbers.
"Why?" I asked shakily. "What's going on?"
"Because--because--" Clarence started rubbing his palms up and down his thighs, gazing wildly around. "Christ! Shit. We may have already set things in motion. Jesus, man. You gotta get through."
"Clarence, why? What's going on?" He kept rubbing his hand up and down his leg. My heart thundered in my chest. Anxiety was contagious.
I grabbed one of his hands. The rubbing stopped. "Tell me."
He stared at me. His eyes looked haunted. "We haven't had time to analyze the system. We don't know how it's programmed. There might be a code that needs to be entered when you disconnect the system." He let out a shuddering breath. "Which means if there is a second device, and if the right code wasn't entered..." His voice trailed off.
"Which means what, Clarence? What are you trying to say?"
He took another long breath. "If those guys mess with the head end at Great Lakes, and they don't have the right code, the device could blow. The other bomb will go off."
Blood shouted in my ears. I whirled around to Nick. His cell was in his ear, his face ashen.
Nobody said a word while we waited for his cell to connect. I held my breath. It felt like hours. We heard fast, repetitive beeps.
A false busy. LeJeune snapped it off and tried again. Snowflakes danced and spun past the car. Again, a false busy.
"Fuck." LeJeune threw the phone down. "We gotta head back to the Loop."
"In this shit?" Clarence motioned outside. "We'll never make it."
"We gotta. We can't let them fuck with the system."
"How much time do we have?" I asked.
LeJeune didn't answer.
"Nick?"
"I don't know. I don't know where they are. They could be in her office already."
Ice replaced blood in my veins.
LeJeune turned to Clarence. "What can you tell me about the source of the signal? Where's it coming from?"
"The range of the Doppler is about two miles. The signal could be anywhere within that radius."
"Can we track it?"
"Hold on." Clarence crawled to the front seat of the van where he picked up a small black box about the size of a cell phone. A stubby antenna extended through its top, and there was a digital panel on the front.
"What's that?" I asked.
"A frequency finder," he said.
"I thought you already knew the frequency."
"We do, but if we happened to be within 100 to 150 feet of the signal, the Scout'll give me a readout." He depressed a few buttons, muttering. "Come on. Come on. Gimme a break." He shook his head. "Nothing." He put it down. "We're gonna need at least one more transmission--probably two--to plot it on the Doppler."
"But that might not be for another six hours," I said. "What if--"
Clarence cut me off. "Keep tryin' that phone, Drummer."
Nick punched Redial. The cords on his neck were stretched taut. Another false busy.
We stared at each other. Panic started to seep through my body. I smelled fear in the van.
LeJeune unzipped his parka. I could see beads of sweat on his brow. He turned to the laptop. "I'm gonna pull up a map."
Clarence started rubbing his thighs again. "If you can't connect on your cell, how you gonna get online?"
LeJeune squeezed his hands into fists. "Shit. Where's the closest exit?"
"Paulina, I think. Near UIC."
"What else is around here? Clarence? Ellie? Come on. Think. What the fuck is around here?"
"Forget it, man." Clarence shot him a resigned look. "There's nothing we can do. Except pray."
"No." LeJeune's face turned hard. "It's not fucking over. We're gonna think our way through this. What's near here? UIC? The United Center? Sears Tower? Come on. Help me out."
Clarence took his time answering. As if he was just humoring LeJeune. Going through the motions. "I don't think it's Sears. The signal looks like it's slightly south of us. Sears is due east."
LeJeune frowned. "But you don't know for sure."
"Not without another plot point."
"But if this one was planted before Nine Eleven like the other, all it would have taken is some guy in a gas or phone company uniform. He could have stowed it in the basement. Or the loading dock. Even a parking lot. Like the first Trade Center." LeJeune bit his lip. "We should send men over there."
He grabbed the phone, punched in more numbers, and shut his eyes while he waited. I heard the fast beeps. "Goddammit."
I sat up and rolled my shoulders, trying to shake off some tension. "You know, the fact they planted it on the crib might be significant."
A small vein on LeJeune's forehead throbbed. "What do you mean?"
"They wanted to sabotage the water supply. To inflict maximum damage to the infrastructure--as well as people."
"Yeah?"
"Maybe they're doing the same thing with the other."
Nick's eyes widened. "That's good, _ch er_." He nodded. "Infrastructure. So what kind of infrastructure is out here?"
"Shit man. Everything," Clarence said. "Electric power. Communications. Highways. Trains." He ticked them off on his fingers.
"You can see the Eisenhower from Dale Reedy's window," I said.
Clarence bolted upright. "Fucking A! The junction! Where the Eisenhower, Dan Ryan, and Kennedy come together!"
"What about it?" LeJeune asked.
"It's the most heavily used access in and out of the Loop. That blows, you got practically no access to downtown."
"And if it happened during rush hour..." LeJeune added, his head nodding, "...with thousands of commuters pouring into the city..."
"My God!" I covered my mouth with my hand.
"That's it." LeJeune's eyes caught fire. "Now it makes sense!"
"What?"
"I've been trying to figure out why they took a chance on the wind."
"The wind?"
He hunched forward. "The crib bomb. If the wind was blowing from west to east when they detonated it, most of the radiation would drift out over the lake, not the Loop. It would poison the water, but fewer people would die."
"So?"
"So...these bastards are vicious. And smart. They'd want to inflict maximum damage." He nodded again, more to himself than us. "I couldn't figure it out. I kept wondering whether they had something else up their sleeve. But they must have taken the wind into account--"
"Cross-contamination," Clarence breathed.
"Yes." LeJeune slammed a fist into his other palm. "With two, on both sides of the Loop, it doesn't matter which way the wind blows."
"The crib's east, the Eisenhower's west," Clarence finished. "They'd have the Loop covered."
LeJeune and he exchanged looks. Clarence moved to the driver's seat.
"Where are you going?" LeJeune asked.
"The signal seemed to be just south of us. We've got to do something. We've got to move."
As we pulled onto the highway, tires swished against snowy asphalt.
LeJeune picked up his cell and dialed one more time. His eyebrows shot up. "It's going through." He thrust it against his ear. "Shit man, it's LeJeune. Where are you?" He listened, then yelled. "No. Don't touch it. You gotta stop. Right now. We got another signal. There may be a second package!"
I heard the exclamations from his phone, but I was only half listening. Part of my brain was tripping over another connection. It was half there. Buried in my subconscious.
LeJeune was watching me. "What is it, _ch er_?"
"I'm not sure. Something you said earlier."
"I said a lot of things." He grinned.
I was astonished he could kid around at a time like this. I tried to call back the conversation. Infrastructure. Power. Electricity. Uniforms.
LeJeune went back to his call. He was frowning when he got off. "We have a problem."
Clarence twisted around.
I sat up straighter.
"You were right, buddy," LeJeune said. "We disconnected the radio temporarily when we disabled the device on the crib. We reconnected it as soon as we could, but our guys are saying there's no way to tell whether that altered the sequence of codes. And there's no time to go into the program to find out."
"What does that mean?" I asked.
"Disconnecting the system--even for a second--may have alerted the computer that something was different. That the system has been penetrated. Tampered with. As a result, the blast we just heard..." He paused. "...may have tripped the code to detonate."
My stomach clutched.
"They said it can take about thirty minutes to run through the codes for detonation." He looked at his watch. "It's one-seventeen now. And we heard the signal about ten minutes ago."
"Are you saying we only have twenty minutes to find it?"
"If we're lucky."
Chapter Forty-five
Despair, not necessity, is the mother of invention. I know that now, because it was while I was staring out the van's window in stunned silence, trying to absorb the fact that life in this city might end, that thousands of people might perish, and that I might be one of them, that it came to me.
It was one of those moments when the future lies in your hands. And you know it. Not one of those times that, in hindsight, turns out to be significant. You know--right as it's happening--that your actions will have a huge impact. That they might even change the world.
Some people welcome those moments. They feel it's their destiny. Not me. I felt nothing but terror. I spoke anyway.
"When I was calling scuba diving schools," I said quietly, "a woman gave me a number for Samir at work. Or at least where he might have worked at one time. When I called it, a 773 area code by the way, the man who answered said it was the maintenance room at People's Edison."
No one said anything. Then Clarence drew in a long, slow breath. "There's a PE substation just south of the junction."
LeJeune's mouth tightened. "Let's go."
***
We veered off at the next exit, crossed over the highway, and reentered the Eisenhower going the other way. Clarence pushed, but in this weather, we couldn't go faster than thirty. I checked the dashboard clock. One twenty-one. Four minutes had passed. Sixteen left. I was desperate to hug Rachel, to brush her hair off her forehead one more time. I wondered if I'd have another chance.
LeJeune spent most of the time on his cell, and Clarence concentrated on the road. I found myself measuring my breaths, trying to space them out evenly. Was I subconsciously devising an internal clock? Or hoarding air while I had it?
After what seemed an eternity, I felt the van turn south. I got to my knees and looked out. The van's headlights caught a small People's Edison sign at the edge of the road, and we pulled into a field that stretched at least two or three acres. A stand of trees, cloaked in an eerie snow-induced twilight, were in front, their bare branches curled upward as if begging for mercy.
We parked and got out. Through the trees was a forest of steel towers, strung together with thickly coiled wires. There must have been nearly a hundred of them, snow falling softly around them. They seemed almost haphazardly placed. Different shapes, too. Some were traditional towers, each side with steel bracing, but others looked like giant monkey bars you find on a playground. Still others were T-shaped poles.
On the ground between some of the towers were boxy units the size of refrigerators, and perched on nearly everything were strings of what looked like super-sized lightbulbs. Insulators, I discovered, that help attach power lines to their structures. Though the snow muffled much of the sound, the hum of a gazillion volts zipped through overhead lines.
A Chicago patrol car pulled up and parked horizontally across the entrance, its revolving Mars light turning the snow into a veil of pink and blue specks. Within minutes, two more sedans pulled up. Men piled out. LeJeune went over to huddle with them. Another van arrived, and half a dozen men climbed out wearing hazmat suits and masks. Two of them were carrying a boxy piece of equipment that was small enough to fit in a backpack. They looked vaguely familiar.
"NEST," Clarence said. "Nuclear Emergency Support Team. They were out on the crib."
"What?"
"They run around patrolling for dirty bombs. They've got their sniffing gear with them."
"Sniffing gear?"
"Gamma ray and neutron flux detectors. Kind of like fancy Geiger counters."
The men split up into teams of twos and threes and scattered through the trees toward the substation. I stamped snow off my feet. "What are they doing?"
"Making sweeps of the area. Looking for the device."
"What time is it?"
Clarence checked his watch. "One twenty-six."
Eleven minutes left. "What--what happens when--if--they find it?"
"They disable it." He started rubbing his hands together. "Actually, that's the easy part," he said. "Or it would be if they had enough time."
"What do you mean?"
"There are a couple ways to go. You could bring in a robot--disable it by remote control. You could also bring in a huge tent and fill it with foam." He blew on his hands. "To contain the radiation in case the bomb blows."
I winced.
"I don't know what they'll do this time. The military's supposed to handle these things. Maybe they'll try to blow the bomb's wiring."
A man ran out, opened a van, grabbed something, and ran back in. I stiffened.
"What time is it?"
"One twenty-eight."
Nine minutes.
Lights flashed. More vehicles converged on the scene. Several men and a woman got out. One of the men was leading a dog. Then another van, filled with Chicago Police Department Bomb Squad personnel, pulled in. They disappeared through the trees.
Suddenly a man's voice shouted through a megaphone. The wind snatched the sound and tossed it around in the air. "All unauthorized personnel must vacate the premises immediately. All unauthorized personnel out. Now."
I grabbed Clarence's arm. "What does that mean?"
He grimaced. "It means they found something. I have to go in. The radio."
"No! Don't leave me!"
But he was already running to the van. I followed him over. He opened the door, grabbed a mask from under the front seat, and headed into the trees.
Alone, I tried to wiggle my fingers, but they were numb. I should never have bitten my nails. It was an annoying habit. Rachel had inherited it from me.
I leaned in to check the time. One thirty-three. Four minutes left.
I started to shiver. The snow was up over my shoes. I wished I had boots that buckled all the way up. Like the shiny pink boots I had as a kid. I never buckled the top strap. Mother always chided me about it.
Suddenly a shout went up. My stomach twisted. I strained to look through the trees, but the falling snow and parked cars blocked my view. LeJeune ran out and started dragging me toward the van.
"They found it! Get out of here!" His face was haggard. "Now!"
Panic radiated out from my stomach. I threw myself into the van. The engine caught right away. I tried to tell myself it would be okay. The bomb squad was handling it. I checked the clock on the dashboard. One thirty-five. Two minutes left. They'd do it. They had to.
I threw the van in reverse. If these were the last two minutes on earth allotted to me, I wanted to be with my family. I started to back up, then stopped, the engine still idling. Dad and Rachel were twenty miles north. I'd never make it. With less than two minutes, I probably wouldn't even make it to the highway. But now what? What should I do?
I was deliberating the absurdity of spending my last two minutes alive with nothing to do and nowhere to go when a dark colored sedan pulled up. Turning into the yard, it slowed to a crawl, and a window rolled down. I looked to see who was driving, but between the snow and the darkness, I couldn't tell. The car rolled a few yards forward, then stopped a few feet from the van. As the driver opened the door and climbed out, I gasped.
It was Abdul.
I cut the engine, my heart banging in my chest. Where was Nick? I had to warn him. That's what I was supposed to do. Make sure he got Abdul. I jumped out of the van and sprinted away from it. I was veering right, angling toward the substation entrance when there was a blinding flash of blue light, and the silence was rent by a roar. A scream tore out of my mouth. I threw myself to the ground.
It took a few seconds to realize I was still alive. There had been no explosion. No fireball. A chopper, its blue lights flashing and its motor whining, had broken through the overcast and was descending. It banked over my head, narrowly missing the utility towers, and landed fifty yards away in the street.
More men poured out, some of them suited up, some in uniform. The military. They ran into the substation.
I got up and brushed off the snow. LeJeune. Abdul. There had to be less than a minute left. I counted steps as I moved through the trees, twisting around and seeing my footprints in the snow. I was just at the entrance to the substation when the megaphone voice barked again.
"Make way...make way! Everybody back! Let's get this sucker out of here."
A knot of men emerged from the substation. In the middle of the group were several men in hazmat suits carrying a steel suitcase on what appeared to be a flat wooden board. They were moving very slowly toward one of the vans; other men surrounded them. I caught a glimpse of Clarence in the group, his mask on. He gave me a thumbs up.
Once the suitcase was in the van and the van had taken off, the men in hazmat suits tore off their masks. Others high-fived each other and laughed. A few wiped tears from their eyes. I looked for LeJeune, but I didn't see him.
I whipped around. I'd confront Abdul myself, although what I would say or do when I found him, I had no idea. I raced back to the van, cold, bone-weary, but resolute.
But when I got there, Abdul had disappeared. His car was gone. The only hint it had been there at all was a set of tire tracks in the snow that were fast filling in.
I checked the clock. One forty-one. More men poured out from the substation.
It was over. With nary more than a whimper.
Chapter Forty-six
I never found LeJeune, and Clarence followed the van with the bomb. Someone else gave me a ride to my car.
When I got home, I took a shower and brewed coffee, then called Dad. Rachel and he were fine; she was still sleeping. I told him I'd pick them up around noon; he should pack an overnight bag so he could spend the weekend with me.
The snow tapered off, and a weak slash of sunlight inched across the kitchen counter, coming to rest on the wall. I prowled around the house, restless and unfocused, too exhausted to sleep. On the surface my world seemed normal and stitched up, but underneath was a crack, a fissure so deep I wasn't sure it would ever mend. I knew I would never look at the world in quite the same way again.
The doorbell chimed around ten. LeJeune. He'd put on a clean shirt, but he needed a shave, and there were dark smudges under his eyes. He kissed me. "What smells so good?"
I'd tried to pretend everything was okay, heating up the oven, putting in the pie, starting to saute celery and onions. "Pecan pie. For tomorrow. I'll get you some coffee."
While I poured, he wandered around the kitchen. I wondered if he was feeling the same way as I. I got out sugar and milk.
He leaned against the counter. "We picked up Hanjour and Reedy."
I spun around. "Both of them?"
"Customs nabbed Reedy trying to hop a flight to Frankfurt. She told us where to find Hanjour. We found him and one of his pals inside a White Hen in Orland Park. Stocking up on donuts and soda."
"Doughnuts? He was buying doughnuts?"
"His pal went for a knife, but we disarmed him. Hanjour just threw up his hands." He stirred his coffee. "Guess we won't need that sequel after all."
"It's really over?"
He hesitated, as if he couldn't quite believe it himself. "We've already cased their apartment, and we're going through their e-mail. They were trying to get as far away as possible before it blew."
"They didn't want to die for the cause?"
"When you've been living stateside for a while, I guess martyrdom loses its appeal."
"And yet they wanted to destroy it."
"Ellie, no one ever said these guys had a tight grip on reality."
I refilled my coffee, then tore open a package of sweetener and dumped it in. "You know, I can't help thinking if I'd found the antenna the first time I met Reedy, none of this would have happened."
"Don't be too hard on yourself, _ch er._ In a way Reedy may have saved your life."
"How do you figure that?"
"When you first met her, you had no reason to link any of the events to Arab terrorists. You thought it was a Mafia scam."
"So?"
"As long as you were running around looking for wise guys, she was able to rein in Hanjour. Convince him to go after the tapes instead of you. Persuade him she'd handle you."
I thought about it. He was right. The last time I'd seen the SUV--until a few days ago--was the day LeJeune and Coates came to the house. I met Dale Reedy the next day.
"She never had any intention of producing a video, did she? She brought me in just to find out about the tape. To play me and see how much I knew."
He nodded.
"Why did things change?"
"There's no guarantee of unanimity when you're dealing with terrorists. There probably was a disagreement over how to deal with you from the beginning. At least that's what she's saying."
"Reedy's talking?"
"Louder and quicker than a scalded cat. She's not stupid." He took a sip of coffee. "She did lay down one condition, though."
"What?"
"That MI5 or Scotland Yard pick up her kids and make sure they're safe."
"Did they?"
"She talked to them on the phone a few hours ago."
I tapped my spoon on the table. "What did she tell you?"
"The bomb was originally supposed to be detonated around the time of September eleventh, but in all the confusion, the final order never came down. Then, afterwards--"
"It was supposed to be part of September eleventh?"
"Apparently. You know how splintered and isolated these cells are." He stared into his coffee cup. "At any rate, with all the attention on security and Arab terrorists, Samir's plans fell into disarray, and he had to abort. Months later, when things calmed down, Reedy got the word to put it back together. That's when they planted the second one."
My stomach twisted. "Got the word? Oh my God--I never got the chance to tell you, Nick. Abdul was there. At the substation. The one I think is in charge."
"Ellie..." He paused. "Abdul's a Saudi intelligence agent. We've been working with him since May. He's been tracking Islamic terrorists for years. He's the one who tipped us to the threat in the first place. Something happening with water. This summer. In the Midwest."
"Abdul's an agent?" I stared. "But he never..."
"He couldn't blow his cover." He grinned. "Of course, you managed to do that for him."
I wrapped my hands around my mug. "But I met him at the Greenbrier. What was he doing there?"
"He was trying to run down a training camp in rural West Virginia. A place where Arab terrorists reportedly train next to white separatists."
"No."
He shrugged. "When you're overthrowing a government, the enemy of my enemy..."
"So that's how he knew the countryside so well."
LeJeune looked puzzled.
"You remember. When he told us about the coal mines--" I stopped. I was confusing LeJeune with David. I bit my lip. I remembered the sheet of paper with Abdul's number at the Four Seasons. "If he was tracking Dale Reedy, why was he calling her at Great Lakes Oil?"
"He was trying to confirm her identity. She'd changed her name, remember? He was sure he would know her voice."
"So the plant acquisition _was_ a pretext."
"You got it."
I sat back in my chair. "So if Abdul wasn't in charge, who gave Reedy the order to put it back together?"
"Aziz. Her husband."
"Why did they need two bombs?"
"To make sure the radiation covered the entire Loop."
"My god!"
"For insurance purposes too. In case the first one didn't work. Remember, it had been underwater for months."
"But they were still getting a radio signal."
He looked grim. "Maybe after they saw the devastation in New York, they decided to up the ante."
"So Samir got a job at PE and cased it on the side."
LeJeune nodded. "Terrorist or not, he needed to make a living. And what better place to rip off supplies?"
I shivered. "Do you think they were planting the crib bomb the night Mary Jo and Rhonda were at Calumet Park?"
"Hard to say. They might have been doing a test run. Or a safety check afterwards. But whatever it was, Samir panicked when he saw the two women."
"And killed Mary Jo."
He nodded again.
"After which he thought everything was under control--until I testified."
"That's why he showed up at the trial. He had to find out how significant that RF damage was. And whether you knew where it came from."
"Which is where he saw Rhonda Disapio. And realized she'd been the one with Mary Jo at the park."
"You got it. She'd always been a loose end."
"How did he kill her--tamper with her brakes or something?"
"Yeah. Reedy wasn't happy after that. She realized he was a loose cannon. That's when she came down on him. Told him to go after the tapes, not people."
"The fire at the studio."
"Yes."
"And Brashares."
"Like the cops say, he was in the wrong place at the wrong time." LeJeune drained his coffee. "They thought they'd patched up all the leaks. Until you screwed things up again."
"Because I saw Reedy's antenna."
"You also had the last copy of the tape." He looked over. "You were in their sights from the beginning, _ch er_."
We were quiet for a moment.
"Do you think she was sending me a message when she canceled the video?"
"A message?" He laughed. "Not hardly, _ch er._ She was busy saving her own skin."
"She didn't have to e-mail me. She could have just disappeared."
"You sound like you're defending her."
I shook my head. "Just trying to understand."
"Don't waste your time." He ran a hand over his face. "You want more coffee?"
"Sure." I handed him my cup.
"Black, right?"
How many times had we drunk coffee together? I pointed to the blue packet of sweetener on the table. He looked embarrassed.
I waited until he sat down again. "What else did Reedy say?"
LeJeune leaned back in the chair. "Well, for one thing, she said it was payback for the Gulf War."
"Huh?"
"When we bombed Iraq the first time, we took out their water treatment plants. With no running water, people hauled buckets in from the Tigris, but it was filled with sewage. Thousands of people died. Typhoid, dysentery, cholera. Even polio. And because of the sanctions, they couldn't import any chlorine."
"Do you believe that?"
He made a noise that was almost a snort. "It makes a convenient excuse. But enough." He raised my hands to his lips. "We'll talk more later. Right now, you and I have some unfinished business." He smiled.
I pulled my hand away.
His smile faded.
"You know," I said slowly, "it's strange what happens when you think you're going to die. A certain clarity emerges." I tucked my hand under my leg. "Tell me something, Nick. The speech about your father losing his leg. And the bit about Huey Long. Was that part of the script?"
He tipped his head to the side. "What are--"
"Don't." I got up and went to the stove. "Don't."
He stood up and straddled a chair. "It wasn't just the job, Ellie. It--it never was. I realized it when I got back from London. I want you. We could mean something to each other."
"But..." I hesitated. "...I don't want you."
He just looked at me.
"As long as we were working toward the same goal, it was easy to act _as if_. To pretend. But that changed when you agreed to use me as bait."
"You made that decision."
"You gave me no choice."
"I never wanted that to happen. I fought it. I nearly lost my job over it."
I bent my head, unsure if it was the truth but not wanting to find out. "It wasn't that it didn't have to be done. It did. But you were the one who talked me into it." I laced my fingers together. "Did you ever read _All the King 's Men_?"
He didn't answer.
"I'll bet you have. Remember how Willy Stark started out as a reformer? Burning with the desire to do good? Along the way, though, he learned how to manipulate people. String them along."
He wouldn't meet my eyes.
"Eventually, of course, it corrupted him." I paused. "The thing is, even so, even with his lust for power, Willy Stark accomplished great things."
He looked up.
"You did, too, Nick. Most people would have given up. But you didn't. And if I ever get in trouble again, there's no one I'd want on my side more than you." I stood up. "But now, I think you'd better leave. And...don't come back."
He ducked his head as if I'd lobbed him a stinging criticism. I turned away and pretended to fuss with the stove. After a moment his steps clicked across the floor and retreated down the hall. The front door opened and closed.
Chapter Forty-seven
"I was wrong."
Dad and I were in the family room, watching flames from the fireplace dance, pop, and float up the chimney. One or two sparks landed on the floor and winked out. Strains of rock music drifted down the stairs from Rachel's room.
"What do you mean?" I picked up my glass of wine, which lay next to a plate of chocolate chip cookies.
"I thought you were on another one of your crusades," he said, balancing a cup of tea in his hand. "You know, trying to fit round pegs into square holes. I was wrong."
"I don't blame you." I twirled my wine, watching the firelight flicker and glint through the glass. "Who would have believed it?"
"I should have trusted you. You're my daughter."
"You know something?" I leaned over and squeezed the hand not holding the teacup. "It takes a _mensch_ to admit he was wrong--especially to his child."
He squeezed back. "So tell me. These--these..." He couldn't seem to bring himself to say the word _bombs_. "They were planted before September eleventh?"
"One was. The other one was added afterward."
My father's brow furrowed. "Are there any more, you think? That haven't been discovered?"
"God forbid."
"Yeah, well, I guess praying is about all we can do." He gazed at me with worried eyes. "So why haven't I seen anything about it on the news?"
"LeJeune said they're keeping a lid on until the government decides how to respond."
"Good luck."
"They could have a shot. The action out at the crib was miles away from anything, and the weather last night was terrible. There weren't a lot of people around."
He sniffed. "That woman--the British one they picked up--she's talking?"
"That's what I hear."
"How did they identify her?"
"Abdul told them about her."
He set down his teacup. "Abdul?"
"The man we met at the Greenbrier."
"The one who's working with David?"
"The same." In fact, he'd dropped by the house that afternoon to apologize and ask my forgiveness. We'd had a long talk. "He's a member of the Saudi intelligence force, Dad. He's been tracking terrorists for years. The oil sheik role was his cover."
"Does David know?"
"Abdul says he explained it to him the other day. He said David wasn't happy about it. But he understands."
Dad bit into a cookie. I flashed back to Abdul's table manners. The match on the silk tablecloth. The caviar on the toast point. Not very royal. It should have been a tip-off.
"Reedy did make a huge blunder, though."
Dad stopped chewing.
"The antenna. Granted, I didn't notice until it was almost too late, but if I hadn't been so unobservant, I might have figured it out sooner. I don't know why she didn't try to disguise it."
"Probably the Empire mentality. You know how arrogant they get."
I remembered the photo of her two boys. "I don't know. I wonder whether, maybe in some subconscious way, she wanted me to figure it out."
Dad cocked his head. "What?"
"She has two boys. The Bureau thinks her husband was holding them over her head. Maybe this was her only way to get them out of his clutches."
"Have they caught him?"
"Aziz? Not yet. But he's not in a position to make many demands."
Dad frowned. "I'm an old man. What do I know?"
That was his way of telling me I was making assumptions again. Maybe I was, but I couldn't help thinking that men just don't get it. The extent to which a mother will go to protect her young. It's hardwired into our psyches. I started to reach for a cookie but stopped. Did that mean there was some unanimity between Dale Reedy and me? A subtle but mutually understood accord? I shifted uneasily.
"How's Sylvia?"
"You'll see for yourself, tomorrow."
"Dad, are you sure you're ready to handle this?"
He wiped his mouth with a napkin. "Listen, Ellie. She's great company, she plays a mean hand of gin, and I like holding her hand almost as much as yours. What more do I need?"
"If you're happy..."
"I'm happy." He covered my hand with his. "What about you?"
"I'm tired. I could sleep for days. But I keep thinking how much I have to be thankful for."
He patted my hand. I curled a leg underneath myself. "There's just one thing. I think Rachel's been calling David with updates about me all fall. The little _dybbuk_ 's been spying on me."
Dad withdrew his hand.
"She and I are going to have one of those boundary discussions after Thanksgiving."
He toyed with his spoon. " I wouldn't do that if I were you."
"What do you mean?"
"You're blaming the wrong person."
"You?"
"Ellie, we're his only family. You may not want to be in touch, but I do. It's the right thing to do."
I scowled, but I wasn't totally surprised. As a young man, my father had fallen in love with David's mother. It hadn't worked out, but it had forged a connection between our families. Indeed, when David and I got together, in part because of the discovery about our parents, my father was convinced it was _bashert_.
"Love is never perfect," he said. "You young people still have a hard time accepting that. You'd rather head for divorce court than work it out."
I started to cut in, but he overrode me. "I'm not saying it wasn't warranted in Barry's case. But if you're lucky enough to have it come your way again, to find someone who loves you more than he loves himself, well..." He threw up his hand. "But what am I telling you this for? You're the hero. You do what you want."
"Labor under correct knowledge," I muttered.
"What is?"
"Luck."
"Exactly what I'm saying."
"You know what I want? I want to get away from here for a while. Go to some hot, dry desert where there's not a drop of water."
"Got any travel companions in mind?"
"I was thinking of you."
"My dance card is all filled up." He motioned toward my bag. "But I bet if you pull out that phone of yours, you might find someone who'd be thrilled to come with you. Especially if you feed him a home-cooked Thanksgiving dinner."
"Dad, I'm sure he's made other plans. He probably won't even take my call. I've--I've been pretty awful to him."
"Why don't you let him decide that?"
"If I were him I wouldn't take the call."
"Then it's good you're not."
"Even if he did talk to me how could he get here in time? It's Wednesday night."
"They have these things called airplanes, Ellie," he said. "Last I heard, they still fly on Thursday morning."
I considered it. Then I stood up. I thought I saw a twinkle in Dad's eye when I pulled out my cell, but with him you can never be sure.
"Maybe you should try him at his office. Maybe he's working late."
I felt my eyes widen and punched in the number.
A female voice answered. "Good evening, Mr. Linden's office."
My eyebrows shot up. I took a breath. "Is--Is he there? It's Ellie Foreman calling."
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The 2011 Finished Vehicle Logistics conference saw more than 200 delegates gather to debate issues around capacity, efficiency and fuel reduction. And in the wake of the Japan crisis and a recovering economy, the market looks set for a capacity challenge later in the year, reports Christopher Ludwig.
Driver shortage solved by co-ops?
The 2011 Finished Vehicle Logistics North America conference in Newport Beach, California revealed an industry moving well along the path of recovery, but which continues to hit multiple speed bumps, including rising fuel prices, restrained production following the Japan crisis and a looming shortage of capacity for trucks and rail wagons.
Continuing a debate that started a year ago at the inaugural event on the West Coast, and mirroring similar debates held on the other side of the Atlantic in Europe at the recent ECG Assembly, carmakers and logistics providers expressed concern that vehicle logistics could become a serious bottleneck for the automotive industry's recovery.
Carmakers including Ford, Toyota, Honda, Chrysler, Hyundai-Kia and others warned that the shortage of transport assets–compounded in recent months by the withdrawal of service in certain locations by the largest US provider, Allied–could reach critical levels, particularly in the fourth quarter of 2011 as production ramps back up for Japanese carmakers.
Mike Nelson, national manager of highway logistics for Toyota Logistics Services, revealed that the US market had seen its car carrier fleet cut from 13,000 in 2006-2007 to 8,000 at the market's low point in 2009, with current levels around 9,000. But the industry does not appear to have plans to add significant capacity to the road as sales increase. Ford's director of North American vehicle logistics, Walter Lowe, said that time was running out to find a solution.
Mercedes-Benz USA's manager of vehicle distribution, Markus Gichert, compared the fourth quarter to an approaching tornado. "We know that it is coming but we do not know exactly where it will touch down, whether it will be rail issues, ro-ro shipping or port congestion on the west coast with more Japanese imports, or trucking capacity in the southeast, for example" he said.
But truck providers at the event argued that current rates and the cost structure of the vehicle logistics industry were simply not sustainable for enough new investment to maintain the service levels required by OEMs. Michael Wysocki, chief executive officer of United Road, one of the largest carriers in the US, said that the problem had been building for 15 years.
"Nobody wants to buy trucks. We need support on the capital side, and I'm not talking about rate guarantees or eliminating risk but simply in economic terms that allow us to get returns on our investments and give our drivers a decent life," he said.
At the same time it would be inaccurate to say that there is currently no investment in the North American vehicle logistics industry. Speaking to several carriers on the sidelines of the event, including Jack Cooper, United Road, and Proficient Auto Transport among others, carriers are once again investing in trucks and trailers (although some appear to be more active on the used truck segment). Dave Floyd, CEO of Professional Auto Transport said that his current investment plan was to grow his fleet 15% per year. The North American railways, meanwhile, have planned massive levels of capital spending, including more than $3 billion for Union Pacific alone.
However, investment in building new equipment is generally muted. Cottrell, one of the two remaining car haul trailer manufacturers in North America (along with Delevan), told Finished Vehicle Logistics that it was operating at about 40% capacity (building about four trailers per day). And according to Keen Doucet, from the Canadian railway CN, the total number of tri-level rail wagons that will be added to the total pool of available wagons would be 341–that is out of a total fleet of 34,845 bi-levels and 14,372 tri-levels, or about 9,000 wagons fewer than the fleet's peak.
That imbalance between bi-level and tri-level could become a serious issue for the sector. While the American shift to small cars has not been as pronounced as might be expected–according to IHS, the current split for sales is 50.8% cars, 49.2% truck/SUV (with hybrids a surprisingly low 2.4% in 2010)–there is nevertheless increasing production and distribution of small cars. Jon Schwartz, manager of procurement and network design for Chrysler, said that his company would be distributing the Fiat 500 by rail out of Toluca, Mexico on tri-level wagons and would add several new models that move on tri-levels over the next year.
WWL's Rocky Luna, vice president of inland distribution, which includes responsibility for procuring Nissan North America's vehicle logistics, said that he was seeing significant shortages related to wagons. "According to the current build plan, even if the industry built a new tri-level railcar everyday until 2013, we would still be short," he said.
Greg May, president of Jack Cooper Transport, warned that the problems would get worse before they improved. "There is a capacity problem and it will likely get worse in the short term. Carriers are doing everything they can to improve efficiency and make the best use of available capacity, but there is a $1 billion solution out there [to replace 4,000 missing trucks]", he said.
Looking at the wider US and North American car market presents a complex picture that is also contributing to the conservative expansion of many vehicle logistics providers. Rebecca Lindland, director of strategic review for IHS Automotive, revealed that the company had reduced its forecast for the US light vehicle market by 600,000 units for 2011 from 13.3m to 12.7m, although she admitted that the figure could be pushed up to 12.8m or 12.9m (11.6m units were sold in 2010). The reduction stemmed from two main issues, one of them short term, the Japanese earthquake, and the other more long term with rising oil prices.
The impact of Japan on global production is staggering, even if recent announcements by Japanese manufacturers suggest a quicker return to full production that previously thought. According to Lindland, about 3.6m units will be lopped off of global light vehicle production as a result of the earthquake in 2011. In North America, the impact is a loss of 454,000 units in the second quarter and a further 90,000 units lost in the third quarter.
It is also of concern to ocean forwarders as Gary Hurley, vice president of NYK Line (NA)'s RoRo division, made clear when he highlighted that there were 20-30 vessels from the various providers serving the market currently sitting in Tokyo harbour waiting for work. "That sounds like great capacity but it's not because a lot of the ships we run services on from the US going to other areas come from Japan. It means tonnage is going to be short in other segments of the market," he said.
He went on to point out that when volumes return ocean providers are going to be thrown into the other extreme a face a serious capacity crunch.
But many signs point to potential bottlenecks in the fourth quarter and beyond (and Davies admitted that her colleague, Mike Nelson, was considerably more worried about capacity). Lindland predicted that the end of the year would see production levels start to make up for the backlog, with 57,000 extra units predicted for the fourth quarter.
Beyond 2011, IHS predicts reasonably strong recovery in the years to come, with 2012 growing to 14.7m, 2013 at 15.8m and 2014 back to pre-crisis levels at 16.5m units.
But while those numbers are encouraging in comparison to the past three years, Lindland stressed that the figures look higher than they actually are. Sales relative to driving age, housing starts and population growth will be at historic lows, even as the industry recovers closer to 17m units. "We are not gaining back the millions of units that we have lost in this crisis period, which means that is simply growth that is gone and not coming back," she said.
The reason for this muted growth is the result of general economic weakness–including a poor housing market, high debt levels, and likely tax rises to address the growing American budget deficit– as well as high oil prices. The price at the pump alone will hold back sales by at least 150,000 units per year, said Lindland.
But not everyone at the event agreed with this restrained forecast for the US and North American market. Mike Riggs, who owns Jack Cooper Transport, was bullish on economic indicators including the scrap rate versus new production, the value of used cars as well as the overall improvements in quality of cars on the market. He suggested that a worst-case scenario for the industry right now would be to underestimate the size of the market, thereby restraining further investment in truck and rail wagon equipment. "We are already seeing capacity issues and if 2012 hits 15m units and goes 300,000 units higher than predicted, that could be a serious problem," he said.
Manufacturers that have seen their production and sales hit hardest by the current crisis are also anticipating that the capacity scenario could get ugly if the industry does not prepare. Toyota's Mike Nelson and American Honda's Scott Crail pointed out that allocation rates were already very high, with most cars sold off the assembly line or on the ship by the time it reaches port. "Toyota has got 50% production right now and that is at a time when our dealers probably need 110%, so we are going to face a lot of challenges in getting cars to market once we come back to full speed later in the year," he said.
Driver shortage or the need for a new business model?
Besides concern for truck and rail wagon capacity shortages, the industry is also facing potential driver shortages. Toyota's Mike Nelson, quoting figures from the Automobile Carriers Conference, an association representing car hauliers, said that more than 60% of drivers were aged 55 years or older. The industry has generally lost drivers in recent years as pay failed to keep pace with other freight trucking sectors. The industry also saw a large exodus or drivers during the 2008-2009 crash.
But at least one newcomer challenged the industry view that companies could not recruit drivers. Steffon Perfect, president of Car-Haul Co-Op, held up a list of about 600 owner-operator drivers that he had signed up over the past four months. Perfect, a native New Zealander and former driver and owner-operator, said he had gathered excitement from drivers based on his plan to create the first cooperative in the car haul sector. Perfect plans to give away nearly 50% of the company after two years, and eventually to move to a 100% employee-owned model. He believed that his focus on treating drivers well would create a more sustainable and service-minded company. "Our people will come first, not the OEMs. We believe that if you put your employees first, the customers and the money will follow," he said.
The company is only four months old and as yet unproven, although Ford's Lowe expressed hope that Perfect would be able to achieve what he said he was going to do. Others expressed worry that the driver base Perfect had recruited would be pulled from other companies and that it did not address the general shortage still facing the industry.
Mike Monell, service director for a major dealer group, Norman Reeves Honda Superstore, said that he worked specifically with his employees and his schedules to help accommodate out-of-hour deliveries. In some instances, the store had switched schedules to Tuesday to Saturday to be able to take Saturday deliveries (and since few deliveries were arriving on Mondays). "I tell my employees to treat the drivers [arriving with the cars] as their customer, and to go out there and help them," Monell said.
But Monell admitted that most dealers, particularly smaller ones, were completely ignorant to the logistics process. For that reason, he emphasised that better communication between the OEMs, logistics providers and dealers would help to better set expectations and estimated arrival times for final customers.
The extent to which OEMs maintain this communication still appears to be inconsistent, with some carmakers providing considerably more visibility and information than others during the delivery process. Steve Seher, from vehicle processor Auto Warehousing Company, said that one specific customer was excellent at communicating and adhering to its forecasts, and to keeping its processing steady in volume and flow, allowing carriers to build the most efficient loads and to help them meet estimated delivery times.
On the other hand, other carmakers appear to communicate much less. Proficient Auto Transport's Williams said that some of its customers do not even communicate which vehicles are stock and which are sold. "That means that sometimes we build our loads as efficiently as we can only to arrive at the dealer and discover that we delivered stock units to it when it was desperate for a specific sold unit," he said.
According to Tom Swennes, vice president of ICL Systems, while most OEMs have good internal visibility they lack awareness outside their own supply chains because of a lack of common standards, entrenched business processes and attitudes, and inflexible legacy IT infrastructure.
Ford's Walter Lowe said that sometimes the problem lies in dealers setting false expectations as well. He related an anecdote in which a dealer had promised its customer a Ford F150 model before Christmas that was not even going to be launched until January. After a delay launch, and several other logistics problems, the customer did not receive his vehicle until March. "There were a lot of failures in that process but the first failure was the dealer giving the end customer false information. We have to work together to make sure we set realistic expectations," he said.
There was some suggestion about how innovative new equipment might help to change the capacity situation. One company, CTM, presented a convertible trailer that can be adjusted to carry vehicles as well as collapse to a flatbed truck that could haul general freight. Likewise, Union Pacific will start production this year of its Autoflex rail wagon that can be converted from a bi-level to a tri-level in 20 man-hours. Linda Brandl, vice president and general manager of UP said that the railway would build 75 wagons to enter the reload fleet by the end of the year and another 25 early in 2012.
But by and large the solution for the capacity issue seemed down to simple investment–the "$1 billion solution" as Greg May said–and carmakers indicated that if the market did not rise up to meet the demand then they would seek other solutions, including making their own investment in transport assets. One carmaker indicated privately that the company was very close to going that route and would likely start its own trucking company.
Lowe said that Ford didn't want to invest in trucking, but that "we will not fail our customers". He said that Ford would step in if necessary to delivery cars itself, but he was clear that he would much prefer that the open market provided the capacity.
Glovis, the logistics arm of Hyundai and Kia, has also recently said that it was exploring the idea of developing its own truck fleet.
Such comments echoed similar sentiments expressed recently in Europe, notably by Daimler's head of worldwide vehicle logistics, Egon Christ, who said that Daimler could resort to building up a distribution company if the market did not step up. But for Christ such a move would also be a reluctant solution.
The railways were relatively quiet on such a proposal, although Norfolk Southern's Richard Kiley indicated that he was not sure that a railway should look to get involved in the trucking industry. "If a guy as smart as Greg May [president of Jack Cooper] is struggling with truck capacity, then why would I be any better at it?" he said.
At the ports, meanwhile, where capacity is likely to become a serious issue when the surge in volumes that is expected by 2012 comes, forward planning and faster throughput involving collaboration between processor, carmaker and ocean forwarder were seen as crucial.
"Dwell is the enemy," said Richard Frick, senior vice president and general manager at Pasha Automotive Services. He admitted that storage was a necessary evil sometimes but it was not ideal. "What is ideal is throughput," he went on, and provided an example of how the company increased turnover and removed four days of dwell time by communicating with the ocean forwarder, the rail provider and the trucking company to get a preferred picture of the load build required.
Of further relevance to ports would be the impact of the strong yen as it spikes to 79 against the dollar, prompting a change in balance from imports from Japan to exports by Japanese manufacturers from the US.
In another sign of the industry's recovery and shifting business model, the conference revealed that the used car market is booming. The lack of inventory for new cars has driven up demand for used cars as well, and according to both Dave Carp, director of fleet and remarketing at Kia Motor America, and Richard Okida, remarketing manager at Toyota Financial Services, used car prices are at all time highs.
This demand is creating opportunities for vehicle transport, according to Carp, as Kia and other brands look to quickly move cars to auction. Okida said that there has been an increase in online traffic, and for demand of single unit moves across the country.
United Road chief Michael Wysocki, whose business is split about 50% between new and used car business, said that the high price of used cars was also increasing the demand for car swaps between dealers.
Wysocki said that United Road's diversification in the used car market had enabled it to grow during the recession, but that is also enhanced the carrier's efficiency in the new car market. "Plugging used cars into our network allows us both to fill trucks and better use our assets, but it also gives us opportunities to reposition our trucks for backhaul moves," he said.
But Wysocki added that transport capacity for the used car market would be subject to the same pressure as the new car market, particularly later in the year. While he said that there was no specific priority for a retail car, manufacturer or used car within the delivery network, he did stress that dealers were often the most concerned about service and delivery time because they needed the cars for their sales. "The cost pressure is the same or greater for a used car or retail car as it is for a manufacturer's car, but dealers often expect a higher service level and are willing to pay for it," he said.
John Felitto, Executive Vice President of WWL Americas.
The high cost of fuel brought into sharper focus the debate on fuel efficiency and 'green logistics'. John Felitto, executive vice president of WWL Americas, reminded delegates of the impending legislation of sulphur emissions for bunker in Emission Control Areas (ECA) in both Europe and along the east coast of the US and Canada that will come into effect in 2012, with even stricter limits slated for 2015. "When these ECAs come into effect we will have to switch from bunker fuel that currently costs $650 per tonne to $950 per tonne," said Felitto.
Felitto admitted that if the market picked up as expected, and particularly as Japanese exports ramped up, that ships may need to run at full speed to help free up capacity. But the industry needed to have a wider debate on speed in the supply chain and the balance between fuel consumption and delivery times.
Such expectations were a point picked up by other modes too in the face of capacity shortages. Pat Donohue from IT provider ICL pointed out that if dealers could live with two days extra lead time, a lot of pressure would be taken off of capacity. Mercedes-Benz's Gichert said he was open to the idea, but admitted that it would be an uphill battle in educating dealers and changing their expectations since most expect vehicles to turn up as soon as they are sold. "It will be very difficult to change the dealer's awareness. Logistics companies get no credit from dealers for doing things right, but they complain a lot about any delay," he said.
In terms of the wider environmental debate, Ford's Lowe as well as Toyota's Davies agreed that the 'green' aspect was simply a label for reducing fuel consumption and cost. "A greener logistics provider would rise to the top, but it would still come down to cheaper rates and better service," Davies said.
But both said that could change as buyers–particularly the younger generation–become more environmentally conscious. "Maybe as consumers look more carefully at the environmental impact of the supply chain we too will change our focus," said Lowe.
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| 8,983
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Guess What's For Dinner!
Dec 10, 2018 | 0
DONE! I have a table filled with cute little plates of Christmas cookies and cards all ready to go out to friends and family. The baby bassets have already delivered the ones that go to friends in the neighborhood. It was really cool to see how excited they were to deliver them. It reminds me of when I was young and delivered the cookies my mom baked at Christmas time. I don't get any help baking, but the boys make up for it with delivery. By the time I finished the last batch around 3pm yesterday I was exhausted, and the last thing I wanted to do was leave the couch. No need! The boys were on the job. I got to relax the rest of the day watching Hallmark Original Christmas movies, and we all had cookies for dinner!
Forbidden Blossom
by Nola Li Barr, Qat Wanders
Can a mysterious inheritance from an eccentric billionaire right a wrong from generations ago?
When Anne is plunged into the lives of the top one percent she finds herself entangled with a cutthroat family who will do anything to keep the past buried. As she uncovers generations of secrets that stir bitter memories she's thrust into the crosshairs of a vindictive scheme. The family lawyer is the only one keeping her strong, and Anne is falling for him, but is he genuine or only interested in her new wealth?
Two generations earlier, Rose has newly immigrated to the United States from Taiwan. In an era when Chinese immigrants are not always welcome, Rose knows she must tread lightly in her new home. However, the temptations in the world of the very rich, the freedom of her new western lifestyle, and her teenage naiveté make her throw caution to the wind. Will she have the strength to walk away when the boss's son turns his attention to her?
Rose's decision could cause ripples through generations of families. Will Anne be affected so many years later or will she set her own path in a new more modern world?
The Templar's Cross: A Medieval Mystery (The Sir Law Kintour Mysteries Book 1)
by J. R. Tomlin
Genre: Historical Fiction | Literature & Fiction
Sir Law Kintour has returned from the war in France injured, broke, and in need of a patron. In desperation, he reluctantly accepts a commission to find a nobleman's runaway wife. He enlists the help of a fellow Scot with whom he escaped after their defeat at the Battle of Verneuil. But his friend is murdered, and Law discovers he has been lied to. As the murders continue to mount, powerful interests come into play. When the Sheriff of Perth considers him a convenient scapegoat, it gives Law no choice but to untangle the lies and find the killer or hang for the murders.
Creatus (The Creatus Series Book 1)
by Carmen DeSousa
As with most of his family, Derrick Ashton knows his future and what position he's destined to fill within his unique society. Everything changes, however, when he breaks one of his family's strictest laws and falls in love — with a human.
In his quest to protect the woman he can never have, a twist of faith propels him into a new role that will cause dissension among his family and endanger the anonymity that they've worked for thousands of years to protect.
Now he will risk everything to save the girl from humans and his own kind. The only thing he can't save her from… is herself
The Creatus Series is not your normal paranormal story… it's a realistic romantic mystery based on the myths you've heard your entire life. Prepare to believe…
Bonus Material! This set includes the prequel, Creatus (They Exist), and a sneak peek at Creatus Rogue.
Chasing the Ghost (The Green Berets Book 7)
by Bob Mayer
Genre: Thrillers | Literature & Fiction | Action & Adventure
Covert operative Horace Chase has been chasing ghosts his entire life.
First, his Medal-of-Honor winner father who died in Vietnam without ever meeting his son. And left him the legacy of an automatic appointment to the Military Academy at West Point which shaped the next thirty-five years of Chase's life. Then, the ghost of his mother, who died while he was at war in Afghanistan and wounded, causing him to resign his commission and return to the United States, a lost soul.
Chase now wears two hats as a Federal counter-terrorism liaison to the local police department in Boulder, Colorado where he becomes embroiled in two seemingly un-related cases. Working as a detective with Boulder PD he chases another death, this one the apparent rape/murder of Rachel Stevens, an upscale housewife attending night classes at the University of Colorado. And with his counter-terrorism team he is embroiled in a series of killings involving a militia group, a rogue ex-Special Forces officer, a psychopathic ex-CIA contract mercenary, and ruthless drug runners.
From the streets of Boulder, to the highest railroad tunnel in the world, to a swingers club hiding in plain site in suburbia, Afghanistan starts to look pretty good to Chase.
Perfect Little Monsters and Other Stories
Genre: Literature & Fiction | Horror | Short Stories
A husband waits until his wife and children are in bed, before inviting a dangerous man into their home…
A girl keeps hold of her mother's necklace, as bloodied hands try to tear it from her grasp…
A gun jams, even as its intended victim begs the universe to let her die…
Perfect Little Monsters and Other Stories is a collection of short stories by Amy Cross. Some of the stories take place in seemingly ordinary towns, whose inhabitants soon discover something truly shocking lurking beneath the veneer of peace and calm. Others show glimpses of vast, barbaric worlds where deadly forces gather to toy with humanity. All the stories in this collection peel back the face of a nightmare, revealing the horror that awaits. And in every one of the stories, some kind of monster lurks…
Perfect Little Monsters and Other Stories contains the new stories Perfect Little Monsters, I Hate You, Meat, Fifty Fifty and Stay Up Late, as well as a revised version of the previously-released story The Scream. This book contains scenes of violence, as well as strong language.
PreviousNot Clear Waters
NextLast of the Breed by Louis L'Amour
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 9,168
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Q: Pagination with HibernateTemplate's findByNamedParam function I've seen lots of examples of how to create pagination with some really simple queries. But I don't see any using HibernateTemplate's findByNamedParam method.
How can I set a query's firstResult and maxResult parameters while also using the findByNamedParam method?
Basically, I'm trying to add pagination to an hql query I'm creating via HibernateTemplate's findByNamedParam method.
A: Ok after a lot of research, I finally got what I wanted.
First, need to create a HibernateCallback implementation:
HibernateCallbackImpl.java:
import java.sql.SQLException;
import java.util.List;
import org.apache.poi.hssf.record.formula.functions.T;
import org.hibernate.HibernateException;
import org.hibernate.Query;
import org.hibernate.Session;
import org.springframework.orm.hibernate3.HibernateCallback;
public class HibernateCallbackImpl
implements HibernateCallback<List<T>> {
private String queryString;
private String[] paramNames;
private Object[] values;
private int firstResult;
private int maxResults;
/**
* Fetches a {@link List} of entities from the database using pagination.
* Execute HQL query, binding a number of values to ":" named parameters in the query string.
*
* @param queryString a query expressed in Hibernate's query language
* @param paramNames the names of the parameters
* @param values the values of the parameters
* @param firstResult a row number, numbered from 0
* @param maxResults the maximum number of rows
*/
public HibernateCallbackImpl(
String queryString,
String[] paramNames,
Object[] values,
int firstResult,
int maxResults) {
this.queryString = queryString;
this.paramNames = paramNames;
this.values = values;
this.firstResult = firstResult;
this.maxResults = maxResults;
}
@Override
public List<T> doInHibernate(Session session) throws HibernateException,
SQLException {
Query query = session.createQuery(queryString);
query.setFirstResult(firstResult);
query.setMaxResults(maxResults);
// TODO: throw proper exception when paramNames.length != values.length
for (int c=0; c<paramNames.length; c++) {
query.setParameter(paramNames[c], values[c]);
}
@SuppressWarnings("unchecked")
List<T> result = query.list();
return result;
}
}
Then, I can just instantiate the new object and it will return what I want:
Example:
@SuppressWarnings("unchecked")
List<TitleProductAccountApproval> tpaas =
getHibernateTemplate().executeFind(
new HibernateCallbackImpl(
hql.toString(),
paramNames.toArray(new String[paramNames.size()]),
values.toArray(),
firstResult,
maxResult
)
);
A: The solution by @Corey works great but it includes a problem inside the for-loop where query.setParameter(...) is called.
The problem is that it doesn't account for parameters which are either a collection or an array and this will result in weired ClassCastExceptions because Hibernate tries to determine the ID by calling getId() on the collection or array (which is wrong). This happens e.g. if you are using an IN-clause (e.g. ...WHERE department IN (:departments) ...) where 'departments' is an array or collection of Department entities.
This is because collections or arrays need to use 'query.setParameterList(paramName, (Object[]) value)' or 'query.setParameterList(paramName, (Collection) value)'
Long story short:
I modified the version by @Corey by adding an 'applyNamedParameterToQuery()' method which I borrowed from org.springframework.orm.hibernate3.HibernateTemplate.applyNamedParameterToQuery(Query, String, Object):
import java.sql.SQLException;
import java.util.List;
import org.apache.poi.hssf.record.formula.functions.T;
import org.hibernate.HibernateException;
import org.hibernate.Query;
import org.hibernate.Session;
import org.springframework.orm.hibernate3.HibernateCallback;
public class HibernateCallbackImpl
implements HibernateCallback<List<T>> {
private String queryString;
private String[] paramNames;
private Object[] values;
private int firstResult;
private int maxResults;
/**
* Fetches a {@link List} of entities from the database using pagination.
* Execute HQL query, binding a number of values to ":" named parameters in the query string.
*
* @param queryString a query expressed in Hibernate's query language
* @param paramNames the names of the parameters
* @param values the values of the parameters
* @param firstResult a row number, numbered from 0
* @param maxResults the maximum number of rows
*/
public HibernateCallbackImpl(
String queryString,
String[] paramNames,
Object[] values,
int firstResult,
int maxResults) {
this.queryString = queryString;
this.paramNames = paramNames;
this.values = values;
this.firstResult = firstResult;
this.maxResults = maxResults;
}
@Override
public List<T> doInHibernate(Session session) throws HibernateException,
SQLException {
Query query = session.createQuery(queryString);
query.setFirstResult(firstResult);
query.setMaxResults(maxResults);
// TODO: throw proper exception when paramNames.length != values.length
for (int c=0; c<paramNames.length; c++) {
applyNamedParameterToQuery(query, paramNames[c], values[c]);
}
@SuppressWarnings("unchecked")
List<T> result = query.list();
return result;
}
/**
* Code borrowed from org.springframework.orm.hibernate3.HibernateTemplate.applyNamedParameterToQuery(Query, String, Object)
*
* Apply the given name parameter to the given Query object.
* @param queryObject the Query object
* @param paramName the name of the parameter
* @param value the value of the parameter
* @throws HibernateException if thrown by the Query object
*/
protected void applyNamedParameterToQuery(Query queryObject, String paramName, Object value)
throws HibernateException {
if (value instanceof Collection) {
queryObject.setParameterList(paramName, (Collection) value);
}
else if (value instanceof Object[]) {
queryObject.setParameterList(paramName, (Object[]) value);
}
else {
queryObject.setParameter(paramName, value);
}
}
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,250
|
Derventa Genocide Blog
Dobrodošli na moj blog ... Bosnian Genocide, 1992-95. Derventa Genocid nad Bosnjacima I Hrvatima u sjeni Srebrenice
KRVOLOCNI LISAC NA SLOBODI I SETA SE KRVAVOM REPUBLIKOM SRPSKOM
rs je genocidna i mora nestati
ZLOCINAC NAJODGOVORNIJI ZA MASAKRE I OKUPACIJU BOS POSAVINE
LIMITED RESPONSIBILITY OF THE ARMY
Although a state of war was not declared in Republika Srpska in 1992, retired VRS general Slavko Lisica 'considered there was a war going on'. This is why everybody, including the police, had to obey his orders 'without demur'. The prosecutor tried to prove that military commanders had 'limited' responsibility
In the cross-examination of retired general Slavko Lisica, prosecutor Korner put it to the witness that he 'wanted' to keep everything in his area of responsibility to be under the military command, but in reality 'it wasn't so'. In 1992, Lisica commanded the 3rd Tactical Group in the VRS 1st Krajina Corps.
According to the prosecutor, there was no state of war in 1992 in the Serbian Republic of Bosnia and Herzegovina and the responsibilities of military commanders, including Lisica, were 'limited'. Lisica didn't have control over 'everyone and everything', contrary to what he claimed to the judges. Lisica explained that once a state of war was declared, this provided a 'legal foundation' for all the resources in an area to be used to support combat. This is why he wanted the legal framework to be set up. This is what 'Karadzic and the Supreme Command' could do, but this was not done, Lisica said. 'In practice' he considered there was 'a war going on', and everyone had to obey his orders 'without demur'.
The prosecutor noted that there were two chains of command in the field – the police and military. As a commander, Lisica had to formulate his orders as requests because the state of war had not been declared. The witness against insisted that 'in my area of responsibility everything was done as if there had been the state of war'. The police probably had to report to 'the interior minister in a non-existent state', Lisica added. The minister, Lisica was convinced, 'probably would have to say to his men, do as the colonel says'.
In his statement to the investigators, Lisica claimed that the police was responsible for the crimes committed by police officers, adding that it was impossible to commit a crime in combat. 'Those who kill civilians had better kill themselves, or I would do that for them', Lisica said then. In the cross-examination Lisica confirmed that police officers, even if they were subordinated to the army, remained police officers, but they were 'in a trench, under a beech tree, not in the village'. Lisica also noted that in a war, military courts tried those who committed crimes; as there were no such courts, then commanders served as 'military judges'.
Replying to Mico Stanisic's defense lawyer, the witness confirmed that the commander of the 1st Krajina Corps had legal authority when he ordered that 'in combat operations, all police forces must be placed under the command of the zone commander, who then decides use'. The witness also confirmed that he 'alone' could order a public security station chief or even a security center chief to subordinate to him some police officers
In the brief cross-examination, Stojan Zupljanin's defense counsel put it to the witness that under the law, 'the armed forces are in a state of war from the moment when war is declared, or from the moment when an armed attack begins'. The armed attack 'occurred in 1991 when the conflict broke out in Croatia', the defense counsel said. The prosecutor noted this was a new line of defense and she sought permission to call an additional expert witness to testify on the issue.
The third and the last witness of the Trial Chamber, Tomislav Kovac, is scheduled to give evidence on 7 and 8 March 2012. In 1992, Kovac served as the deputy to Mico Stanisic. Stanisic, former interior minister, is on trial with Stojan Zupljanin, former chief of the Banja Luka region police, for crimes against Croats and Muslims in 20 municipalities in Bosnia and Herzegovina in 1992.
Objavio/la uskok2 u 11:25, 0 komentar(a), print, #
Srpski genocid u BiH nesmije biti zaboravljen
Jednostavno
KRIMINAL U VLASTI
Michael Tsarion blog by VolimSvojuBosnu ♥
PosavskiDomobrani
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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{"url":"https:\/\/www.physicsforums.com\/threads\/markov-chain-sum-of-n-dice-rolls.565476\/","text":"# Markov chain, sum of N dice rolls\n\n1. Jan 6, 2012\n\n### simba_\n\nQuestion : Let Xn be the maximum score obtained after n throws of a fair dice\n\na) Prove that Xn is a markov chain and write down the transition matrix\n\nIm having a problem starting the transition matrix\n\nim assuming the states are meant to be the sum. then do you write out the transition matrix for the first 2 throws and have this matrix to the power of n-1?\n\n2. Jan 6, 2012\n\n### Stephen Tashi\n\nI suppose if you are studying markov chains with an infinite number of states, you could try interpreting \"maximum score\" to mean some sort of sum. However, it seems to me that the problem intends the state of the process on the nth roll to be $max \\{ R_1,R_2,...R_n\\}$ and not $R_1 + R_2 + ... + R_n$. So if make 3 rolls and they are {3,5,4} the state of the process is $X_3 = 5$\n\n3. Jan 7, 2012\n\n### simba_\n\nSo the transition matrix is an upper triangular matrix to the power of n-1 with the diagonal entries 1\/6, 2\/6, 3\/6, 4\/6, 5\/6, 6\/6 respectively?\n\n4. Jan 7, 2012\n\n### Stephen Tashi\n\nThat is incorrect terminology. To compute things about the state at step n in the process, one may raise the transition matrix to a power, but the transition matrix itself, in simple examples, is not a function of n.\n\nYes.\n\n5. Jan 7, 2012","date":"2017-09-24 00:19:40","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8119714260101318, \"perplexity\": 399.65589190106044}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-39\/segments\/1505818689806.55\/warc\/CC-MAIN-20170923231842-20170924011842-00096.warc.gz\"}"}
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Q: Pocketsphinx android demo with arabic model doesn't work as the pc model I'm working on pocket-sphinx android demo .. I have tried my model on the PC .. but on the mobile it is not as good as the PC .. I thought of modifing the code to insert an audio file instead of recording from the mic of the mobile . Can anyone tellme what can I change in the code to do that? Thanks in advance
A: You need to dump audio files and listen for them and try to recognize them. If there are breaks it means your recognizer is not fast enough, you need to optimize the model size.
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"redpajama_set_name": "RedPajamaStackExchange"
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\section{Acknowledgements}\label{sec:Acknowledgement}
The authors gratefully acknowledge the support of the United States
National Science Foundation for the construction and operation of
the LIGO Laboratory and the Particle Physics and Astronomy Research
Council of the United Kingdom, the Max-Planck-Society and the State
of Niedersachsen/Germany for support of the construction and
operation of the GEO600 detector. The authors also gratefully
acknowledge the support of the research by these agencies and by the
Australian Research Council, the Natural Sciences and Engineering
Research Council of Canada, the Council of Scientific and Industrial
Research of India, the Department of Science and Technology of
India, the Spanish Ministerio de Ciencia y Tecnologia, the John
Simon Guggenheim Foundation, the Leverhulme Trust, the David and
Lucile Packard Foundation, the Research Corporation, and the Alfred
P. Sloan Foundation. We are grateful to Scott Barthelmy and the GCN
network and Kevin Hurley and the IPN network for providing us with
near real time GRB triggers and to the Ulysses, Konus, SAX, and HETE
experiments, who detect and generate the events distributed by GCN
and IPN. This research has made use of data obtained from the HETE
science team via the website http://space.mit.edu/HETE/Bursts/Data.
HETE is an international mission of the NASA Explorer program, run
by the Massachusetts Institute of Technology.
\section{Analysis}\label{sec:Analysis}
The goal of the analysis is either to identify significant events
in the signal region or, in the absence of significant events, to
set a limit on the strength of the associated gravitational wave
signal. Simulations and background data were used to determine the
detection efficiency for various ad-hoc and model-based waveforms
(Section~\ref{sec:Efficiencies}) and the false alarm rate of the
detection algorithm respectively.
The analysis takes advantage of the information provided by the
astrophysical trigger. The trigger time determined when to perform
the analysis. As discussed below, the time window to be analyzed
around the trigger time was chosen to accommodate most current
theoretical predictions and timing uncertainties. The source
direction was needed to calculate the attenuation due to the LIGO
detector's antenna pattern for the astrophysical interpretation.
The two co-located and co-aligned Hanford detectors had very similar
frequency-dependent response functions at the time of the trigger.
Consequently, the detected arrival time and recorded waveforms of a
gravitational wave signal should be essentially the same in both
detectors. It is natural then to consider cross-correlation of the
two data streams as the basis of a search algorithm. This conclusion
can also be reached via a more formal argument based on the maximum
log-Likelihood ratio test~\cite{Anderson01,Mohanty04}.
The schematic of the full analysis pipeline is shown in
Figure~\ref{fig:PIPE}. The underlying analysis algorithm is
described in detail in Ref.~\cite{Mohanty04}.
\begin{figure}[!t]
\includegraphics[angle=0,width=84mm]{PIPE.eps}
\caption{The schematic of the analysis pipeline } \label{fig:PIPE}
\end{figure}
The background data, the signal region data and the simulations are
all processed identically. The background region consists of the
data where we do not expect to have a gravitational wave signal
associated with the GRB. We scan the background to determine the
false alarm distribution and to set a threshold on the event
strength that will yield an acceptable false alarm rate. This
threshold is used when scanning the signal region and simulations.
In order to estimate our sensitivity to gravitational waves,
simulated signals of varying strength are added to the detector data
streams.
The signal region around the GRB trigger is scanned to identify
outstanding signals. If events were detected above threshold, in
this region, their properties would be tested against those
expected from gravitational waves. If no events were found above
threshold, we would use the estimated sensitivity to set an upper
limit on the gravitational wave strain at the detector.
The output from each interferometer is divided into 330~sec long
segments with a 15~second overlap between consecutive segments (both
ends), providing a tiling of the data with 300~second long segments.
In order to avoid edge effects, the 180~sec long signal region lies
in the middle of one such 300~sec long segment. This tiling method
also allows for adaptive data conditioning and places the
conditioning filter (see Sec.~\ref{sec:DataConditioning} below)
transients well outside of the 300 second long segment containing
the signal region.
\subsection{Choice of signal region}\label{sec:Signal_region}
Current models suggest~\cite{Meszaros03} that the gravitational wave
signature should appear close to the GRB trigger time. We
conservatively chose the duration and position of the signal region
to over-cover most predictions and to allow for the expected
uncertainties associated with the GRB trigger timing. A 180 second
long window (see Figure~\ref{fig:HETE}), starting 120 seconds before
the GRB trigger time is sufficient; roughly ten times wider than the
GRB light curve features, and wide enough to include most
astrophysical predictions. Most models favor an ordering where the
arrival of the gravitational wave precedes the GRB
trigger~\cite{Meszaros03}, but in a few other cases the
gravitational wave arrival is predicted to be
contemporaneous~\cite{Araya-Gochez03,Putten04} to the arrival and
duration of the gamma rays (i.e after the GRB trigger). The 60
second region after the GRB trigger time, is sufficient to cover
these predictions and also contains allowance for up to 30 seconds
uncertainty on trigger timing, which is a reasonable choice in the
context of the HETE light curve. Figure~(\ref{fig:HETE}) shows a
signal rise time of order $10~s$, precursor signals separated from
the main peak, and significant structure within the main signal
itself. Effects due to the beaming dynamics of the GRB and the
instrumental definition of the trigger time can also be significant
contributors to the timing uncertainty.
\subsection{Search algorithm}\label{sec:SearchAlgorithm}
\subsubsection{Data Conditioning}\label{sec:DataConditioning}
The data-conditioning step was designed to remove instrumental
artifacts from the data streams. We used an identical data
conditioning procedure when processing the background, the signal
region and the simulations.
The raw data streams have narrowband lines, associated with the
power line harmonics at multiples of 60Hz, the violin modes of the
mirror suspension wires and other narrow band noise sources. The
presence of lines has a detrimental effect on our sensitivity
because lines can produce spurious correlations between detectors.
In addition, the broadband noise shows significant variations over
timescales of hours and smaller variations over timescales of
minutes and seconds due to alignment drift and fluctuations. The
background data must portray a representative sample of the
detector behavior around the time of the trigger. Broadband
non-stationarity can limit the duration of this useful background
data and hence the reliability of our estimated false alarm rate.
Our cross correlation based algorithm performs best on white
spectra without line features. We use notch filters to remove the
well-known lines, such as power line and violin mode harmonics
from both data streams. Strong lines of unknown origin with
stationary mean frequency are also removed at this point. We also
apply a small correction to mitigate the difference between the
phase and amplitude response of the two Hanford detectors.
We bandpass filter and decimate the data to a sampling rate of
4096~Hz to restrict the frequency content to the ${\simeq}80$~Hz to
${\simeq}2048$~Hz region, which was the most sensitive band for both
LIGO Hanford detectors during the S2 run.
In order to properly remove weaker stationary lines and the small
residuals of notched strong lines, correct for small slow changes
in the spectral sensitivity and whiten the spectrum of the data,
we use adaptive line removal and whitening. As all strong lines
are removed before the adaptive whitening, we avoid potential
problems due to non-stationary lines and enhance the efficiency of
the follow up adaptive filtering stage. The conditioned data has a
consistent white spectrum without major lines and sufficient
stationarity, from segment to segment, throughout the background
and signal regions.
The end result of the pre-processing is a data segment with a flat
power spectral density (white noise), between $\simeq$80~Hz and
$\simeq$2048~Hz. The data conditioning was applied consistently
after the signal injections. This ensures that any change in
detection efficiency due to the pre-processing is properly taken
into account.
\subsubsection{Gravitational Wave Search Algorithm}\label{sec:Algorithm}
The test statistics for a pair of data streams are constructed as
follows. We take pairs of short segments, one from each stretch,
and compute their cross-correlation function. The actual form of
the cross-correlation used ($\widetilde{C}^{m,n}_{k,p,j}$) is
identical to the common Euclidean inner product:
\begin{equation}
\widetilde{C}^{m,n}_{k,p,j}= \sum_{i=-j}^{j} H_m[k+i]
H_n[k+p+i]~,~
\end{equation}
where the pre-conditioned time series from detector ``x" is ${\bf
H}_x=\{H_x[0],H_x[1],\ldots\}$ and i,k,p and j are all integers
indexing the data time series, with each datum being (1/4096)~s
long. As we now only consider the two Hanford detectors ``m" and
``n" can only assume values of 1 ($H_1$) or 2 ($H_2$). There are
therefore three free parameters to scan when searching for
coherent segments of data between a pair of interferometers (m,n):
1. the center time of the segment from the first detector (k); 2.
the relative time lag between the segments from the two detectors
(p); and 3. the common duration of segments (2j+1) called the
integration length.
The optimum integration length to use for computing the
cross-correlation depends on the duration of the signal and its
signal-to-noise ratio, neither of which is {\em a priori} known.
Therefore the cross-correlation should be computed from segment
pairs with start times and lengths varying over values, which
should, respectively, cover the expected arrival times (signal
region) and consider durations of the gravitational wave burst
signals~\cite{Muller2004,Fryer2004,Fryer2002,Zwerger1997,Dimmelmeier2002,Burrows1996}
($\sim$O(1-128ms)).
Hence we apply a search algorithm~\cite{Mohanty04} that processes
the data in the following way.
(1) A three dimensional quantity (${\cal C}_{k,j}[p]$) is
constructed:
\begin{equation}\label{CCStat}
{\cal C}_{k,j}[p] =
\left[\left({\widetilde{C}^{1,2}_{k,p,j}}\right)^2 +
\left({\widetilde{C}^{2,1}_{k,-p,j}}\right)^2\right]^{1/2}\;,
\end{equation}
scanning the range of segment center times (k), integration
lengths (2j+1) and relative time shifts ($p =
0,\pm1,\pm2,\ldots$). A coherent and coincident signal is expected
to leave its localized signature within this three dimensional
quantity.
We use a fine rectangular grid in relative time shift (p) and
integration length (2j+1) space. The spacing between grid points is
$\simeq$1~ms for the segment center time (k) and (1/4096)~s for the
relative time shift. The spacing of the integration lengths is
approximately logarithmic. Each consecutive integration length is
${\simeq}$50\% longer than the previous one, covering integration
lengths from ${\simeq}$1~ms to ${\simeq}$128~ms.
Introducing small, non-physical relative time shifts (much larger
than the expected signal duration) between the two data streams
before computing the cross-correlation matrix suppresses the average
contribution from a GW signal. This property can be used to estimate
the local noise properties, thereby mitigating the effects of
non-stationarity in the interferometer outputs. Accordingly, ${\cal
C}_{k,j}[p]$ contains the autocorrelation of the coherent signal for
relative time shifts at and near p~$=$~0 (called ``\emph{core}"),
while far away, in the ``side \emph{lobes}", the contribution from
the signal autocorrelation is absent, sampling only the random
contributions to the cross-correlation arising from the noise. The
optimal choice of the core size depends on the expected signal
duration (integration length), the underlying detector noise and it
cannot be smaller than the relative phase uncertainty of the
datastreams. The core region can reach as far as 5~ms, as it
increases with increasing integration length. The size of each side
lobe is twice the size of the core region and the median time shift
associated with the side lobes can be as large as 325~ms as it is
also increasing with increasing integration length. We use the side
lobes of ${\cal C}_{k,j}[p]$ to estimate the mean
($\widehat{\mu}_{k,j}$) and variance ($\widehat{\sigma}_{k,j}$) of
the local noise distribution, which is also useful in countering the
effects of non-stationarity.
\begin{figure}[!t]
\includegraphics[angle=0,width=87mm]{Corrgram.eps}
\caption{Examples of corrgram images. The horizontal axes are time
(linearly scaled) and the vertical axes are integration length
(logarithmically scaled). The color axis, an indicator of the excess
correlation, is independently auto-scaled for each quadrant for
better visibility, therefore the meaning of colors differ from
quadrant to quadrant. The time ticks also change from quadrant to
quadrant for better visibility. The rainbow type color scale goes
from blue to red, dark red marking the most significant points
within a quadrant. The upper two quadrants show the corrgram image
of injected Sine-Gaussians (250~Hz,~Q~$=$~8.9). The bottom quadrants
are examples of noise. The maximum of the intensity scale is
significantly higher for both quadrants with injections, when
compared to the noise examples. The top left injection is strong
enough to be significantly above the preset detection threshold,
while the top right injection is weak enough to fall significantly
below the detection threshold.} \label{fig:Corrgram}
\end{figure}
(2) The three dimensional quantity is reduced to a two dimensional
image (see Fig.~\ref{fig:Corrgram}), called a {\em corrgram}, as
follows. The values of ${\cal C}_{k,j}[p]$ in the core region are
standardized by subtracting $\widehat{\mu}_{k,j}$ and then dividing
by $\widehat{\sigma}_{k,j}$. {\em Positive} standardized values in
the core region are summed over $p$ to determine the value of the
corrgram pixel. Each pixel is a measure of the excess
cross-correlation in the core region when compared to the expected
distribution characterized by the side lobes for the given (k,j)
combination.
(3) A list of events is found by recursively identifying and
characterizing significant regions (called ``clusters") in the
corrgram image. Each event is described by its arrival time, its
optimal integration length and its strength (ES). The event's
arrival time and its optimal integration length correspond to the
most significant pixel of the cluster. The event strength is
determined by averaging the five most significant pixels of the
cluster, as this is helpful in discriminating against random
fluctuations of the background noise.
The strength of each event is then compared to a preset detection
threshold corresponding to the desired false alarm rate. This
detection threshold is determined via extensive scans of the
background region.
\section{Summary}\label{sec:Discussion}
\subsection{Comparison with previous searches for gravitational waves from GRBs}\label{sec:Comparison}
Our result is comparable to the best published results searching for
association between gravitational waves and GRBs~\cite{Astone2004},
however these studies differ in their most sensitive frequency.
Tricarico et al.~\cite{Tricarico01} used a single resonant mass
detector, AURIGA~\cite{Prodi98} , to look for an excess in
coincidences between the arrival times of GRBs in the BATSE 4B
catalog. They used two different methods. They searched for events
identified above a certain threshold in the gravitational wave data,
and also attempted to establish a statistical association between
GRBs and gravitational waves. No significant excess was found with
the former method. The latter used a variant of the correlation
based Finn-Mohanty-Romano (FMR) method~\cite{FMR99}. However,
instead of using the cross-correlation of two detectors, as proposed
in the FMR method, only the variance of the single detector output
was used. A sample of variances from times when there were no GRBs
was compared with a corresponding sample from data that spanned the
arrival times of the GRBs. An upper limit on the source-averaged
gravitational wave signal root mean square value of
$1.5\times10^{-18}$ was found using 120 GRBs. This limit applies at
the AURIGA resonant frequencies of 913 and 931 Hz, which are very
far from the most sensitive frequency of the LIGO detectors
($\simeq$250~Hz). This work ~\cite{Tricarico01} was later extended
~\cite{Tricarico03}, which led to an improved upper limit.
The data analysis method employed in Modestino \&
Moleti~\cite{Modestino02} is another variant of the FMR method.
Instead of constructing off-source samples from data segments that
are far removed from the GRB trigger, the off-source samples are
constructed by introducing non-zero time shifts between the two
detector data streams and computing their cross-correlation. For
narrowband resonant mass detectors, the directional information of
a GRB cannot be exploited to discriminate against incorrect
relative timing since the signal in the output of the detector is
spread out by the detector response over time scales much larger
than the light travel time between the detectors.
Astone et al.~\cite{Astone99,Astone02} report on a search for a
statistical association between GRBs and gravitational waves using
data from the resonant mass detectors EXPLORER~\cite{Astone93} and
NAUTILUS~\cite{Astone97}. They report a Bayesian upper limit on
gravitational wave signal amplitudes of 1.2$\times$10$^{-18}$, at
95\% probability, when the maximum delay between the GRB and
gravitational wave is kept at 400 sec. The upper limit improves to
6.5$\times$10$^{-19}$ when the maximum delay is reduced to 5 sec.
However, the absence of directional and/or distance information for
most of these GRBs precluded accounting for source variations; the
gravitational wave signal amplitude was assumed to be the same for
all of the GRBs.
Astone et al.~\cite{Astone99B} report on the operation of the
resonant mass detectors EXPLORER during the closest ever gamma ray
burst (GRB980425) with known redshift and direction. At the time of
the burst, EXPLORER was taking data with close to optimal
orientation. GRB980425 was $\simeq$23 times closer to Earth than
GRB030329 giving a $\simeq$520 increase in energy sensitivity. Based
on their sensitivity and the loudest event within $\pm$5~minutes of
the GRB980425 trigger the authors quote a limit of
$\simeq$1600~M$_\odot$ for a simple model assuming isotropic
gravitational wave emission.
Recently, Astone et al.~\cite{Astone2004} executed a search aiming
to detect a statistical association between the GRBs detected by the
satellite experiments BATSE and BeppoSAX, and the EXPLORER and
NAUTILUS gravitational wave detectors. No association was uncovered.
Their upper limit is the lowest published result, which is based on
bar-detector gravitational wave data.
\subsection{Conclusion}\label{sec:Conclusion}
We have executed a cross-correlation-based search for possible
gravitational wave signatures around the GRB030329 trigger, which
occurred during the Second Science Run (S2) of the LIGO detectors.
We analyzed a 180 second signal region around the GRB and 4.5 hours
of background data, surrounding the signal region, corresponding to
a single coincident lock stretch. These data were sufficient to
characterize the background, scan the signal region and estimate our
efficiency. We used the same procedure, based on cross correlation,
for each of these studies. We evaluated the sensitivity of the
search to a large number of broad and narrow band waveforms.
We observed no candidates with gravitational wave signal strength
larger than a pre-determined threshold, therefore we set upper
limits on the associated gravitational wave strength at the
detectors. The present analysis covers the most sensitive frequency
range of the Hanford detectors, approximately from 80~Hz to 2048~Hz.
The frequency dependent sensitivity of our search was h$_{RSS}
{\simeq}$O(6$\times$10$^{-21}$)~Hz$^{-1/2}$.
The prospect for future searches is promising, as the sensitivity of
the instruments improves with further commissioning.
Once operating at target sensitivity, the detectors will be more
sensitive to strain than they were during S2 by factors of 10~-~100,
depending on frequency (see Figure~\ref{fig:LIGOEFF}.). This implies
an improvement of a factor of $\sim$1000 in sensitivity to E$_{GW}$,
since E$_{GW}$ scales like $\sim h_{RSS}^{2}$ (see for example
Eq.~\ref{Int5}).
Detection of GRBs with measured redshifts significantly smaller
than GRB030329's is certainly possible. GRB030329's
electromagnetic brightness was due to a favorable combination of
distance and our position in its beam. One year of observation
will incorporate hundreds of GRBs with LIGO data coverage and some
of these GRBs, even though
fainter~\cite{Woosley2004,Sazonov2004,Soderberg2004} than
GRB030329, could be significantly closer, as was 1998bw. We can
also hope for sources with more optimal direction and coincidence
between three or four observing interferometers.
\section{Introduction}\label{sec:Introduction}
Gamma Ray Bursts (GRBs) are short but very energetic pulses of gamma
rays from astrophysical sources, with duration ranging between 10~ms
and 100~s. GRBs are historically divided into two
classes~\cite{Kouveliotou93,Meszaros03} based on their duration:
``short" ($<$~2~s) and ``long" ($>$~2~s). Both classes are
isotropically distributed and their detection rate can be as large
as one event per day. The present consensus is that long
GRBs~\cite{Meszaros03} are the result of the core collapse of
massive stars resulting in black hole formation. The violent
formation of black holes has long been proposed as a potential
source of gravitational waves. Therefore, we have reason to expect
strong association between GRBs and gravitational
waves~\cite{Fryer01,Davies02,Putten04}. In this paper, we report on
a search for a possible short burst of gravitational waves
associated with GRB030329 using data collected by the Laser
Interferometer Gravitational Wave Observatory (LIGO).
On March 29, 2003, instrumentation aboard the HETE-2
satellite~\cite{HETE04} detected a very bright GRB, designated
GRB030329. The GRB was followed by a bright and well-measured
afterglow from which a redshift~\cite{Price03} of $z=$0.1685
(distance~$\simeq$800~Mpc~\cite{Matheson03}) was determined. After
approximately 10 days, the afterglow faded to reveal an underlying
supernova (SN) spectrum, SN2003dh~\cite{Sokolov03}. This GRB is the
best studied to date, and confirms the link between long GRBs and
supernovae.
At the time of GRB030329, LIGO was engaged in a 2-month long data
run. The LIGO detector array consists of three interferometers, two
at the Hanford, WA site and one at the Livingston, LA site.
Unfortunately, the Livingston interferometer was not operating at
the time of the GRB; therefore, the results presented here are based
on the data from only the two Hanford interferometers. The LIGO
detectors are still undergoing commissioning, but at the time of
GRB030329, their sensitivity over the frequency band 80 to 2048~Hz
exceeded that of any previous gravitational wave search, with the
lowest strain noise of ${\simeq}$6$\times$10$^{-22}$~Hz$^{-1/2}$
around 250~Hz.
A number of long GRBs have been associated with X-ray, radio and/or
optical afterglows, and the cosmological origin of the host galaxies
of their afterglows has been unambiguously established by their
observed redshifts, which are of order unity~\cite{Meszaros03}. The
smallest observed redshift of an optical afterglow associated with a
detected GRB (GRB980425~\cite{Kulkarni98,Iwamoto98,Galama98}) is
$z$=0.0085 ($\simeq$35~Mpc). GRB emissions are very likely strongly
beamed~\cite{Frail01,Putten03}, a factor that affects estimates of
the energy released in gamma rays (a few times 10$^{50}$ erg), and
their local true event rate (about 1 per year within a distance of
100Mpc).
In this search, we have chosen to look for a burst of gravitational
waves in a model independent way. Core collapse~\cite{Davies02},
black hole formation~\cite{Putten04,Bulik04} and black hole
ringdown~\cite{Hughes98,Jolien99} may each produce gravitational
wave emissions, but there are no accurate or comprehensive
predictions describing the gravitational wave signals that might be
associated with GRB type sources. Thus, a traditional matched
filtering approach~\cite{Helstrom68,InspiralS1} is not possible in
this case. To circumvent the uncertainties in the waveforms, our
algorithm does not presume any detailed knowledge of the
gravitational waveform and we only apply general bounds on the
waveform parameters. Based on current theoretical considerations, we
anticipate the signals in our detectors to be weak, comparable to or
less than the detector's noise~\cite{Muller2004,Fryer2004,Ott2004}.
This paper is organized as follows: Section II summarizes the
currently favored theories of GRBs and their consequences for
gravitational wave detection. Section III provides observational
details pertinent to GRB030329. Section IV briefly describes the
LIGO detectors and their data. Section V discusses the method of
analysis of the LIGO data. In Section VI we compare the events in
the signal region with expectations and we use simulated signal
waveforms to determine detection efficiencies. We also present and
interpret the results in this section. Section VII offers a
comparison with previous analyses, a conclusion, and an outlook
for future searches of this type.
\section{OVERVIEW OF THE LIGO DETECTORS}\label{sec:LIGO_Detectors}
The three LIGO detectors are orthogonal arm Michelson laser
interferometers, aiming to detect gravitational waves by
interferometrically monitoring the relative (differential)
separation of mirrors, which play the role of test masses. The LIGO
Hanford Observatory (LHO) operates two identically oriented
interferometric detectors, which share a common vacuum envelope: one
having 4 km long arms (H1), and one having 2 km long arms (H2). The
LIGO Livingston Observatory operates a single 4 km long detector
(L1). The two sites are separated by $\simeq$3000~km, representing a
maximum arrival time difference of ${\simeq}\pm$10~ms.
A complete description of the LIGO interferometers as they were
configured during LIGO's first Science Run (S1) can be found in Ref
~\cite{Stan03}.
\subsection{Detector calibration and configuration}\label{sec:Configurations}
To calibrate the error signal, the response to a known
differential arm strain is measured, and the frequency-dependent
effect of the feedback loop gain is measured and compensated for.
During detector operation, changes in calibration are tracked by
injecting continuous, fixed-amplitude sinusoidal excitations into
the end test mass control systems, and monitoring the amplitude of
these signals at the measurement (error) point. Calibration
uncertainties at the Hanford detectors were estimated to be
$<11\%$.
Significant improvements were made to the LIGO detectors following
the S1 run, held in early fall of 2002:
\begin{enumerate}
\item {The analog suspension controllers on the H2 and L1
interferometers were replaced with digital suspension controllers
of the type installed on H1 during S1, resulting in lower
electronics noise.}
\item {The noise from the optical lever servo that damps the angular
excitations of the interferometer optics was reduced.}
\item {The wavefront sensing system for the H1 interferometer was
used to control 8 of 10 alignment degrees of freedom for the main
interferometer. As a result, it maintained a much more uniform
operating point over the run.}
\item {The high frequency sensitivity was improved by operating the
interferometers with higher effective power, about 1.5 W.}
\end{enumerate}
These changes led to a significant improvement in detector
sensitivity. Figure~\ref{fig:LIGOEFF} shows typical spectra
achieved by the LIGO interferometers during the S2 run.
The differences among the three LIGO spectra reflect differences in
the operating parameters and hardware implementations of the three
instruments which are in various stages of reaching the final design
configuration.
\begin{figure}[!t]
\includegraphics[angle=0,width=94mm]{LIGO_EFF.eps}
\caption{Typical LIGO Hanford sensitivity curves during the S2 Run
[strain~Hz$^{-1/2}$] (black and grey lines). The LIGO design
sensitivity goals (SRD) are also indicated (dotted and dashed
lines).} \label{fig:LIGOEFF}
\end{figure}
\subsection{The second science run}\label{sec:S2Run}
The data analyzed in this paper were taken during LIGO's second
Science Run (S2), which spanned approximately 60 days from
February 14 to April 14, 2003. During this time, operators and
scientific monitors attempted to maintain continuous low noise
operation. The duty cycle for the interferometers, defined as the
fraction of the total run time when the interferometer was locked
and in its low noise configuration, was approximately 74\% for H1
and 58\% for H2. The longest continuous locked stretch for any
interferometer during S2 was 66 hours for H1.
At the time of the GRB030329 both Hanford interferometers were
locked and taking science mode data. For this analysis we relied on
the single, ${\simeq}4.5$ hours long coincident lock stretch, which
started ${\simeq}3.5$ hours before the trigger time. With the
exception of the signal region, we utilized $\simeq$98\% of the data
within this lock stretch as the \emph{background} region (defined in
section V). 60~seconds of data before and after the signal region
were not included in the background region. Data from the beginning
and from the end of the lock stretch were not included in the
background region to avoid using possibly non-stationary data, which
might be associated with these regions.
As described below, the false alarm rate estimate, based on
background data, must be applicable to the data within the signal
region. We made a conservative choice and avoided using background
data outside of the lock stretch containing the GRB trigger time.
This is important when considering the present non-stationary
behavior of the interferometric detectors.
\section{GRB030329 RELATED OBSERVATIONAL RESULTS }\label{sec:Observations}
\subsection{Discovery of GRB 030329 and its afterglow}\label{sec:Afterglow}
On March 29, 2003 at 11:37:14.67 UTC, a GRB triggered the FREGATE
instrument on board the HETE-2
satellite~\cite{HETE04,Vanderspek03,Ricker03,Vander2004}. The GRB
had an effective duration of $\simeq$50~s, and a fluence of
1.08$\times$10$^{-4}$~erg/cm$^{2}$ in the 30-400~keV
band~\cite{Vander2004}. The KONUS detector on board the Wind
satellite also detected it~\cite{Golenetskii03}, triggering about
15~seconds after HETE-2. KONUS observed the GRB for about 35
seconds, and measured a fluence of
1.6$\times$10$^{-4}$~erg~/~cm$^{2}$ in the 15-5000~keV band. The
measured gamma ray fluences place this burst among the brightest
GRBs. Figure~\ref{fig:HETE} shows the HETE-2 light curve for
GRB030329~\cite{HETE030329}.
\begin{figure}[!t]
\includegraphics[angle=0,width=84mm]{HETE.eps}
\caption{The GRB030329 light curve as measured by the HETE-2 FREGATE
B instrument. The arrow indicates the HETE trigger time. The signal
region analyzed in this study is indicated by the horizontal bar at
the top. This figure is the courtesy of the HETE collaboration. }
\label{fig:HETE}
\end{figure}
The rapid localization of the GRB by HETE ground analysis gave an
accurate position which was distributed about 73 minutes after the
original trigger. A few hours later, an optical
afterglow~\cite{Lipkin03,Price03} was discovered with magnitude
R=12.4, making it the brightest optical counterpart to any GRB
detected to date. The RXTE~\cite{Bradt93} satellite measured a X-ray
flux of 1.4$\times$10$^{-10}$~erg~s$^{-1}$~cm$^{-2}$ in the 2-10 keV
band about 4h51m after the HETE trigger, making this one of the
brightest X-ray afterglows detected by RXTE~\cite{Marshall03}. The
National Radio Astronomy Observatory (NRAO) observed ~\cite{GCN2014}
the radio afterglow, which was the brightest radio afterglow
detected to date~\cite{Berger03}. Spectroscopic measurements of the
bright optical afterglow~\cite{Greiner03} revealed emission and
absorption lines, and the inferred redshift ($z$~=~0.1685,
luminosity distance $D_L~\cong~$800~Mpc) made this the second
nearest GRB with a measured distance. To date, no host galaxy has
been identified. It is likely that numerous other GRBs have been
closer than GRB030329, but the lack of identified optical
counterparts has left their distances undetermined.
Spectroscopic measurements~\cite{Stanek03,Mazzali03,Matheson03},
about a week after the GRB trigger, revealed evidence of a
supernova spectrum emerging from the light of the bright optical
afterglow, which was designated SN2003dh. The emerging supernova
spectrum was similar to the spectrum of SN1998bw a week before its
brightness maximum~\cite{Hjorth03,Greiner03N}.
SN1998bw was a supernova that has been spatially and temporally
associated with GRB980425~\cite{Kulkarni98,Iwamoto98,Galama98}, and
was located in a spiral arm of the barred spiral galaxy ESO 184-G82
at a redshift of $z~=$~0.0085 ($\simeq$35~Mpc), making it the
nearest GRB with a measured distance. The observed spectra of
SN2003dh and SN1998bw, with their lack of hydrogen and helium
features, place them in the Type Ic supernova class. These
observations, together with the observations linking GRB980425
(which had a duration of ${\simeq}$23~s) to SN1998bw, make the case
that collapsars are progenitors for long GRBs more convincing. In
the case of SN1998bw, Woosley \emph{et al.}~\cite{Woosley99} and
Iwamoto \emph{et al.}~\cite{Iwamoto98} found that its observed
optical properties can be well modeled by the core collapse of a C+O
core of mass 6~M$_{\odot}$ (main sequence mass of 25~M$_{\odot}$)
with a kinetic energy of ${\simeq}$2$\times$10$^{52}$~ergs. This
energy release is about an order of magnitude larger than the
energies associated with typical supernovae.
\subsection{GRB030329 energetics}\label{sec:Energetics}
A widely used albeit naive quantity to describe the energy emitted
by GRBs is the total isotropic equivalent energy in gamma rays:
\begin{equation}\label{IEE}
E_{iso} = 4 \pi (BC) D_{L}^{2} f / (1+z) \approx
2\times10^{52}~erg~.~
\end{equation}
where $f$ is the measured fluence in the HETE-2 waveband and BC is
the approximate bolometric correction for HETE-2 for long GRBs.
Using a ``Band spectrum"~\cite{Band93} with a single power law to
model the gamma ray spectrum, and using a spectral index,
$\beta=-$2.5, gives that the GRB's total energy integrated from 1
keV to 5 GeV is greater than that present in the band 30-400 keV by
a factor 2.2.
However, it is generally believed that GRBs are strongly beamed, and
that the change in slope in the afterglow light curve corresponds to
the time when enough deceleration has occurred so that relativistic
beaming is diminished to the point at which we ``see" the edge of
the jet. This occurs during the time in which the relativistic
ejecta associated with the GRB plows through the interstellar
medium, and the beaming factor $\Gamma^{-1}$, where $\Gamma$ is the
bulk Lorentz factor of the flow, increases from a value smaller than
the beaming angle $\Theta_j$, to a value larger than $\Theta_j$.
Effectively, prior to this time the relativistic ejecta appears to
be part of a spherical expansion, the edges of which cannot be seen
because they are outside of the beam, while after this time the
observer perceives a jet of finite width.
This leads to a faster decline in the light curves. Zeh \emph{et
al.} and Li \emph{et al.}~\cite{Zeh03,Li03} show that the initial
``break" or strong steepness in the light curve occurs at about 10
hours after the initial HETE-2 detection.
Frail \emph{et al.}~\cite{Frail01} give a parametric relation
between beaming angle $\Theta_{j}$, break time t$_{j}$, and
E$_{iso}$ as:
\begin{eqnarray}\label{Theta}
\Theta_{j} \approx 0.057 ~{\left(\frac{t_{j}}{24
hours}\right)}^{3/8} {\left(\frac{1+z}{2}\right)}^{-3/8}\times\nonumber\\
\times~{~} {\left(\frac{E_{iso}}{10^{53}
ergs}\right)}^{-1/8}{\left(\frac{\eta_\gamma}{0.2}\right)}^{1/8}{\left(\frac{n}{0.1cm^{-3}}\right)}^{1/8}~{~{~{~}}}~.~
\end{eqnarray}
where $\Theta_{j}$ is measured in radians. It was argued that the
fireball converts the energy in the ejecta into gamma rays
efficiently~\cite{Beloborodov00} (${\eta_\gamma}\approx${0.2}), and
that the mean circumburst density is {n}$\approx${0.1~cm$^{-3}$}
~\cite{Frail00}. Evaluating equation ~\ref{Theta} for the parameters
of GRB030329 (t$_{j}\approx$~10~hours, $z$=0.1685, and
E$_{iso}$=2$\times$10$^{52}$~erg) gives $\Theta_{j}\approx$0.07~
rad.
Therefore the beaming factor that relates the actual energy released
in gamma rays (E$_\gamma$) to the isotropic equivalent energy is
{${\Theta_{j}^{2}}$/2}$\approx$1/400, so that
E$_\gamma\approx$5$\times$10$^{49}$~erg. Comparing E$_{iso}$ and
E$_\gamma$ with the histograms in Fig. 2 of Frail \emph{et
al.}~\cite{Frail01} , GRB030329 resides at the lower end of the
energy distributions. The calculated isotropic energy from
GRB980425, the GRB associated with SN1998bw, is also low
(${\simeq}$10$^{48}$~erg).
\section{Production of gravitational waves in massive core collapses}\label{sec:Production_of_GW}
The apparent spatial association of GRB afterglows with spiral arms,
and by implication star formation regions in remote galaxies, has
lead to the current ``collapsar'' or ``hypernova'' scenario
~\cite{MacFadyen99,Heger2003} in which the collapse of a rotating,
massive star to a Kerr black hole can lead to relativistic ejecta
emitted along a rotation axis and the associated production of a GRB
jet. The identification of GRB030329 with the supernova SN2003dh
(section 3 below) gives further support to this association. This
observation is consistent with the theory that the GRB itself is
produced by an ultra-relativistic jet associated with a central
black hole. Stellar mass black holes in supernovae must come from
more massive stars. Ref. ~\cite{Heger2003} presents ``maps'' in the
metallicity-progenitor mass plane of the end-states of stellar
evolution and shows that progenitors with 25 $M_{\odot}$ can produce
black holes by fall-back accretion.
The observed pulsar kick velocities of $\simeq$500~km/s hint at a
strong asymmetry around the time of maximum compression, which may
indicate deviations from spherical symmetry in the progenitor. The
resulting back reaction on the core from the neutrino heating
provides yet another potential physical mechanism for generating a
gravitational wave signal. In the model of ~\cite{Burrows1996} it
imparts a kick of 400-600 km/s and an induced gravitational wave
strain roughly an order of magnitude larger than in
~\cite{Muller2004} and an order of magnitude smaller than
~\cite{Dimmelmeier2002}.
Theoretical work on gravitational wave (GW) signals in the process
of core-collapse in massive stars has advanced much in recent years,
but still does not provide detailed waveforms. Current models take
advantage of the increase in computational power and more
sophisticated input physics to include both 2D and 3D calculations,
utilizing realistic pre-collapse core models and a detailed, complex
equation of state of supernovae that produce neutron stars. The most
recent studies by independent groups give predictions for the strain
amplitude within a similar range, despite the fact that the dominant
physical mechanisms for gravitational wave emission in these studies
are different ~\cite{Muller2004, Fryer2004, Fryer2002, Zwerger1997,
Dimmelmeier2002}. The calculations of ~\cite{Muller2004} are
qualitatively different from previous core collapse simulations in
that the dominant contribution to the gravitational wave signal is
neutrino-driven convection, about 20 times larger than the
axisymmetric core bounce gravitational wave signal.
The applicability of the above models to GRBs is not clear, since
the model endpoints are generally neutron stars, rather than black
holes. Another recent model involves accretion disks around Kerr
black holes~\cite{vanPutten2003}, subject to non-axisymmetric
Papaloizou-Pringle instabilities~\cite{Papaloizou1984} in which an
acoustic wave propagates toroidally within the fallback material.
They are very interesting since they predict much higher
amplitudes for the gravitational wave emission.
For our search, the main conclusion to draw is that in spite of the
dramatic improvement in the theoretical models, there are no
gravitational waveforms that could be reliably used as templates for
a matched filter search, and that any search for gravitational waves
should ideally be as waveform independent as practical. Conversely,
detection of gravitational waves associated with a GRB would almost
certainly provide crucial new input for GRB/SN astrophysics. It is
also clear that the predictions of gravitational wave amplitudes are
uncertain by several orders of magnitude, making it difficult to
predict the probability to observe the gravitational wave signature
of distant GRBs.
The timeliness of searching for a gravitational wave signal
associated with GRBs is keen in light of the recent work by
~\cite{Fryer2004} and \cite{Muller2004}. ~\cite{Muller2004} finds
that the signal due to neutrino convection exceeds that due to the
core bounce and therefore a chaotic signal would be expected.
Studies with simplified or no neutrino transport (e.g.,
~\cite{Fryer2004}, \cite{Ott2004}) find the core-bounce to be the
dominant contributor to the GW signal. The large-scale, coherent
mass motions involved in the core bounce leads to a predicted
gravitational wave signal resembling a damped sinusoid.
\section{Results}\label{sec:Results}
\subsection{False alarm rate measurements}\label{sec:FAR}
In order to assess the significance of the cross-correlated power
of an event, we determined the false alarm rate versus event
strength distribution. We used the full background data stretch
for this measurement.
\begin{figure}[!t]
\includegraphics[angle=0,width=92mm]{FAR.eps}
\caption{False alarm rate as a function of the event strength
threshold as determined from background data. The error bars reflect
90\% CL Poisson errors, based on the the number of events within the
given bin. The pointer indicates the event strength threshold used
for the analysis, which corresponds to an interpolated false alarm
rate of less than $5\times10^{-4}$~Hz. Note that the signal region
data is not included in this calculation. The position of symbols
correspond to the center of the bins.} \label{fig:FAR}
\end{figure}
Figure~\ref{fig:FAR} shows the event rate as a function of the
event strength threshold for the background region. The error bars
reflect 90\% CL Poisson errors, based on the the number of events
within the given bin. We used this distribution to fix the event
strength threshold used in the subsequent analysis.
We chose an event strength threshold with an associated false alarm
rate of less than ${\simeq }$5$\times$10$^{-4}$~Hz, equivalent to
less than ${\simeq}$9\% chance for a false alarm within the 180
second long signal region.
\subsection{Efficiency determination}\label{sec:Efficiencies}
The detection sensitivity of the analysis was determined by
simultaneously adding simulated signals of various amplitudes and
waveforms to both data streams in the background region and
evaluating the efficiency of their detection as a function of the
injected amplitude and waveform type.
The waveforms we considered include Sine-Gaussians to emulate
short narrow-band bursts, Gaussians to emulate short broad-band
signals, and Dimmelmeier-Font-M\"{u}ller numerical
waveforms~\cite{Dimmelmeier2002}, as examples of astrophysically
motivated signals.
Calibration of the waveforms from strain to ADC counts was
performed in the frequency domain, and was done separately for
each interferometer. Calibration procedures of the LIGO detectors
are described in Ref.~\cite{Stan03}. The transformed signals, now
in units of counts of raw interferometer noise, were then simply
added to the raw data stream.
The amplitudes and the times of the injections were randomly
varied. In this way we ensured that each amplitude region sampled
the full, representative range of noise variations and that we had
no systematic effects, for example, due to a regular spacing in
time.
To a reasonable approximation the sensitivity of our analysis
pipeline can be expressed in terms of the frequency content, the
duration and the strength of the gravitational wave signal.
Therefore, it is sufficient to estimate the sensitivity of our
search for a representative set of broad and narrow band waveforms,
which span the range of frequencies, bandwidth, and duration we wish
to search.
We characterize the strength of an arbitrary waveform by its
root-sum-square amplitude ($h_{RSS}$), which is defined
as~\cite{Burstpaper}:
\begin{equation}\label{hRSS}
h_{RSS} = \sqrt{\int_{-\infty}^{\infty} |~h(t)~|^2~ \,dt }~.~
\end{equation}
The above definition of $h_{RSS}$ includes all frequencies, while
the gravitational wave detectors and search algorithms are only
sensitive in a restricted frequency band. In principle, one can
analogously define a ``band-limited" $h_{RSS}$, in which only the
sensitive frequency band of the analysis is taken into account.
Within this paper we choose to adopt the Eq.~\ref{hRSS} definition
of h$_{RSS}$ for historical reasons.
The extracted sensitivities (see the examples in
Figures~\ref{fig:EFF_250_8.9} and ~\ref{fig:CAL_250_8.9}) can be
used to generalize our measurements and estimate the pipeline's
sensitivity for other similar band-limited waveforms.
\begin{figure}[!t]
\includegraphics[angle=0,width=84mm]{EFF_250_9.eps}
\caption{Efficiency of the detection algorithm for a sample waveform
as a function of signal strength ($h_{RSS}$); in this case a
Sine-Gaussian of $f_0=250$~Hz and Q~$=$~8.9. To extract this curve
numerous simulated waveforms were embedded in a representative
fraction of the background data at random times with randomly
varying signal strength. The plot shows the fraction of signals
detected as the function of amplitude and a sigmoid function fit.
The reconstructed signal onset times were required to fall within
$\pm$60~ms of the true onsets, which also explains why the low
$h_{RSS}$ end of the curve falls near zero. This is a typical plot
and in general, the agreement between the measured values, and the
fit is better than ${\simeq}5\%$. We relied on the fit to extract
our upper limits for an optimally oriented and polarized source.
Section~\ref{sec:Interpretation} below describes the corrections due
to non-optimal source direction and polarization.}
\label{fig:EFF_250_8.9}
\end{figure}
\begin{figure}[!t]
\includegraphics[angle=0,width=84mm]{CAL_250_9.eps}
\caption{Detected event strength versus $h_{RSS}$ of the injected
Sine-Gaussian waveform with $f_0=250$~Hz and $Q~\approx~8.9$. The
dots indicate the scatter of the distribution of raw measurements.
The gray band shows the quadratic polynomial fit, which allows us to
convert the strength of an observed event into the equivalent
$h_{RSS}$ value and determine the associated $90\%$ CL error bars.
The markers with error bars represent the $90\%$ CL regions for
subsets of the data. For each marker, $90\%$ of the measurements
used were within the horizontal error bars and $90\%$ of the
detected event strengths values fell within the vertical error bars.
The vertical dash-dot line represents the $50\%$ detection
efficiency associated with the waveform type and the chosen
detection threshold (horizontal dotted line). As expected, the
crossing of the threshold and the $50\%$ efficiency lines agree well
with the fit and the center of the corresponding marker. The
vertical dashed line represents the boundary of the region where we
have better than $90\%$ detection efficiency. The ``corner" defined
by the event strength threshold and the $90\%$ detection efficiency
boundary (dashed lines) agrees well with the curve outlined by the
lower end of the vertical error bars of the markers. All events in
the upper right corner of the plot (above and beyond the dashed
lines) are detectable with high confidence. This plot is typical for
different waveforms considered in the analysis.}
\label{fig:CAL_250_8.9}
\end{figure}
\begin{figure}[!t]
\includegraphics[angle=0,width=94mm]{SENS.eps}
\caption{Sensitivity of the detection algorithm for detecting
Sine-Gaussian waveforms versus characteristic frequency. The plot
shows the strength necessary for $90\%$ detection efficiency. The
grey spectra illustrate the sensitivity of the 2K and 4K Hanford
detectors during the time surrounding the GRB030329 trigger. The
error bars reflect a total $15\%$ error.} \label{fig:SENS}
\end{figure}
To assess the sensitivity for relatively narrow band waveforms, we
used Sine-Gaussian injections of the form:
\begin{equation}\label{SineGauss}
h(t) = h_\circ \sin(\omega_\circ\,t)e^{-t^2/2\sigma^2} \;,
\end{equation}
with a central angular frequency of~~$\omega_\circ = 2\pi
f_\circ$, and $Q=\omega_\circ \sigma = 2\pi f_\circ \sigma $. The
relation between $h_\circ$ and $h_{RSS}$ is given as:
\begin{eqnarray}\label{SGhRSS}
h_{RSS}^{SG} = h_\circ~\sqrt{\frac{\sqrt{\pi}~\sigma}{2} \left(
1-e^{-Q^2}\right)}\, ~{\approx}~\nonumber\\
~{\approx}~ h_\circ~\sqrt{\frac{Q}{4\sqrt{\pi}~f_\circ}}~ \times
~\left\{
\begin{array}{cc}
1 & Q\gg1 \\
0.8 & Q\simeq1 \\
\end{array}
\right\}\;.
\end{eqnarray}
\\
The injected signals covered the frequency range between 100 and
1850 Hz with 13 values of $f_{\circ}$. To test the dependence of the
sensitivity on signal duration, we used three values of Q (4.5, 8.9
and 18) for each frequency (see Table~\ref{t:NBET}). Near the most
sensitive frequency region, around $\simeq$250~Hz, our gravitational
wave strain sensitivity for optimally polarized bursts was better
than h$_{RSS}{\simeq}$5$\times$10$^{-21}$~Hz$^{-1/2}$.
Figure~\ref{fig:SENS} shows the sensitivity for these narrow band
waveforms. The symbols mark the simulated event strength ($h_{RSS}$)
necessary to achieve $90\%$ detection efficiency for each waveform.
We quote the gravitational wave signal strength associated with the
$90\%$ detection efficiency, as this can be related to the upper
limits on the gravitational wave strength associated with the
source. Figure~\ref{fig:SENS} also illustrates the insensitivity of
the detection efficiency to the Q of the Sine-Gaussian waveforms
with the same central frequency, as these reach their $90\%$
efficiency levels at similar gravitational wave strengths, even
though their Q differ by a factor of ${\simeq}4$; for a given
h$_{RSS}$, a longer signal (higher Q) would of course, have a
smaller h$_{PEAK}$. This strength is frequency dependent, naturally
following the frequency dependence of the detector sensitivities,
which are also indicated in Figure~\ref{fig:SENS}.
Table~\ref{t:BBET} shows a similar set of efficiencies estimated
using broad-band simulated signals. We used two types of broadband
waveforms, Sine-Gaussians with unity quality factor and Gaussians.
Both are short bursts, however, the Gaussians are even functions
while the Sine-Gaussians are odd, leading to different peak
amplitudes with the same $h_{RSS}$ value. Gaussians were
parametrized as:
\begin{equation}\label{Gaussian}
h(t) = h_\circ e^{-t^2/2\sigma^2} \;.
\end{equation}
The relationship between $h_\circ$ and $h_{RSS}$ for a Gaussian
is:
\begin{equation}\label{GaussianhRSS}
h_{RSS}^{GA} = h_\circ~\sqrt{ \sqrt{\pi}\>\sigma}\;.
\end{equation}
The estimated sensitivities indicate that the 90$\%$ detection
efficiency limits for short bursts are similar to those obtained for
the narrow band waveforms when one takes into account that only part
of the power of the broad-band waveforms is confined to the analysis
frequency band. Longer Gaussian bursts are more difficult to detect,
as their spectrum has a significant low frequency component, outside
the sensitive band of our analysis.
We have also estimated our efficiency for a set of astrophysically
motivated burst waveforms~\cite{Dimmelmeier2002} (see Table
~\ref{t:DFMT}). These simulated waveforms are not expected to be
necessarily associated with GRBs, rather these results are presented
here to further illustrate the waveform independence of the
analysis.
\begin{table} [htb]
\begin{center}
\caption[]{\label{t:NBET} $h_{RSS}$ [$Hz^{-1/2}$] for 90\% detection
efficiency for Sine-Gaussians (SG) waveforms at various frequencies
($f_{\circ}$) and Q (see eq.~\ref{SineGauss}). The quoted values are
the results of simulations and are subject to a total of
${\simeq}~15\%$ statistical and systematic errors, which are taken
into account when quoting the ${UL}^{90\%CL}_{h_{RSS}}$ values. Note
that at the low and at the high frequency end, the low Q waveforms
have significant power outside of the analysis frequency band.}
\small
\begin{tabular}{lccccc}
\hline\hline
Waveform & $\sigma$ [ms] & Q & $f_{\circ} [Hz]$ & $h_{RSS}^{90\%} [Hz^{-1/2}]$ & ${UL}^{90\%CL}_{h_{RSS}} [Hz^{-1/2}]$ \\
\hline
SG & 7.2 & 4.5 & 100 & $17\times 10^{-21}$ & $20\times 10^{-21}$ \\
SG & 2.9 & 4.5 & 250 & $4.8\times 10^{-21}$ & $5.6\times 10^{-21}$ \\
SG & 2 & 4.5 & 361 & $5.8\times 10^{-21}$ & $6.7\times 10^{-21}$ \\
SG & 1.6 & 4.5 & 458 & $ 7.0\times 10^{-21}$ & $ 8.0\times 10^{-21}$ \\
SG & 1.3 & 4.5 & 554 & $7.9\times 10^{-21}$ & $9.1\times 10^{-21}$ \\
SG & 1 & 4.5 & 702 & $10\times 10^{-21}$ & $11\times 10^{-21}$ \\
SG & 0.84 & 4.5 & 850 & $12\times 10^{-21}$ & $14\times 10^{-21}$ \\
SG & 0.72 & 4.5 & 1000 & $15\times 10^{-21}$ & $17\times 10^{-21}$ \\
SG & 0.53 & 4.5 & 1361 & $27\times 10^{-21}$ & $31\times 10^{-21}$ \\
SG & 0.49 & 4.5 & 1458 & $30\times 10^{-21}$ & $34\times 10^{-21}$ \\
SG & 0.46 & 4.5 & 1554 & $37\times 10^{-21}$ & $43\times 10^{-21}$ \\
SG & 0.42 & 4.5 & 1702 & $43\times 10^{-21}$ & $50\times 10^{-21}$ \\
SG & 0.39 & 4.5 & 1850 & $50\times 10^{-21}$ & $58\times 10^{-21}$ \\
\hline
SG & 14 & 8.9 & 100 & $18\times 10^{-21}$ & $21\times 10^{-21}$ \\
SG & 5.7 & 8.9 & 250 & $4.6\times 10^{-21}$ & $5.3\times 10^{-21}$ \\
SG & 3.9 & 8.9 & 361 & $ 6.0\times 10^{-21}$ & $6.9\times 10^{-21}$ \\
SG & 3.1 & 8.9 & 458 & $7.1\times 10^{-21}$ & $8.1\times 10^{-21}$ \\
SG & 2.6 & 8.9 & 554 & $7.3\times 10^{-21}$ & $8.4\times 10^{-21}$ \\
SG & 2 & 8.9 & 702 & $8.9\times 10^{-21}$ & $10\times 10^{-21}$ \\
SG & 1.7 & 8.9 & 850 & $10\times 10^{-21}$ & $12\times 10^{-21}$ \\
SG & 1.4 & 8.9 & 1000 & $13\times 10^{-21}$ & $15\times 10^{-21}$ \\
SG & 1 & 8.9 & 1361 & $20\times 10^{-21}$ & $23\times 10^{-21}$ \\
SG & 0.97 & 8.9 & 1458 & $23\times 10^{-21}$ & $27\times 10^{-21}$ \\
SG & 0.91 & 8.9 & 1554 & $26\times 10^{-21}$ & $30\times 10^{-21}$ \\
SG & 0.83 & 8.9 & 1702 & $32\times 10^{-21}$ & $37\times 10^{-21}$ \\
SG & 0.77 & 8.9 & 1850 & $38\times 10^{-21}$ & $44\times 10^{-21}$ \\
\hline
SG & 29 & 18 & 100 & $23\times 10^{-21}$ & $26\times 10^{-21}$ \\
SG & 11 & 18 & 250 & $ 5.0\times 10^{-21}$ & $5.7\times 10^{-21}$ \\
SG & 7.9 & 18 & 361 & $6.4\times 10^{-21}$ & $7.4\times 10^{-21}$ \\
SG & 6.3 & 18 & 458 & $7.9\times 10^{-21}$ & $9.1\times 10^{-21}$ \\
SG & 5.2 & 18 & 554 & $7.7\times 10^{-21}$ & $8.9\times 10^{-21}$ \\
SG & 4.1 & 18 & 702 & $9.8\times 10^{-21}$ & $11\times 10^{-21}$ \\
SG & 3.4 & 18 & 850 & $10\times 10^{-21}$ & $12\times 10^{-21}$ \\
SG & 2.9 & 18 & 1000 & $12\times 10^{-21}$ & $14\times 10^{-21}$ \\
SG & 2.1 & 18 & 1361 & $19\times 10^{-21}$ & $21\times 10^{-21}$ \\
SG & 2 & 18 & 1458 & $21\times 10^{-21}$ & $24\times 10^{-21}$ \\
SG & 1.8 & 18 & 1554 & $22\times 10^{-21}$ & $25\times 10^{-21}$ \\
SG & 1.7 & 18 & 1702 & $29\times 10^{-21}$ & $33\times 10^{-21}$ \\
SG & 1.5 & 18 & 1850 & $34\times 10^{-21}$ & $39\times 10^{-21}$ \\
\hline\hline
\end{tabular}
\end{center}
\end{table}
\begin{table} [htb]
\begin{center}
\caption[]{\label{t:BBET} As in Table~\ref{t:NBET}, $h_{RSS}$
[$Hz^{-1/2}$] for 90\% detection efficiency for Gaussian (GA)
waveforms of various durations ($\sigma$) (see eq.~\ref{Gaussian})
and for Sine-Gaussians (SG) waveforms at various frequencies
($f_{\circ}$) and $Q=1$ (see eq.~\ref{SineGauss}). Note that these
broadband waveforms have significant power outside of the analysis
frequency band.} \small
\begin{tabular}{lccccc}
\hline\hline
Waveform & $\sigma$ [ms] & Q & $f_{\circ} [Hz]$ & $h_{RSS}^{90\%} [Hz^{-1/2}]$ & ${UL}^{90\%CL}_{h_{RSS}} [Hz^{-1/2}]$ \\
\hline
SG & 1.6 & 1 & 100 & $10\times 10^{-21}$ & $12\times 10^{-21}$ \\
SG & 0.64 & 1 & 250 & $6.5\times 10^{-21}$ & $7.4\times 10^{-21}$ \\
SG & 0.44 & 1 & 361 & $8.4\times 10^{-21}$ & $9.7\times 10^{-21}$ \\
SG & 0.35 & 1 & 458 & $10\times 10^{-21}$ & $12\times 10^{-21}$ \\
SG & 0.29 & 1 & 554 & $13\times 10^{-21}$ & $14\times 10^{-21}$ \\
SG & 0.23 & 1 & 702 & $18\times 10^{-21}$ & $20\times 10^{-21}$ \\
SG & 0.19 & 1 & 850 & $23\times 10^{-21}$ & $26\times 10^{-21}$ \\
SG & 0.16 & 1 & 1000 & $26\times 10^{-21}$ & $30\times 10^{-21}$ \\
SG & 0.12 & 1 & 1361 & $39\times 10^{-21}$ & $45\times 10^{-21}$ \\
SG & 0.11 & 1 & 1458 & $44\times 10^{-21}$ & $51\times 10^{-21}$ \\
SG & 0.1 & 1 & 1554 & $46\times 10^{-21}$ & $52\times 10^{-21}$ \\
SG & 0.094 & 1 & 1702 & $55\times 10^{-21}$ & $63\times 10^{-21}$ \\
SG & 0.086 & 1 & 1850 & $61\times 10^{-21}$ & $70\times 10^{-21}$ \\
\hline
GA & 0.5 & & & $8.3\times 10^{-21}$ & $9.6\times 10^{-21}$ \\
GA & 0.75 & & & $9.6\times 10^{-21}$ & $1.1\times 10^{-20}$ \\
GA & 1 & & & $1.3\times 10^{-20}$ & $1.5\times 10^{-20}$ \\
GA & 2 & & & $3.3\times 10^{-20}$ & $3.8\times 10^{-20}$ \\
GA & 3 & & & $8.2\times 10^{-20}$ & $9.5\times 10^{-20}$ \\
GA & 4 & & & $1.9\times 10^{-19}$ & $2.2\times 10^{-19}$ \\
GA & 5.5 & & & $8.5\times 10^{-19}$ & $9.8\times 10^{-19}$ \\
GA & 8 & & & $1.3\times 10^{-17}$ & $1.5\times 10^{-17}$ \\
GA & 10 & & & $1.0\times 10^{-16}$ & $1.2\times 10^{-16}$ \\
\hline\hline
\end{tabular}
\end{center}
\end{table}
\begin{table} [htb]
\begin{center}
\caption[]{\label{t:DFMT} As in Table~\ref{t:NBET}, $h_{RSS}$
[$Hz^{-1/2}$] for 90\% detection efficiency for astrophysically
motivated waveforms. These waveforms are described in detail in
Ref.~\cite{Dimmelmeier2002}. Note that most of these waveforms
have significant power outside of the analysis frequency band.}
\small
\begin{tabular}{lccccc}
\hline\hline
Simulation & Waveform & $h_{RSS}^{90\%} [Hz^{-1/2}]$ & ${UL}^{90\%CL}_{h_{RSS}} [Hz^{-1/2}]$ \\
\hline
DFM & A1B1G1 & $12\times 10^{-21}$ & $14\times 10^{-21}$ \\
DFM & A1B2G1 & $13\times 10^{-21}$ & $15\times 10^{-21}$ \\
DFM & A1B3G1 & $12\times 10^{-21}$ & $14\times 10^{-21}$ \\
DFM & A1B3G2 & $12\times 10^{-21}$ & $14\times 10^{-21}$ \\
DFM & A1B3G3 & $12\times 10^{-21}$ & $14\times 10^{-21}$ \\
DFM & A1B3G5 & $34\times 10^{-21}$ & $39\times 10^{-21}$ \\
DFM & A2B4G1 & $24\times 10^{-21}$ & $27\times 10^{-21}$ \\
DFM & A3B1G1 & $19\times 10^{-21}$ & $21\times 10^{-21}$ \\
DFM & A3B2G1 & $20\times 10^{-21}$ & $23\times 10^{-21}$ \\
DFM & A3B2G2 & $15\times 10^{-21}$ & $17\times 10^{-21}$ \\
DFM & A3B2G4 & $14\times 10^{-21}$ & $16\times 10^{-21}$ \\
DFM & A3B3G1 & $28\times 10^{-21}$ & $33\times 10^{-21}$ \\
DFM & A3B3G2 & $17\times 10^{-21}$ & $20\times 10^{-21}$ \\
DFM & A3B3G3 & $12\times 10^{-21}$ & $14\times 10^{-21}$ \\
DFM & A3B3G5 & $30\times 10^{-21}$ & $34\times 10^{-21}$ \\
DFM & A3B4G2 & $23\times 10^{-21}$ & $27\times 10^{-21}$ \\
DFM & A3B5G4 & $26\times 10^{-21}$ & $29\times 10^{-21}$ \\
DFM & A4B1G1 & $38\times 10^{-21}$ & $44\times 10^{-21}$ \\
DFM & A4B1G2 & $32\times 10^{-21}$ & $36\times 10^{-21}$ \\
DFM & A4B2G2 & $42\times 10^{-21}$ & $48\times 10^{-21}$ \\
DFM & A4B2G3 & $39\times 10^{-21}$ & $45\times 10^{-21}$ \\
DFM & A4B4G4 & $17\times 10^{-21}$ & $19\times 10^{-21}$ \\
DFM & A4B4G5 & $12\times 10^{-21}$ & $13\times 10^{-21}$ \\
DFM & A4B5G4 & $21\times 10^{-21}$ & $25\times 10^{-21}$ \\
DFM & A4B5G5 & $19\times 10^{-21}$ & $22\times 10^{-21}$ \\
\hline\hline
\end{tabular}
\end{center}
\end{table}
\subsection{Signal region}\label{sec:SignalRegion}
The analysis of the signal region (Fig.~\ref{fig:SRH}) yielded
only events well below the predetermined event strength threshold
($\lesssim$60\% of threshold). Since we had no candidate event, we
placed an upper limit on the detected strength of gravitational
waves associated to GRB030329. Our fixed false alarm rate
permitted the results of simulations to be used directly in
setting upper limits.
\begin{figure}[!t]
\includegraphics[angle=0,width=84mm]{SRH.eps}
\caption{Number of events versus event strength in the signal region
(circle). The diamonds show the expected distribution based on the
background region. The squares mark the expected distribution based
on non-physical time shifts (ranging from 2 to 9 seconds) between
the H1 and H2 datastreams in the background region. The error bars
reflect $90\%$ CL Poisson errors. The position of the symbols
correspond to the center of the bins.} \label{fig:SRH}
\end{figure}
The upper limits on $h_{RSS}$ for narrow band waveforms are given
in Table~\ref{t:NBET}. Tables~\ref{t:BBET}~and~\ref{t:DFMT} show
the upper limits for the broadband simulations and astrophysically
motivated waveforms, respectively.
\subsection{Errors}\label{sec:Errors}
The analysis method, the procedures used to determine the
efficiencies, and the non-stationary nature of the data, all
contribute to the uncertainty associated with the results.
The efficiency (versus $h_{RSS}$) values have an estimated
${\simeq}11\%$ uncertainty due to our limited knowledge of the
calibrated response of our detectors. This estimate also accounts
for the slight difference in calibrated response between the signal
region and background data used for the simulations.
An additional uncertainty arises from the non-stationarity of the
data. The results of the simulations exhibit a slight dependence on
the choice of the actual data segments (``base" data) used for the
injections. This dependence was characterized via simulations using
numerous different sub-segments of the background data. We repeated
the full efficiency estimation process several times for the same
waveform, while injecting into various base data stretches. The
variation in the measured upper limits indicated ${\simeq}10\%$
uncertainty due to the dependency of our upper limits on the base
data. This uncertainty shall also account for the statistical error
due to the finite number of simulations used.
We characterized the detection efficiencies for each waveform
considered via fits of sigmoid functions (see for example
Figure~\ref{fig:EFF_250_8.9}). The fits agree well with the data,
but small differences are occasionally observed in the $\gtrsim90\%$
efficiency region. We estimate that using these fits can
underestimate the 90\% limits by $\lesssim5\%$.
The uncertainties listed above are taken into account by
specifying a total 15\% uncertainty for each measurement in
Figure~\ref{fig:SENS} and in all Tables.
The false alarm rate associated with the results was also measured.
The false alarm rate limit is based on the measurement with zero lag
data plus the 90\% confidence Poisson error bars. We have checked
the assumption of Poisson background statistics by examining the
time intervals between consecutive triggers and the variance in
trigger counts for varying ES thresholds when the background sample
is divided into 50 equal-length intervals. Good agreement with the
Poisson expectation is observed. This choice provides a conservative
estimate of our associated (${\simeq }$5$\times$10$^{-4}$~Hz) false
alarm rate.
\subsection{Astrophysical interpretation}\label{sec:Interpretation}
GRB030329 has a well-determined redshift, therefore we can relate
our observed limits on strain to a measure of the total
gravitational wave energy emission. For a strain $h(t)$ at
distance $D_L$ from a source of gravitational radiation, the
associated power is proportional to $\dot{h}^2$ ($\dot{h}=dh/dt$),
though the proportionality constant will depend on the (unknown)
emission pattern of the source and the antenna pattern of the
detector (for the known source position, but unknown polarization
angle).
In general, it is not possible to relate our upper limit on the
strain from a particular waveform to a limit on the energy radiated
by the source, without assuming a model. Sources that radiate energy
$E_{\mathrm{GW}}$ might produce an arbitrarily small signal $h(t)$
in the detector, e.g., if the dynamics in the source were purely
axi-symmetric with the detector located on the axis. Nevertheless,
we can associate a strain $h(t)$ in the detector with some minimum
amount of gravitational-wave energy radiated by the source by
choosing an ``optimistic'' emission pattern, thereby obtaining a
measure of the minimum amount of energy that would need to be
radiated in order to obtain a detectable signal. We will show that
the progenitor of GRB030329 is not expected to have produced a
detectable signal.
We are interested in a ``plausible case scenario'' of gravitational
wave emission in order to obtain the minimum (plausible) amount of
gravitational-wave energy radiated that could be associated with a
detector signal $h(t)$. We do not expect the gravitational waves to
be strongly beamed, and we expect that we are observing the GRB
progenitor along some preferred axis. We take a model best case
scenario to be that of gravitational wave emission from a triaxial
ellipsoid rotating about the same axis as the GRB (i.e., the
direction to the Earth). If we assume quadrupolar gravitational wave
emission, the plus- and cross-polarization waveforms, emitted at a
polar angle $\theta$ from the axis of rotation to be:
\begin{eqnarray}
h_+ &=& \frac12 (1 + \cos^2\theta)\, h_{+,0} \\
h_\times &=& \cos\theta\, h_{\times,0}
\end{eqnarray}
where $h_{+,0}$ and $h_{\times,0}$ are two orthogonal waveforms
(e.g., a Sine-Gaussian and a Cosine-Gaussian), each containing the
same amount of radiative power. That is, we assume that the same
amount of gravitational-wave energy is carried in the two
polarizations and that they are orthogonal:
\begin{equation}
\int_{-\infty}^\infty \dot{h}_{+,0}^2 \, dt
= \int_{-\infty}^\infty \dot{h}_{\times,0}^2 \, dt
\quad\textrm{and}\quad
\int_{-\infty}^\infty
\dot{h}_{+,0}
\dot{h}_{\times,0} \, dt = 0.
\end{equation}
Thus, we would expect that the gravitational waves travelling
along the rotational axis (toward the Earth) would be circularly
polarized, and that the detector would receive the signal
\begin{equation}
h = F_+ h_{+,0} + F_\times h_{\times,0}
\end{equation}
where $F_+$ and $F_\times$ represent the detector responses to the
polarization components $h_{+,0}$ and $h_{\times,0}$
\cite{Thorne-300}, and depend on the position of the source in the
sky and on a polarization angle. The radiated energy from such a
system is calculated to be
\begin{equation}
E_{\mathrm{GW}} = \frac{c^3}{16\pi G}\int dA \int_{-\infty}^\infty
(\dot{h}^2_+ + \dot{h}^2_\times) dt
= \frac{c^3}{5G}\frac{D_L^2}{\eta^2} \int_{-\infty}^\infty \dot{h}^2 dt
\end{equation}
where $\eta^2=F_+^2+F_\times^2$ (which depends only on the
position of the source on the sky) and where we are integrating
over a spherical shell around the source with radius $D_L$ (the
distance to the Earth). Alternatively, using Parseval's identity,
we have
\begin{equation}
E_{\mathrm{GW}} = \frac{8\pi^2c^3}{5G}\frac{D_L^2}{\eta^2}
\int_0^\infty |f\tilde{h}|^2 df
\end{equation}
where
\begin{equation}
\tilde{h}(f)=\int_{-\infty}^\infty h(t) e^{-2\pi ift}\, dt.
\end{equation}
Whereas optimal orientation gives $\eta=1$ for a source at zenith,
the position of GRB030329 was far from optimal. The angle with
respect to zenith was $68^\circ$ and the azimuth with respect to
the $x$-arm was $45^\circ$, which yields $\eta=0.37$.
We now relate $E_{\mathrm{GW}}$ to the strain upper limits using
the specific waveforms used in the analysis. For a Gaussian
waveform [see Eq.~(\ref{Gaussian})]:
\begin{equation}\label{Int5}
E_{\mathrm{GW}} = \left(\frac{\sqrt{\pi}c^3}{10G}\right)
\left(\frac{D_L^2 h_\circ^2}{\sigma}\right)
\end{equation}
and for a sine-Gaussian waveform [see Eq.~(\ref{SineGauss})]:
\begin{equation}
E_{\mathrm{GW}} = \left(\frac{\sqrt{\pi}c^3}{20G}\right)
\left(\frac{D_L^2 h_\circ^2}{\sigma}\right)
( 1 + 2Q^2 - e^{-Q^2} )
\end{equation}
where $Q=\omega_\circ\sigma=2\pi f_\circ\sigma$. The relation
between $h_\circ$ and $h_{RSS}$ is given in
Eqs.~(\ref{GaussianhRSS}) and~(\ref{SGhRSS}).
We can relate the observed limit on $h_{RSS}$ to an equivalent mass
$M_{\mathrm{EQ}}$ which is converted to gravitational radiation with
$100\%$ efficiency, $E_{\mathrm{GW}}=M_{\mathrm{EQ}}c^2$, at a
luminosity distance $D_L\approx 800\,\textrm{Mpc}$. For
sine-Gaussian waveforms with $f_\circ=250\,\textrm{Hz}$ and $Q=8.9$,
$M_{\mathrm{EQ}}=1.9\times10^4\eta^{-2}M_\odot$. For Gaussian
waveforms with $\sigma=1\,\textrm{ms}$,
$M_{\mathrm{EQ}}=3.1\times10^4\eta^{-2}M_\odot$. However, we would
not expect that the gravitational-wave luminosity of the source
could exceed $\simeq c^5/G=2\times10^5M_\odot c^2$ per
second~\cite{misner73a}, so we would not expect an energy in
gravitational waves much more than $\simeq 2\times10^3~M_\odot c^2$
in the $\simeq10\,\textrm{ms}$ Sine-Gaussian waveform, or an energy
of much more than $\simeq 3\times10^4 M_\odot c^2$ in the maximum
duration ($150\,\textrm{ms}$) of the search; far below the limits on
$M_{\mathrm{EQ}}c^2$ that we find in this analysis. Present
theoretical expectations on the gravitational wave energy emitted
range from 10$^{-6}$ M$_\odot$ c$^2$ - 10$^{-4}$ M$_\odot$ c$^2$ to
10$^{-1}$ M$_\odot$ c$^2$ - M$_\odot$ c$^2$ for some of the most
optimistic models [see e.g.~\cite{Mosquera02, Ruffini01, Putten04,
Araya-Gochez03}]. Nonetheless, these scalings indicate how we can
probe well below these energetic limits with future analyses. For
example, assuming similar detector performance for an optimally
oriented trigger like GRB980425 (D$_L\approx$35~Mpc) the limit on
the equivalent mass would be $M_{\mathrm{EQ}}\approx$60~M$_\odot$
for the Gaussian waveforms mentioned above with
$\sigma=1\,\textrm{ms}$.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,741
|
Q: Plotting Date with Matplotlib and Pandas I am trying to plot number of sales on Y axis against the dates on the X axis. I can't seem to work out a way to do the dates on the X axis. If i change the dates to a float, it I can plot it but I still cannot figure out a way to plot the date format.
import matplotlib.pyplot as plt
import csv
x = []
y = []
with open('book1.csv','r') as csvfile:
plots = csv.reader(csvfile, delimiter=',')
for row in plots:
x.append(float(row[0]))
y.append(float(row[1]))
plt.plot(x,y, label='Loaded from file!')
plt.xlabel('x')
plt.ylabel('y')
plt.title('Interesting Graph\nCheck it out')
plt.legend()
plt.show()
My csv looks something like:
1/1/2019,7980.185714
2/1/2019,9478.297619
3/1/2019,9282.166667
4/1/2019,6900.833333
5/1/2019,5563.716667
If I change the dates to a float format, I get this graph:
A: You need change the type of x to datetime :
import matplotlib.pyplot as plt
import csv
import datetime as dt
x = []
y = []
with open('book1.csv','r') as csvfile:
plots = csv.reader(csvfile, delimiter=',')
for row in plots:
x.append(dt.datetime.strptime(row[0],'%m/%d/%Y').date())
y.append(float(row[1]))
plt.plot(x,y, label='Loaded from file!')
plt.xlabel('x')
plt.xticks(x, x, rotation=90)
plt.ylabel('y')
plt.title('Interesting Graph\nCheck it out')
plt.legend()
plt.show()
A: sales are declining. Set the date as an index then plot the Sales
txt="""1/1/2019,7980.185714
2/1/2019,9478.297619
3/1/2019,9282.166667
4/1/2019,6900.833333
5/1/2019,5563.716667"""
from io import StringIO
f = StringIO(txt)
df = pd.read_table(f,sep =',')
df.columns=['Date','Sales']
df['Date']=pd.to_datetime(df['Date'])
df.set_index('Date',inplace=True)
plt.title("Sales")
plt.plot(df)
plt.xticks(rotation=90)
plt.show()
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,666
|
El Atlántico Medio es una región de Estados Unidos que comprende los siguientes estados:
Delaware
Maryland
Nueva Jersey
Nueva York
Pensilvania
También incluye a Washington D.C., que no es un estado sino un distrito federal.
En ocasiones puede incluir a:
Virginia
Virginia Occidental
Este término se empezó a emplear ya con los primeros pobladores europeos tras el asentamiento inglés en las colonias (Middle Colonies). Estos estados se caracterizaron por su industria pesada tras la Declaración de Independencia del Reino Unido. En esta región se mezclaron inmigrantes de todo el mundo en lo que pasó a llamarse el melting pot (crisol) de culturas.
Principales ciudades del Atlántico Medio
Albany, Nueva York
Baltimore, Maryland
Búfalo, Nueva York
Filadelfia, Pensilvania
Harrisburg, Pensilvania
Jersey City, Nueva Jersey
Ciudad de Nueva York, Nueva York
Newark, Nueva Jersey
Pittsburgh, Pensilvania
Rochester, Nueva York
Trenton, Nueva Jersey
Véase también
Economía de los Estados Unidos
Trece Colonias
Rust Belt
Regiones de Estados Unidos
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 567
|
The Whitewater Valley Conference was an IHSAA-sanctioned conference based in Fayette, Franklin, Union, and later Henry and Wayne counties in East Central Indiana. The conference was founded in 1940 as a merger of the Franklin County Conference and Union County Conference, though because two of the FCC schools were not able to play a full conference schedule in the 1940-41 school year, two Fayette County Conference schools were added. The conference's last season was in 1967-68, as the consolidation wave of the 1950s and 1960s would leave the conference with three schools and no suitable replacements in the area, as Lewisville and Straughn became part of Tri in 1968. College Corner, whose location on the border of Indiana and Ohio allowed them to play in both the WVC and the Preble County League in Ohio, would continue to play in the PCL until joining with Short in Liberty to form Union County High School in 1974. Whitewater Township would merge into Brookville that same year. Laurel struggled on as an independent for two decades, as they were too far from the two conferences in the general region that featured schools of a similar size and sports offering, the Mid-Hoosier and Ohio River Valley conferences. The school eventually consolidated with Brookville to form Franklin County High School in 1989.
Membership
Alquina and Harrisburg played concurrently in the WVC and FCC 1940-58.
Brookville and Springfield Township played concurrently in the WVC and FCC 1940-41.
College Corner played in both the WVC and Ohio's Preble County League throughout the WVC's existence. When the WVC folded, they remained in the PCL until consolidating into Union County in 1974.
Liberty played concurrently in the WVC and the ECC from 1947 to 1962.
Membership timeline
References
Indiana high school athletic conferences
High school sports conferences and leagues in the United States
Indiana High School Athletic Association disestablished conferences
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,910
|
If you like a fun quirky bag on your shoulder and have a penchant for foxes you are probably going to love this fox bag from Temporary:Secretary. It is kind of a satchel but with a giant fox face peering out.
I don't actually own a grown-up sensible bag at all. Most of mine are found at the charity shop and come in variety of bright colours. So why change a good thing? I might as well throw my money onto the table, bite the bullet and welcome my new foxy friend into my wardrobe.
There is a choice of either light or dark brown. If you are looking for an affordable addition to your handbag collection then at £18 you can't go much wrong with this delightful little fella.
I love this, but I am 32 now. Too old?
you're never too old for awesomeness.
Correct answer. I'm going to come to you with more important life questipns.
I have one 'sensible' bag (a black leather Kelly bag, very good if I must look Very Grown Up). Everything else is bought for colour/pattern/sparkle.
I think I need a fox bag!
I ordered this a couple of weeks ago, but off ebay... a mite cheaper!
This is so cool. Never too old to look cunning and sneaky, I say!
I have one of these! My family were convinced it was meant to be an owl and not a fox however and spent all of christmas teasing me about it.
My only problem is that I normally carry 10 tonnes of stuff in my handbag and this bag is too small to do that!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,321
|
Q: How to use Heroku Review Apps with Twilio? When a new Pull Request is created for our app, a new Heroku instance is spun up, with a new, unique URL each time.
However, the app needs to handle incoming text messages, which requires that we specify a callback URL in the Twilio dashboard.
Is there a way to solve this catch 22?
A: Twilio developer evangelist here.
I have not worked with Heroku Review apps myself, so I'm not sure, but here is an idea to follow up on.
The Heroku Review app allows you to specify a post-deploy script that runs once the app is deployed. I don't know if you can get the URL for the new application within that post-deploy script or the environment, but if you can, you can then make a call to the Twilio API to either create a new phone number and set it's SmsUrl or update an existing phone number's SmsUrl to the new application's URL.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,127
|
North Royalton's Joey Marousek commits to Akron: Football recruiting roundup
North Royalton's Joe Marousek rolls out of the pocket against Hudson in the first half of play Friday, Sept. 29, 2017 in North Royalton. (Joshua Gunter, cleveland.com)cleveland.com
By Matt Goul, cleveland.com
CLEVELAND, Ohio — The anticipation of his next quarterback room made Joey Marousek's eyes light up.
One of the area's most prolific passers in high school football, Marousek announced Tuesday that committed to the University of Akron for next year. A senior this fall at North Royalton, Marousek threw for 2,422 yards and 29 touchdowns with nine interceptions as a junior.
At 6-foot-3 and 200 pounds, Marousek also can run. He had 575 yards and nine more TDs with his feet, which led to a slew of mid-major offers. However, Akron didn't show interest until Tom Arth took over the Zips for Terry Bowden in December.
The former John Carroll coach played at St. Ignatius and JCU before an NFL career that including backing up Peyton Manning on the Indianapolis Colts.
"He's really trying to set the standard and it's something that excited me," Marousek said Tuesday evening. "He worked his way up. He's a top-notch guy. Being able to play under a guy like that is an opportunity you don't get everywhere."
Staying Close To Home 📍🦘🔵 #COMMITTED #GoZips #2IPC0DE #OurWay pic.twitter.com/ysVyonQdqc
— Joey Marousek (@JoeyMar17) June 25, 2019
Marousek's other top offers included Buffalo, Robert Morris and Ashland. Bucknell, Youngstown State, Georgetown, Valparaiso, Sacred Heart and Ohio Dominican also extended offers, he said.
But that interest shown by Akron, once Arth took over the program, won him over.
Marousek is Akron's second Northeast Ohio commitment in a week. Benedictine receiver Charles Rosser announced Friday he also is joining the Zips.
At 6-3 and 195 pounds, Rosser caught 42 passes last season for 783 yards and nine touchdowns at Benedictine.
Marousek said he has not spoken with Rosser yet, but hopes to soon change that.
"We're going to start recruiting anybody else who's on the fence," said Marousek, who plans to major in engineering. "It's going to be fun, going from being recruited to the recruiter."
pic.twitter.com/p2X5axG3Gp
— Charles Rosser (@Charlesrosser_) June 21, 2019
Michigan State scored two additions from the area on defense in the last week. First, St. Edward's Jeff Pietrowski announced he is committed to the Spartans. Within a few days, Archbishop Hoban's Devin Hightower also made it known his next destination is East Lansing.
Pietrowski, a 6-foot-1, 235-pound defensive end, played linebacker last year for the Eagles before a midseason positional flip with then-sophomore teammate C.J. Hankins. As a result, St. Edward entered the playoffs as a seventh seed in Division I, Region 1 and made it all the way to Canton to win the state title. Pietrowski led the Eagles with 11.5 sacks and 27 quarterback hurries.
He also had 114 tackles.
Go Green! pic.twitter.com/bYD3NytsXE
— Jeff Pietrowski Jr (@Jptrow47) June 18, 2019
Once Pietrowski reaches Michigan State next year, he will be reunited with Hightower.
A 6-1, 218-pound linebacker, Hightower transferred last year from St. Edward to Hoban. He is one of a few linebackers returning for Hoban, which is fresh off its fourth straight state championship and second in Division II.
Hightower had seven sacks and two interceptions.
C O M M I T T E D. 🙏🏽 @DantonioMark @CoachMikeTress pic.twitter.com/Jg0iWIUIhA
— Devin Hightower (@1devinhightower) June 24, 2019
Trayanum update
Hightower's current teammate, DeaMonte Trayanum, could soon decide his college destination. The Hoban senior, who plays running back, linebacker, safety and even some cornerback — which he did in last year's state title game vs. Massillon — told 247Sports.com that he will soon decide between Ohio State, Penn State, Arizona State and Wisconsin.
The 6-1, 215-pound Trayanum has said for more than a year that he wants to play running back at the next level, despite his versatility and prowess on defense.
As a junior, Trayanum rushed for 1,313 yards and 26 touchdowns on just 106 carries while splitting the backfield with senior teammate Tyris Dickerson. With Dickerson now at Eastern Kentucky, the backfield appears to be all Trayanum's this fall.
That should be a scary proposition for opposing defenses. Trayanum is ranked fifth in the state among top seniors, according to 247Sports.com's rankings, and is the No. 1 returning player in the area, according to cleveland.com's signing day watch list from February.
Contact sports reporter Matt Goul on Twitter (@mgoul) or email (mgoul@cleveland.com). Or log in and leave a message in the comments section.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 1,732
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Замок Трив () — средневековый шотландский за́мок, который расположен в области Дамфрис-энд-Галловей, в Шотландии. Считается, что замок ранее был домом для древних правителей Галлоуэйя, но к нашему времени не осталось доказательств подтверждающих это.
История
Сэр Арчибальд Дуглас, более известный как Арчибальд Грим, построил высокую, неприступную башню в 1369 году. Он лишь с недавних пор считается лордом Галлоуэя. К тому времени, как он умер в 1400 году, он стал 3-м графом Дуглас, и был одним из самых могущественных вельмож на юге Шотландии.
К 1450 году, из-за того что Чёрные Дугласы стали невероятно могущественными, Яков II желал, чтобы их семья была свергнута. 8-й граф Дуглас принял меры, чтобы защитить себя и свою семью, уничтожив все вспомогательные пристройки Трива, чтобы освободить место для специально построенной после артиллерийской стены (это защитное сооружение неплохо сохранилось и до наших времён).
В 1455 году произошла двухмесячная осада островной крепости. Гарнизон замка сдался только после того, как король Яков II подкупил их, для быстрого взятия укрепления. После успешной осады крепость вернулась к правительству и с тех пор была не очень значимой в истории Шотландии.
Архитектура
Крепость Трив ранее была домом-башней высотой почти 30 метров, окружённый большим комплексом других зданий — одна из первых в своем роде.
Всего постройка имела пять этажей, а толщина стен достигала 3 метров, с небольшими окнами, построенными в сторону к острову. Сооружение также имело и укреплённые зубцы, как и большинство замков в те времена.
Комнаты и пристройки
Склад и служебные помещения замка находились на нижних этажах, а личные комнаты Арчибальда Дугласа были расположены на этажах повыше, для лучшего обзора прилегающих территорий.
Большинство пристроек крепости не возможно теперь увидеть, так как со временем замок превратился в руины. В 1970-х были проведены археологические раскопки, во время которых были обнаружены: внешний зал, где Арчибальд проводил суд, дополнительные жилые помещения для слуг замка и древние остатки порта.
См. также
Список замков Шотландии
Список замков Дамфрис-энд-Галловея
Арчибальд Дуглас, 3-й граф Дуглас
Примечания
Ссылки
Информация про замок на Historicenvironmental.scot
Статья о замке на Undiscoveredscotland.co.uk
Статья о замке на Electricscotland.com
Замки Дамфрис-энд-Галловея
Руины замков в Шотландии
Замки, построенные в XIV веке
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{
"redpajama_set_name": "RedPajamaWikipedia"
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| 7,974
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RESIST ELECTRICITY TARIFF HIKES AND POOR SUPPLY
Posted in Electricity, Socialist Democracy May - June 2022, Economy and economic crisis, Trade Unions
Public Ownership of Power Sector Under Workers and Consumers' Democratic Control is the Way Out
By Chinedu Bosah
The Minister of Finance, Mrs. Zainab Ahmed disclosed on Thursday, March 10, 2022, that the federal government had quietly removed subsidy on electricity tariff. The implication of this policy is that working people and the poor masses will be forced to pay higher tariff twice every year. The only achievement of the Buhari government is the imposition of more hardship on the working masses and the poor. Coalition for Affordable and Regular Electricity (CARE) condemns the policy that will allow the power companies to hike tariff increasingly beyond affordability for the majority of consumers. We call on the working people to organize to resist this action of the government that wants consumers to pay more for poor supply or darkness.
Besides, the government claimed that it was subsidizing the power sector to the tune of N1.7 billion daily but in reality, the government was only bailing out the corruption, expensive lifestyle of the directors and top managers and unsustainable debts incurred by the power companies. For instance, N6 billion released by the Central Bank of Nigeria (CBN) to the Ibadan Electricity Distribution Company (IBEDC) was subjected to insiders abuse and sharp practices in 2017 wherein IBEDC gave a loan of about N6 billion to its parent company (Integrated Energy Distribution and Marketing Group). This is one example of how privileged private companies mismanage the so-called subsidy and bailout funds. Hence, hikes in tariff and constant bailouts are jumbo rewards to the power companies for their obvious failure and inefficiency.
Since the so-called subsidy was removed early this year, electricity supply has got much worse in widespread darkness which has manifested through incessant system collapse and heavy load shedding. Another contributing factor to the poor electricity supply aside from the weak electricity infrastructure is the fact that the Distribution Companies regularly rejects electricity wheeled to them by the Transmission Company of Nigeria (TCN) and the Generating Companies. According to Punch newspaper of April 25, 2022, DISCOS rejected 2495.3MW in just one week in the month of April 2022. It has become a standard practice of the DISCOS to reject power wheeled to it in order to make more profit considering the fact that there are overwhelmingly more consumers on the estimated billing system and the power company deliberately refused to issue prepaid meters to theses consumers because billing are usually outrageous and fraudulent. the distribution companies reject more electricity load because they can manoeuvre with estimated billing and load shedding and still make more money despite supplying little or no electricity. It is clear that the more the capitalist ruling elite heightens the exploitation, the less electricity supply the masses get. We are going to witness outrageous tariff hikes more frequently.
In a profit-first system, there will always be unjust hike in the prices of petrol and electricity just like other essential commodities. The capitalist government and their private collaborators always clamour for higher rates and prices to guarantee super profit. The power companies have shown obvious incapacity to develop the sector, they are only interested in super profit with higher tariff and exploitative billing methodology (estimated billing) while the power infrastructure and facilities are in decrepit conditions triggering incessant system collapse and leaving communities mostly in darkness. The power sector has recorded more system collapse since privatization than in the previous 40 years before privatization and considering that governments have spent over $15 billion on electricity in the last 15 years and nothing fundamental to show for it except darkness. Since the power sector was privatized in November 2013, electricity tariff has been steadily hiked from about N12.50 to about N48 representing a whopping 280% increment. The majority of consumers are yet to be metered and are forced to pay outrageous (crazy) bills and those who have been metered were forced to pay outrageous amounts to procure them. In other words, the electricity privatization has not only proved to be a monumental failure in term of service but also a major fraudulent arrangement for super exploitation of the masses.
Labour must act and not keep quiet
The most worrisome is the fact that both Nigeria Labour Congress (NLC), and Trade Union Congress (TUC), have kept quiet while the government to quietly gave the go-ahead to the power companies to increase tariff under the guise of the removal of the so-called subsidy. Labour leaders threatened to embark on strike on September 28, 2020, over increments in the price of electricity and petrol only to back down and resorted to talks with the government. As a matter of fact, it was the botched strike action and the refusal of NLC and TUC to lead a mass struggle that provided a vacuum filled by the ENDSARS mass movement that protested police brutality and rising poverty. The talks with the government led to the setting up of a Technical Committee comprising 6 representatives of government/DISCOS and 3 representatives of NLC and TUC. The only thing labour leaders got from the talks was a delay in the implementation of the then tariff hike for about a month and government acceptance to have labour represented in the NERC. Considering labour's strategic alliance with the self-serving ruling elite, it appears the natural course of action or inaction of labour leaders is to allow anti-people policies unchallenged and unchecked after 'board-room' discussions.
Implicitly, the labour leaders have agreed with the government to have electricity tariff increased so often despite growing poverty and cost of living. The Labour leaders went into talks with the government but never reported back to Nigerian workers on what was agreed upon as they have chosen to remain quiet. It is the same manner NLC and TUC on a few occasions bark for a struggle but refuse to bite and undemocratically call off strikes. In October 2020, labour unfortunately and openly endorsed privatization and deregulation and so, it is not shocking to clandestinely endorse tariff hikes.
The Labour leaders have left the flood gate open for numerous attacks on the working people. N30,000 minimum wage has not been implemented by many state governments despite having been even eroded by the rising cost of living and labour leaders stand akimbo. Diesel and Kerosene have been deregulated and the prices have steadily moved up and currently, about N800 per litre and over N500 per litre respectively There is rising cost of living, high inflation and rising poverty; there is a massive infrastructural deficit and unemployment and labour leaders are less concerned. There have been three electricity tariff hikes since NLC threatened to go on strike in September 2020 but labour leaders have quietly looked the other way. The anti-labour policies such as casualization are endemic in workplaces but labour leaders look the other way and in some cases like what we have seen in the food industry, NUFBTE (Food Union) leaders themselves set up outsourcing companies for the purpose of casualizing workers.
These privileged right-wing labour leaders are comfortable with the present socio-economic crises because they are now aristocratic and the capitalist system has made them privileged such that they do not feel the impact of the economic crises ravaging the working people. All this has called for workers to demand a fighting labour leadership that is run democratically and consistently defend the interest of workers and the poor masses. Any labour leader who cannot do this should be asked to step down.
What the power sector requires like any other key sector of the economy is a massive public investment to expand infrastructure and facilities and to motivate workers via decent working conditions in order to guarantee uninterrupted electricity at an affordable tariff rate. So far, the capitalist government and the so-called private sector have proved incapable of achieving this despite huge public fund that has been spent on this sector.
Therefore, we call on the labour movement and consumers to build a mass movement to reject the high electricity tariff and also demand the return of the power sector to public ownership under democratic control of workers and consumers. This is to ensure the judicious use of finances of the sector and its democratic running. However, workers may have to mount pressure on the labour leadership to support this demand.
By and large, privatisation, deregulation and other anti-people capitalist policies are responsible for the growing socio-economic crises. Instead of doing away with these policies, the capitalist ruling elite is sustaining them for its self-serving profit interest. Therefore, the labour movement and the working class need a mass working people party on a socialist programme to wrest political power in order to form a socialist government wherein the commanding heights of the economy are nationalized and democratically controlled by the working class in order to mobilise adequate resources for the benefit of the society and the vast majority as against the greed of a few.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 4,004
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Q: Pivot Control ManipulationCompleted ManipulationCompleted event doesn't work in WP8 SDK. What would be an alternative way to solve this problem?
Including dectecting swipe right or left, or choose other pivot item by clicking on the pivot header.
A: You can use SelectedIndex to identified selected pivot page. For instance, you can use as:
private void Pivot_SelectionChanged(object sender, SelectionChangedEventArgs e)
{
switch (((Pivot)sender).SelectedIndex)
{
case 0:
ApplicationBar = ((ApplicationBar)Resources["AppBar1"]);
break;
case 1:
ApplicationBar = ((ApplicationBar)Resources["AppBar2"]);
break;
case 2:
ApplicationBar = ((ApplicationBar)Resources["AppBar3"]);
break;
case 3:
ApplicationBar = ((ApplicationBar)Resources["AppBar4"]);
break;
}
}
Hope this help
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,777
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The campground got one last shot at us. In the pouring rain, we hooked up the toad and started out, only to find that the narrow and twisting road downstream of the assigned "Big Rig" spot was not able to handle the combined length of the vehicles and so we had to disconnect the toad and actually go to the entrance road to hook back up. Yep-still in the rain.
Today, we also completed our second transit within three days of the Atlanta highway system. If you have made this journey, I need say no more. If you have not made this journey at rush hour, I am at a loss for words to explain how entertaining it is.
We are now overnighting in Montgomery, AL. When we checked in we received a lot of weather related information concerning what to do if???
So checking the weather channels and internet, we caught up on the huge storm heading north to remind everyone that winter in already here in many places. It looks like we might get lucky here and just get some high winds and hopefully non-damaging thunderstorms. We arrived here on the day that tied the record for high temperature that was set in 1929. We enjoyed the 79 degrees and sat in the shadow of the motorhome and enjoyed a pleasant sunset that will be the last with these temperatures for perhaps months to come in this area.
Today we traveled from Charlotte, NC to Resaca, GA to visit with Gretchen and BO. We selected a KOA very nearby their home as a good base for the two days. It had glowing literary prose in the guide about their wooded sites with privacy and suitable for big rigs. Imagine our surprise when we arrived and found this very small and less than adequate campground. With considerable difficulty we found a space to park that had the coach hanging out at both ends. The only thing big about the place was the very long list of rules. Rather than a sign of welcome, you were greeted by a large sign of multiple warnings about the penalties of breaking the long list of rules. It turns out that, at least for us, KOA stood for Kampground of annoyance.
But we put this frustration behind and had a great evening with Gretchen and Bo and caught up on events since our last visit in March. We saw first hand some of the big changes since our last visit. Bo has been converted into a cat lover and they have gone from keeping the neighbor cats at bay to owning five very cute cats. Four of them are Bengal cats, which are a breed that is relatively new, and look very much like little tigers. They have made a special addition on the porch which gives the cats a great place to safely enjoy the outdoors.
Of course, the discussions would not be complete without sharing our excitement about the arrival of the new addition to the family in the spring. We are all looking forward to this big event. There will be many changes to come in the future months as preparations are made for the new arrival.
While Gretchen and Bo worked, we headed out to see Ruby falls and checking in with an Itasca dealer about our leaking slide. No luck with the slide, but the trip to Ruby Falls cave and underground water fall was definitely worth the trip.
We got to spend another enjoyable evening with Gretchen and Bo. Regretfully, we have to move on to meet a too tight schedule for a planned get together in Las Vegas for the holidays. We will be back in the spring.
Sunday is our last day in this State Park which, except for the rain, we have enjoyed very much. I can highly recommend it to anyone wanting a rustic setting with good facilities. Katie developed this strong urge to have a game called the Wii, so she arrived at the appropriate store much before the crack of dawn to stand in line. She sent a text message to say she was successful and would arrive to show us her prize after she had a nap. Thus, we decided to spend some time hiking in the woods waiting for our daughter to arrive. It was a great day and good exercise. After our return, we prepared a campfire to end all campfires to celebrate our time here with Katie and to await her arrival. She appeared in the mid-afternoon with her prize game. Before we could start the campfire, we had to take in the gear she wished us to transport back to Las Vegas and, of course, try the game. I must admit it was good with real action with the whole body being involved. I think I hurt myself playing tennis. Just like real life! Just like the old days, she managed to trash the coach with her stuff, and yes, she left us to clean it up after her departure. AAHH-fond memories.
The campfire was indeed great and signaled an end to our time in Virginia.
On Monday we traveled to Charlotte, NC and found a very nice layover spot at the Field Ridge acres Campground. Tomorrow, we will arrive in GA to visit with our oldest daughter and her husband- Gretchen and Bo.
Okay. No one attempted to answer quiz one. There can be several reasons for this. Zip interest or no one knew the answer. The answer is: The White House of the Confederacy in Richmond, Va, where President Jefferson Davis and his family resided for most of the Civil War period. This house is on the property of the Museum of the Confederacy noted in our blog.
This time we will try something less historical. The pictures in this post have something in common. What is it?
Today we ventured to Newport News,VA to visit the mariner's Museum and visit a newly found favorite store-Trader Joe's.
I am glad to report that the Turkey and all the rest of the food for Thanksgiving came out great. So we are now apparently qualified to prepare food in a convection oven. We ate our fill and enjoyed the day. Today we worked in our wet compartments as the sun was out with rising temperatures. The outside storage areas dried quickly, but we continued to attempt to get the heavily padded carpet on the inside storage dry. We purchased a small wet Vac and pulled some additional water out of the carpet, so hopefully that will speed up the drying process.
We took this opportunity of good weather to take care of some basic necessity chores, which we generally do not discuss in polite society.
We enjoyed a pleasant day around the park and built a campfire and cooked dinner and marshmallows over the open fire under a star filled sky.
It rained all day the 21st, 22nd and all this morning for a grand total of three days with high winds. A good size tree across from our spot failed during the wind storm and luckily went into the woods and not towards us. This on top of the big storm last week that gave us four inches of rain in only a little over 8 hours.
Like a number of holidays in the past, while camping or boating, there are issues that always seem to arise on the holiday. This time it was weather related and dealing with the bugs in a new motorhome. By chance we looked in some out of the out of the way storage areas of the coach and discovered a lot of moisture. So the hunt was on. One area was easily discovered and repaired with a bit more conduit putty. However, the leak inside the coach in the bedroom was both more troubling and uncertain as to source. I believe I have finally isolated it to the rear seal of the bedroom slide. A fix on this is pending support advice from our dealer via e-mail.
We are just trying for the first time our convection open. We managed to get the pumpkin pie baked correctly and will find out how the Turkey turns out in a few hours. I wonder how a sandwich will sit with Katie if this project goes south?
Despite these minor annoyances, we have a lot to be thankful for including our families and good health ( for our age-as my doctor likes to say). We have two children in the military, our daughter Katie, who we are here to visit, in this rain infested area and our son-in-law who is stationed in FL, but soon will be sent to Iraq. So we are always very aware of the risks of life. We are also thankful that our oldest daughter and her husband, Gretchen and Bo, are expecting their first child in the spring.
We also wish all our friends and family a very happy and safe (and dry) thanksgiving!!
We headed out to the James River plantation row on the road between Richmond and Jamestown/Williamsburg. With a number of plantations to choose from, we decided on the Shirley Plantation, which is the oldest settled plantation in VA. What makes it even more interesting is the fact that it has been continuously owned and operated by the same family since the mid 1600's. Certainly, that alone is more than can be said about most modern businesses and may well be the oldest continuous business in the country. While the physical facilities do not leave you with any real WOW factor, what is very impressive is the plantation's long history both in length of time and the part it enjoyed in the history of this young country. Due to the wealth and influence of the various owners (in one family) and its strategic location on the James River, the plantation hosted many of the signers of the Declaration of Independence and other notables such as George Washington, Thomas Jefferson, and even Robert E. Lee. Lee's mother was born in this home and where she married Lee's father. Lee spent time here both as a child and after the Civil War was over. It has continued to be an operating plantation throughout its history. Indeed, while we were visiting, the harvesting of the cotton crop was in progress.
At its peak, it was over 4000 acres and had up to 150 slaves working the fields. The eldest of the family which oversaw this operation also owned a number of other plantations in a number of counties and reportedly had over 800 slaves. He was referred to as King, since he was thought to be the wealthiest man in the colonies.
When you reflect on the history that is embedded in this relatively small part of this country and the detail that is known about places such as Shirley Plantation, you realize how little you really know about the history of this country. I have always amused myself with the idea that I was fairly well informed about certain parts of our history, but I am forced to admit that I have never heard of this influential family or their relationship to many of the well known figures in US history. Looks like I need to keep traveling to improve myself!
Today was a second trip to Williamsburg, this time with our daughter, Katie. We got a late start as we had a large campfire last night with Katie and one of her classmates that went on past midnight. It was a great time and brought back many memories our camping trips when the kids were small.
This trip started with one of those life reversals that I guess one expects as both you and your children get older. We had agreed to meet and start the trip at 1000. Now much of my life, or so it seems, has been taken up waiting for our children to get ready to go any place. So, while I acknowledge we were maybe 15 minutes late starting, I was still amazed by the call that went something like-Where are you- I have gotten up and had breakfast, coffee and gassed up the car and I am ready to go. You are late! And I am not happy. HA HA, I loved it! I slowed my preparations down just a bit.
The rest of the day went smoothly and we enjoyed a walk around the colonial area and lunch in the King's Arms again. Of course, no road trip like this would be complete without the apparently obligatory stop at the outlet mall where of course my budget suffered a few more blows.
We have been led to believe that folks that read this kind of log like to participate by showing their knowledge in various areas. So we begin today with this question. What is the building in the picture and its significance in history (if any)? You guessed it -there is no prize but the satisfaction of being correct.
Today was a road trip to Richmond, VA to see the sights. First, we went to the historical area know as Shockoe Slip. It is a small area of shops and businesses near the river. We had lunch in the Tobacco Company Restaurant, which was decorated in an elaborate 19th century style and the food was good, as well. We then checked out the 17th street farmer's market and found that this was not much, at least on this Saturday.
Then we searched and finally found the Museum of the Confederacy. It had been completely swallowed up by a new hospital and university project. This museum contains a good number of original artifacts and art work form the civil war period and appears to attract many history buffs.
We returned in time to meet Katie and a classmate for an evening star gazing program at the Pocahontas State Park. Unfortunately, this was canceled due to clouds. We repaired back to the campsite and built a roaring fire that entertained us for more than 6 hours along with roasting hot dogs and marshmallows. We secured when we ran out of wood.
|
{
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}
| 577
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Die Niederländische Badmintonmeisterschaft 2012 fand vom 3. bis zum 5. Februar 2012 in Almere statt.
Austragungsort
Topsportcentrum Almere, Almere
Medaillengewinner
Weblinks
https://nl.tournamentsoftware.com/sport/tournament?id=060C9963-E5B5-4C4C-B989-6275DAC99CF9
2012
Badminton 2012
Badmintonwettbewerb in Almere
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 647
|
layout: guide
---
Objects can be used in many different ways in JavaScript. There are a number of
different ways to type them in order to support all the different use cases.
In Flow, there are two different kinds of object types: exact object types and inexact object types.
In general, we recommend using [exact object types](#toc-exact-object-types) whenever possible. Exact object types are more
precise and interact better with other type system features, like spreads.
## Object type syntax <a class="toc" id="toc-object-type-syntax" href="#toc-object-type-syntax"></a>
Object types try to match the syntax for objects in JavaScript as much as
possible. Using curly braces `{}` and name-value pairs using a colon `:` split
by commas `,`.
```js
// @flow
var obj1: { foo: boolean } = { foo: true };
var obj2: {
foo: number,
bar: boolean,
baz: string,
} = {
foo: 1,
bar: true,
baz: 'three',
};
```
> **Note:** Previously object types used semicolons `;` for splitting
> name-value pairs. While the syntax is still valid, you should use commas `,`.
#### Optional object type properties <a class="toc" id="toc-optional-object-type-properties" href="#toc-optional-object-type-properties"></a>
In JavaScript, accessing a property that doesn't exist evaluates to
`undefined`. This is a common source of errors in JavaScript programs, so Flow
turns these into type errors.
```js
// @flow
var obj = { foo: "bar" };
// $ExpectError
obj.bar; // Error!
```
If you have an object that sometimes does not have a property you can make it
an _optional property_ by adding a question mark `?` after the property name in
the object type.
```js
// @flow
var obj: { foo?: boolean } = {};
obj.foo = true; // Works!
// $ExpectError
obj.foo = 'hello'; // Error!
```
In addition to their set value type, these optional properties can either be
`void` or omitted altogether. However, they cannot be `null`.
```js
// @flow
function acceptsObject(value: { foo?: string }) {
// ...
}
acceptsObject({ foo: "bar" }); // Works!
acceptsObject({ foo: undefined }); // Works!
// $ExpectError
acceptsObject({ foo: null }); // Error!
acceptsObject({}); // Works!
```
## Object type inference <a class="toc" id="toc-object-type-inference" href="#toc-object-type-inference"></a>
Flow can infer the type of object literals in two different ways depending on
how they are used.
### Sealed objects <a class="toc" id="toc-sealed-objects" href="#toc-sealed-objects"></a>
When you create an object with its properties, you create a _sealed_ object
type in Flow. These sealed objects will know all of the properties you declared
them with and the types of their values.
```js
// @flow
var obj = {
foo: 1,
bar: true,
baz: 'three'
};
var foo: number = obj.foo; // Works!
var bar: boolean = obj.bar; // Works!
// $ExpectError
var baz: null = obj.baz; // Error!
var bat: string = obj.bat; // Error!
```
But when objects are sealed, Flow will not allow you to add new properties to
them.
```js
// @flow
var obj = {
foo: 1
};
// $ExpectError
obj.bar = true; // Error!
// $ExpectError
obj.baz = 'three'; // Error!
```
The workaround here might be to turn your object into an _unsealed object_.
### Unsealed objects <a class="toc" id="toc-unsealed-objects" href="#toc-unsealed-objects"></a>
When you create an object without any properties, you create an _unsealed_
object type in Flow. These unsealed objects will not know all of their
properties and will allow you to add new ones.
```js
// @flow
var obj = {};
obj.foo = 1; // Works!
obj.bar = true; // Works!
obj.baz = 'three'; // Works!
```
The inferred type of the property becomes what you set it to.
```js
// @flow
var obj = {};
obj.foo = 42;
var num: number = obj.foo;
```
##### Reassigning unsealed object properties <a class="toc" id="toc-reassigning-unsealed-object-properties" href="#toc-reassigning-unsealed-object-properties"></a>
Similar to [`var` and `let` variables](../variables/#toc-reassigning-variables)
if you reassign a property of an unsealed object, by default Flow will give it
the type of all possible assignments.
```js
// @flow
var obj = {};
if (Math.random()) obj.prop = true;
else obj.prop = "hello";
// $ExpectError
var val1: boolean = obj.prop; // Error!
// $ExpectError
var val2: string = obj.prop; // Error!
var val3: boolean | string = obj.prop; // Works!
```
Sometimes Flow is able to figure out (with certainty) the type of a property
after reassignment. In that case, Flow will give it the known type.
```js
// @flow
var obj = {};
obj.prop = true;
obj.prop = "hello";
// $ExpectError
var val1: boolean = obj.prop; // Error!
var val2: string = obj.prop; // Works!
```
As Flow gets smarter and smarter, it will figure out the types of properties in more scenarios.
##### Unknown property lookup on unsealed objects is unsafe <a class="toc" id="toc-unknown-property-lookup-on-unsealed-objects-is-unsafe" href="#toc-unknown-property-lookup-on-unsealed-objects-is-unsafe"></a>
Unsealed objects allow new properties to be written at any time. Flow ensures
that reads are compatible with writes, but does not ensure that writes happen
before reads (in the order of execution).
This means that reads from unsealed objects with no matching writes are never
checked. This is an unsafe behavior of Flow which may be improved in the
future.
```js
var obj = {};
obj.foo = 1;
obj.bar = true;
var foo: number = obj.foo; // Works!
var bar: boolean = obj.bar; // Works!
var baz: string = obj.baz; // Works?
```
## Exact object types <a class="toc" id="toc-exact-object-types" href="#toc-exact-object-types"></a>
In Flow, it is considered safe to pass an object with extra properties where
a normal object type is expected.
```js
// @flow
function method(obj: { foo: string }) {
// ...
}
method({
foo: "test", // Works!
bar: 42 // Works!
});
```
> **Note:** This is because of ["width subtyping"](../../lang/width-subtyping/).
Sometimes it is useful to disable this behavior and only allow a specific set
of properties. For this, Flow supports "exact" object types.
```js
{| foo: string, bar: number |}
```
Unlike regular object types, it is not valid to pass an object with "extra"
properties to an exact object type.
```js
// @flow
var foo: {| foo: string |} = { foo: "Hello", bar: "World!" }; // Error!
```
Intersections of exact object types may not work as you expect. If you need to combine exact object types, use object type spread:
```js
// @flow
type FooT = {| foo: string |};
type BarT = {| bar: number |};
type FooBarFailT = FooT & BarT;
type FooBarT = {| ...FooT, ...BarT |};
const fooBarFail: FooBarFailT = { foo: '123', bar: 12 }; // Error!
const fooBar: FooBarT = { foo: '123', bar: 12 }; // Works!
```
## Explicit inexact object types <a class="toc" id="toc-explicit-inexact-object-types" href="#toc-explicit-inexact-object-types"></a>
In addition to the default `{}` syntax, you can explicitly indicate an inexact
object by using an ellipsis at the end of your property list:
```js
// @flow
type Inexact = {foo: number, ...};
```
[Flow is planning to make object types exact by default](https://medium.com/flow-type/on-the-roadmap-exact-objects-by-default-16b72933c5cf).
This is available via an [option in your flowconfig](../../config/options/#toc-exact-by-default-boolean).
You can also read our [upgrade guide](https://medium.com/flow-type/how-to-upgrade-to-exact-by-default-object-type-syntax-7aa44b4d08ab)
for steps to enable this option in your own project.
In a project using exact-by-default syntax, the explicit inexact object type syntax is the only way to express an inexact object type.
## Objects as maps <a class="toc" id="toc-objects-as-maps" href="#toc-objects-as-maps"></a>
Newer versions of the JavaScript standard include a `Map` class, but it is
still very common to use objects as maps as well. In this use case, an object
will likely have properties added to it and retrieved throughout its lifecycle.
Furthermore, the property keys may not even be known statically, so writing out
a type annotation would not be possible.
For objects like these, Flow provides a special kind of property, called an
"indexer property." An indexer property allows reads and writes using any key
that matches the indexer key type.
```js
// @flow
var o: { [string]: number } = {};
o["foo"] = 0;
o["bar"] = 1;
var foo: number = o["foo"];
```
An indexer can be optionally named, for documentation purposes:
```js
// @flow
var obj: { [user_id: number]: string } = {};
obj[1] = "Julia";
obj[2] = "Camille";
obj[3] = "Justin";
obj[4] = "Mark";
```
When an object type has an indexer property, property accesses are assumed to
have the annotated type, even if the object does not have a value in that slot
at runtime. It is the programmer's responsibility to ensure the access is safe,
as with arrays.
```js
var obj: { [number]: string } = {};
obj[42].length; // No type error, but will throw at runtime
```
Indexer properties can be mixed with named properties:
```js
// @flow
var obj: {
size: number,
[id: number]: string
} = {
size: 0
};
function add(id: number, name: string) {
obj[id] = name;
obj.size++;
}
```
### `Object` Type <a class="toc" id="toc-object-type" href="#toc-object-type"></a>
> NOTE: For new code, prefer `any` or `{ [key: string]: any}`. `Object` is an alias to [`any`](../any/) and will
> be deprecated and removed in a future version of Flow.
Sometimes it is useful to write types that accept arbitrary objects, for
those you should write `{}` like this:
```js
function method(obj: {}) {
// ...
}
```
However, if you need to opt-out of the type checker, and don't want to go all
the way to `any`, you could use `{ [key: string]: any}`. (Note that [`any`](../any/) is unsafe and
should be avoided). For historical reasons, the `Object` keyword is still available.
In previous versions of Flow, `Object` was the same
as `{ [key: string]: any}`.
For example, the following code will not report any errors:
```js
function method(obj: { [key: string]: any }) {
obj.foo = 42; // Works.
let bar: boolean = obj.bar; // Works.
obj.baz.bat.bam.bop; // Works.
}
method({ baz: 3.14, bar: "hello" });
```
Neither will this:
```js
function method(obj: Object) {
obj = 10;
}
method({ baz: 3.14, bar: "hello" });
```
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,618
|
{"url":"http:\/\/math.harvard.edu\/~wboney\/seminar\/","text":"# Harvard Logic Seminar\n\nThe Harvard Logic Seminar meets Tuesdays from 5:15pm to 6:15pm in Science Center 507. Email Will Boney (wboney@math.harvard.edu) if you are interested in speaking or being added to the mailing list.\n\nThe schedule for the Logic Colloquium can be found here.\n\nNext talk:\n1. November 14: Linda Westrick (University of Connecticut), \"Towards a notion of computable reducibility for discontinuous functions\"\nAbstract: If X and Y are computably presented uncountable metric spaces, the collection of all functions from X to Y has cardinality too large to allow such functions to be represented as elements of Baire space. Nevertheless, we have some intuitive idea of what it should mean for one discontinuous function to compute another. I will discuss the problem of defining an appropriate notion of computable reducibility on this space. Joint work with Adam Day, Rod Downey and Takayuki Kihara.\n\nNovember 15: Steve Jackson will be giving the Logic Colloquium.\n\nFuture talks:\n1. November 21: No seminar, \"Cancelled\"\nAbstract: Thanksgiving break is a common tradition in American universities that cancels class for two or three days the week of Thanksgiving. In this cancelled talk, we explore how this tradition creeps earlier into the week and causes earlier meetings to be cancelled as well.\n\n2. November 28: Gabe Conant (University of Notre Dame), \"A group version of stable regularity\"\nAbstract: Given an abelian group G and a subset A of G, one can construct a graph on G in which distinct elements x,y in G are connected if x+y is in A. If this graph is stable, then work of Malliaris and Shelah implies that it satisfies a strong form of Szemeredi's regularity lemma (and this has nothing to do with groups). A corollary of recent work of Terry and Wolf is that if G is a finite dimensional vector space over a prime field, then the regular partition of such a stable graph can be obtained using cosets of a subgroup. This motivates a statement of coset regularity\" for subsets A of arbitrary finite groups G, such that \"xy in A\" is a stable binary relation. We prove this statement using local stable group theory and an ultraproduct construction. Joint with A. Pillay and C. Terry.\n\n3. February 6: Jesse Han (McMaster University), \"Strong conceptual completeness for \\omega-categorical theories\"\nAbstract: Suppose we have some process to attach to every model of a first-order theory some (permutation) representation of its automorphism group, compatible with elementary embeddings. How can we tell if this is really an imaginary sort of our theory?\n\nIn the '80s, Michael Makkai proved that the answer to our question is yes if and only if our given process is compatible with all ultraproducts and all \"formal comparison maps\" between them (generalizing e.g. the diagonal embedding into an ultrapower). This is known as \/strong conceptual completeness\/; formally, the statement is that the category Def(T) of definable sets can be reconstructed up to bi-interpretability as the category of \"ultrafunctors\" Mod(T) \\to Set.\n\n\\omega-categorical structures, having few definable sets, are exceptionally simple to understand, and in fact are determined up to bi-interpretability by the action of their automorphism groups. Any general framework which reconstructs theories from their categories of models should therefore be considerably simplified for \\omega-categorical theories.\n\nIndeed, we show:\n\n1. If T is \u03c9-categorical, then X : Mod(T) \u2192 Set is definable, i.e. isomorphic to (M \\mapsto \u03c8(M)) for some formula \u03c8 \u2208 T, if and only if X preserves ultraproducts and diagonal embeddings into ultrapowers. This means that all the preservation requirements for ultramorphisms, which a priori get unboundedly complicated, collapse to just diagonal embeddings when T is \u03c9-categorical.\n\n2. This definability criterion fails if we remove the \u03c9-categoricity assumption. We construct examples of theories and non-definable functors Mod(T) \\to Set exhibiting this.\n\nPast talks:\n1. November 7: CANCELLED\n\nNovember 8:Victoria Gitman will be giving the Logic Colloquium on November 8.\n\n2. October 31: Sebastien Vasey, \"Non-elementary classification theory\"\nAbstract: The classification theory of elementary classes was started by Michael Morley in the early sixties, when he proved that a countable first-order theory with a single model in some uncountable cardinal has a single model in all uncountable cardinals. The proof of this result, now called Morley's categoricity theorem, led to the development of forking, a joint generalization of linear independence in vector spaces and algebraic independence of fields, which is now a central pillar of modern model theory.\n\nIn recent years, it has become apparent that the theory of forking can also be developed in several non-elementary contexts. Prime among those is the axiomatic framework of abstract elementary classes (AECs), encompassing the class of models of any L_{infinity, omega}-theory and closely connected to the more general framework accessible categories. A test question to judge progress in this direction is the forty year old eventual categoricity conjecture of Shelah, which says that a version of Morley's categoricity theorem should hold of any AEC. I will survey recent developments, including the connections with category theory and large cardinals as well as my resolution of the eventual categoricity conjecture for classes of models axiomatized by a universal L_{infty, omega}-theory.\n\n3. October 24: Will Boney, \"Interpolation beyond $\\mathbb{L}_{\\omega_1, \\omega}\" Abstract: For a logic$\\mathcal{L}$, interpolation holds in$\\mathcal{L}$iff for every implication$\\phi \\to \\psi$, there is a sentence$\\chi$in their common language (in$\\mathcal{L}$) such that$\\phi \\to \\chi$and$\\chi \\to \\psi$. It is well-known that$\\mathbb{L}_{\\omega, \\omega}$and$\\mathbb{L}_{\\omega_1, \\omega}$satisfy interpolation, but no other$\\mathbb{L}_{\\kappa, \\lambda}$does. We discuss a logic$\\mathbb{L}^1_\\kappa$developed by Shelah that (for$\\kappa = \\beth_\\kappa$) is intermediate between$\\mathbb{L}_{\\kappa, \\omega}$and$\\mathbb{L}_{\\kappa, \\kappa}$and satisfies interpolation. 4. October 17: Cameron Freer (Borelian and Remine), \"Feedback Computability\" Abstract: The notion of a feedback query is a natural generalization of choosing for an oracle the set of indices of halting computations. Notice that, in that setting, the computations being run are different from the computations in the oracle: the former can query an oracle, whereas the latter cannot. A feedback computation is one that can query an oracle, which itself contains the halting information about all feedback computations. Although this is self-referential, sense can be made of at least some such computations. We'll discuss feedback around Turing computability. In one direction, we examine feedback Turing machines, and show that they provide exactly hyperarithmetic computability. In the other direction, Turing computability is itself feedback primitive recursion (at least, one version thereof). We'll also briefly consider notions for parallel computation, and for Borel maps on Cantor space. Joint work with Nate Ackerman and Bob Lubarsky. 5. October 10: Nate Ackerman, \"Trees, Sheaves and Definition by Recursion\" Abstract: We will show there is a topological space for which presheaves are the same thing as trees. We will further show that there is a sheaf on this topological space which has an important relationship with Baire space. We will then use these connections to show how a definition by transfinite recursion can be thought of as an operation on sheaves, and how the well-definedness of such a definition can be thought of as a property of the sheaf we are working on. This will then allow us to define a second order tree as a sheaf on a tree and to expand our notion of definition by transfinite recursion to all well-founded second order trees (even those which are ill-founded as normal trees). We will then mention how these techniques can be used to prove a variant of the Suslin-Kleene Separation theorem. 6. October 3: Jason Rute, \"A uniform reducibility in computably presented Polish spaces\" Abstract: (Joint work with Tim McNicholl.) In this talk we will define a new uniform computable reducibility between computable Polish spaces. No specialized knowledge of computability theory is required. Given computably presented Polish spaces X and Y, we say x in X is reducible to y in Y if there is a Pi^0_1 subset P of Y and a computable map f : P -> X such that f(y)=x. For each space X one may consider the corresponding degree structure deg(X). For example, deg(2^omega) is (isomorphic to) the truth-table degrees, whereas both deg(omega^omega) and deg(reals) are proper extensions of deg(2^omega). This new reducibility has many motivations. First, it is based on the notion of truth-table reducibility (which we will define). Truth-table reducibility on 2^omega is too restrictive of a setting for working within Baire space or the real numbers. For example, there are functions f in omega^omega not truth-table reducible to any X in 2^omega and sequences X in 2^omega such that X\/3 is not truth-table reducible to X. Our reducibility gives the correct generalization of truth-table reducibility to these spaces. Second, this project mirrors Miller's non-trivial work extending Turing reducibility to computably presented Polish spaces. Last, our reducibility grew naturally out of work of the first author on computable arcs and the second author on Schnorr randomness. For example, we show that, for the vector space R^d, every Schnorr random is found in some computable arc. 7. September 26: Gabriel Goldberg, \"The least strongly compact cardinal and the Ultrapower Axiom\" Abstract: It is consistent that the least strongly compact cardinal is the least supercompact cardinal, but it is also consistent that the least strongly compact cardinal is the least measurable cardinal. Which is it? The Ultrapower Axiom is an abstract comparison principle motivated by inner model theory that roughly states that any pair of ultrapowers can be ultrapowered to a common ultrapower. We give a characterization of supercompact cardinals in terms of the Mitchell order and use this to prove that the least strongly compact cardinal is supercompact assuming the Ultrapower Axiom and the GCH. 8. September 19: Will Boney, \"Model-theoretic characterizations of large cardinals\" Abstract: Compact cardinals get their names from a characterization in terms of the compactness of L_{\\kappa, \\kappa}. Measurable and supercompact cardinals also have characterizations in these terms, and Magidor has used second-order logic to characterize supercompacts and extendible cardinals in this way. We will continue this line of model-theoretic characterizations and discuss the characterizations of large cardinals in terms of compactness for omitting types focusing on three logics: L_{\\kappa, \\kappa}, second-order, and sort logic. 9. September 12: Sebastien Vasey, \"Internal sizes in$\\mu$-abstract elementary classes\" Abstract: The internal size of an object$M$inside a given category is, roughly, the least infinite cardinal$\\lambda$such that any morphism from M into the colimit of a$\\lambda^+$-directed system factors through one of the components of the system. In the category of set, the internal size of an object is its cardinality. In the category of vector spaces, the internal size is the dimension, and in the category of metric spaces, the internal size is the least cardinality of a dense subset. We will discuss questions around internal sizes in the framework of$\\mu$-abstract elementary classes ($\\\u03bc$-AECs), which are, up to equivalence of categories, the same as accessible categories with all morphisms monomorphisms. We will in particular examine an example of Shelah---a certain class of sufficiently-closed constructible models of set theory---which shows that the categoricity spectrum can behave very differently depending on whether we look at categoricity in cardinalities or in internal sizes. This is joint work with Michael Lieberman and Ji\u0159\u00ed Rosick\u00fd. 10. September 5: Nate Ackerman, \"Vaught's Conjecture for a Grothendieck topos\" Abstract: In this talk we will give background on$\\mathcal{L}_{\\infty, \\omega}(L)$, categorical logic as well as Grothendieck toposes. We will then show how to make precise a version of Vaught's conjecture for a Grothendieck topos as well as discuss various analogs of Morley's theorem which hold in all Grothendeick toposes (under mild set theoretic assumptions). If we have time we will also discuss analogs of other theorems of$\\mathcal{L}_{\\infty, \\omega}\\$ for Grothendieck toposes.\n\nYou also might be interested in some nearby events:\n\u2022 UConn is having a special semester in logic with several speakers coming in. The schedule is here.\nBack to Harvard Logic.\nBack to my homepage.","date":"2017-11-23 21:57:58","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7519608736038208, \"perplexity\": 1226.424173857725}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-47\/segments\/1510934806979.99\/warc\/CC-MAIN-20171123214752-20171123234752-00247.warc.gz\"}"}
| null | null |
The new and ambitious residential project of Sierra Blanca Estates has chosen Malaga for its location, becoming a project that due to its characteristics and location has attracted the gaze of investors from all over the world. Sierra Blanca Tower is a residential complex of luxury homes and penthouses-duplexes, whose owners will be able to exclusively enjoy common areas and services at the level of a 5-star hotel.
The project sponsored by Antonio Banderas, Sierra Blanca Tower, will redefine the concept of luxury in front of the sea in Malaga capital by Sierra Blanca Estates, the leading developer of luxury homes on the Costa del Sol whose objective is focused on offering an unmatched lifestyle to its customers.
Its height of 71 meters and its 21 floors open to the sea will be conformed by 69 apartments and 2 duplex penthouses with between 1 and 4 bedrooms with views of the Mediterranean Sea. The architecture of Sierra Blanca Tower comes from the firm of Estudio Lamela Arquitectos, the renowned architecture studio led by Carlos Lamela, who has made more than 2,000 projects come true since 1954.
The interior design of the first 69 apartments has been directed by González Casaus, an architecture and interior design studio founded by Lucía Casaus and Alexis González in 2017. Likewise, the design and decoration of the Sierra Blanca Tower penthouses has been conceived by the renowned international brand The One Atelier.
In short, Sierra Blanca Tower is and will undoubtedly be the benchmark residential project that will define true luxury in Malaga.
For more info, visit the website of the project: www.sierrablancatower.com
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 2,614
|
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<span id="prop_tgt:VISIBILITY_INLINES_HIDDEN"></span><h1>VISIBILITY_INLINES_HIDDEN<a class="headerlink" href="#visibility-inlines-hidden" title="Permalink to this headline">¶</a></h1>
<p>Whether to add a compile flag to hide symbols of inline functions</p>
<p>The <code class="docutils literal"><span class="pre">VISIBILITY_INLINES_HIDDEN</span></code> property determines whether a flag for
hiding symbols for inline functions, such as <code class="docutils literal"><span class="pre">-fvisibility-inlines-hidden</span></code>,
should be used when invoking the compiler. This property affects compilation
in sources of all types of targets (subject to policy <span class="target" id="index-0-policy:CMP0063"></span><a class="reference internal" href="../policy/CMP0063.html#policy:CMP0063" title="CMP0063"><code class="xref cmake cmake-policy docutils literal"><span class="pre">CMP0063</span></code></a>).</p>
<p>This property is initialized by
the value of the <span class="target" id="index-0-variable:CMAKE_VISIBILITY_INLINES_HIDDEN"></span><a class="reference internal" href="../variable/CMAKE_VISIBILITY_INLINES_HIDDEN.html#variable:CMAKE_VISIBILITY_INLINES_HIDDEN" title="CMAKE_VISIBILITY_INLINES_HIDDEN"><code class="xref cmake cmake-variable docutils literal"><span class="pre">CMAKE_VISIBILITY_INLINES_HIDDEN</span></code></a> variable if it
is set when a target is created.</p>
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{"url":"https:\/\/www.physicsforums.com\/threads\/how-do-we-cut-spending-us.501806\/page-4","text":"# News How do we cut spending? (US)\n\n#### turbo\n\nGold Member\nThis is something that Government clearly doesn't understand - or we wouldn't have a $14Trillion national debt growing at OVER 10% per year ongoing for decades - AS IT'S PLAN. The subject of this thread is \"how do we cut spending\" - my response is account for EVERY dollar. It has now become acceptable for politicians to round bills to the nearest hundred-billion and talk in terms of trillions - it seems just a few years ago$1Billion was a lot of money - now it's a waste of a politicians time to discuss? (IMO)\nOne reason for our excessive spending IMO is that huge omnibus bills are cobbled together, loaded with pork that might not pass if they were considered on their own merits. Perhaps we could get congressional rules changed so that every spending bill is considered separately, with only relevant amendments allowed. That would create more transparency. As things stand now, only a handful of staffers and lobbyists are likely to know what is in these monster bills.\n\n#### WhoWee\n\nOne reason for our excessive spending IMO is that huge omnibus bills are cobbled together, loaded with pork that might not pass if they were considered on their own merits. Perhaps we could get congressional rules changed so that every spending bill is considered separately, with only relevant amendments allowed. That would create more transparency. As things stand now, only a handful of staffers and lobbyists are likely to know what is in these monster bills.\nI agree - smaller focused Bills would expose waste and drive accountability on every level.\n\n#### Al68\n\nMemic is the state's worker's comp provider.\nSeriously, that explains a lot.\nThe playing field is highly skewed in favor of Memic, despite the claims from the right that our state's laws are \"anti-business\".\nCould it be that the businesses the right was referring to are not those businesses in bed with the state, getting their business from the state, like Memic? :uhh:\n\n#### turbo\n\nGold Member\nSeriously, that explains a lot.\nYou don't know the back-story. My uncle and other business-owners in the forest-products industry in this region banded together to self-insure for workers comp. When you have a lot of guys out in the woods with chain-saws, cable skidders, etc, there is potential for some serious injury, and the comp coverage available was too expensive. My uncle and his partners found out that self-insurance not only saved them money, the premiums that they paid to their consortium earned them some serious interest, since they were quite cognizant of worker safety and kept injuries as low as possible. Get caught running a chain-saw without cut-resistant chaps, gloves, helmet with face-shield, etc? Down the road you go.\n\nEnter the state insurance board, which in its \"wisdom\" decreed that such self-insurance plans had to be dissolved, putting those business owners at the mercy of the costly insurance market, which already had deemed the logging and forest-product trucking industries to be high-risk. The big insurance companies and their lackeys in Augusta win again. In this case, the state was anti-business for real. Just anti-modest-sized entrepreneurial businesses. The insurance giants made out just fine, thank you.\n\n#### Al68\n\nYou don't know the back-story.....Enter the state insurance board, which in its \"wisdom\" decreed that such self-insurance plans had to be dissolved, putting those business owners at the mercy of the costly insurance market, which already had deemed the logging and forest-product trucking industries to be high-risk. The big insurance companies and their lackeys in Augusta win again. In this case, the state was anti-business for real. Just anti-modest-sized entrepreneurial businesses. The insurance giants made out just fine, thank you.\nThat was my point. The anti-business nature of the state's laws referred to by the right are clearly not referring to businesses in bed with the state, that benefit from such policies.\n\nThey're talking about private businesses that want only to be left alone by the state, not businesses that profit from state intervention in the economy.\n\n#### mege\n\nThat was my point. The anti-business nature of the state's laws referred to by the right are clearly not referring to businesses in bed with the state, that benefit from such policies.\n\nThey're talking about private businesses that want only to be left alone by the state, not businesses that profit from state intervention in the economy.\nAs an aside - this is exactly part of why the Affordable Care Act scares me - the insurance companies are for it.\n\nOne reason for our excessive spending IMO is that huge omnibus bills are cobbled together, loaded with pork that might not pass if they were considered on their own merits. Perhaps we could get congressional rules changed so that every spending bill is considered separately, with only relevant amendments allowed. That would create more transparency. As things stand now, only a handful of staffers and lobbyists are likely to know what is in these monster bills.\nI would go one step further that each bill needs to have a neccessary end date and be reinstituted (think about how they implemented the PATRIOT Act). Shouldn't policies be placed in mind to solve something? (solving indicating a resolution) Too many bills become self-serving loads on the government's books because noone has the heart to say 'wait, why do we still need this?' The way bills are passed now, force politicing and don't have long term goals in mind other than 'solve right now!'\n\n#### Al68\n\nAs an aside - this is exactly part of why the Affordable Care Act scares me - the insurance companies are for it\nYes, plus the name. A general rule of thumb is that the name of a law promoted by the left tends to be the opposite of its contents. The use of the word \"reform\" always means it's the epitome of corruption. So of course they're going to put the word \"affordable\" in the name of a law that does the opposite. Just to be sarcastic and rub it in, you know.\n\n#### WhoWee\n\nIMO, the only way to effectively cut our deficit is to cut spending ASAP and to keep all future spending on-budget and transparent.\n\nWe can get rid of a lot of spending with a few moves. First, close at least 600 foreign military bases. They are not necessary to the security of our country. Foreign bases are a drain on our treasury and a direct monetary transfer to the countries that host those bases. Bring home the military personnel, administrators, and their families. We will all be better off.\n\nNext, drop a couple of carrier groups. Our Navy does not need to be able to project aggression all over the world. No other country has this force.\n\nNext, drop the support for ethanol, and all other price-supports for agricultural programs that steer so many millions of dollars to ADM, Monsanto, etc, while making basic foods more expensive for US citizens.\n\nThis is not an inclusive list. I hope others can chime in.\n\nThe GOP fantasy that we must attack deficits by cutting Medicare, Medicaid, and Social Security, is just that. A fantasy that cannot work while keeping our most vulnerable citizens covered by at least a minimum social net.\nI think we can all agree there are basically 2 ways to cut spending - immediately or in phases.\n\nWith that said, it might be wise to engage in a policy of non-renewal or non-replace whenever possible?\n\nAs for the \"GOP fantasy that we must attack deficits by cutting Medicare, Medicaid, and Social Security\" - how can we not make cuts and other adjustments somewhere in these areas? As the economy tanked, Medicaid has been expanded very rapidly over the past few years - does anyone think this expansion is sustainable at the state level?\n\n#### turbo\n\nGold Member\nAs for the \"GOP fantasy that we must attack deficits by cutting Medicare, Medicaid, and Social Security\" - how can we not make cuts and other adjustments somewhere in these areas? As the economy tanked, Medicaid has been expanded very rapidly over the past few years - does anyone think this expansion is sustainable at the state level?\nSS is self-sustaining, and can continue to be so for many decades. Just lift the cap on earnings.\n\nMedicaid and Medicare costs are increasing not because the programs are broken and need to be slashed, but because of our out-of-control health-care costs. Health insurance is far too expensive in the US. Other industrialized countries do not pay anywhere what we have to in order to get coverage for their citizens. Institute a true public option, perhaps by allowing younger, healthier workers to opt in to Medicare.\n\n#### WhoWee\n\nSS is self-sustaining, and can continue to be so for many decades. Just lift the cap on earnings.\n\nMedicaid and Medicare costs are increasing not because the programs are broken and need to be slashed, but because of our out-of-control health-care costs. Health insurance is far too expensive in the US. Other industrialized countries do not pay anywhere what we have to in order to get coverage for their citizens. Institute a true public option, perhaps by allowing younger, healthier workers to opt in to Medicare.\nIMO - as we speak, Social Security is under attack from the \"generosity\" of our politicians. We've discussed this previously:\n\nMedicaid costs are increasing because the program is expanding - and it's putting a great deal of stress on the states. Isn't Medicare 100% under the control of Government already?\n\n#### WhoWee\n\nThis sums up the problem - IMO.\n\nhttp:\/\/www.usatoday.com\/news\/washington\/2011-06-06-us-owes-62-trillion-in-debt_n.htm\n\n\"The government added $5.3 trillion in new financial obligations in 2010, largely for retirement programs such as Medicare and Social Security. That brings to a record$61.6 trillion the total of financial promises not paid for.\nThis gap between spending commitments and revenue last year equals more than one-third of the nation's gross domestic product.\nMedicare alone took on $1.8 trillion in new liabilities, more than the record deficit prompting heated debate between Congress and the White House over lifting the debt ceiling. STORY: Government's mountain of debt Social Security added$1.4 trillion in obligations, partly reflecting longer life expectancies. Federal and military retirement programs added more to the financial hole, too.\nCorporations would be required to count these new liabilities when they are taken on \u2014 and report a big loss to shareholders. Unlike businesses, however, Congress postpones recording spending commitments until it writes a check.\nThe $61.6 trillion in unfunded obligations amounts to$527,000 per household.\"\n\n#### Al68\n\n\"The government added $5.3 trillion in new financial obligations in 2010, largely for retirement programs such as Medicare and Social Security. That brings to a record$61.6 trillion the total of financial promises not paid for.\nDoes anyone really think our children and grandchildren should assume such a debt, when they made no such promise? A debt of over $500,000.00 per household? Are they really expected to buy the argument that they owe that debt on the basis that they will in return receive a promise that their children and grandchildren will assume a debt to them in return? Only complete sheep would buy such a Ponzi scheme argument and assume a debt they do not in fact owe. And the amount is far beyond what even the sheepiest of socialist Ponzi drones could assume even if they wanted to. There are only three alternatives: massive changes to the Ponzi schemes in question, abandonment of the Ponzi schemes in question, or collapse of the federal government. At this point, if so many people have such a narrow and delusional worldview as to consider the Ryan plan \"far right\", that last option seems inevitable. That's the normal fate of Ponzi schemes, anyway, and a federal government could be re-instated sans unconstitutional Ponzi schemes. #### WhoWee Does anyone really think our children and grandchildren should assume such a debt, when they made no such promise? A debt of over$500,000.00 per household?\n\nAre they really expected to buy the argument that they owe that debt on the basis that they will in return receive a promise that their children and grandchildren will assume a debt to them in return? Only complete sheep would buy such a Ponzi scheme argument and assume a debt they do not in fact owe. And the amount is far beyond what even the sheepiest of socialist Ponzi drones could assume even if they wanted to.\n\nThere are only three alternatives: massive changes to the Ponzi schemes in question, abandonment of the Ponzi schemes in question, or collapse of the federal government.\n\nAt this point, if so many people have such a narrow and delusional worldview as to consider the Ryan plan \"far right\", that last option seems inevitable. That's the normal fate of Ponzi schemes, anyway, and a federal government could be re-instated sans unconstitutional Ponzi schemes.\nmy bold\n\nIs it any wonder the Left hates Glen Beck?\n\nhttp:\/\/www.foxnews.com\/story\/0,2933,542521,00.html\n\"BECK (voice-over): The Cloward-Piven strategy in a nutshell.\nLeft-wing radicals Richard Cloward and Frances Fox Piven advance their strategy to end poverty in a May 2, 1966 article in The Nation.\nCloward-Piven starts with the idea that there is a wide gap between the entitlements that poor people are eligible for and the entitlements that they are actually collecting. They say getting everybody on to the welfare rolls will wipe out poverty and the only way to accomplish that is through a massive multi-city education campaign making heavy use of the media.\n\nSo, once the poor know that they're eligible and start to sign up, what happens next? In their own words, \"a crisis.\" A crisis that starts in the cities that would rapidly spread to a nationwide level and force the government to move quickly to create a new program for direct income distribution: Marxism.\nWait a minute \u2014 community organizations overwhelming the system that would lead to a massive redistribution of wealth? Gee, where have I heard that before?\nThat's Cloward-Piven strategy in a nutshell.\"\n\nLast edited by a moderator:\n\n#### Zarqon\n\nOne the one hand I can see how all this extra spending lead to economic issues, but on the other hand I just don't get how our society can \"run out of money\" to provide a good life for everyone when we're so much more efficient at producing stuff than every before in our history.\n\nCompared to let's say 100 years ago, our capability for producing things like food and houses are vastly superior. For example, even in the face of a increasing life expectancy, and thus more older people to care for, how can we not have become so much more efficient that it is a a simple matter to provide all basic needs to these people?\n\nWhat in the society is eating up all the efficiency we have developed over the last century?\n\n#### mege\n\nWhat in the society is eating up all the efficiency we have developed over the last century?\nhttp:\/\/en.wikipedia.org\/wiki\/File:United_States_Capitol_-_west_front.jpg\" [Broken]\n\nLast edited by a moderator:\n\n#### WhoWee\n\n1.) K-12 schools have a break in the summer. The tradition began when children were needed in years gone by to help on the family farm. Now it seems the tradition continues to facilitate family vacations.\n\n2.) Lower income families are typically eligible for food assistance and there are in-school \"free lunch\" (and\/or breakfast) programs in many schools.\n\nIMO - both of those programs are needed. (Although, I would like to see the purchasing addressed - another conversation.)\n\nHowever, this program does not make sense to me:\nhttp:\/\/www.ode.state.oh.us\/GD\/Templates\/Pages\/ODE\/ODEPrimary.aspx?page=2&TopicRelationID=835 [Broken]\n\n\"Just as learning does not end when school lets out, neither does a child's need for good nutrition end. The Summer Food Service Program (SFSP) provides free, nutritious meals and snacks to help children in low-income areas get the nutrition they need to learn, play and grow throughout the summer months when they are out of school.\nThe SFSP was created to ensure that children in low-income areas can continue to receive nutritious meals during long school vacations when they do not have access to school lunches or breakfasts. But, although millions of children depend on nutritious, free and reduced-price meals and snacks at schools during nine months of the year, just a fraction of that number receive the free meals provided by the SFSP during the summer months.\"\n\nWhy is this necessary? Where is parental responsibility?\n\nLast edited by a moderator:\n\n#### mege\n\nIMO - both of those programs are needed. (Although, I would like to see the purchasing addressed - another conversation.)\nPurchasing? Did I miss part of the conversation? :p\n\nWhy is this necessary? Where is parental responsibility?\nIMO this is one of the worst side-effects of any nanny-policy, the waning of responsibilities. And of course is one of the key arguements for free market principles - individuals taking responsibility for themselves are stronger than those that do not take responsibility for themselves. One of my biggest issues with these types of policies is they're band-aids - there is no attempt to actually RESOLVE the impoverishment. We don't need safety nets, we need safety trampolenes.\n\n#### Al68\n\nIMO this is one of the worst side-effects of any nanny-policy, the waning of responsibilities. And of course is one of the key arguements for free market principles - individuals taking responsibility for themselves are stronger than those that do not take responsibility for themselves. One of my biggest issues with these types of policies is they're band-aids - there is no attempt to actually RESOLVE the impoverishment.\nThat's a key point. It's obvious from the design of the programs advocated by the left, and simple political logic, that resolving the problem is not the goal. Their permanent solution is the perpetuation of the problem, and the perpetuation of their \"band-aid\" policies.\n\nAnd like any group of mobsters, it's hopeless to try to sway them with reason and logic. To paraphrase Ben Franklin: Force poops on the back of reason. We will never change the minds of the power hungry.\n\nOf course they have gotten so greedy the last couple of years that their system will likely crash. And they will predictably try to blame those they pillage from instead of their pillaging, just like they do now.\n\n#### WhoWee\n\nPurchasing? Did I miss part of the conversation? :p\nI don't want to divert this thread. In other threads, I've noted the purchasing power the Government would have if they bought generic staples and paid distributors on a cost plus basis - rather than allow consumers to pay retail for processed foods of their choice - again, IMO - poverty is too comfortable.\n\n#### talk2glenn\n\nMedicaid and Medicare costs are increasing not because the programs are broken and need to be slashed, but because of our out-of-control health-care costs. Health insurance is far too expensive in the US. Other industrialized countries do not pay anywhere what we have to in order to get coverage for their citizens. Institute a true public option, perhaps by allowing younger, healthier workers to opt in to Medicare.\nWhat does this even mean?\n\nCurrently, the Department of Health decides how much to reimburse providers for procedures performed under Medicare. These rates are already significantly below the going market rate - so much so, that the system suffers from provider flight. Your argument is that these costs are still too high. If so, the Medicare advisory board already has the authority to negotiate lower reimbursement rates - why don't they?\n\nBecause, of course, reimbursement rates aren't \"too high\"; in fact, they're too low, from the markets perspective. Other industrial countries pay less because they use non-price determinants of supply; specifically, they make do with less access (first-come first-serve rationing rather than price-rationing) and fewer procedures (the most expensive treatments are priced out of the market and totally unavailable outside the United States). This is less of an option here because the government must compete with private consumers for access to providers - something illegal in places like England and Canada (though I think Canada is flirting with privatization).\n\nSo, you've got two options:\n\nNationalization, so that the government becomes a monopsonist (the sole consumer of the good). It can dictate lower prices to providers (they won't be able to simply turn away the government pool and service a private pool). Many providers will be priced out of the market, reducing the supply and resulting in lengthy wait times for access (or eliminating access altogether, if reimbursement rates won't cover operating costs at even the most efficient firms for the most exotic procedures). This is the British model.\n\nPrivatization. Reduce the benefits provided by the Medicare program, and encourage customers to supplement with private coverage, or pay out of pocket. So, for example, Medicare might provide catastrophic coverage, with routine, long-term, maintenance and prescription coverage obtained privately or on a for-cash basis. This is the Ryan model.\n\nIn neither case is the existing model sustainable, let alone expandable. Note also, before pushing for the former, that the vast majority of healthcare innovation occures in the United States. Routine access to new procedures becomes commonplace in foreign markets only years after procedure introduction here, when costs have fallen to workable levels, given the lower international reimbursement schedules. It is possible that healthcare innovation would mostly stop if the incentive to innovate - a rich and willing to pay American market - were eliminated.\n\n#### CAC1001\n\nOn the issue of servicing the national debt, I think it was mentioned earlier that we are at a double-digit percentage of the budget being used to pay for servicing the debt. However, this is incorrect, the percentage of the federal budget used to service the debt is still in single digits, however, we are at a double-digit percentage of the federal revenues being used to service the debt. Right now, it's about 13% of the federal revenues, which is a red flag. If it gets up to 18% or over, that is a major red flag that can cause a downgrade to the bonds.\n\nOn the issue of the entities that buy U.S. debt being willing to buy it at very low rates right now, what happens if the interest rate has to increase? With a moderately-size level of national debt, this isn't a problem, but with a massive level of national debt, this can create a huge problem, because at that point, even an uptick in the interest rate of one percentage point could mean a massive increase in the amount of money it takes to service the debt. In other words, we could go from 13% of the federal revenues to 18% or higher being used to service the debt really quickly. Japan for example has a national debt that is 230% of their GDP. They have a AA bond rating. Their national debt is so high that if they were to increase their interest rate by one percentage point, it would wipe out whole sections of the Japanese government. Probably one way in which Japan is able to get around this is that they hold 90% of their debt.\n\nOn making Social Security more solvent by raising the cap on income taxed, the argument against this is that it would turn SS into a de-facto welfare program. SS is (or was) \"supposed\" to be a program in which you get paid out what you paid in. So you cap how much income is taxed, because otherwise the guy making $500,000 a year would end up getting massive payouts from SS while the person who made$30,000 a year wouldget much smaller payouts, which would lead to a lot of populist screaming. If you make the higher-earners pay into SS but only get a tiny amount paid out, if anything at all even, it is now a welfare program. One thing that could be done for SS is to raise the retirement age a bit more.\n\nOn the issue of a minimum wage, you raise the minimum wage too high, and you'll hike up the unemployment rate. I have actually wondered how the unemployment rate currently would respond if both the national and all state-level minimum wages were suspended for a couple of years. One of the first things the Democrats did under Nancy Pelosi upon taking control of the Congress in 2006 was to vote in an increase in the minimum wage. A healthy economy could maybe withstand that (although we had around 5% unemployment under Bush, at the limit for being considered full employment), but with it having been increased and then with the real-estate industry crashing, and then the financial crisis in 2008, I'd imagine the higher minimum wage is probably keeping unemployment higher than what it otherwise might be as the economy is very sensitive.\n\nAs for cutting spending, on the issue of waste and corruption, a problem with much of that is you just can't cut it, because it's like the marbling in a steak (marbling is the fat webbed throughout the meat). If there are big pieces of fat on the ends of the steak, you can take a knife and just cut them off, but it's pretty impossible to cut out the fat webbed throughout the meat, and that is how a lot of the corruption that occurs in the government is.\n\n#### WhoWee\n\nIMO, the only way to effectively cut our deficit is to cut spending ASAP and to keep all future spending on-budget and transparent.\n\nWe can get rid of a lot of spending with a few moves. First, close at least 600 foreign military bases. They are not necessary to the security of our country. Foreign bases are a drain on our treasury and a direct monetary transfer to the countries that host those bases. Bring home the military personnel, administrators, and their families. We will all be better off.\n\nNext, drop a couple of carrier groups. Our Navy does not need to be able to project aggression all over the world. No other country has this force.\n\nNext, drop the support for ethanol, and all other price-supports for agricultural programs that steer so many millions of dollars to ADM, Monsanto, etc, while making basic foods more expensive for US citizens.\n\nThis is not an inclusive list. I hope others can chime in.\n\nThe GOP fantasy that we must attack deficits by cutting Medicare, Medicaid, and Social Security, is just that. A fantasy that cannot work while keeping our most vulnerable citizens covered by at least a minimum social net.\nTo re-address the OP - I'm in favor of a phased-in percentage reduction across the board starting now with a goal of paying down the national debt over 40 years - perhaps to a 90 day working capital limit. If the budget cut is 20% or 30% - let the administrators of the specific departments sort out the details.\n\nIMO - when Government has a lean budget - waste will become quite obvious and any cuts in service favored over cuts in staff (Government size) will also be obvious.\n\n#### Char. Limit\n\nGold Member\nThe http:\/\/www.cnn.com\/2011\/US\/06\/02\/georgia.immigration.farm.workers\/index.html?iref=allsearch\" there are several states in the mid-$8 range. The article must be mistaken, but that's where I got my number. Is that before taxes or after taxes? I specifically mentioned \"taking home\"$7.80 an hour.\n\nCurrent binary employment issues aside, my point is that actual take-home isn't the issue for those that are working full time. Working 50 hours at a mill or farm isn't out of the ordinary - unskilled labor for road construction, county crews, etc, generally pay much higher (esspecially for unionized employees). The problem is really WHAT jobs people are doing. It's easy to be willing to work at WalMart or Target, and you're getting paid accordingly - lots of people love retail jobs because of the extreme flexability and possible discouts\/perks. It's also a job that can be done by someone with on the job training, making these jobs also good for young folks in college or just starting out. The retail environment is sketchy for the employer - you cannot give all of their staff full time hours. I was a supervisor at a big box store for about 2 years - out of 25 employees in my department during christmas, I had 3 that were full time. 6-8 each day during the week and everyone on Saturday and Sunday. In the non-holiday times that scaled down to a total of 8 with 2 full time. 3 per day during the week, and again, everyone on the weekends. How can I employ more people - the business couldn't handle it? Mandating that a company has x% full-time employees is lunacy and will spell doom for the retail industry - or prices will neccessarilly go up accordingly.\nAh, I see what you mean now. See, explanations are good. They help me understand things.\n\nOne could argue that the issues that General Motors faced in the past decade were because of their generous hiring practices. As a rule through the 60s and 70s, GM didn't hire temporary workers, it didn't hire part time workers - it didn't need to. It could afford the inefficiencies with the excuse that they had a huge available talent pool (ever heard the phrase 'Generous Motors'?). As their market share shrunk through the 80s and into the 90s they had to start cutting chaff and become more lean and mobile. They voluntarilly chose to be a respite, of sorts, for the community and had very loose hiring standards and practices - but that ended up biting them in the rear. Now, except for some highly specialized engineering positions, they're not hiring for permanent positions and they've started contracting more temps than ever. They got bit by the excess on their employee's behalf.\n\nFinally, and I generally avoid any ad hominem discussions - but if you're only working 20 hours\/week, that's your own fault. There may not be jobs you want to do out there to make minimum wage or better with full time hours, but they are out there. I've always had a job where I was getting 40+ hours. For the first time, come this spring, in almost 10 years I will not have a full time job (by my own choosing, to focus on school). I've jumped between jobs as I've moved around, but a month or two of unemployement aside - I've worked 40-60hrs\/week for a decade. Most have been indoor desk or retail jobs, but some of the best paying jobs I've had were labor (I did road construction for 3 summers between my first stint at college). Most of that time was spent living in the Detroit area... so arguebly an area that has been worse for longer as far as employement has concerned. Point being - jobs are there, it just may not be precicely what you want to do.\nI'm just going to ignore this part; I don't want to get into a flame war right now. Allow me just to mention that it's a trifle difficult to work 40+ hours a week and still manage to get 15-18 hours of class and 45-54 hours of studying in, per week. And still have time to sleep.\n\nAnd before you say anything, yes it took me this long to get back to you.\n\nLast edited by a moderator:\n\n#### mege\n\nI'm just going to ignore this part; I don't want to get into a flame war right now. Allow me just to mention that it's a trifle difficult to work 40+ hours a week and still manage to get 15-18 hours of class and 45-54 hours of studying in, per week. And still have time to sleep.\n\nAnd before you say anything, yes it took me this long to get back to you.\nThen you have extra circumstances that prevent you from working 40+hrs\/week. That's a different situation than jobs being available for someone looking for a career to live off of.\n\nI moved a year ago and it took me 3-4 job interviews and offers before I found one compatable with going back to school. I eventually did find something compatable (I work nights doing server-watching which gives me a lot of homework time!). I was close to giving up my search for a fulltime job and accepting part time somewhere for the flexability, but I consider myself lucky.\n\n#### Char. Limit\n\nGold Member\nThen you have extra circumstances that prevent you from working 40+hrs\/week. That's a different situation than jobs being available for someone looking for a career to live off of.\n\nI moved a year ago and it took me 3-4 job interviews and offers before I found one compatable with going back to school. I eventually did find something compatable (I work nights doing server-watching which gives me a lot of homework time!). I was close to giving up my search for a fulltime job and accepting part time somewhere for the flexability, but I consider myself lucky.\nHmm, that actually sounds pretty interesting (and useful). Nice catch!","date":"2019-11-19 19:57:58","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.20822152495384216, \"perplexity\": 2434.3064469035853}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-47\/segments\/1573496670255.18\/warc\/CC-MAIN-20191119195450-20191119223450-00449.warc.gz\"}"}
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- Kim Hyung-gon
- Park Shin-sung
- Kim Hyung-woo
- Kim Dong-wan
- Oh Heung-su
- Kim Ji-young
- Jeon Jae-woo
- Jung Seol-hwa
- Hwang Yoon - Mi
- Kim Hyun-Ah
- Jang Woo-jin
- Park Ji-young
- Yu Min-young
- Song Hye-Won
- Yuhn Cecilia
- Kaylin Ro
- Joo Brian
- Kwon Lina
- Wang soo-jung
- Jeon Sung-cho
A father and daughter move into a new home and soon, the father begins to hear strange noises coming from the closet in his daughter's room.
Soon, the daughter begins to show strange symptoms and a few days later, she disappears into thin air! The father begins searching for his daughter when he's approached by a mysterious man who says he knows the secret behind the daughter's disappearance. It's the new movie "The Closet." We asked the director what prompted him to make this film.
Park Hyuk-kwon and Ryu Hyun-kyung
Actors Park Hyuk-kwon and Ryu Hyun-kyung are loved for their flawless performances and distinct qualities. They are showing off an impeccable chemistry in the new movie "Pray." Their appealing qualities stand out in all of their works so we wish them only the best in their upcoming activities.
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Haganai (Boku wa Tomodachi ga Sukanai) is een Japanse animanga-boekenserie uitgegeven door Media Factory en geschreven door Yomi Hirasaka. De reeks begon als stripreeks op 27 maart 2010 en wordt sindsdien gepubliceerd door Media Factory. De serie is later gesplitst in twee animatieseries door AIC Build. Het dertien afleveringen tellende seizoen 1 liep van 7 oktober tot 23 december 2011 en seizoen 2 van 11 januari tot 29 maart 2013.
Seinen
Japanse stripreeks
Light novel
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"redpajama_set_name": "RedPajamaWikipedia"
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"Far Northern Memories" by Bert Rowe
Far-Northern Memories by H. (Bert) Rowe 31st July, 1973.
After many years of talking and thinking about it, I definitely made up my mind that I would go back to the district where I was born. I was born in Farina, in the Far North of South Australia, in the year 1891: there were three others of my family born there.
The only distinction Farina could lay claim to, at that time, was that it was the gateway to what was known as the Innamincka Track, where all the stations on the track used to send their teams (both bullock and horse teams) to deliver their wool clip and to take back their provisions, which meant that, when the teams were in, the town was lively and prosperous.
At that time, the town consisted of two hotels, butcher and baker shops, a saddler shop, a big store and a blacksmith shop, which was owned by my father. He was the only blacksmith north of Hawker, at the time, and he was kept busy.
Recently, a holiday weekend gave us the opportunity to make the trip. Our destination was Marree (known as Hergott Springs, in my day). As we had only three days to make the trip, we decided to leave on the Friday night, and make Quorn our object that night – two hundred miles away. My son and his family, with my wife and myself, left our Ascot Park home at 6.30 pm on a cold and frosty night.
After wending our way through heavy traffic, we eventually bypassed Gawler – one hour later (25 miles): we were still a long way from Quorn. As our car had air-conditioning, it was quite cosy inside, but very cold outside.
On we went through Rhynie, Auburn and Watervale, which brought back to me many memories of the cricket matches I had played on their ovals, in the Mid-North Cricket Association.
We arrived at that lovely Mid-North town of Clare before 9 pm, with its shops in the main street all ablaze with lights and its citizens doing their weekend shopping. After leaving Clare, we passed through Yacka and Georgetown, and took a wrong turn. By some instinct I was certain we were on the wrong road, so we turned back and were again on the right road to Wilmington, where we arrived at about 10.30 pm. At last we were headed for Quorn, 24 miles away, where we arrived at 11 pm. We stayed at the Old Mill Motel, which was supplied with electric wall-heaters and electric blankets, for which we were thankful, as it was still very cold.
Before breakfast next morning, I went for a walk around the town and was agreeably surprised to see that Quorn was not the ghost town that I thought it would be. Before the advent of Leigh Creek coalfields, we had to stop at Quorn for the night whenever we travelled on the train to the city: but, thanks to the tourist trade and being the gateway to the Flinders Ranges, Quorn is still very much alive, with four hotels and its nice motel and its shopping centre. It suffered a big loss in business and population when the railway bypassed it from Leigh Creek to Port Augusta, but it has survived.
After viewing the museum at the Old Mill Motel, we drove around the town, filled up with petrol and headed for the wide open spaces of the North. We crossed the Willochra Plain, passing the deserted towns of Gordon and Wilson, and arriving at Hawker about 10.45 am. Hawker has a population of about 500 and is the centre of pastoral and farming interests, set against the magnificent background of the Flinders Ranges.
We looked around the town, taking a few moving pictures, and then headed for the last town in which I had lived in the Far North – Parachilna – and what a town! Today, it consists of the hotel, post office and railway station. I walked around to see if I could find any old landmarks, but they had all disappeared. Even the house I had lived had gone, so I had my photo taken on that spot, with the Flinders Ranges in the background.
Parachilna, in my day, was a very busy railhead for the Blinman copper mine and smelters, with its dozens of teams of all descriptions – horses, donkeys, mules and even camels – carting coke to the smelters and copper-ore to the railhead.
It brought back old memories of hot nights, nights when it was too hot to drink in the bar of the pub. The teamsters used to have a kerosene-tin bucket filled with beer and would drink it out on the street. They would start singing and step-dancing and sometimes it would finish up in a fight – but everybody was happy. They were wild and woolly days, with plenty of money to spend.
There were not many women in the town, then – not more than a dozen – so you can imagine how they were in great demand when a dance was held in the pub dining room. The only other outings they used to have were the picnics which were arranged on a Sunday, up in the gap in the ranges. Everyone would go and enjoy a good day's outing.
Motor cars, in those days, were non-existent in the district: our only means of transport were horses and traps, or on horseback. My father, at that time, owned a well-bred racing mare and foal, which I always rode to the picnics. Once, somehow, she broke loose from the other horses and galloped off into the ranges. It was a very frightened and unhappy lad who went home and reported her loss.
Next morning, I was told to borrow another horse and not come back until I had found her. After about an hour's riding across the plain to the gap, I could see a cloud of dust ahead and, to my delight, it was Mavis and her foal. I called her and she came to. We returned home in triumph – an experience I will never forget.
As a man grows older, he begins to understand some of the things he didn't understand when he was young. In these days, we hear a lot about the loneliness of old people in our society. It was also very much in evidence in those days when I live in Parachilna.
The hotel was owned by an old man named Mr. W. Darmody, who was retired, and the actual management was done by his niece and nephew, Miss Annie Darmody and Jim Darmody, and they employed two waitresses and a cook, who were generally engaged through an employment agency in the city.
One cook had lived there for about three months when she suddenly died – she was about sixty and a widow. She knew no one in the town, only the hotel staff. As there were no doctors or police in the town, they had to get in touch with them in Blinman before she could be buried. The police tried, through the agent, to get in touch with relatives or friends, but they failed to find any and they gave the order to go ahead with the funeral, which became the most unfuneral-like funeral I have ever seen.
Old Charlie Brooks, the boots at the hotel, was sent to the local cemetery, which was about a mile along the Blinman Road, to dig the grave. My father, being the only skilled tradesman in the town, had to make the coffin – he did this for all the Far-Northern towns in which he lived. As there was no such thing as a hearse in any town in the North, the only means of conveying the coffin to the cemetery was a horse and dray belonging to the hotel.
On the day of the funeral, the cortege consisted of Jim Darmody and my father walking ahead, myself sitting on the front rail of the old tip-dray driving an old white horse (which was blind) following with the coffin, and Charlie Brooks walking behind with a long-handled shovel over his shoulder. Jim Darmody read the burial service. It was one of the loneliest funerals I ever saw – and yet another of the experiences I will never forget.
Whilst on the subject of the Darmody family, I recall another memory of a dog named Lassie, which was owned by them. One of old Mr Darmody's friends had been paying him a visit and, as he was an old drover, he wanted a sheep dog to take with him up the Innamincka track. He was given Lassie – much to my sorrow, and that of everybody in the town, as she was a great favourite with everyone. Lassie was put into the dog-box of the brake-van on the train and taken to Farina, eighty miles further north, where she arrived in the dark.
About a fortnight later, after we had finished work, I happened to glance up the Northern Road and I could see something coming towards me: I waited awhile and, much to my surprise and delight, it was Lassie – footsore, weary and hungry. I took her over to old Mr. Darmody and, with tears in his eyes, he said: "I will never do that again". Lassie had come home and she was still there when I left Parachilna.
The main source of milk in Blinman, in those days, was goats. Nearly every family had a flock of them, so naturally there was a demand for billy-goat carts as a means of carting wood and water from the local well. The homemade variety of a box on two wheels wasn't big or strong enough for the job, so they wanted something bigger and stronger made on the model of the old English wagon, with four wheels and an undercarriage, and with two pairs of shafts, so that they could yoke up anything from two to eight goats, as a team.
The first order for one of these wagons to be built was for one of my cousins, Ted Roberts, and my father turned the job over to me, to help me to learn the trade. The second one was built for another cousin of mine, Jack Whitford, who was later well-known in the North as Superintendent of Roads. The third was for Lance Nicholls, whose father, at that time, had the mail service between Parachilna and Blinman. I have still got the photo of myself and that wagon, taken in Parachilna about the year 1904. Lance Nicholls still lives on Warraweena Station in the Flinders Ranges, I believe.
In those days at Parachilna, my father was a friend of Mr. George McDonald, who was the manager of Lake Torrens sheep station and each year at shearing time, he wanted extra hands to do the mustering and to be handy around the place. He always asked if I could help, and as I loved riding horses, I was very eager to go.
Lake Torrens Station was owned by Mr. Frank Whyte, who also owned another property name Deep Well, out from Farina, and they brought the sheep from there to be shorn at Lake Torrens Station. Of course, they had to be taken back again – a distance of about 100 miles, and, one year, I two other lads and Mr. Brown were the party to take them back. I was appointed cook.
My first experience at cooking a damper was a great success, because I followed exactly the instructions on the baking powder tin; there was no self-raising flour, then. My second attempt was a complete flop, because I tried to take a large quantity and somehow it became unbalanced. No one could eat it, but it did come in handy as ammunition, the next day.
Our route was through the Flinders Ranges, which we entered on the third day, in beautiful weather, amid beautiful scenery. We camped that night at the old Leigh Creek Station, where the vermin-proof fence crossed the main road. We were about to turn in for the night, about 10 o'clock, when we heard a terrible clatter and some profane language. Presently, a man came up to the campfire, wheeling a bike with a buckled front wheel: he had run into the gate which he had not known was there. He was the last of a gang of shearers who had been shearing at Wooltana Station and, as he was in a hurry to get to the Leigh Creek pub before it closed, his language was not too polite.
After a wonderful trip in the fresh air, sleeping under the stars, we eventually arrived at our destination. Deep Well Station took its name from a very deep well at the homestead: I can't remember the exact depth, but, when you dropped a stone into it, you have to wait a considerable time before it hit the water. It was the deepest well in the North. Bores were the main source of supply of water in those days.
On Sunday, we three lads were taken into Farina to stop at the pub, so that we could catch the Monday morning train home. At dusk, we noticed a man coming down the Marree Road, wheeling a bike with a broken pedal. We were amazed as, at first, we thought he was an aboriginal, the first we had ever seen with a bike; however, he had been burnt black by the sun. He was Francis Birtles, the overland cyclist, a man who was making history. He was on his way from Port Darwin to Adelaide by bike, the first man to cross Australia from North to South in this way – a distance of over 2,000 miles, which was a wonderful feat.
That night, as we had an empty bed in our room, he was put in with us, and what a tale he had to tell! We sat up half the night to listen to his adventures; he told of his troubles with the hostile tribes in the Northern Territory, when he was lucky to escape with his life; the difficulties he had had crossing creeks and rivers; the problems of riding a bike with no roads, through sand and gibbers, and only the overland telegraph line to guide him. He was a brave man who, after finishing that trip, went on to other cross-country feats and later became an author, writing several books.
We left next morning for home – I for Parachilna and the other two lads for Hawker. Our droving for that year was over, much to our disgust.
I don't know whether meeting Francis Birtles had any effect, or not, but I still ride a bike.
The Blinman mine was discovered in the middle of the nineteenth century by a man named Blinman. After it was put into operation by several companies, it was closed down for a long period, but was finally reopened in 1902 by the Tasmanian Copper Company, with Mr. C.M. Henrie as manager. After a few years of prosperity, it was decided to open a smelter at Leigh Creek, so that they could utilise the coalfields for their steam-engines – but they never turned a wheel.
On a tour of inspection of the plant, Mr. Henrie died suddenly at the Copley Hotel. A new manager was sent, but he promptly closed everything down. Another bubble had burst! That meant that Parachilna, as a place of business, was finished and it has never recovered.
My father had taken on a business in the Barossa Valley and I was asked to ride the mare, Mavis, down there. At the age of 16 I left Parachilna on a Wednesday morning, and rode to Hawker, 57 miles away. I stayed at the hotels in the various towns overnight, and received many queries as to my journey.
The next day I rode on to Carrieton and left on the following morning for Black Rock. This was a truly memorable day as, on the way, I was dive-bombed by magpies (it was springtime and the nesting season). I saw my first motor car, which the horse would not pass – she shied and the motorist stopped the car to allow us to pass, then Mavis bolted; to finish an eventful day, I was told, on arrival at the Black Rock Hotel, that the publican had the D.T's! I was scared, and locked myself into my room that night.
Saturday night was spent at Yarcowie and, as I sat on the pub verandah, I could hear many raised voices in the bar. At last, one of the blokes came out and said that they had been discussing a horse out in the stable. He asked if it was mine and if it had been a racehorse: he also said that they were betting on its name. I answered 'yes' to the first two questions and said that we called the horse 'Mavis'. All abets were cancelled as the names that the men had been arguing over were all those of stallions!!
I rode on to Burra and, after dinner in the hotel, spent Sunday evening with the aborigines from one of Kidman's droving plants, who were on their way back North. Racialism didn't seem to be a factor, at that time' certainly not in my life.
On Monday I rode to Kapunda – 50 miles, the second longest ride of the journey – and, next day (the seventh) rode the last nine miles to Greenock where, after nearly 300 miles, the journey ended. I sent a telegram to Adelaide – 'Arrived Greenock be home tonight' – and caught the afternoon coach and train to Adelaide, where I was met by one of my sisters…..
Back to our journey of 1972:
We left Parachilna about 1.30 p.m. for Beltana, another of the deserted towns of the North which in the old days, was a prosperous and pretty town. We arrived at the old railway station, where we met Mr. Hull, who had purchased the old buildings to live in, so he could study old relics in the district for the Historical Society. We drove across the town, which is in ruins, and met a busload of Scouts from Adelaide, who were re-building a stone wall.
A function in connection with the Smith of Dunesk Mission was to be held in the hall, the next day (Sunday), when they expected a large crowd.
This Mission led to the foundation of the Australian Inland Mission and the Flying Doctor Service, as John Flynn served as a missioner, there.
We asked the Scouts where the old blacksmith's shop was, as my father had been the last blacksmith to work there, nearly a hundred years ago. Needless to say, it was in ruins, but the bench and lower part of a vice were still there.
After the importation of camels into Australia, an attempt was made to breed them at the old Beltana Station, as it was thought they were the best means of transport in the dry North. Teamsters and drivers of horses hated them because, when they met them, the horses were frightened and quite a few accidents were caused. However, the camels proved themselves in the dry, empty, northern parts of the State.
We headed North again for Copley and Leigh Creek, not stopping on the way, as we thought we would have more time on the return journey. We were disappointed with what we had seen; apart from a modern and well laid-out town, the coalfields were just a big hole in the ground, with a large machine extracting coal from it. The most interesting part was the huge mounds of overburden taken from the top of the coal-seam and scattered all over the place, like Miniature Mountains.
We passed the small town of Lyndhurst and on to Farina, the town of my birth, and how different it was from the town of my birth! We passed down the main street and I could recognise the old hotels, shops and other buildings, but, of course, they were in ruins.
The only things which were the same were the gibbers, which Farina was always noted for. There was only one house which we thought looked as if it could be inhabited. Farina was a ghost town.
We now headed for Marree, thirty-two miles away, and, after a very dusty trip, we arrived at 4.30 p.m. The dust was very bad on the road; when we met a truck or car, we had to stop to allow dust to settle before we could see the road ahead.
Like other towns in the North, Marree was not the bush town I remembered as Hergott Springs. Except for the pub, post office and one or two houses, the old lay-out of the town was altered, with all the most fascinating features of the old town gone. To my mind, it was too modernised, with its motor cars and electric lights.
In the old days, the town had three distinct features: on the west side of the line was the European section and all the business sector: on the eastern side was the Afghan town: and, further out, were the camps of the natives, which consisted of a number of wurlies made of old bags, flattened kerosene tins and a number of sticks. The camps were shifted every few months for the purpose of hygiene. The wurlies were round, about four feet high, and, to enter, you had to get down on hands and knees and crawl in – everybody in together, dogs and all. It seemed, at times, that there were more dogs than people.
One of my brothers, John Guy Rowe, was born on 5th November (Guy Fawkes Day) and all of us lads used to work for months ahead to prepare a big bonfire to celebrate both events, burning the guy and letting off fireworks. It was a gala night, attended by everybody – black, white and brown.
Our main supply of bags and rubbish came from the abandoned camps and, after one strenuous Saturday spent collecting fuel, we arrived home for tea with my brother squirming and scratching all the time. When asked what was wrong with him, he said "nothing", but, later, when we had our Saturday bath (water was scarce), he was found to be alive with lice and his clothes had to be burnt. I was lucky, I escaped. What great days they were! By a coincidence, on this trip it happened to be cracker night and it brought back old memories of the past.
The aborigines of those days were a lot different from what they are today: they were a happy, simple people and always grateful for anything given to them. In dry times, when game was scarce, they had a struggle to exist and the Government used to supply them, once a month, with rations.
Every Monday morning, we used to have a couple come to do our washing and chop wood. At the end of the day they would be supplied with rations and a few sticks of tobacco and, of course, they were fed during the day. They generally brought all their relations. They were said to be of royal blood, so we were honoured.
I cannot remember any of the local aborigines being in trouble with the law, but quite often, when the Oodnadatta train arrive, there was a van-load of wild aborigines from the Northern Territory being taken to Port Augusta gaol for spearing cattle, which they needed for food. They were generally chained together like animals, with one white constable and a black tracker in charge.
The Australian aboriginal is gifted with wonderful eyesight. I remember once my mother and I visited one of our family graves in the cemetery, about a mile and a half away over stony ground, and when we returned home, she found she had lost her gold brooch. I was sent over to the police station to see if the tracker would help find it. He came over and picked up our tracks at our house; we followed them and when we came to gravel, I would say, "Where tracks now, Fred?" and he would point to a small stone that had been slightly moved – only very keen eyesight could have noticed it. Happily, we found my mother's brooch not far from the grave we had visited.
The natives never seemed to need a doctor: I suppose they had their own remedies. There was one old lubra who used to come into the town, always accompanied by another carrying a tin of water from the bore. When she took a fit, she was brought around by the water. You could always trace her movements by the damp spots on the ground. The remedy was rough, but effective.
There were a few of the aborigines employed usefully about town and outlying stations, but they never seemed to visit the camp. Racialism, in those days, was not a problem: as lads, we all played together – black, white and brindle – with never a care in the world. There was no effort made to educate the native children in the local school, as none attended whilst I was there, but, in later years, a bigger school was built and I believe a lot attend, now.
On the Saturday night of our trip, we stayed at the Marree Hotel and, after dinner, visited the front bar and parlours, and it is quite safe to say that the majority was aboriginal: wives and girl-friends were in the parlours, all were smartly dressed in the latest fashions and seemed to be enjoying themselves, with plenty of money to spend. The younger generation, through education and mingling with the white settlers, had easily adapted to the white man's environment, which is not always in their best interest.
I was disappointed to find that the Afghan side of the town had gone. It used to be the most fascinating feature of the old town, with its Eastern influence, its two mosques, camels and three date-plantations. Every morning and night, at sunrise and sunset, we would hear their priest, the imam, with a very mournful cry, calling the faithful to prayer.
In front of the mosques there was always a small dam, with bore water running into it, neatly covered around the edges with boards to eliminate the mud, and before they were allowed to enter they had to take their boots off and wash their feet, leaving their footwear outside. Often there was a strange collection of footwear outside the mosques, some with the toes turned up, in the Eastern tradition.
The mosques were built by the Afghans themselves, mostly of mud and straw, with a thatched roof, the walls about four feet six inches high, with a space of about six feet before the roof began, wide open to the elements and sightseers. The walls and floor were spotless, clean and white; there were no seats of any description, and the worshippers stood for a while, then knelt with their heads touching the floor, and always facing the sun. We were often spectators.
There was a fairly large population of Afghans in Marree, in those days, as it was the main centre for the camel-trains, and long strings of camels moved out to cart provisions and all sorts of merchandise to stations along the Cooper and over the Queensland border, also supplying those along the Birdsville Track.
It was amazing to see the variety of goods carried on camels, at times – anything from small rainwater tanks to telegraph poles: there was nothing they couldn't carry. The load had to be tied on each side of pack-saddles with ropes, and had to be evenly balanced. It was a job that called for good judgment and experience. The goods were tied into bundles in the railway yard a couple of days before they departed, and weighed, on an average, about 7 cwt.
When the camels were eventually brought to be loaded, it used to cause a lot of interest and sometimes amusement, especially when they were breaking-in young camels, as they would get savage and try to bite the legs of the men loading them. Someone would have to stand in front of them, with a big stick in his hand, and give them a crack on the head when they tried to bite. They have long necks and can reach a long way back.
Many times we have seen the main street of Marree littered with broken cases of jam, pickles, tomato sauce and other goods, because the young camels had thrown off their loads and caused confusion amongst the others. Camels are slow-moving animals and, with their heavy loads, do not travel very far in a day, and the monotonous task of unloading and loading has to be done every day.
There is a mistaken idea that camels can go a long way without water, but this is not so; it is only those that have been trained to do so that can do this. They need water like other animals, but they do eat almost anything and keep alive where other animals would die. It was in the 1920's that the motor trucks started to take over their work and, by the middle of the thirties, they had died out altogether.
The Afghans, being Mohammedans, were very strict in their religion; they never patronised the hotel or drank any alcohol. Every night, when the butcher was going to kill meat, he would go over to the Afghan town and pick up one of the men in his cart, to take to the killing-yard, as they would not buy meat if one of them had not stuck a knife into the beast. They were very fond of fruit and vegetables, but the local shops had to be careful that their smallgoods and bacon were not on display, otherwise they would not enter the shop. Pork is very much against their religion and they won't eat it.
On the whole, the Afghan people were a quiet and peaceful group, but there were times when violence broke out and resulted in the use of firearms. One man was shot dead one night, and when the police and the tracker went to find the culprit, next day, they discovered that he had walked along the metal on the railway line towards Farina, but when he left the line, the tracker immediately saw his track and they caught him that night in the bush.
With the disappearance of the camels and Afghans, with their colourful dress of baggy calico pants, fancy waistcoats and long turbans, Marree lost one of its most interesting features and it is now just an ordinary Australian small town – much to my disappointment.
Marree stands on the edge of that geological phenomenon, the artesian basin, which reaches almost to the Queensland border on the Birdsville Track. The bore used to be situated in the railway yard and flowed down through the fence into a long trough outside, where all the stock used to drink; then it would run away in a stream for about a mile and a half and disappear into a swamp. The water was slightly brackish but good stock water. When rainwater was scarce, it was used to wash clothes, after it had been boiled and skimmed!
According to a geologist, the basin was like a huge pie in a dish, with a hard crust on top and, once the crust was pierced, the pressure would force the water into the air. There was a Government bore at Lake Harry, twenty miles from Marree, where the water was hot, but as you advanced up the track, the bores got deeper and hotter. I have heard teamsters say that, at some bores, it was possible to make tea with water straight from the bore. It had to be run off into cooling ponds before the stock could drink it………..
On the Sunday morning, we went to see if the bore and the date-plantations were still there, but I was told they had disappeared years ago. The date-plantations were a real oasis in the desert, in the old days, with the stream from the bore running along one side of them. When necessary, it would be used to irrigate the date palms. The Government controlled one, with a gardener, but the other two were owned by the Afghans, with their mosque in the centre.
The dates picked fresh from the palm had a different from those we buy, preserved, from the shops today. Sometimes we would buy them dried and, when we used to shake them, we could hear the stone rattle inside. They had a distinct nutty flavour.
What a different scene it is, today; with the bore and plantations gone, it is a desert. It is not the old town that I knew.
After filling with petrol, we decided to visit the cemetery where my youngest brother was buried in 1898 – seventy-four years ago. We were amazed that the lettering on his tombstone was as good as the day it was put there. I saw a lot of other names of people well-known to me in the old days – European and Afghan.
When I was about four years old, we had moved to Hergott Springs (Marree) from Farina, as my father had decided that business prospects were much brighter there. After living in the town for about five years, where some of us went to school, he decided to move the entire family to Adelaide, where facilities for education were better, as one of my sisters wanted to become a school teacher.
I and my brothers attended the Norwood Public School, but, after about a year, when I was so fed up with city life – which I hated – and had worried my parents so much that they eventually agreed to send me back "bush" with my father, I returned to Marree, much to my satisfaction. It was at Marree that I finished my schooling and also started to work.
The school, in those days, was a single room built of wood and iron. It was used for all purposes – dances, concerts, and all denominations used it for their services. On this trip, I notice that they have a much larger and more modern school, now, but not on the old site.
My father was one of the best-known men in the North and he was generally the instigator of all sporting and social functions. Many a time I have heard men from drover's plants or teamsters say to him, when they came to town: "What about a dance, tonight, Bill?" I would be told to ride a horse around to all the houses to inform them that there would be a dance that night. Everybody would turn up, and all the ladies would bring a basket of food for supper; sometimes the dance would go on until daylight.
Friday afternoon was generally the day when wild cattle from Kidman's stations were yarded, for trucking to the Adelaide market next day. Our shop, being opposite the yards, was a good place to see all the excitement of the yarding, as it took a lot of hard riding before they were all safely inside. Often a beast would break away and get tangled up in, the railway wire-fence and had to be shot. There would be free beef around the town, next day.
About that time, a family in Quorn owned a dog known throughout the North as "Bob, the railway dog". He very often used to take trips up North on the trains, looked after by the railway men. When North and South-bound trains left Quorn about the same time in the mornings, both pointing East, he had enough intuition to know which train to catch – which was more than some humans knew. He always travelled North.
A few years later, another "railway Bob" appeared, in the form of the Rev. R. Wilkinson, the Church of England minister of a large Northern parish. His only means of transport was a bike or the train – he always seemed to be on a train, hence his nickname.
On one occasion, he was in Farina and wanted to hold a service in Marree on a Sunday night. His only means of transport, that day, was his bike, and my father decided to go in a buggy and pair to meet him. As there were no such things as grids in the roads, those days, I had to go with him to open and shut the gates. To leave a gate open was – and is now – a crime.
About half a mile out of town, we had to pass through a big paddock belonging to one of the land-owning Afghan gentry who used to breed camels. He had imported, from Arabia, a large, double-humped camel to be used as a stud – the only one of its kind in the North. We noticed him feeding with others out on the plain, when we were going through.
After we had picked up the minister and were on our way back into town, the camel noticed us; he came towards us at an angle to cut us off at a smart trot. Not knowing what to expect, the three of us, as well as the horses, were a bit scared. When he got about twenty yards from us, my father handed me the reins and got out of the buggy, picking up a couple of stones. When the camel got closer, he let fly with a stone and was lucky enough to hit him on the side of the head. The camel let out a bellow and "turned tail" where he got another stone, and then galloped away, very much to our relief. We never did know what his intentions were. Camels have been known to attack men, at certain seasons.
I pointed out the spot where it had occurred to my family, on this trip – a spot I shall never forget.
The weather in the North in the winter time is the best in the world – beautiful sunny days and no wind. The only suitable outdoor sport to play was tennis: also, there were no facilities for other sports.
There was great rivalry between the Marree and Farina teams. Twice a year, they would meet in home-and-away matches and the excitement and barracking would do credit to a grand-final football match. There would be six men in each side and the match would last all day, with the rival barrackers wearing their team's colours in ribbons and rosettes. At night, there would be a grand concert and ball, which would last till daylight. It would be a subject for conversation for a long time, and also good business for the local hotels, as it would bring a lot of people from the district into the town.
Another event of great importance was the annual race meeting, which was supposed to last for one day but was generally celebrated for a week, with something on every day and night – good for all-round business, as money flowed pretty freely. People would come from stations along the Birdsville Track, and other outlying stations, and make it their annual holiday.
I was always pleased when the meetings were over, as my father, in his spare time, used to train some horses for Mundowdna Station, to try out at the local races, and I had to get up at daylight, in the cold, and help him ride them on the track, which I didn't appreciate very much.
The outback bushman has always been noted for his ingenuity when he gets into strife in the bush. He has had to be inventive because sometimes his life depended on it. I remember one time the Birdsville mail, which used to arrive once a fortnight on a Saturday at noon, was very late. It was only a large buggy, drawn by five mules or horses and driven by one of the toughest men in the bush, Billy James. When it eventually arrived, it had only three wheels – and a mulga stick! The back axle had broken off at the shoulder and, of course, the stick had to be renewed frequently: needless to say, we had to repair the axle. Only a tough man would have got the mail through.
At times, we would get word from Farina asking if we would go down and open the blacksmith's shop there for a few weeks, as several bullock-teams were expected to come in from the Innamincka Track and would want a lot of repairs done. Bullock teams in a district meant a lot of work for a blacksmith, because all their equipment was repaired and made by the smith. All the saddler got out of them was a tail for the whip.
In most towns, there would be a public cueing-pen for nailing the cues on the bullocks. It consisted of three posts in the ground and two rails on each side, to which the legs were tied with a rope. We used to make hundreds of cues, as it took eight for a set, one for each claw. It used to create a lot of interest among the public, when the cueing was on, but those days and events are gone forever.
I mentioned before that ingenuity is an essential trait for the bushman, and recall two other examples of this. In the early days in Farina, my father employed a young lad named Joe Wright, who wanted to learn the trade. He lived with us as one of the family and was a brother to us, becoming a very good tradesman.
After we had shifted to Marree, the South Australian Government wanted to put down a line of bores from Coward Springs, eighty miles north of Marree, to the Western Australian border. It was necessary for them to have a blacksmith on the boring plant and, as smiths were scarce, they applied to my father to release Joe, which he did. The first bore; they attempted to put down was about sixty miles west of Coward Springs, in the middle of nowhere. As internal-combustion engines did not exist, in those days, it meant they had to use a steam-boiler for power which also meant water. They had to cart water from Coward Springs in a wagon drawn by camels, and every drop of water was precious.
Later on, they realised that a lot of water was being wasted in the steam from the exhaust, and they asked Joe if he could bend a length of bore-casing, sixteen feet long with four inch diameter, into a "U" shape, so it would return the steam back to the boiler and eventually condense back to water. What a job! Only a tradesman can appreciate the magnitude of the task even in a well-equipped workshop it would not be easy.
Fortunately, they were situated close to some scrub and there was plenty of dry timber about. First, Joe put three posts into the ground and secured one end of the casing between two of them with some chains; he then lit a series of fires along the casing, using plenty of wood. He hooked camels on to the other end of the casing and pulled it against the third post, gradually shifting the post along a few times until the casing was pulled into the desired shape.
We saw a photograph of it when he came home, and it looked a good job. However, as it turned out, after drilling down nearly a mile (over five thousand feet) they did not strike water of sufficient quantity to go any deeper and the whole scheme was abandoned.
Joe got good recommendations from his boss and later in life, he went back to the Government to work until he retired.
The other incident also was concerned with a steam-boiler, which was being carted bay bullock-teams to somewhere in the outback, near the Queensland border, when they discovered that it was gradually sliding off the back of the wagon. As it weighed nearly four tons, they were in a quandary to know how to get it back in its place. If it fell off, they would never get it back on to the wagon, so they decided that, as they could not shift the boiler forward, they would have to shift the wagon backwards.
They cut down some long poles and built tripods at each end of the boiler, then secured the boiler to the tripods with bullock chains, which were always in good supply with a bullock-team. They then jacked up each wheel separately and dug out a trench under it; in this way, they lowered the wagon. Everything was back to normal, the job had taken a week, but it was worth it. We are told that "necessity is the mother of invention" and nowhere does it apply more than in the bush.
The bush is also responsible for a variety of characters. A blacksmith's shop, in the country, was a real meeting place for men who had some spare time to spend, and you heard all sorts of stories, tall and otherwise. Even the great Sir Sidney Kidman, sometimes accompanied by his wife, would call into our shop for horses to be shod or buggy repairs when he was visiting some of his stations up the Birdsville Track. He would tell a few stories himself. He used to tell of his early life, when he left his Kapunda Home, as a lad, with a one-eyed horse and one pound in his pocket. Later, he bought and sold another horse and made one pound profit – the start of his huge fortune.
In his early days, he concentrated more on buying and selling horses; it was later that he turned to sheep and cattle, to build up the Kidman empire. In those days, he was always known as Sid, never Mr Kidman. It was true, natural democracy in the bush.
He was a tall, upright man of over six feet, a typical bushman with his elastic-sided boots and wide-brimmed, felt hat. He was a strict teetotaller who never smoked or swore. When he was really angry with a man or a beast, he would refer to him or it as "a jolly tinker". All his business dealings didn't meet with the approval of the business people in the North, and sometimes, when he bought a property in a district, it was referred to as "the Queensland blight has come over the land". He was supposed to have a wonderful memory and to be able to tell, to a day, where any of his mobs of cattle and droving plants would be. He was finally granted a knighthood and died a rich man, and parts of his vast empire still exist.
Another character I remember was an old man named Dan Dwyer, who always visited our shop once a year. He was the typical old, Australian swagman, with his spotless white moleskin pants and bowyangs, his white beard and huge swag, with his old, black billy. He used to walk all the way from Gawler, over 400 miles, because he hated the wet weather in the South. Dan always camped in the corner of our shop and cooked his meals on the forge. He would wait until the weather down South had cleared up, then he would walk back again. He always said, when he left us, "I'll see you next year", which he did for years. The last time we saw old Dan he became very ill in our shop, and my mother had to nurse him back to health for a month. She tried to persuade him to travel South by train, but he refused. He had taken his last walk: we never saw him again.
A well-known character in the North was an Indian hawker named Artmour Singh, with his big, covered van and four horses. He used to do all his trading up the Birdsville Track and, when he came back to replenish his stock, he always did his cooking on the forge. I can still smell his curries, for which he was famous. He later opened an emporium in Kingscote, Kangaroo Island, and retired a rich man.
Some of the most colourful characters were sons of well-to-do English families; known as remittance men, they were generally sent out to Australia because they lived too wild a life at home. Most of them used to work out on the stations until their remittance arrived, then they would come to the town and have a wild time. Often some would say that they were off to the city, but they never got past the first pub. They were always well-educated men, with all sorts of talents – some as artists and some as poets – and, if anything unusual happened, it was generally written up in verse by one of the poets, usually in humourous style.
Here is an example of the verses we would hear – evidently written by the unknown "poet" after hearing two men in a pub discussing the relative merits of their sheep-dogs. Just why this particular one has remained in my memory, I don't know, as we heard so many which I have forgotten:
You can talk about your sheep dog, said a man from Coopers Creek,
I know a dog that would simply knock you ratty.
He was owned by Daly, a drover cove, from Oodnadatee.
We were talking in the pub, on a Summer's afternoon
When a blowfly comes a-buzzing around the ceiling.
Now up got Mike and said, 'Look here, Pat,
I'll show you what this dog is at heeling.'
There was an empty pickle-bottle standing on the shelf;
He took it down and put it on the table.
He showed the dog the bottle and pointed to the fly
And, in less than fifteen seconds – Gor, spare me days,
It ain't a lie –
That dog had got the blowfly in the bottle. (Some dog!)
We would hear a lot of stories, from both black and white men, about the Gammon Ranges and how the name originated. According to the blackfellows' legend, in the "dreamtime" it was the home of a huge snake that was supposed to keep out evil spirits. There used to be all sorts of rumbles, like thunder, and other weird noises due to the lack of earth amongst the rocks – not having anything to bed themselves in, they rolled down the side of the hills and caused those rumbles.
It is rather difficult to know where the Flinders Ranges end and the Gammon Range begins – it is somewhere North-East of Leigh Creek, in very barren country. In my days in the North, the word "gammon" was used a lot, by both black and white, young and old. If you were telling a story to anyone and he said that it was only gammon, it meant that it was a tall story and wasn't true – that you were trying to deceive him. I have never heard the word used, down in the Southern part of the State, yet it is part of the English language.
I think the name originated in this way: someone, hearing the legend of the snake, called it "gammon". I'm sure the sophisticated natives around Marree, today, would say it was only gammon.
When I set out to write this, I intended it to be about memories of my days in the North, but there is one tragic experience of my father's I would like to put on record. It happened before I was born and was told to me by my mother. In those days, my father owned two horses and a covered-in buggy, which he used to travel around to the various towns in the North, staying as long as work was available.
About 1884, he shifted to Innamincka and, as he always wanted someone to help in his shop, his youngest brother came up from Adelaide to help him. He had never been in the bush before, and after he had been there some time, he was taken suddenly ill, with supposed typhoid fever. There were no doctors for hundreds of miles. My mother nursed him, but eventually he died; he was only twenty. My father, as the only skilled tradesman in the district, had to dig his grave, make the coffin and read the burial service over him – a task that no one would envy him. They were tough days in the bush: you had to be tough to survive.
Returning to our modern journey: we left Marree on Sunday morning at about 11.30, and headed for Parachilna, where we had to meet some friends. We called in to see the Aroona dam, which is three miles off the main road and was built by the Government to supply water for the Leigh Creek coalfields. It is a picturesque spot, with the blue hills in the background, and was well worth the visit.
There was a race meeting in progress at Parachilna when we arrived. After the race meeting, we, together with our friends, were invited to tea at a station between Parachilna and Blinman. Later, we drove on to Blinman, where we spent the night.
Blinman, like other towns in the Far North, has deteriorated: all its past glories have faded. Many times, when the mine was booming and the main mail coach was over-crowded on Saturdays, Joe Kipling used to call on me to drive another one around the old road, to deliver the Moolooloo Station mail-bag. As it is eight miles further, we generally arrived at the post office after dark. We carried no lights, so the road was left to the horses to follow. Thanks to the surrounding pastoral interests and tourism, the town has survived. We had passed many camp-fires and tents in the ranges as we drove along the road on Sunday night.
The next morning, at breakfast in the hotel dining-room, we struck up conversation with two other guests, who were visiting the town of their birth. One of them turned out to be a long-lost cousin of mine, young Joe Kipling from Port Augusta. His mother was my cousin, Ivy, a daughter of my uncle, Dick Roberts. In the old days, I had a lot of relations living in Blinman, but they have all drifted away.
After taking on more petrol and saying goodbye to our friends, we at last headed for home about nine o'clock on Monday morning. After a pleasant drive in lovely weather down through the ranges, we called in to see the Wilpena Pound Motel, which has been enlarged considerably. It would be a nice place to spend a healthy and quiet holiday.
We continued on the road to Hawker and, about halfway there, we struck the new bitumen road which is being built as far as Wilpena Pound. We arrived back at Quorn and lunched at one of the hotels.
It was a very interesting and pleasant journey, but I was somewhat disappointed to see the decay of some of the towns. After so many years, it is hard for me to realise that I have been back.
We arrived in the city, amidst the noise and bustle of traffic, at about 5 o'clock, and have settled down again to the humdrum existence of suburban life.
Our journey into the past is now behind us.
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Tag Archives: Afghans
Evacuation flights restart from Kabul after Afghans desperate to flee cleared from airfield
KABUL (Reuters) -U.S. military flights evacuating diplomats and civilians from Afghanistan restarted on Tuesday after the runway at Kabul airport was cleared of thousands desperate to flee following the Taliban's sudden takeover of the capital.
The number of civilians had thinned out, a Western security official at the airport told Reuters, a day after chaotic scenes in which U.S. troops fired to disperse crowds and people clung to a U.S. military transport plane as it taxied for take-off.
"Runway in Kabul international airport is open. I see airplanes landing and taking off," Stefano Pontecorvo, NATO's civilian representative, said on Twitter.
At least 12 military flights had taken off, a diplomat at the airport said. Planes were due to arrive from countries including Australia and Poland to pick up their nationals and Afghan colleagues.
As they rush to evacuate civilians, foreign powers are also assessing how to respond to the new rulers in Kabul and also how to deal with refugees trying to flee the country.
Former U.S. President George W. Bush, who launched a "war on terror" in Afghanistan following the Sept. 11, 2001, attacks, said the United States must move quickly to help Afghan refugees.
Turkish Foreign Minister Mevlut Cavusoglu said the country was in talks with all parties in Afghanistan, including the Taliban, and was positive on their statements since they took control.
Under a U.S. troops withdrawal pact struck last year, the Taliban agreed not to attack foreign forces as they leave.
U.S. forces took charge of the airport – their only way to fly out of Afghanistan – on Sunday, as the militants wound up a week of rapid advances by taking over Kabul without a fight, 20 years after they were ousted by a U.S.-led invasion.
Flights were suspended for much of Monday, when at least five people were killed, witnesses said. Media reported two people fell to their deaths from the underside of a U.S. military aircraft after it took off.
U.S. troops killed two gunmen who appeared to have fired into the crowd at the airport, a U.S. official said.
U.S. President Joe Biden defended his decision to withdraw U.S. forces after 20 years of war – the nation's longest – which he described as costing more than $1 trillion.
But a video of hundreds of desperate Afghans trying to clamber on to a U.S. military plane as it was about to take off could haunt the United States, just as a photograph in 1975 of people trying to get on a helicopter on a roof in Saigon became emblematic of the humiliating withdrawal from Vietnam.
Biden said he had to decide between asking U.S. forces to fight endlessly or follow through on a withdrawal agreement negotiated by his predecessor, Republican Donald Trump.
"I stand squarely behind my decision," Biden said. "After 20 years I've learned the hard way that there was never a good time to withdraw U.S. forces. That's why we're still there."
Facing criticism from even his own diplomats, he blamed the Taliban's takeover on Afghan political leaders who fled and its army's unwillingness to fight.
The Taliban captured Afghanistan's biggest cities in days rather than the months predicted by U.S. intelligence. In many cases, demoralised government forces surrendered despite years of training and equipping by the United States and others.
The Taliban began their push in the spring with attacks on government positions in the countryside and targeted killings in cities. The International Committee of the Red Cross said more than 40,000 people with wounds caused by weapons had been treated at facilities it supports in June, July and August, 7,600 of them since Aug. 1.
Chinese Foreign Minister Wang Yi told U.S. Secretary of State Antony Blinken that the hasty pullout of U.S. troops had a "serious negative impact", China's state broadcaster CCTV reported, adding that Wang pledged to work with Washington to promote stability.
U.S. forces are due to complete their withdrawal by the end of this month under the deal with the Taliban that hinged on their promise not to let Afghanistan be used for international terrorism.
President Ashraf Ghani left Afghanistan on Sunday as the Islamist militants entered Kabul, saying he wanted to avoid bloodshed.
That day, some 640 Afghans crammed into a U.S. C-17 transport aircraft to fly to Qatar, a photo taken inside the plane showed.
The U.N. Security Council called for talks to create a new government in Afghanistan after Secretary General Antonio Guterres warned of "chilling curbs" on human rights and violations against women and girls.
During the Taliban's 1996-2001 rule, women could not work and punishments such as public stoning, whipping and hanging were administered.
The Taliban have said there will be no retribution against opponents and promised to respect the rights of women, minorities and foreigners, but many Afghans are skeptical and fear old enemies and activists will be rounded up.
The top U.N. human rights official voiced concern about the safety of thousands of Afghans who have worked on human rights. The U.N. refugee agency called for a halt to forced returns of Afghans including asylum seekers whose requests had been rejected.
(Reporting by Kabul and other bureaus; Writing by Jane Wardell, Robert Birsel; Editing by Simon Cameron-Moore, Timothy Heritage and Nick Macfie)
Tagged Afghans, Evacuation flights, Kabul, Refugees, Taliban, U.S. troops, War on Terror | Leave a comment
Thousands more Afghans can resettle in U.S. as refugees, says State Dept
WASHINGTON (Reuters) -Thousands more Afghans who may be targets of Taliban violence due to their U.S. affiliations will have the opportunity to resettle as refugees in the United States under a new program announced by the State Department on Monday.
Reuters exclusively reported on plans to set up the "Priority Two" refugee program, covering Afghans who worked for U.S.-funded projects and for U.S.-based non-government bodies and media outlets, earlier on Monday.
"In light of increased levels of Taliban violence, the U.S. government is working to provide certain Afghans, including those who worked with the United States, the opportunity for refugee resettlement to the United States," the State Department said in the announcement.
"This designation expands the opportunity to permanently resettle in the United States to many thousands of Afghans and their immediate family members who may be at risk."
The announcement of the program comes as fighting surges in Afghanistan ahead of the formal completion of the U.S. troop withdrawal at the end of this month, with the Taliban pushing to capture key provincial capitals.
U.S. President Joe Biden has faced pressure from lawmakers and advocacy groups to aid Afghans at risk of Taliban retaliation because of their association with the United States during the 20-year war.
Those who worked as employees of contractors, locally employed staff and interpreters and translators for the U.S. government or armed forces are eligible for the new designation, as well as Afghans employed by a U.S.-based media organization or non-governmental organization (NGO), the State Department said.
Secretary of State Antony Blinken will deliver remarks on the program at 2 p.m. (1800 GMT), the department said.
The new program for Afghans requires applicants to be referred by a U.S. agency or for the senior-most U.S. citizen employee of an NGO or media organization headquartered in the United States.
Once they have applied, they will be contacted by email to let them know they are in the system and will then have to make their own way out of Afghanistan to a third country, a senior State Department official said.
The process from that point involves security screening and can take from 12 to 14 months, said the official, who briefed reporters on condition of anonymity.
"We've already been in discussion with neighboring countries, as well as (the U.N. refugee agency), to be prepared for potential outflows," said another official who briefed reporters, adding that it was important that Pakistan's borders with Afghanistan remained open, while others might travel to Turkey via Iran.
That requirement differs from an existing refugee program for Iraqis, which allows Iraqis to apply directly but has been indefinitely suspended while U.S. officials pursue a sweeping fraud investigation.
The Priority Two program applies to Afghans who do not qualify for the Special Immigration Visa (SIV) program that covers interpreters and others who worked for the U.S. government, and their families.
About 200 SIV applicants whose visas are in the final stages of processing and family members flew into the United States last week at the start of an evacuation effort dubbed "Operation Allies Refuge" that could include as many as 50,000 people or more.
A second plane carrying several hundred more Afghan SIV applicants arrived overnight in the United States and they will join the first arrivals at Fort Lee, Virginia, a U.S. official said on Monday.
(Reporting by Lisa Lambert, Ted Hesson, Jonathan Landay and Idrees Ali, Simon Lewis and Daphne Psaledakis; Editing by Andrew Cawthorne, Timothy Heritage, Giles Elgood and Jonathan Oatis)
Tagged "Priority Two" refugee program, Afghans, permanently resettle, Refugees, targets of Taliban, United States | Leave a comment
Police find 41 migrants alive in truck in northern Greece
ATHENS (Reuters) – Greek police found 41 migrants, mostly Afghans, hiding in a refrigerated truck at a motorway in northern Greece on Monday, officials said.
The discovery came 10 days after 39 bodies, all believed to be Vietnamese migrants, were discovered in the back of a refrigerated truck near London. Two people have been charged in Britain and eight in Vietnam over the deaths.
The refrigeration system in the truck where the migrants were found in northern Greece had not been turned on, and none of the migrants was injured, though some asked for medical assistance, a Greek police official said.
Police had stopped the truck near the city of Xanthi for a routine check, arresting the driver and taking him and the migrants to a nearby police station for identification.
Greece is currently struggling with the biggest resurgence in arrivals of migrants and refugees since 2015, when more than a million crossed into Europe from Turkey via Greece.
Most of them are reaching Greek Aegean islands close to the Turkish coast via boats but a large number also come overland, using a river border crossing with Turkey.
Road accidents, mainly in northern Greece, involving migrants trying to cross into other countries have become more frequent in recent years. Police have arrested dozens of people believed to be involved in human trafficking so far in 2019.
About 34,000 asylum seekers and refugees are being held in overcrowded camps on the Aegean islands under conditions which human rights groups have slammed as appalling.
The conservative government that came to power in July has vowed to move up to 20,000 off the islands and deport 10,000 people who do not qualify for asylum by the end of 2020.
Arrivals of unaccompanied children have also increased. About 1,000 minors have arrived since July, the Greek labor ministry said, with the total number estimated at over 5,000.
A fifth of them are now missing, the ministry said, pledging to build more facilities and shelters for migrant children.
(Reporting by Lefteris Papadimas and Angeliki Koutantou; Writing by Renee Maltezou; Editing by Mark Heinrich)
Tagged Afghans, Britain, Greece, Human Trafficking, London, Migrants, refrigerated truck, vietnamese migrants | Leave a comment
Taliban team at Afghan peace talks in Qatar to include women: spokesman
By Abdul Qadir Sediqi
KABUL (Reuters) – Women will be included for the first time in the Taliban delegation to peace talks in Qatar this month, the movement's main spokesman said on Monday, ahead of the latest round of meetings aimed at ending the war in Afghanistan.
For a group notorious for its strictly conservative attitude to women's rights, the move represents a step towards addressing demands that women be included in the talks, intended to lay the foundations for a future peace settlement.
The April 19-21 meeting in Doha will be between the Taliban and a delegation comprising prominent Afghans, including opposition politicians and civil society activists. It follows similar talks between the two sides in Moscow in February.
The non-Taliban delegation that was in Moscow could be expanded next week to include some government officials, but acting in their private capacities as the insurgents have refused to hold formal talks with Kabul.
"There will be women among Taliban delegation members in the Doha, Qatar meeting," Zabihullah Mujahid, the Taliban's main spokesman, said by telephone.
He did not name the women, but added, "These women have no family relationship with the senior members of the Taliban, they are normal Afghans, from inside and outside the country, who have been supporters and part of the struggle of the Islamic Emirate".
In a tweet, he specified that the women would only join the discussions with Afghan civil society and political representatives, not in the main negotiations with American officials, led by U.S. special peace envoy Zalmay Khalilzad.
Khalilzad has not yet set dates for the next round of talks with the Taliban, a State Department spokesperson said.
"We do not have new U.S. talks with the Taliban to announce at this time. Before additional talks, we look forward to knowing the outcome of the intra-Afghan dialogue," the spokesperson said in an email.
Khalilzad, a veteran Afghan-born U.S. diplomat, is working to secure an accord with the Taliban on a U.S. troop pullout, measures to prevent al Qaeda and other extremists from using Afghanistan as a springboard for attacks, a ceasefire and inter-Afghan talks that include the government on the country's political future.
The Taliban have rejected formal talks with the government, which they dismiss as a "puppet" regime controlled by the United States.
While Afghanistan remains a deeply conservative country, especially in rural areas, there have been major advances in women's rights since the U.S-led campaign of 2001 that toppled the Taliban government. Many women fear that if the group regains some power, many of these gains could be erased.
The movement gained worldwide notoriety when it came to power in the 1990s by forcing women to wear full facial covering and imposing severe restrictions including banning girls from school and forbidding women from working outside the home.
However, Taliban spokesmen say the group has changed and it encourages girls' education and other women's rights within an Islamic Sharia system.
Civil society groups, the Western-backed government and Afghanistan's international partners have pressed for women to take part in the talks and news of the Taliban delegation was welcomed. Fawzia Koofi, a former member of parliament who took part in the meetings in Moscow, said the presence of women in the Taliban team was a "good step".
"Only women can feel the pain and miseries that Afghan women have suffered. The presence of women among the Taliban negotiators shows that the Taliban's ideology has changed."
Jeanne Shaheen, a member of the U.S. Senate Armed Services Committee, who has been pressing for women to play a role in the peace talks said the process should be inclusive.
Future international support for Afghanistan could be affected by whether women's rights were properly respected in any settlement, she said.
"There are certain levers that we have, that the Taliban are interested in," she told reporters in Kabul, where she was visiting as part of a Congressional delegation. "There is going to be an interest in economic support after the conflict ends."
"I think if the Taliban has any interest in getting international support … it would be in their interest to recognize the importance of including women and including human rights as part of any settlement that happens."
(Additional reporting by James Mackenzie; Editing by Frances Kerry, Toby Chopra and Neil Fullick)
Tagged Afghanistan, Afghans, Islamic Emirate, Peace Talks, Qatar, Taliban delegation | Leave a comment
Drought drives desperate Afghans to marry off children for money – U.N
By Jared Ferrie
PHNOM PENH (Thomson Reuters Foundation) – Afghanistan's worst drought in decades has driven tens of thousands of people from their homes and is pushing families to marry off their children in exchange for dowries in order to survive, the United Nations said on Tuesday.
About 223,000 people have been uprooted from their homes in the drought-hit western provinces of Herat, Badghis and Ghor this year, according to the U.N. children's agency (UNICEF).
The U.N. Food and Agriculture Organization (FAO) said Afghan families have been skipping meals, selling off livestock and moving to cities where it is easier to access aid and services.
Some displaced families are taking even more drastic measures, according to UNICEF, which documented 161 child betrothals or marriages in Herat and Badghis between July and October. Of those, 155 were girls and six were boys.
"The drought is the worst in decades," UNICEF spokeswoman Alison Parker told the Thomson Reuters Foundation.
"Children are becoming the collateral."
Families receive a bride price that can ease their financial woes, having lost their livelihoods and assets, said Parker.
Many drought-hit families have had to borrow money to pay for transport, food or healthcare, the United Nations said.
The charity World Vision reported that half of households it surveyed in Badghis in September said child marriage was a measure taken to put food on the table in times of drought.
About 11 million people – almost half of Afghanistan's rural population – will be facing "severe acute food insecurity" until February, according to the Integrated Food Security Phase Classification (IPC) system used by charities to measure hunger.
"Years of civil conflict and instability, as well as the severely degraded condition of much of the land, have compounded the impacts of the drought," said an IPC report from August.
In addition to those forced by drought to leave their homes, conflict between the government and an array of armed groups, including the Taliban, has uprooted at least 282,000 people so far this year, according to the United Nations.
The 17-year war has also devastated Afghanistan's education system, according to a report released on Tuesday by the Global Coalition to Protect Education from Attack, an alliance of aid agencies that includes UNICEF and Save the Children.
With a rising number of attacks on schools, teachers and students, the number of children who are not in education is increasing for the first time since 2002, the agencies said.
(Reporting by Jared Ferrie @jaredferrie; Editing Kieran Guilbert. Please credit the Thomson Reuters Foundation, the charitable arm of Thomson Reuters, that covers humanitarian news, women's and LGBT+ rights, human trafficking, property rights, and climate change. Visit http://news.trust.org)
Tagged Afghanistan, Afghans, Children, Money, U.N. Food and Agriculture Organization, UNICEF, United Nations | Leave a comment
In Greece, refugee women and children live in limbo
y Zohra Bensemra
CHIOS, Greece (Reuters) – Thousands of refugee woman and children are living in limbo in Greece, waiting for the day they will be reunited with their families in other European countries.
The United Nations Children's Fund (UNICEF) says nearly 75,000 refugees and migrants stranded in Greece, Bulgaria, Hungary and the Western Balkans are at risk of "psychological distress" caused by existing in a prolonged state of transit.
About 60,000 refugees and migrants, mostly Syrians, Afghans and Iraqis, have been stuck in Greece for over a year after border closures in the Balkans halted the onward journey many planned to take to central and western Europe.
More than a quarter are children and over half the new arrivals have been women and children, according to U.N. data. Men were the first family members to flee to Europe in previous years, leaving others to follow.
"Despair is haunting me at the moment," said Soha, a 23-year-old Syrian who lives in a tent on the island of Chios with two her two-year-old daughter and other Syrian women.
In the camp, next to the ruins of an ancient castle, overcrowded tents are pitched on the edge of the pebbled shore, and rats roam among the garbage. Women say they are too scared to leave their tents at night, fearing harassment.
Like other women, Soha declined to give her last name or be identified in photographs, fearing it could affect her application to join her husband in Germany.
Family reunification can take between 10 months and two years, UNICEF says, making life particularly hard those left behind.
The uncertainty caused "significant psychological distress and anxiety for children and their families, setting them back for years to come", UNICEF Regional Director Afshan Khan said.
A one-year-old girl smiles as she sits with her mother, Ibtissam, 22, at the Souda Refugee Camp in Chios island, Greece, June 10, 2017. "I was one month pregnant with my daughter and my son was one year old when my husband migrated to Germany." Ibtissam, who is from Raqqa said. "I feel devastated, at the moment I can't apply for family reunification because I have to wait until my husband gets his asylum document…. I feel depressed but I have to keep holding on for my children." REUTERS/Zohra Bensemra
"I spend most of the day alone," said Farhiya, a 23-year-old Somali who lives in a volunteer-run camp on Lesbos island.
"The other refugees don't speak English and I don't speak Arabic. It's hard to live alone," she said. Farhiya applied to join her husband in Austria seven months ago while still pregnant, but has not heard back, she said.
In Athens, 36-year-old Khalissa, who fled Syria with her three young children, spends her days in a drop-in center run by a UNICEF partner, a brief respite from her problems.
She colors in hearts representing her feelings about the past, present and future. The past is blue for sadness, the present brown for fear and the future, in which she hopes to reunite with her husband after two years, yellow for happiness.
Ultimately, she longs to go home.
"If Syria becomes as before the war, I will return home," she said. "We must return home."
(Writing by Karolina Tagaris, editing by Ed Osmond)
Tagged Afghans, Bulgaria, Greece, Hungary, Iraqis, refugee women and children, reunited, Syrians, UNICEF, Western Balkans | Leave a comment
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,491
|
A heart afraid of breaking.
A child afraid of waking.
The sadness won't find you.
You're burning all the bridges.
Where anger writes the pages.
The answers you'll find at the end.
So this is my goodbye.
Why you love to hate?
Turning the blade on you.
Oh, how does it feel?
Couldn't get out of your own mind.
I walked down the plank as the curtains fell down on you.
Where no one can see.
And down below, forever, oh how sad.
And down below forever, oh so sad.
Roam our circle, come home.
On my ocean, are you shadowing me?
So clear, and I'm with you again.
Only few won't fade away.
When sunrise outshines the grey.
Noone could hear the silence calling her.
Like ice flowers trapped in time.
No one could read the silence in his eyes.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 513
|
\section{Introduction}
Collective movements describe the tendency of a group of individuals to coordinate
their motion in a manner that generates net flow of the entire population. Examples
range from cells to animals, from migrating cell clusters during development and
cancer invasion \cite{haeger2015,mayor2016} to shifting bird flocks and fish shoals; the
latter extend to the kilometre-spanning shoals formed from hundreds of millions of
herrings \cite{makris2009}. A point of recent interest concerns the potential
division of a population into ``leaders'' and ``followers'' and, consequently, how
leaders influence swarm dynamics. Clearcut leadership could result from
experience, age or prior knowledge: the presence of older birds
improves path efficiency in migrating cranes \cite{mueller2013}; post-menopausal orcas adopt leading
positions during pod foraging \cite{brent2015}; only the ``househunting'' scouts
know the final destination of a new honeybee nest \cite{seeley}. Subtler leaders
arise within superficially identical populations, such as the presence of
faster or ``braver'' individuals in fish shoals and bird flocks \cite{reebs2000,nagy2010}.
Leader/follower statuses also occur in a host of cellular systems, ranging from
collective movements of aggregated amoebae to sprouting blood capillaries during
development, physiology and disease \cite{mayor2016}.
Most theoretical descriptions of collective migration have employed
agent/particle-based approaches, e.g. \cite{vicsek,Couzin2,Couzin,Cucker,Cucker2},
see also \cite{berdahl2018}. Despite the plethora of models, they typically share a
so-called set of ``first principles of swarming'' \cite{Carrillo}: specifically,
particle trajectories governed by a combination of repulsion (preventing collisions),
attraction/cohesion (preventing dispersal) and alignment of direction/velocity
according to neighbour positions, each operating over specified interaction ranges. Models
based on these features reproduce a wide variety of collective migration
phenomenologies, e.g. see \cite{berdahl2018}. In the context of follower-leader systems,
a key finding has been that swarms can be efficiently guided by a small number of
anonymous (i.e. not clearly distinct from the crowd) informed individuals \cite{Couzin}.
These individuals influence their neighbours, which in turn influence further followers
and knowledge is relayed through the swarm. Surprisingly, as the swarm population
increases a diminishing fraction of leaders is needed to achieve the
same level of guidance efficiency \cite{Couzin}. Follower-leader alignment strategies have also
been incorporated within other swarming studies. For example, in
\cite{jadbabaie2003coordination} a ``transient leadership'' model was considered to
imitate bird flocks while ``hierarchical leadership'' was studied in \cite{shen2007cucker}
within the framework of a Cucker-Smale model.
Beyond agent-based models, a plethora of continuous models have been proposed,
for example see \cite{mogilner1999,topaz2006,Eftimie,eftimie2012,painter2015}. In
common with their individual-based counterparts, movement is governed by a set or
subset of attracting/repelling/aligning interactions, typically generating
integro-differential equations of parabolic (e.g. \cite{mogilner1999,painter2015}) or hyperbolic (e.g.
\cite{eftimie2012}) form. While these models gain analytical tractability, their
connection to individual-level behaviour is inevitably blurred.
In this paper we derive macroscopic partial differential equations (PDEs) to describe
follower-leader interactions within
swarming populations, starting from an underlying particle description for movement. In
particular, we exploit the well-studied exemplar of honeybee swarming to frame our
modelling assumptions. In the next section we summarise key properties and prior modelling in this
system, subsequently refining to a relatively minimal set of assumptions to test
whether a simple process of velocity alignment can generate coherent and guided swarm
movement. We proceed to examine hyperbolic and parabolic scaling limits under
homogeneous and inhomogeneous interaction (alignment) kernels and discuss the new modelling challenges to describe realistic follower-leader dynamcis.
\section{Honeybee swarming}
\subsection{Biological background}
When a colony of honeybees outgrows its home, two-thirds of the population
(including the queen) departs. While the remaining population raises a
new queen and forms a daughter colony, the homeless bees bivouac nearby while
scouts scour the countryside for a new site. Reappearances at the colony are
accompanied by a ``waggle dance'', communicating distance, direction and
quality of possible new nests to other scouts. Proposals are analysed
before consensus is achieved, at which point the population takes to the air and
the scouts guide the ensuing swarm to the new nest, potentially several
kilometres away \cite{seeley}. Given that only a small (3\%-5\%) proportion of the
swarm are scouts, successful translocation relies on the ability of a small
percentage of informed individuals -- from a few hundred to a thousand for the
10,000+ swarm \cite{seeley1979natural,seeley1999group} -- to impart their knowledge.
Early observations of ``high-speed'' bees, flying in the upper portion of the swarm
and towards the nest site, led to the suggestion that this population coincided with the scouts and
that the behaviour allowed followers to infer nest
direction \cite{lindauer1955}. Subsequent studies have tested this
``streaker hypothesis'' against alternative theories of swarm guidance
\cite{seeley1979natural,beekman2006does,schultz,greggers2013}. For example, thorough
analysis of individual trajectories in videoed swarms supports the presence of
streakers, along with their localisation towards the upper swarm
\cite{beekman2006does,schultz}. Harmonic radars applied to scouts prior to the swarm's
movement revealed a subsequent streaker-like behaviour, with
fast bursts of nest-directed movements interspersed with periods of negligible
or slower movements in the opposite direction \cite{greggers2013}. A fast-flying line
of foragers across a swarm's path disrupts swarm guidance, consistent with the theory
that fast-flying behaviour impacts on a bee's flight trajectory \cite{latty}.
Inevitably, streaking sends scouts to the leading swarm edge, so
questions arise on their subsequent behaviour. Two proposals,
suggested in \cite{seeley}, are: (i) slowly travelling backwards,
perhaps along the bottom or sides to minimise their influence, or (ii)
``waiting out'' the swarm's passage, before rejoining
and streaking again. Harmonic radar experiments provide some support for either
mechanism \cite{greggers2013}, though technical limitations are unable to
give precise answers at present. Questions also arise on just how much guidance
information is transmitted from
streakers to uninformed bees. Flying fast at the top clearly raises visibility,
but despite growing understanding of their visual abilities (e.g. \cite{rigosi2017}),
the extent, range and manner of bee alignment to their conspecifics remains
unclear. Do followers synchronise movements to all their near neighbours or just
the fastest? Over what range can streaker bees be identified? In short,
what is the extra weighting generated through performing a streak?
\subsection{Prior modelling}
Among numerous models of collective movement, several have specifically
focussed on bee swarms. For example, Fetecau and Guo \cite{Fetecau} implement a
first-order model involving attraction, repulsion (described
by a Morse potential) and alignment, along with a random component
activated only under low interactions between an individual and the rest of the swarm.
Each bee's visual field corresponds to a planar cone
aligned in their direction of motion and formed from two regions: individuals in a
central cone are fully seen, while those in the peripheral region are partially seen (and
ascribed lower weight). Further, a leader subpopulation ignores other swarm members
and moves in oscillatory fashion, with fast/high-visibility movements towards the nest
until they reach the front edge, followed by slower/low-visibility movements
towards the rear. Attraction, repulsion and alignment also lie at the heart of a
first-order model in \cite{Janson}. Alignment follows a Euclidean metric-based assumption,
with (follower) bees synchronising their movements according to all visible neighbours,
regardless of their speed. As in \cite{Fetecau}, a set of leaders is given a
back-and-forth motion.
The model in \cite{Diwold} incorporates both site decision-making and subsequent
swarm guidance. Regarding the latter, a cohesion term attracts each bee
according to the barycenter of a set of sufficiently fast individuals, along with
their topological metric-based alignment. The resulting model was applied to compare swarming of
two honeybee species, {\em Apis mellifera} and {\em Apis florea}.
The same basic rules are implemented in \cite{bernardi2018discrete} to study
whether a single leader can transmit guidance to an entire swarm, testing both
topological (a fixed number of bees, regardless of distance)
and metric-based (all bees within a set region) alignments. Efficient
guidance occurred for either a sufficiently large region (encompassing the
entire bee cloud) under metric alignment or for a high enough number of groupmates
(i.e. $\geq$ 13) under topological alignment. Extensions in \cite{Bernardi2018particle}
considered how the subtlety of scout visibility and behaviour impacted on swarm efficiency.
Further, the scout percentage to total swarm size relationship was addressed, with larger
swarms demanding a smaller fraction of scouts to achieve comparable targeting, echoing the
results of earlier models \cite{Couzin, Fetecau}. Further additions included
greater environmental complexity, e.g. forcing swarms to circumnavigate structural elements.
Finally, in \cite{Bernardi2018} models were tested against experiments on swarm disruption
due to high velocity forager highways \cite{latty}. Based on fitting against data, the
leading plausible set of model assumptions involved follower bees synchronising
movement according to all sufficiently close insects (regardless of their status),
provided that passive leaders slowly returned from the front to the rear edge of the swarm.
\section{Swarm description}\label{sec: swarm description}
We assume the swarm is divided into two main subpopulations: followers ($f$) and leaders, with the latter further subdivided into those engaging as streakers ($s$) and those behaving passively ($p$). The total number of leaders is considered small (for bees, $3\%-5\%$ of the total population e.g. \cite{seeley}) compared to the followers. We assume that the leaders have knowledge of the nest location while the followers are completely uninformed. Streakers correspond to those leaders flying fast in the nest direction, while passive leaders are those behaving otherwise. Specifically, we assume that (i) a streaker flies
fast to the swarm's front edge, (ii) switches to a passive role and returns more slowly towards
the rear, and (iii) closes the cycle by switching back to a streaker role. Streaking is
marked by the distinctive velocity, so that streakers lead the swarm through flying with fixed
maximum speed $c_s$ in the direction of the nest and alerting followers of the nest direction in the process. Followers and passive leaders, meanwhile, are presumed to fly with fixed speeds $c_f$ and $c_p$ respectively.
An obviously important quantity in the model is the position of the swarm with respect to the nest. Given
a generic bee located at $\mathbf{x}_k$, its distance to the nest at $\mathbf{x}_{\textup{nest}}$ is
denoted by $I_{\textnormal{nest}}(\mathbf{x}_k(t))=|\mathbf{x}_k(t) - \mathbf{x}_{\textup{nest}}|$.
We fix a corresponding field $\mathbf{b} \simeq \nabla I_{\textnormal{nest}}$ within the domain outside the nest, such that $- \mathbf{b}$ defines the direction for the leaders to the nest. Specifically, we choose $\mathbf{b}=\nabla I_{\textnormal{nest}}$ to be divergence free at the nest and tangent to the boundary of the domain. As $|\mathbf{x}| \to \infty$, we impose $\mathbf{b}(\mathbf{x}) \to \nabla I_{\textnormal{nest}}$.
We define $\bar{\sigma}_i (\mathbf{x},t,\theta,\tau)$, for $i \in \left\{f,p,s\right\}$, as the {\em microscopic densities} of followers, passive leaders and streakers at position $\mathbf{x}$, time $t$, moving in direction $\theta$ for some time $\tau$. Integrating with respect to $\tau$, we
denote $\sigma_i (\mathbf{x},t,\theta) = \int_0^t \bar{\sigma}_i (\mathbf{x},t,\theta,\tau)d\tau$
as the density of each subpopulation at position $\mathbf{x}$, time $t$ and moving in
direction $\theta$. Subsequently integrating over $\theta$ generates the {\em macroscopic densities}
\[
\rho_i (\mathbf{x},t) = \frac{1}{|S|}\int_0^t \int_S \bar{\sigma}_i(\cdot,\theta,\tau)d\theta d\tau\,,
\]
where $S=\{\mathbf{x}\in \mathds{R}^n:\ |\mathbf{x}|=1\}$ is the unit sphere in $\mathds{R}^n$ with surface area $|S|$. Dropped subscripts are used to denote total population densities, i.e.
$\bar{\sigma} =\bar{\sigma}_f+\bar{\sigma}_p+\bar{\sigma}_s$, $\sigma =\sigma_f+\sigma_p+\sigma_s$
and $\rho =\rho_f+\rho_p+\rho_s$.
The dynamics of each subpopulation are described according to a set of rules, minimally chosen to sufficiently describe swarm behaviour while restricting excess complexity. For the followers we assume:
\begin{enumerate}
\item[1F.] Trajectories comprise of straight line motions interrupted by (effectively) instantaneous reorientations, where the new direction of motion is randomly chosen. This movement is called a velocity-jump process \cite{othmer1988}. Individuals stop (i.e. reorient) with a rate given by a fixed parameter $\beta$.
\item[2F.] At each reorientation, with probability $\zeta\in(0,1)$ it selects a new direction of motion $\eta$, taken to be symmetrically distributed with respect to the previous one according to
\[k(\mathbf{x},t,\mathbf{\theta};\mathbf{\eta})=\tilde{k}(\mathbf{x},t,|\eta-\theta|)\ .\]
Because $\tilde{k}$ is a probability distribution, it is normalized to $\int_S\tilde{k}(\mathbf{x},t,|\theta-e_1|)d\theta=1$ where $e_1 = (1, 0, . . . , 0)$. The turn angle operator $T$ is defined as
\begin{equation}
T\phi(\eta)=\int_Sk(\mathbf{x},t,\theta;\eta)\phi(\theta)d\theta\ .\label{eq: turning angle simple}
\end{equation}
Trivially, $\tilde{k}$ could be a uniform distribution; more generally, a bias according to the previous orientation would incorporate an element of persistence of orientation, so we consider $\tilde{k}(|\eta-\theta|)$.
\item[3F.] With probability $1-\zeta$ the follower instead aligns with the orientation of the local population, according to
\begin{equation}
\Lambda(\mathbf{x},\theta,t)=\frac{\mathcal{J}(\mathbf{x},t)}{|\mathcal{J}(\mathbf{x},t)|}\ ,\ \mathcal{J}(\mathbf{x},t)=\int_{\mathbb{R}^n}\int_SK(|\mathbf{y}-\mathbf{x}|)\sigma(\mathbf{y},t,\theta)\theta d\mathbf{y}d\theta\ .\label{eq: mean direction}
\end{equation}
If the flux $\mathcal{J}(\mathbf{x},t)=0$, we assume that $\Lambda(\mathbf{x},\theta,t)$ takes the value $\theta$ \cite{degond2008continuum}.
A slight generalisation of the above would be to choose something of the form:
\begin{equation}
\Lambda(\mathbf{x},\theta,t)=\frac{\mathcal{J}(\mathbf{x},t)}{|\mathcal{J}(\mathbf{x},t)|}\ ,\ \mathcal{J}(\mathbf{x},t)=\int_{\mathbb{R}^n}\int_SK(|\mathbf{y}-\mathbf{x}|)\left(\sigma_f+\sigma_p+\lambda \sigma_s\right) d\mathbf{y}d\theta\ .\label{eq: mean direction alt form}
\end{equation}
where $\lambda \ge 1$ would reflect that streakers have higher visibility and therefore provide greater weighting to the orientation of the followers. We note that the limitations of the above homogeneous choice will be discussed in Section \ref{sec:macrodiscussion}. In particular, an inhomogeneous variant, given by $\Lambda(\mathbf{x},t)=\nu\mathcal{J}(\mathbf{x},t)$, will subsequently be considered in Section \ref{sec: inhom}.
\item[4F.] $c_f\lesssim c_s$.
\end{enumerate}
\noindent
Rules for the passive leaders are given by:
\begin{enumerate}
\item[1P.] They also move according to a velocity jump process as described in 1F, with the same stopping rate $\beta$.
\item[2P.] No alignment according to other individuals.
\item[3P.] Passive leaders move in the direction of the rear of the swarm (effectively, in the direction of $\mathbf{b}$).
\item[4P.] The new direction of motion $\eta$ is then biased by these two quantities and is given by the turn angle operator $$ B(\mathbf{x},\eta)=B^0(\eta)+B^1(\mathbf{x},\eta)=T+ T_{\rho_f}\ .$$
To be specific, we fix $T_{\rho_f}=(\mathbf{b}\cdot\nabla\rho_f)\nabla\rho_f$. Note that this operator does not depend on the incoming direction.
\item[5P.] $c_p\lesssim c_s$.
\item[6P.] Transitions between streaker and passive leader take place at the front and rear edges of the swarm, as described below.
\end{enumerate}
Finally, streakers move according to:
\begin{enumerate}
\item[1S.] Given their knowledge of their nest, they move as ballistic particles in direction $-\mathbf{b}$ (towards the front of the swarm as represented in Figure \ref{fig: swarm1}), with a fixed maximum speed given by $c_s$. Thus, no randomness is assumed.
\item[2S.] Transitions between streaker and passive leader take place at the front and rear edges of the swarm, as described below.
\end{enumerate}
We note that as movements into orientations outside $-\mathbf{b}$ are not permitted for leaders within this model, evolution equations for the streakers will be written directly at the level of
the macroscopic density $\rho_s$. Microscopic densities can then be inferred as singular distributions, where the local macroscopic density is concentrated into orientation $-\mathbf{b}$: i.e.,
$\sigma_s(\mathbf{x},t,\theta) = \rho_s(\mathbf{x},t) \delta (\theta+\mathbf{b})$ where $\delta$ is the Dirac delta function.
\begin{figure}
\centering
\includegraphics[scale=0.4]{Fig1.png}
\caption{Illustration of switching between streakers and passive leaders.}
\label{fig: swarm1}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.25]{Fig2.png}\hspace{0.3cm}\includegraphics[scale=0.25]{Fig3.png}
\caption{Illustration of different swarm shapes.}
\label{fig: swarm23}
\end{figure}
Suppose a Gaussian-type curve for the swarm profile. Physically, the transition rate from streaker to passive leader, $R_{sp}$, should be localised to the front edge of a swarm: in a region where
$(\nabla\rho_f,\mathbf{b})>0$, as illustrated in Figure \ref{fig: swarm1}. As follower densities
decrease, i.e. $\rho_f\rightarrow 0$, $R_{sp}$ should be bounded away from zero: $R_{sp}\geq r_0>0$ as
$\rho_f\rightarrow 0$, where $r_0$ is the minimal conversion rate outside the swarm.
The rules for transition rate from passive leaders to streakers, $R_{ps}$, obey similar rules.
Transitions should be concentrated to the rear edge of the swarm, i.e. where $(\nabla\rho_f,\mathbf{b}) <0$. Again, as follower densities decrease we require $R_{ps}\geq r_0>0$.
For more complicated swarm profiles, e.g. dual peaked as in Figure \ref{fig: swarm23}, let $c_s/r_0$ define a length scale on which the density of streakers (and hence also that of passive leaders)
varies. If the spacing between the two peaks is sufficiently larger than $c_s/r_0$ then streakers
consider the blobs as separate swarms and there are separate fronts and backs for each swarm. For
peaks spaced below this distance, the majority of streakers fly towards the leading edge
before converting to passive leaders and it can be considered a single swarm with a single leader population.
We note that while a change from passive to streaker leader involves transition into a population with fixed orientation, a streaker to passive change demands transition into a population distributed over the unit sphere. We define $R_{sp}=\frac{1}{|S|}\int_SR_{sp}(\theta)d\theta$.
\subsection{Microscopic description}
Following the swarm description given in Section \ref{sec: swarm description}, the
population densities satisfy systems of integro-differential equations as follows. For the streakers and passive leaders,
\begin{equation}
\partial_t\rho_s-c_s\mathbf{b}\cdot\nabla\rho_s =-R_{sp}\rho_s+R_{ps}\rho_p\ ,\label{eq: first s}
\end{equation}
\begin{empheq}[left=\empheqlbrace]{align}
\left(\partial_{\tau}+\partial_t+c_p\theta\cdot\nabla \right)\bar{\sigma}_p(\cdot,\theta,\tau)-R_{sp}(\theta)\rho_s+R_{ps}\bar{\sigma}_p(\cdot,\theta,\tau)=-\beta\bar{\sigma}_p(\cdot,\theta,\tau),\label{eq:governing_eq1new}\\ \bar{\sigma}_p(\cdot,\eta,\tau=0)=\int_{0}^{t}\int_{S}B(\mathbf{x},\eta)\beta\bar{\sigma}_p(\cdot,\theta,\tau)d\theta d\tau\ .\label{eq:governing_eq2new}
\end{empheq}
For the followers, following the approach in \cite{estrada2019interacting}, we have
\begin{empheq}[left=\empheqlbrace]{align}
\left(\partial_{\tau}+\partial_t+c_f\theta\cdot\nabla \right)\bar{\sigma}_f(\cdot,\theta,\tau)=-\beta\bar{\sigma}_f(\cdot,\theta,\tau)\label{eq: alignment first} \ ,\\
\bar{\sigma}_f(\cdot,\eta,\tau=0) =\int_SQ(\eta,\theta)\int_0^t\beta\bar{\sigma}_f(\cdot,\theta,\tau)d\tau d\theta\ ,\label{eq: alignment first first}
\end{empheq}
where
\begin{equation}
Q(\eta,\theta)=\zeta \tilde{k}(|\eta-\theta|)+(1-\zeta)\Phi(\Lambda\cdot\eta)\ .\label{eq: alignment}
\end{equation}
$\Phi(\Lambda\cdot\eta)$ is the distribution of the new aligned direction and satisfies $\int_S\Phi(\Lambda\cdot\eta)d\eta=1$.
Integrating with respect to $\tau$ the system (\ref{eq:governing_eq1new})-(\ref{eq:governing_eq2new})
we obtain, for $\sigma_p(\cdot,\theta)=\int_0^t\bar{\sigma}_p(\cdot,\theta,\tau)d\tau$,
\begin{align}
\partial_t\sigma_p+c_p\theta\cdot\nabla\sigma_p-R_{sp}(\theta)\rho_s+R_{ps}\sigma_p &=-\beta\sigma_p+\bar{\sigma}_p(\cdot,\theta,0)\nonumber\\ &=-\beta\sigma_p+B(\mathbf{x},\theta)\beta\int_0^t\int_S\bar{\sigma}_p(\cdot,\eta,\tau)d\eta d\tau\ .
\end{align}
Defining the macroscopic density
$
\rho_p(\mathbf{x},t)=\frac{1}{|S|}\int_0^t\int_S\bar{\sigma}_p(\cdot,\eta,\tau)d\eta d\tau\,,
$
we obtain the mesoscopic equation
\begin{equation}
\partial_t\sigma_p+c_p\theta\cdot\nabla\sigma_p=R_{sp}(\theta)\rho_s-R_{ps}\sigma_p-\beta\sigma_p+|S|\beta B\rho_p\ .\label{eq: first p}
\end{equation}
Similarly, integrating system (\ref{eq: alignment first})-(\ref{eq: alignment first first}) with respect to $\tau$ and using the definition of $T$ given in (\ref{eq: turning angle simple}) we obtain
\begin{equation}
\partial_t\sigma_f+c_f\theta\cdot\nabla\sigma_f=-\beta\sigma_f+\zeta\beta T\sigma_f+(1-\zeta)\beta\Phi(\Lambda\cdot\theta)\rho_f\ .\label{eq: first f}
\end{equation}
\section{Macroscopic PDE description}\label{sec: macroscopic PDE}
As shown in the previous section, the densities $\rho_s(\mathbf{x},t)$, $\sigma_p(\mathbf{x},t,\theta)$ and $\sigma_f(\mathbf{x},t,\theta)$ satisfy the following system of kinetic equations,
\begin{align}
\partial_t\rho_s-c_s\mathbf{b}\cdot\nabla\rho_s & =-R_{sp}\rho_s+R_{ps}\rho_p\label{eq: streakers}\ ,\\ \partial_t\sigma_p+c_p\theta\cdot\nabla\sigma_p & =R_{sp}(\theta)\rho_s-R_{ps}\sigma_p-\beta\sigma_p+|S|\beta B\rho_p \label{eq: passive}\ ,\\ \partial_t\sigma_f+c_f\theta\cdot\nabla\sigma_f & = -\beta\sigma_f+\zeta\beta T\sigma_f+(1-\zeta)\beta\Phi(\Lambda\cdot\theta)\rho_f \label{eq: followers} \ .
\end{align}
Treating the population of streakers and passive leaders as the total population of leaders, $\rho_\ell=\rho_s+\rho_p$, we derive a conservation equation for $\rho_\ell$.
First, integrating (\ref{eq: passive}) with respect to $\theta$ we obtain
\begin{equation}
\partial_t\rho_p+nc_p\nabla\cdot w_p=R_{sp}\rho_s-R_{ps}\rho_p\ ,\label{eq: money}
\end{equation}
where
\begin{equation}
\rho_p=\frac{1}{|S|}\int_S\sigma_pd\theta\ \ \textnormal{and}\ \ w_p=\frac{1}{n|S|}\int_S\theta\sigma_pd\theta\ ,\label{eq: macroscopic}
\end{equation}
and where we assumed that the turn angle operator $B$ preserves the number of particles, i.e., $\int_SBd\theta=1$.
Adding the resulting equation (\ref{eq: money}) together with (\ref{eq: streakers}) we obtain
\begin{equation}
\partial_t\rho_\ell+nc_p\nabla\cdot w_p-c_s\nabla\cdot (\mathbf{b}\rho_s)=0\label{eq: conservation of leaders}\ ,
\end{equation}
where $\nabla\cdot(\mathbf{b}\rho_s)=\mathbf{b}\cdot\nabla\rho_s$, provided that $\nabla\cdot\mathbf{b}=0$ as follows from the definition of $\mathbf{b}$ in Section \ref{sec: swarm description}.
Similarly, we control that the population of the followers (\ref{eq: followers}) is conserved:
\begin{align}
\partial_t\int_S\sigma_fd\theta+c_f\int_S\theta\cdot\nabla\sigma_fd\theta=-\beta\int_S\sigma_fd\theta+\zeta\beta\int_ST\sigma_fd\theta+(1-\zeta)\beta\rho_f=0 , \label{eq: conservation followers}
\end{align}
since $\int_S\Phi(\Lambda\cdot\theta)d\theta=1$ and $\int_S Td\theta=1$ by definition.
\subsection{Diffusion limit}
Next we turn to determining macroscopic equations that describe the interactions between the three different populations over long time regimes. Introducing a parabolic scaling $(\mathbf{x},t)\mapsto (\mathbf{x}/\varepsilon,\ t/\varepsilon^2 )$, transition rates $R^\varepsilon_{sp}, R^\varepsilon_{ps}$ are multiplied by a factor of $\varepsilon$ and $T_{\rho_f}^\varepsilon=\varepsilon^2(\mathbf{b}\cdot\nabla\rho_f)\nabla\rho_f$. {The operator} $B^\varepsilon(\mathbf{x},\theta)$ {satisfies} $\int_SB^\varepsilon(\mathbf{x},\theta)\theta d\theta=\varepsilon C(\nabla\rho_f)\nabla\rho_f$ {to leading order in $\varepsilon$.}
Equations (\ref{eq: streakers}), (\ref{eq: passive}) and (\ref{eq: followers}) then become
\begin{align}
\varepsilon^2\partial_t\rho_s-\varepsilon c_s\mathbf{b}\cdot\nabla\rho_s & =-\varepsilon R_{sp}^\varepsilon\rho_s+\varepsilon R_{ps}^\varepsilon\rho_p\ \label{eq: streakers original},\\
\varepsilon^2\partial_t\sigma_p+\varepsilon c_p\theta\cdot\nabla\sigma_p & =\varepsilon R_{sp}^\varepsilon(\theta)\rho_s-\varepsilon R_{ps}^\varepsilon\sigma_p-\beta\sigma_p+|S|\beta B^\varepsilon\rho_p\label{eq: passive scaled}\ ,\\
\varepsilon^2\partial_t\sigma_f+\varepsilon c_f\theta\cdot\nabla\sigma_f & =-\beta\sigma_f+\zeta\beta T\sigma_f+(1-\zeta)\beta\Phi^\varepsilon(\Lambda\cdot\theta)\rho_f\ .\label{eq: diffusion followers}
\end{align}
Let us suppose that the total population of passive leaders, $\sigma_p$, can be written in terms of the following expansion:
\begin{equation}
\sigma_p=\sigma_p^0+\varepsilon\sigma_p^1+\mathcal{O}(\varepsilon^2)\label{eq: passive expansion}\ .
\end{equation}
Substituting the above expression into (\ref{eq: passive scaled}) and letting $\varepsilon\to 0$,
\begin{align}
\varepsilon^{0}:\ \ \sigma_p^0 & =|S|B^0 \rho_p\ ,\label{eq: epsilon-1}\\\varepsilon^{1}:\ \ \sigma_p^1 & =\frac{1}{\beta}(-c_p\theta\cdot\nabla\sigma_p^0+R_{sp}^0(\theta)\rho_s-R_{ps}^0\sigma_p^0)\ .\label{eq: epsilon0}
\end{align}
We obtain a Chapman-Enskog expansion from (\ref{eq: passive expansion}):
\begin{equation}
\sigma_p=|S|B^0\rho_p+\varepsilon\sigma_p^1+\mathcal{O}(\varepsilon^2)\ .
\end{equation}
Substituting the above expression into the mean direction of the passive leaders, $w_p$ in (\ref{eq: macroscopic}), we obtain
\begin{align*}
w_p & =\frac{1}{n}\int_S\theta B^0(\theta)d\theta \rho_p+\frac{\varepsilon}{n|S|}\int_S\theta\sigma_p^1d\theta\ , \\ & =\frac{1}{n}\mathcal{B}^0_\theta\rho_p+\frac{\varepsilon}{n|S|}\int_S\theta\sigma_p^1d\theta\ ,
\end{align*}
where $\mathcal{B}_\theta^0=\int_S\theta B^0d\theta$. The conservation equation for the leaders (\ref{eq: conservation of leaders}), is then
\begin{equation}
\varepsilon^2\partial_t\rho_\ell+\varepsilon c_p\nabla\cdot \Bigl(\mathcal{B}^0_\theta\rho_p+\frac{\varepsilon}{|S|}\int_S\theta\sigma_p^1d\theta\Bigr)-\varepsilon c_s\nabla\cdot (\mathbf{b}\rho_s)=0 \label{eq: final conservation}\ .
\end{equation}
The term $\nabla\cdot\int_S\theta\sigma^1_pd\theta$ can be explicitly computed using (\ref{eq: epsilon-1}) and (\ref{eq: epsilon0}) as follows:
\begin{align*}
\frac{1}{|S|}\nabla\cdot\int_S\theta\sigma_p^1d\theta & =\frac{-c_p}{|S|\beta}\int_S(\nabla\cdot\theta)(\theta\cdot\nabla)B^0\rho_pd\theta+\frac{1}{|S|}\nabla\cdot\int_S\theta\Bigl(R^\varepsilon_{sp}(\theta)\rho_s+|S|R^\varepsilon_{ps}B^0\rho_p\Bigr)d\theta\ ,\\& =-\Delta (D\rho_p) +\nabla\cdot(\mathcal{R}^\varepsilon_{sp}(\theta)\rho_s)+R^\varepsilon_{ps}\nabla\cdot(\mathcal{B}^0_\theta\rho_p)\ ,
\end{align*}
where $$D=\frac{c_p}{|S|\beta}\int_S\theta\theta^TB^0 d\theta\ \ \textnormal{and}\ \ \mathcal{R}^\varepsilon_{sp}(\theta)=\frac{1}{|S|}\int_S\theta R_{sp}^\varepsilon(\theta)d\theta\ .$$ $D$ is the diffusion coefficient and $\mathcal{R}^\varepsilon_{sp}$ describes some turning of the streaker population $\rho_s$.
If we substitute $\rho_\ell=\rho_s+\rho_p$ into the scaled conservation equation (\ref{eq: final conservation}) and $\partial_t\rho_s$ from (\ref{eq: streakers original}), we observe that as $\varepsilon\rightarrow 0$ we obtain from the leading order
$
c_p\nabla\cdot(\mathcal{B}_\theta^0\rho_p)=R_{sp}^0\rho_s-R_{ps}^0\rho_p\ .
$
Finally, for the passive leaders we have
\begin{equation}
\partial_t\rho_p-nc_p\Delta(D\rho_p)+nc_p\nabla\cdot(\mathcal{R}^0_{sp}\rho_s)+nc_pR_{ps}^0\nabla\cdot(\mathcal{B}^0_\theta\rho_p)=0\ .
\end{equation}
The conservation equation (\ref{eq: conservation followers}) for the follower population is
\begin{equation}
\partial_t\rho_f+nc_f\nabla\cdot w_f=0\ ,
\end{equation}
where $w_f=\frac{1}{n\varepsilon|S|}\int_S\theta\sigma_fd\theta$. To compute the mean direction of the followers, $w_f$, we multiply (\ref{eq: diffusion followers}) by $\theta$ and integrate over $S$,
\begin{align}
\varepsilon^2\int_S\theta\sigma_fd\theta+\varepsilon c_f\nabla\int_S\theta\theta^T\sigma_fd\theta= & -\beta\int_S\theta\sigma_fd\theta+\zeta\beta\int_S\theta T\sigma_fd\theta\nonumber\\ & +(1-\zeta)\beta \rho_f\int_S\theta\Phi^\varepsilon(\Lambda\cdot\theta)d\theta\ .\label{eq: followers diffusion 1}
\end{align}
Using the expansion $\sigma_f=|S|^{-1}(\rho_f+\varepsilon n\theta \cdot w_f)$ and letting $\varepsilon\to 0$ we obtain
\begin{equation}
w_f=\frac{1-\zeta}{n(1-\zeta\nu_1)}\rho_f\int_S\theta\Phi^0(\Lambda^W\cdot\theta)d\theta -\frac{c_f}{n\beta(1-\zeta\nu_1)}\nabla\rho_f\ ,\label{eq: mean direction followers}
\end{equation}
where $\Phi^0(\Lambda^W\cdot\theta)=\lim_{\varepsilon\to 0}\Phi^\varepsilon(\Lambda^W\cdot\theta)$, $W$ is the total mean direction of the whole population and
\[
\Lambda^W=\frac{\mathcal{J}^W}{|\mathcal{J}^W|}
\ \ \textnormal{for}\ \
\mathcal{J}^W=\frac{n\varepsilon}{|S|}\int_\mathbf{y}K^\varepsilon\Bigl(\frac{|\mathbf{y}-\mathbf{x}|}{\varepsilon} \Bigr)W(\mathbf{y},t)d\mathbf{y}\ .
\]
If we consider $\mathcal{J}$ as in (\ref{eq: mean direction alt form}) then the mean direction, $\Lambda$, will depend only on the mean direction of the streakers, $w_s$, as follows
\[
\Lambda^{s}=\frac{\mathcal{J}^s}{|\mathcal{J}^s|}\ \textnormal{where}\ \mathcal{J}^s=\frac{n\varepsilon\lambda}{|S|}\int_{\mathbf{y}}K^\varepsilon\Bigl( \frac{|\mathbf{y}-\mathbf{x}|}{\varepsilon}\Bigr)w_sd\mathbf{y}\ .
\]
Integrating over $S$, $\int_S\theta\Phi^0(\Lambda^W\cdot\theta)d\theta=z\Lambda^W$. $z$ can be computed using polar coordinates $\theta=(\cos(s),\sin(s))$ for $n=2$ or spherical coordinates $\theta=(\cos\phi\sin(s),\sin\phi\sin(s),\cos(s))$ for $n=3$, and is given by \cite{dimarco2016self}
\begin{equation}
z=\begin{cases}
\int_{0}^{2\pi}\Phi^0(\cos(s))\cos(s) ds, & \text{if $n=2$},\\
2\pi\int_{0}^{\pi}\Phi^0(\cos(s))\cos(s)\sin(s) ds, & \text{if $n=3$}.\label{eq: constant z}
\end{cases}
\end{equation}
We can then write
\begin{equation}
w_f=D\Lambda^W\rho_f-C_f\nabla\rho_f\ ,
\end{equation}
where the alignment and diffusion coefficients are respectively given by
\[
D=\frac{z(1-\zeta)}{(1-\zeta\nu_1)} \ \ \textnormal{and}\ \ C_f=\frac{c_f}{\beta(1-\zeta\nu_1)}\ .
\]
The macroscopic PDE description for the follower population is
\begin{equation}
\partial_t\rho_f+c_f\nabla\cdot (D\Lambda^W\rho_f-C_f\nabla\rho_f)=0\ .\label{eq: followers diffusion final}
\end{equation}
Finally, the system describing the macroscopic densities of followers and leaders, in the diffusion limit, reads as follows
\begin{align}
c_s\mathbf{b}\cdot\nabla\rho_s- R_{sp}^0\rho_s+ R_{ps}^0\rho_p & =0\ , \label{parstreakers}\\
\partial_t\rho_p-nc_p\Delta(D\rho_p)+nc_p\nabla\cdot(\mathcal{R}^0_{sp}\rho_s)+nc_pR_{ps}^0\nabla\cdot(\mathcal{B}^0_\theta\rho_p)& =0\ , \label{parpassive}\\ \partial_t\rho_f+c_f\nabla\cdot(D\Lambda^W\rho_f-C_f\nabla\rho_f) & =0\ . \label{parfollow}
\end{align}
\subsection{Hyperbolic limit}
In this section we investigate the dynamics of the swarm over shorter time scales. Consider the following scaling $(\mathbf{x},t)\mapsto(\mathbf{x}/\varepsilon,t/\varepsilon)$, then we write
\begin{align}
\varepsilon\partial_t\rho_s-\varepsilon c_s\mathbf{b}\cdot\nabla\rho_s & =-\varepsilon R_{sp}^\varepsilon\rho_s+\varepsilon R^\varepsilon_{ps}\rho_p\label{eq: streakers hyperbolic}\ ,\\
\varepsilon\partial_t\sigma_p+\varepsilon c_p\theta\cdot\nabla\sigma_p & =\varepsilon R^\varepsilon_{sp}(\theta)\rho_s-\varepsilon R_{ps}^\varepsilon\sigma_p-\beta\sigma_p+|S|\beta B^\varepsilon\rho_p\label{eq: passive hyperbolic}\ ,\\
\varepsilon\partial_t\sigma_f+\varepsilon c_f\theta\cdot\nabla\sigma_f & =-\beta\sigma_f+\zeta\beta T\sigma_f+(1-\zeta)\beta\Phi^\varepsilon(\Lambda\cdot\theta)\rho_f\label{eq: followers hyperb}\ .
\end{align}
The conservation equation (\ref{eq: conservation of leaders}) in this case is given by
\begin{equation}
\varepsilon\partial_t\rho_\ell+\varepsilon nc_p\nabla\cdot w_p-\varepsilon c_s\nabla\cdot(\mathbf{b}\rho_s)=0\ .
\end{equation}
Following the same procedure as before we consider the expansion (\ref{eq: passive expansion}) and from (\ref{eq: passive hyperbolic}) we obtain
\begin{align}
\sigma_p^0 & =|S|B^0 \rho_p \ .
\end{align}
Hence, substituting $\sigma_p=|S|B^0\rho_p+\varepsilon\sigma_p^1+\mathcal{O}(\varepsilon^2)$ into the mean direction $w_p$, as defined in (\ref{eq: macroscopic}), the conservation equation reads
\begin{equation}
\partial_t(\rho_p+\rho_s)+nc_p\nabla\cdot(\mathcal{B}_\theta^0\rho_p+\frac{\varepsilon}{|S|}\int_S\theta\sigma_p^1d\theta)-c_s\nabla\cdot(\mathbf{b}\rho_s)=0\ .\label{eq: conservation hyperbolic}
\end{equation}
The next step is to find the term $\sigma_p^1$. Substituting the expansion for $\sigma_p$ back into (\ref{eq: passive hyperbolic}) and letting $\varepsilon \to 0$ we get
\begin{align}
-|S|^{-1}\beta\sigma_p^1 & =\rho_p\partial_tB^0+B^0\partial_t\rho_p+c_p(\theta\cdot\nabla)(B^0\rho_p)-|S|^{-1}R_{sp}^0(\theta)\rho_s+R_{ps}^0 B^0\rho_p\ .
\end{align}
Substituting $\partial_t\rho_p$ from the conservation equation (\ref{eq: conservation hyperbolic}) we obtain
\begin{align}
-|S|^{-1}\beta\sigma_p^1 & =\rho_p\partial_tB^0+B^0\Bigl(-\partial_t\rho_s-nc_p\nabla\cdot(\mathcal{B}^0_\theta\rho_p) +c_s\nabla\cdot(\mathbf{b}\rho_s)\Bigr)\nonumber\\ & \ \ \ \ +c_p(\theta\cdot\nabla)(B^0\rho_p)-|S|^{-1}R_{sp}^0(\theta)\rho_s+R_{ps}^0 B^0\rho_p\nonumber\ .
\end{align}
We note that from (\ref{eq: streakers hyperbolic}) we can substitute $-\partial_t\rho_s+c_s\nabla\cdot(\mathbf{b}\rho_s)$ to obtain
\begin{align}
-|S|^{-1}\beta\sigma_p^1 &=-nc_pB^0\nabla\cdot(\mathcal{B}^0_\theta\rho_p)+\rho_p\partial_tB^0+c_p(\theta\cdot\nabla)(B^0\rho_p)+B^0|S|R_{sp}^0\rho_s\nonumber\\ &\ \ \ -|S|^{-1}R_{sp}^0(\theta)\rho_s+B^0(1-|S|)R_{ps}^0\rho_p\nonumber\\ & = c_pB^0(\theta-n\mathcal{B}^0_\theta)\cdot\nabla\rho_p+c_p(\theta\cdot\nabla B^0-nB^0\nabla\cdot \mathcal{B}^0_\theta)\rho_p+\rho_p\partial_t B^0 \nonumber\\ &\ \ \ \ +B^0|S|R_{sp}^0\rho_s-|S|^{-1}R_{sp}^0(\theta)\rho_s\ .
\end{align}
Letting
\[
q_1=c_pB^0(\theta-n\mathcal{B}^0_\theta)\ ,\ \ q_2=c_p(\theta\cdot\nabla B^0-nB^0\nabla\cdot \mathcal{B}^0_\theta)+\partial_t B^0\ ,
\]
we can write
\begin{equation}
\sigma_p^1=\frac{-|S|}{\beta}\Bigl(q_1\cdot\nabla\rho_p+q_2\rho_p+B^0|S|R_{sp}^0\rho_s-|S|^{-1}R_{sp}^0(\theta)\rho_s\Bigr)\ .
\end{equation}
Finally, by substituting $\sigma_p^1$ back into the conservation equation (\ref{eq: conservation hyperbolic}), we obtain
\begin{align}
\partial_t\rho_\ell+nc_p\nabla\cdot (\mathcal{B}_\theta^0\rho_p)-c_s\nabla\cdot(\mathbf{b}\rho_s)& =\varepsilon nc_p\Bigl[Q_1\nabla\rho_p-Q_2\rho_p+\Bigl(|S|\mathcal{B}^0_\theta R_{sp}^0-\mathcal{R}_{sp}^0(\theta)\Bigr)\rho_s \Bigr], \label{eq: hyperbolicccc}
\end{align}
where
\[
Q_1=\frac{1}{\beta}\int_S\theta\cdot q_1d\theta\ ,\ \ Q_2=\frac{-1}{\beta}\int_S\theta q_2d\theta\ .
\]
Splitting again the leaders' population into $\rho_p$ and $\rho_s$ and substituting $\partial_t\rho_s$ from (\ref{eq: streakers hyperbolic}) into (\ref{eq: hyperbolicccc}) we obtain
\begin{align}
\partial_t\rho_p+nc_p\nabla\cdot (\mathcal{B}_\theta^0\rho_p)-R_{sp}^0\rho_s +R^0_{ps}\rho_p &=\varepsilon nc_p\Bigl[Q_1\nabla\rho_p-Q_2\rho_p\nonumber\\ & \ \ \ \ +\Bigl(|S|\mathcal{B}^0_\theta R_{sp}^0-\mathcal{R}_{sp}^0(\theta)\Bigr)\rho_s \Bigr]\ .
\end{align}
For the case of the follower population and using the hyperbolic scaling we can rewrite (\ref{eq: followers hyperb}) as
\begin{equation}
\partial_t\sigma_f+c_f\theta\cdot\nabla\sigma_f=\frac{\beta}{\varepsilon}\Bigl(-\sigma_f+\zeta T\sigma_f+(1-\zeta)\Phi^\varepsilon(\Lambda\cdot\theta)\rho_f \Bigr)\ ,\label{eq: right hand side}
\end{equation}
and, as $\varepsilon\rightarrow 0$, we obtain
\begin{equation}
\sigma_f^0=(\zeta+(1-\zeta)\Phi^0(\Lambda\cdot\theta))\rho_f^0\label{eq: expansion followers}\ .
\end{equation}
Here we considered only the first eigenvalue of $T$, i.e. $T\sigma_f=\rho_f^0$ as in Appendix \ref{sec: turn_angle_properties}.
Substituting (\ref{eq: expansion followers}) into (\ref{eq: followers hyperb}) and integrating with respect to $S$ gives
\begin{align}
\partial_t\rho_f^0\int_S(\zeta+(1-\zeta)\Phi^0(\Lambda\cdot\theta))d\theta+c_f\int_S\theta\cdot\nabla(\zeta+(1-\zeta)\Phi^0(\Lambda\cdot\theta))\rho_f^0d\theta=0 \ ,
\end{align}
where the right hand side is zero using the approximation $T\sigma_f=T_0\sigma_f$. In the left hand side we note that $\int_S\Phi^0(\Lambda\cdot\theta)d\theta=1$ and, as before, $\int_S\theta\Phi^0(\Lambda\cdot\theta)d\theta=z\Lambda$ where $z$ is given by (\ref{eq: constant z}). Hence the conservation equation for the followers reads
\begin{equation}
\partial_t\rho_f^0+c_fz(1-\zeta)\nabla\cdot(\rho_f^0\Lambda)=0\ .
\end{equation}
Next we need to find the mean direction $\rho_f^0\Lambda$. Following the same analysis as in \cite{dimarco2016self,estrada2019interacting} we substitute the expansion for $\sigma_f^0$ given in (\ref{eq: expansion followers}) into (\ref{eq: followers hyperb}) and multiply by $\theta\cdot v$, where $v\in\mathds{R}^n$ is orthogonal to $\Lambda$. Integrating over $S$ then gives
\begin{equation}
\Bigl(\partial_t\int_S\theta\Psi(\theta)\rho_f^0d\theta+c_f\int_S\theta\cdot\nabla(\Psi(\theta)\rho_f^0)\theta d\theta\Bigr)\cdot v=\mathcal{O}(\varepsilon)\ .
\end{equation}
Here $\Psi(\theta)=\zeta+(1-\zeta)\Phi^0(\Lambda\cdot\theta)$, and letting $\varepsilon\rightarrow 0$ in the right hand side we obtain
\[
\left(z(1-\zeta)\partial_t(\rho_f^0\Lambda)+c_f\int_S\theta\cdot\nabla(\rho_f^0\Psi(\theta))\theta d\theta \right)\cdot v=0\ .
\]
Using the fact that $v \perp \Lambda$, we can reformulate the above expression in terms of the orthogonal projection $P_\perp=\mathds{1}-\Lambda\otimes\Lambda$ onto $\Lambda^\perp$,
\begin{equation}
P_\perp\Bigl( z(1-\zeta)\partial_t(\rho_f^0\Lambda)+c_f\nabla\cdot\rho_f^0\int_S(\theta\otimes\theta)\Psi(\theta)d\theta\Bigr)=0\ .
\end{equation}
For the first term of the above expression we can write
\begin{equation}
z(1-\zeta)P_\perp(\rho_f^0\partial_t\Lambda+\Lambda\partial_t\rho_f^0)=z(1-\zeta)\rho_f^0\partial_t\Lambda\ ,
\end{equation}
since $\langle\partial_t\Lambda,\Lambda \rangle=\frac{1}{2}\partial_t|\Lambda|^2=0$, i.e., $\Lambda\perp\partial_t\Lambda$. For the second term we must compute the integral $\int_S(\theta\otimes\theta)\Psi(\theta)d\theta$, where we use $\theta=\cos(s)\Lambda+\sin(s)\Lambda^\perp$ in polar coordinates for $n=2$ and spherical coordinates for $n=3$ as in \cite{dimarco2016self}. Finally, we obtain
\begin{equation}
\rho_f^0(z(1-\zeta)\partial_t\Lambda+C_1\Lambda\cdot\nabla\Lambda)+C_2P_\perp\nabla\rho_f^0=0\ ,
\end{equation}
where we have used $\Lambda^\perp\otimes\Lambda^\perp=\mathds{1}-\Lambda\otimes\Lambda$. Here $C_1=c_f(1-\zeta)a_3$ and $C_2=c_f(1-\zeta)\mathds{1}a_1+c_f\mathds{1}\pi\zeta$ for, $a_3=a_0-a_1$ and
\begin{equation}
a_0=\begin{cases}
\int_{0}^{2\pi}\Phi^0(\cos(s))\cos(s)^2 ds, & \text{if $n=2$},\\
2\pi\int_{0}^{\pi}\Phi^0(\cos(s))\cos(s)^2\sin(s) ds, & \text{if $n=3$},\label{eq: constant a0}
\end{cases}
\end{equation}
\begin{equation}
a_1=\begin{cases}
\int_{0}^{2\pi}\Phi^0(\cos(s))\sin(s)^2 ds, & \text{if $n=2$},\\
\pi\int_{0}^{\pi}\Phi^0(\cos(s))\sin(s)^3\sin(s) ds, & \text{if $n=3$}.\label{eq: constant a1}
\end{cases}
\end{equation}
The final system of equations is given as
\begin{align}
\partial_t\rho_s- c_s\mathbf{b}\cdot\nabla\rho_s + R_{sp}^0\rho_s-R^0_{ps}\rho_p & =0\ , \label{eq: hyperbolic streakers}\\\
\partial_t\rho_p+nc_p\nabla\cdot (\mathcal{B}_\theta^0\rho_p) -R_{sp}^0\rho_s +R^0_{ps}\rho_p& =\varepsilon nc_p\Bigl[Q_1\nabla\rho_p-Q_2\rho_p\nonumber\\ & \ \ \ \ +\Bigl(|S|\mathcal{B}^0_\theta R_{sp}^0-\mathcal{R}_{sp}^0(\theta)\Bigr)\rho_s \Bigr]\ , \label{eq: hyperbolic passive}\\
\partial_t\rho_f+c_fz(1-\zeta)\nabla\cdot(\rho_f\Lambda) &=0\ ,\label{eq: hyperbolic followers 1}\\ \rho_f(z(1-\zeta)\partial_t\Lambda +C_1\Lambda\cdot\nabla\Lambda)+C_2P_\perp\nabla\rho_f &=0\ .\label{eq: hyperbolic followers 2}
\end{align}
\subsection{Discussion of macroscopic equations} \label{sec:macrodiscussion}
The macroscopic equations obtained above capture behaviours depicted in Figures \ref{fig: swarm1} and
\ref{fig: swarm23} according to their respective scales. From Equation \eqref{parstreakers}, we observe that the distribution of streakers on long (parabolic) time scales is constant along the vector field $\mathbf{b}$ whenever $R^{0}_{sp}$ and $R^{0}_{ps}$ vanish, i.e.~within the interior of the swarm. At the front of the swarm, the lower bound $R^{0}_{sp}\geq r_0$ forces $\rho_s$ to decay exponentially fast to zero over a characteristic length scale $\frac{c_s}{r_0}$, as streakers convert to passive leaders.
Equation \eqref{parpassive} reflects an undirected diffusion of the passive leaders through the swarm, with a transport term $\nabla\cdot(\mathcal{B}^0_\theta\rho_p)$ corresponding to the velocity opposite
those of the streakers. As passive leaders reach areas of large $R^{0}_{ps}$, corresponding to the rear boundary of the swarm, they convert to streakers and head again to the swarm front. The characteristic length scale is $\frac{c_p}{r_0}$. Note that the movement of streakers and passive leaders only depends on the followers through the transition rates $R_{sp}^0$ and $R_{ps}^0$: this is logical enough, given that leaders have knowledge of the nest site and should not be swayed by the uninformed. A
more detailed model may also incorporate some influence of the followers on the leader direction (e.g. due to avoiding collisions), however we have neglected that here for simplicity.
From \eqref{parfollow}, the density of followers also diffuses over parabolic time scales. The directional movement of the swarm depends on the leaders only through $\Lambda^W$. This highlights the limitations of modelling via the homogeneous alignment kernel \eqref{eq: mean direction}: since
$\Lambda^W$ is a unit vector, only a single leader is required to direct the swarm to its nest,
a result which clearly stretched credulity for large swarms. This pathology motivates the study of inhomogeneous alignment kernels in Section \ref{sec: inhom}, where the size of the orientation vector
$\Lambda^W$ is taken to increase both with the number of leaders moving along $\mathbf{b}$ and with the strength of the interactions.
A basic question concerns whether two follower populations, as in Figure \ref{fig: swarm23}, behave as one joint swarm or two separate swarms. From previous discussions in Section \ref{sec: swarm description}, for the model here the distinction lies with the separation of the two populations: for separations much larger than $\frac{c_s}{r_0}$, leaders remain confined to their separate follower swarms; for
separations smaller than $\frac{c_s}{r_0}$, the leader populations bridge the gap that separates the peaks to join the swarms together; streakers to passive leader conversion occurs predominantly within a single zone at the swarm front.
Equations \eqref{eq: hyperbolic streakers} to \eqref{eq: hyperbolic followers 2} for shorter,
hyperbolic time scales support the general description above, with a few notable modifications. First,
over this shorter time scale the streakers follow a transport equation \eqref{eq: hyperbolic streakers}
that includes a time derivative absent from the parabolic limit, allowing a time-dependent
description of streaker movement through the swarm. Second, the passive leader distribution in
\eqref{eq: hyperbolic passive} does not undergo undirected diffusion over hyperbolic time scales, rather
it is simply transported in the opposite direction to the streakers with its respective velocity. The
terms of order $\varepsilon$ account for secondary effects in the collective movement. Finally, the
equations for the followers, \eqref{eq: hyperbolic followers 1} and \eqref{eq: hyperbolic followers 2}, correspond to classical equations for swarming particles \cite{estrada2019interacting}. Again, the directional movement of the swarm depends on the leaders only through $\Lambda^W$, and the swarm
impacts on the movement of the leaders only through the switching rates $R_{sp}^0$ and $R_{ps}^0$.
\section{Inhomogeneous alignment kernel}\label{sec: inhom}
In this section we consider the alignment kernel given by
\begin{equation}
\Lambda^*=\nu\mathcal{J}(\mathbf{x},t)\label{eq: new alignemnt}
\end{equation}
where we have removed the normalization and $\mathcal{J}(\mathbf{x},t)$ is given as in (\ref{eq: mean direction}). Here we limit our discussion to the case $n=2$. Parameter $\nu$ is the relaxation frequency, previously assumed constant but in this approach taken to depend on the norm of $\mathcal{J}(\mathbf{x},t)$. Here we study the diffusion and hyperbolic limit of the system for follower-leader interactions, as previously done in Section \ref{sec: macroscopic PDE}. In particular, we focus on the equation for followers (\ref{eq: diffusion followers}) and (\ref{eq: followers hyperb}) under an alignment
given by (\ref{eq: new alignemnt}).
The distribution of aligned directions $\Phi(\Lambda^*\cdot\theta)$ in (\ref{eq: alignment}) is replaced by
\begin{equation}
\bar{\Phi}(\Lambda^*\cdot\theta)=\frac{\Phi(\Lambda^*\cdot\theta)}{\int_0^{2\pi}\Phi(|\Lambda^*|\cos\theta)d\theta}
\end{equation}
such that $\int_S\bar{\Phi}(\Lambda^*\cdot\theta)d\theta=1$.
In the diffusion limit, we replace $\Phi^0(\Lambda\cdot\theta)$ by $\bar{\Phi}^0(\Lambda^*\cdot\theta)$ in equations (\ref{eq: followers diffusion 1}) and (\ref{eq: mean direction followers}). Noting that
\begin{equation}
\int_S\theta\bar{\Phi}^0(\Lambda^{*W}\cdot\theta)d\theta=\bar{z}\Lambda^{*W}\ ,\ \ \textnormal{where}\ \ \ \bar{z}=|\Lambda^*|\int_0^{2\pi}\bar{\Phi}^0(|\Lambda^*|\cos(s))\cos(s) ds\ ,\label{eq: new z}
\end{equation}
for $\theta=\cos(s)\Lambda^*+\sin(s)\Lambda^{*\perp}$ we again obtain (\ref{eq: followers diffusion final}), where in this case
\[
D=\frac{\bar{z}(1-\zeta)}{n(1-\zeta\nu_1)}\ .
\]
For the hyperbolic limit, let us first define the right hand side of (\ref{eq: right hand side}) as
$$
L(\sigma_f)=-\beta\sigma_f+\beta\zeta T\sigma_f+(1-\zeta)\beta\bar{\Phi}^\varepsilon(\Lambda^*\cdot\theta)\rho_f\ .
$$
We know that as $\varepsilon\rightarrow 0$ the solution $\sigma_f^0=\bar{\Psi}(\theta)\rho_f^0$, where $\bar{\Psi}(\theta)=\zeta+(1-\zeta)\bar{\Phi}^0(\Lambda^*\cdot\theta)$. The new operator $\bar{\Psi}(\theta)$ needs to be a Generalized Collisional Invariant of the operator $L(\sigma_f)$, as in the following sense \cite{degond2013macroscopic,degond2008continuum,dimarco2016self}.
\begin{definition}
A function $\bar{\Psi}(\theta)$ is a Generalized Collisional Invariant of $Q$ if it satisfies
\[
\int_S L(\sigma_f)\bar{\Psi}(\theta)d\theta=0\ ,
\]
for any $\sigma_f$. Equivalently, $\sigma_f$ satisfies
$
P_\perp\Bigl(\int_S\sigma_f(\mathbf{x},t,\theta)\theta d\theta \Bigr)=0
$, where $P_\perp=\textnormal{Id}-\Lambda^*\otimes\Lambda^*$ is an orthogonal projection to $\Lambda^*$.\label{def: GCI}
\end{definition}
Note that for the case $\zeta=0$, i.e. when only alignment is considered, from Definition \ref{def: GCI} we conclude that $\int_SL(\sigma_f)\bar{\Phi}^0(\Lambda^*\cdot\theta)d\theta=0$, where $\bar{\Phi}^0$ can be taken as the von Mises-Fisher distribution. Then, the analysis in \cite{degond2013macroscopic} follows.
The system (\ref{eq: hyperbolic followers 1})-(\ref{eq: hyperbolic followers 2}) can be written now as
\begin{align}
\partial_t\rho_f+c_f\bar{z}(1-\zeta)\nabla\cdot(\rho_f\Lambda^*) & =0\ ,\\ \rho_f(\bar{z}(1-\zeta)\partial_t\Lambda^*+\bar{C}_1\Lambda^*\cdot\nabla\Lambda^*)+\bar{C}_2P_\perp\nabla\rho_f & =0\ ,
\end{align}
where $\bar{z}$ is given by (\ref{eq: new z}) and, similar to previous derivation, $\bar{C}_1=c_f(1-\zeta)\bar{a}_3$, $\bar{C}_2=c_f(1-\zeta)\mathds{1}\bar{a}_1+c_f\mathds{1}\pi\zeta$ for, $\bar{a}_3=\bar{a}_0-\bar{a}_1$ with
\[
\bar{a}_0=|\Lambda^*|^2\int_0^{2\pi}\bar{\Phi}^0(|\Lambda^*|\cos(s))\cos^2(s)ds\ ,\ \ \bar{a}_1=|\Lambda^*|^2\int_0^{2\pi}\bar{\Phi}^0(|\Lambda^*|\cos(s))\sin^2(s)ds\ .
\]
\section{Discussion}
The capacity of individuals to coordinate their movement to generate collective movement
is a phenomenon that has attracted significant interest, in both cellular and animal systems.
Much of the progress in this area has been facilitated through modelling studies, particularly via
the employment of agent-based (or particle) descriptions that consider the movement
response of each single member according to its neighbours. Yet the largest swarms can
extend over kilometres and contain millions (e.g. herrings, \cite{makris2009}) or even hundreds of
billions (e.g. desert locusts, \cite{rainey1967,skaf1990}) of members. At such numbers and scales,
continuous modelling approaches become necessary for their efficiency and increased tractability. Consequently, there is a clear interest in clarifying the relevant form of continuous models
for swarming systems.
Much recent interest has focussed on follower-leader systems, where the population is
decomposed into a leader population which somehow guides a population of followers. Honeybee
swarms offer an ideal case study for exploring the continuous modelling of follower-leader
systems: given their large size ($O(10^4)$ individuals),
a continuous approach is relevant and a significant body of work has elucidated key
insights into how the distinct leader population guides the uninformed
followers \cite{seeley}. Inspired by this system we have formulated a minimalistic microscopic
description for a population of informed scouts and the followers,
deriving the ensuing macroscopic models under distinct scaling limits. To test
the extent to which velocity alignment by itself can propel a guided and coherent
swarm, follower orientation is limited to the interaction
choice (\ref{eq: mean direction alt form}): alignment according to
the velocity direction. Under both hyperbolic and parabolic scaling, macroscopic models
feature drift-terms with advection in the streaking direction. Thus,
alignment to velocity alone yields translocation of the swarm towards the nest.
Under the hyperbolic scaling a pure-drift equation is generated, implying the
potential for travelling-pulses with movement of a cohesive colony towards the
nest site. The parabolic limit, on the other hand, generates a drift-diffusion equation.
While drift is in the direction of the nest site, the additional dispersion leads to
swarm spreading with time. Early stretching of the colony is to be expected, as the
initially tight cluster morphs into a migrating swarm. Once a swarm is established, however,
it appears to retain a relatively stable speed and shape: any continued dispersal would be
far from optimal, leading to colony loss (bees losing contact and becoming ``lost'') and placing the
all-important queen at risk of exposure. Of course, the question of which is the appropriate
limit boils down to the relevant spatio-temporal scales. Straight-line motions of follower bees
within a swarm is unlikely to be longer than a few seconds or so, longer runs would move
them beyond swarm boundaries. At a macroscopic scale, swarms travel from a few hundred metres
(over a few minutes) to multiple kilometres (up to an hour) to new nest locations \cite{beekman2006does}.
While the hyperbolic model may be relevant for shorter swarm migrations, over larger
migrations the parabolic scaling would be necessary.
The above must be viewed in light of our intentionally simple modelling approach, where
we have specifically tested the practicalities of an ``alignment-only'' mechanism.
Agent-based approaches for modelling bee swarms (and, indeed, swarming phenomena
in general) are typically augmented by additional attractive/cohesion behaviours, where
individuals are also pulled in the direction of those in their neighbourhood; in non-local
continuous models, a similar effect is gained through a nonlocal attraction term that
biases movement direction. As stressed above, we have reasonably excluded such considerations
from the present model for simplicity, however including nonlocal attractions could clearly
counteract swarm dispersal. For example, following \cite{bellomo2008modeling}, attraction (or
repulsion) between individuals can be considered by
\[
\mathcal{A}=\Bigl(\frac{d_c-\textnormal{dist} (\mathbf{x},\mathbf{y})}{\textnormal{dist} (\mathbf{x},\mathbf{y})} \Bigr)e^{-(d_c-\textnormal{dist}(\mathbf{x},\mathbf{y}))^2}\vec{e}(\mathbf{x},\mathbf{y})\ .
\]
Here $\textnormal{dist}(\mathbf{x},\mathbf{y})$ denotes the distance between two individuals $\mathbf{x}$ and $\mathbf{y}$, along the direction of the unit vector $\vec{e}(\mathbf{x},\mathbf{y})$. If $\textnormal{dist}(\mathbf{x},\mathbf{y})<d_c$, where $d_c$ is a critical distance, then the action of $\mathcal{A}$ is repulsive and if $\textnormal{dist}(\mathbf{x},\mathbf{y})>d_c$ we have attraction.
Recently, there has been considerable interest in the composition and structuring of
swarming populations. For example, in the case of bird flocks and fish shoals, the existence
of faster and/or braver individuals can lead to hierarchical swarm arrangements
\cite{reebs2000,nagy2010} and the question is raised as to how much swarm movement is
dominated by the choice
of a few. For simplicity, the movements of our leaders has been set here
somewhat naively: during streaks, they operate as ballistic particles adopting fast
movements towards the nest. While this may be a reasonable approximation
for bee swarms, where leaders have specific {\em a priori} knowledge, more general ``leaders''
may be more subtle and get influenced by neighbour movements. Consequently, a
logical extension would be to also adopt a velocity jump model for the leader population, where one
of their movement contributions stems from an interaction function similar to
(\ref{eq: mean direction alt form}), distinctly weighted. In other instances, leader/follower
statuses may be transient, for example resulting from spatial position within the swarm, and it
may be necessary to include switching terms between follower and leader populations.
The study of population dynamics, where a few discrete agents act as leaders,
provides an excellent scenario to derive specific control laws and interactions that drive
self-organisation, leading the swarm optimally to a desired outcome. This
will allow an analytic understanding of follower-leader interactions, with applications
not only to understanding the collective dynamics of biological populations but also outside biology,
for example to hierarchical swarms of robots.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 457
|
\section{Introduction}
The compass model is one of the simplest models possessing orbital
degenerate states. Originally developed\cite{kugel82} as a model for
Mott insulators it has recently seen renewed
interest\cite{nussinov_orbitalorder,khomskii-2003,mostovoy:167201} in
connection with orbital-order in materials like transition metal (TM)
compounds. Despite its closeness to ordinary models of quantum
magnetism, like the Heisenberg model, there is no ordered phase
characterized by magnetization properties. This means that the ordered
phase appearing in the model is especially interesting in that it
cannot be classified according to the Mermin-Wagner criterion.
A competition
of interactions in different directions rather results in a special
long-range ordered state\cite{mishra:207201} possessing a sense of
orientation,\footnote{Hence the name compass model.} and the transition
is at the same time accompanied by dimensional
reduction.\cite{Batista_dimred2005} The current interest in this model
is furthermore triggered by the recent discovery that it
describes arrays of superconducting Josephson junctions and because of a
possible realization of a system which protects qubits against
unwanted decay in quantum
computation.\cite{doucot:024505,milman:020503}
The compass model is a spin model on simple-cubic lattices in $d$ dimensions of size $N=L^d$ defined by the Hamiltonian
\begin{equation}
\label{eq:hamiltonian1}
\mathcal{H}=\sum_{i} \sum^d_{k} J_k S^k_i S^k_{i+e_k}\,,
\end{equation}
where $S^k_i$ represents the $k$-th component of a spin $\mathbf{S}$
at site $i$ and $i+e_k$ is the nearest neighbor of $i$ in the $k$
direction. In the classical case we have $\mathbf{S}\in O(d)$, or in a
more explicit vector representation with $\varphi$ and $\theta$ being
angles on the sphere, we use the expression
$\mathbf{S}^{\mathsf{T}}=\left(\cos(\varphi),\sin(\varphi)\right)$ and
$\mathbf{S}^{\mathsf{T}}=\left(\cos(\varphi)\sin(\theta),\sin(\varphi)\sin(\theta),\cos(\theta)\right)$
in two and three dimensions, respectively. In the two-dimensional
(2D) quantum case $\mathbf{S}$ represents a spin-$1/2$ operator
$\mathbf{S}=(1/2)\,(\sigma_x,\sigma_z)$ and the Hamiltonian assumes the form
\begin{equation}
\label{eq:hamiltonian2}
\mathcal{H}=(1/4)\,\sum_{i}\left( J_x \sigma^x_i \sigma^x_{i+e_x} + J_z \sigma^z_i \sigma^z_{i+e_z}\right)\,,
\end{equation}
where we have chosen the $z$ instead of the $y$ direction as a matter
of convenience (usually we take $S^z$ as the quantization component
in quantum Monte Carlo). In this work the coupling constants
are taken to be equal, $J_k=J$, and positive although the sign plays no role
since it can be transformed away on bipartite lattices ($L$ must be even).
Recent contributions in the literature have explicitly investigated
the properties of the 2D compass model for both the classical and
quantum Hamiltonian. Analytical and Monte Carlo
work on the classical case proved the existence of
a directional-ordering transition at finite-temperatures and it was
argued that this transition belongs to the 2D Ising universality
class.\cite{mishra:207201} Using exact diagonalization techniques and Green-function Monte
Carlo the energy spectrum of low lying states was analyzed for the
quantum model in detail.\cite{doucot:024505,dorier:024448} These
studies provided the key result that the ground state is exponentially
degenerate possessing a degeneracy of $2\times2^L$. This turns the
relatively simple Hamiltonian into a hard problem comparable to
frustrated magnets.
Later work\cite{chen:144401} determined the nature
of the quantum phase transition to be of first order when driving the
system by changing the coupling ratio $J_x/J_z$. A variant of the
model possessing a similar quantum phase transition was finally
analyzed in one dimension.\cite{brzezicki:134415} In a recent Letter
\cite{tanaka:256402} the finite temperature properties of the quantum
compass model were analyzed for the first time by means of a world
line quantum Monte Carlo scheme based on the Suzuki-Trotter
discretization. The authors conclude with the intriguing effect, that
the presence of random site dilution has much weaker effects on
criticality for quantum degrees of freedom than for classical ones.
The numerical analysis supporting this conclusion is, however, based
on rather small lattice sizes and the quality of the quantitative
results is modest and in view of our results reported below would
need further investigations.
Due to the relevance of the model and the potential implications for
future applications it would be desirable to have a more precise
understanding of the critical behavior at the directional-ordering
transition in the quantum compass model. The purpose of this work is
to tackle this problem with a comprehensive Monte Carlo study for both
the classical and quantum case where we will focus here on the
non-disordered case. Our motivation to restudy the classical case is
to gain as much experience as possible about the transition and
difficulties that may arise in the Monte Carlo sampling and data analysis.
Using this experience a large-scale simulation of the quantum compass
model in 2D will follow in the second part. The next section
introduces the methods and tools we used to accomplish this. Section
\ref{class2d} contains our results for the classical compass model and
Sec.~\ref{quantum2d} the respective analysis for the quantum case. We
close in Sec.~\ref{sec:summary} with a summary and our conclusions.
\section{\label{sec:methods}Observables and Methods}
\begin{figure}[b]
\begin{minipage}{0.49\columnwidth}
\includegraphics[bb=132 360 475 707, clip, width=0.95\textwidth]{pictures/L12disordered.eps}
\end{minipage}
\begin{minipage}{0.49\columnwidth}
\includegraphics[bb=132 360 475 707, clip, width=0.95\textwidth]{pictures/L12ordered.eps}
\end{minipage}
\caption{\label{fig:disorderorder}(Color online) Visualization of
different phases in the 2D compass model. \textit{Left:} For
$T>T_\mathrm{c}$ the system is disordered and the distribution of bonds
possessing less than average bond energy (thick lines) is rather random.
\textit{Right:} For $T<T_\mathrm{c}$ the prevalent correlations order into
one direction, i.e. the system is in a directionally-ordered state.
The pictures are snapshots of a Monte Carlo simulation of the
classical model with $L=12$ at $T=0.3J$ and $T=0.10J$ respectively
(ferromagnetic representation). The small arrows indicate the spin
degree of freedom.}
\end{figure}
\subsection{Observables}
In this section we describe the observables that are used to
characterize the phases and to probe the phase boundaries of the
compass model. The basic quantity is the total energy $E = H =\sum_k E_k$ and the corresponding heat capacity $C=\partial{E}/\partial
T$. With $E_k=J_k S^k_i S^k_{i+e_k}$ we denote the energy along the
$k$-th direction or on $k$-bonds in the system. Using this definition a
useful order parameter in 2D can
then be defined as\cite{mishra:207201,tanaka:256402}
\begin{align}
\label{eq:orderparameter}
D&=\frac{1}{N} \left| \sum_{i} \left( J_x S^x_i S^x_{i+e_x} - J_z S^z_i S^z_{i+e_z}\right) \right|\,,\\
\nonumber &=\frac{1}{N} \left| E_x - E_z \right|\,,
\end{align}
where $N=L^2$. The quantity
$D$ measures the excess energy in one direction compared
to the other direction. If $D>0$ the system is said to
possess long-ranged orbital or directional order whereas for
${D=0}$ the system is disordered.
An alternative definition for the order parameter
\begin{equation}
D^\prime=\frac{1}{N} \left\langle \min{E_k} - \sum_{k=1}^{d=2} E_k / d \right\rangle\,,
\label{eq:ordergeneral}
\end{equation}
can be used to give a visualization and characterization of the
different phases as in Fig.~\ref{fig:disorderorder}. On the lattice we
thereby mark all bonds which have less than the average bond energy
(those that contribute most to the partition function) and look at the
global structure of the resulting bond clusters. In the disordered phase we expect
rather random clusters whereas the ordered phase is characterized by
clusters which are directionally ordered and independent of each other
(dimensional reduction). Note, that in two dimensions $D$ and
$D^\prime$ are actually the same quantity up to a constant
factor, because $D=2D^\prime$. However, Eq.~\eqref{eq:ordergeneral}
provides the general possibility to define an order parameter in any
dimension $d$, which might be useful for future studies. In order to
investigate the universality class of the phase transition we further
look at the susceptibility $\chi$ and Binder parameter $Q_2$ which are
respectively defined as
\begin{equation}
\chi=N\left(\langle D^2 \rangle - \langle D \rangle^2 \right)\,,\quad Q_2=1-\frac{1}{3}\frac{ \langle D^4 \rangle}{\langle D^2 \rangle^2}\,,
\end{equation}
where $\langle D^n \rangle$ denotes an average of the $n$-th moment computed from the time series of $D$.
For the susceptibility we expect a finite-size scaling behavior of the form
\begin{equation}
\label{eqn:ffsgamma}
\chi\sim L^{\gamma/\nu}\,,
\end{equation}
at the critical point with $\nu$ being the correlation length
critical exponent and $\gamma$ the exponent for the susceptibility.
Neglecting corrections to scaling, the Binder parameters for different lattices sizes $L$ should ideally
cross at the critical temperature $T_\mathrm{c}$. In any case, the behavior of the
crossing points of $Q_2$ at lattice sizes $L$ and $2L$ should approach $T_\mathrm{c}$ like
$L^{-1/\nu-\omega}$ if we have corrections to scaling ($w<\infty$).
\subsection{Monte Carlo methods}
Ordinary Metropolis Monte Carlo simulations are used for the classical
model where we update each spin sequentially. During the
thermalization procedure we adjust the proposed moves such that
an average acceptance rate of about $50\%$ is obtained at each
temperature. As it already becomes apparent from simulations on very
small lattice sizes $L$ that the system suffers from huge
autocorrelation times we add a parallel tempering (PT)
scheme\cite{geyerPT,hukushimaPT}, where we propose to exchange spin
configurations between simulation threads at different temperatures
$T_i$. This exchange is attempted every $n$ sweeps, where $n$ is
typically in the range $2$ to $20$. By tracking individual
configurations we make sure all temperatures are seen and that
sufficient diffusion through temperature space is performed. For
simplicity the simplest PT scheme is used, meaning that an equidistant
temperature spacing between neighboring processes is chosen. As a
result a reduction of autocorrelation times by two orders of magnitude
is achieved which pays off in comparison to little longer simulation
times.
In case of quantum spin degrees of freedom, we employ a quantum Monte
Carlo (QMC) procedure based on the stochastic series expansion
(SSE)\cite{PhysRevB.43.5950} technique originally developed by Sandvik. Our own implementation is based on the
(directed) loop scheme\cite{PhysRevE.66.046701} supplemented by ideas of Ref.~\onlinecite{alet:036706}. Recall that the principle of SSE is sampling the
series expansion of the quantum partition function
\begin{align}
\label{eq:z}
\nonumber
Z&=\mathrm{tr}\left(\exp(-\beta \mathcal{H})\right)=\sum_\alpha \sum_n \frac{(-\beta)^n \langle \alpha| \mathcal{H}^n |\alpha\rangle}{n!}\,,\\
&=\sum_{b_i\in S_n} \sum_\alpha \sum_n \frac{\beta^n}{n!}\langle \alpha | \prod_i^n \mathcal{H}_{b_i} | \alpha \rangle\,.
\end{align}
by a Markov chain stochastic process, where $\beta=1/k_\mathrm{B}T$ is the inverse temperature. The last line of Eq.~\eqref{eq:z}
is the central starting point\cite{PhysRevB.43.5950} of the method because it
specifies the configuration space (and the weights) in which the
sampling takes place. A configuration lives in the product space of spin configurations
$|\alpha\rangle$ times the space of all possible sequences (or
permutations) $S_n$ of $n$ bond operators (or vertices) $\mathcal{H}_{b_i}$. The
degrees of freedom are thus $|\alpha\rangle$, $n$, and $S_n$, which
are sampled by the usual combination of diagonal, non-diagonal, and spin flip
updates.\cite{PhysRevE.66.046701}
In the case of the compass model the bond operators $\mathcal{H}_{b_i}$ can be derived from
the Hamiltonian \eqref{eq:hamiltonian2} as
\begin{equation*}
\mathcal{H}_{b}\in
\begin{cases}
S^z_i S^z_j & \text{if $b$ is a $z$ bond}
\\
\{S_i^+S_j^+,S_i^-S_j^-,S_i^+S_j^-,S_i^-S_j^+\} & \text{if $b$ is a $x$ bond}\,,
\end{cases}
\end{equation*}
where the appearance of pure $S_i^{-}S_j^{-}$, and $S_i^{+}S_j^{+}$
terms are a notable difference to an ordinary Heisenberg model. Here
$S^+$ and $S^-$ refer to creation and annihilation operators and the
subscripts $i,j$ are the two sites of the bond $b$. Simulations of
the quantum compass model are furthermore more involved since the
Hamiltonian dictates an asymmetry between bonds in $x$ and $z$
direction, allowing no spin flip operators of type $S^{\pm}S^{\pm}$ to
reside on $z$-bonds. On the other hand, there are {\it a priori} no
diagonal terms $S^zS^z$ on $x$-bonds. However, since non-diagonal
terms can only be introduced into the SSE configuration space after
the diagonal-update (non-diagonal operators must be present!) we are
therefore forced to introduce a positive non-zero energy shift
$\epsilon$ into the Hamiltonian of Eq. \eqref{eq:hamiltonian2}. As a
consequence both non-diagonal and diagonal terms may reside on
$x$-bonds. On $z$-bonds only diagonal terms are allowed.
Unfortunately, this asymmetry in the operator representation cannot be
transformed away by a simple ``symmetrizing'' rotation of the
Hamiltonian because of emerging minus sign problems. Note finally that
the non-zero energy shift $\epsilon$ has an effect on the order
parameter $D$ since it influences the number and the distribution of
bond operators in the operator sequence.\footnote{This is an effect caused by defining the quantity $D$ as in Eq.~\eqref{eq:orderparameter} and by taking the absolute value $||$ explicitly from the time series of $E_x-E_y$.
A better way would be to take $D^\star=\sqrt{\langle (E_x-E_y)^2\rangle}$ which is not dependent on $\epsilon$.
Then $D^\star$ is the true quantum estimate but is much harder to sample in SSE resulting in larger error bars.
$D$ is therefore an approximation to $D^\star$ which becomes almost perfect in the ordered phase. For our main objective of criticality this is, however, irrelevant and we choose a quantity which is more accurately to measure.}
This effect can cause
additional finite-size contributions also in the susceptibility and
the Binder parameter which vanish in the thermodynamic limit and for
$T\to 0$. We have checked that at $L=16$ no difference could be
detected in the susceptibility maxima locations for $\epsilon=0.1,
0.5, 0.9$ within error bars. Here we work with $\epsilon=0.5$ at all
lattices sizes.
Since simulations of the quantum model display the same rapid
critical slowing down as the classical model we perform additional
quantum PT updates\cite{PhysRevB.65.155113} in the same manner as described
above.
We implemented both the classical and quantum Monte Carlo PT
scheme on parallel architectures with the restriction of shared memory
access for fast communication between processes. This is essential
since PT updates are done rather often.
For data analysis purposes we use well-known multi-histogram
techniques to optimally combine simulations at different temperatures.
Those techniques are available for both the
classical\cite{ferrenberg:multi} and quantum
cases\cite{PhysRevLett.90.120201,Troyer_multihist}. In combination
with optimization routines like the Brent method\cite{brent} they
allow rather systematic and unbiased estimation of pseudocritical
temperatures from peaks in the susceptibility.
\subsection{Boundary conditions}
\begin{figure}[b]
\begin{minipage}{0.2\textwidth}
\psfrag{l}{{\rm pbc}}
\includegraphics[width=\textwidth]{pictures/pbc.eps}
\end{minipage}\makebox[0.5cm]{}
\begin{minipage}{0.2\textwidth}
\psfrag{l}{{\rm abc}}
\includegraphics[width=\textwidth]{pictures/abc.eps}
\end{minipage}
\caption{\label{fig:bc}(Color online) Visualization of the different
boundary conditions used in this work. {\it Left:} Ordinary
periodic boundary conditions. All bonds carry the same coupling and
the dashed bonds connect the spins across boundaries. The topology is a
torus. We refer to this case as pbc. {\it Right:} So called
``annealed'' boundary conditions (abc).\cite{mishra:207201} Here
the sign of the couplings on the dashed boundary bonds may fluctuate
dynamically resulting in an additional degree of freedom. As an
example we draw some thick bonds indicating a negative coupling.}
\end{figure}
\begin{figure*}
\begin{minipage}{0.33\textwidth}
\includegraphics[width=\textwidth]{pictures/PICorder_pbc.ps}
\includegraphics[width=\textwidth]{pictures/PICorder_abc.ps}
\end{minipage}
\begin{minipage}{0.33\textwidth}
\includegraphics[width=\textwidth]{pictures/PICordersus_pbc.ps}
\includegraphics[width=\textwidth]{pictures/PICordersus_abc.ps}
\end{minipage}
\begin{minipage}{0.33\textwidth}
\includegraphics[width=\textwidth]{pictures/PICorderQ2_pbc.ps}
\includegraphics[width=\textwidth]{pictures/PICorderQ2_abc.ps}
\end{minipage}
\caption{\label{fig:result2d} Data for the 2D classical compass model
obtained by Monte Carlo simulations. The top row displays the
results for periodic boundary conditions and the bottom row for
annealed boundary conditions. Note the the temperature ranges are
different for both cases and that not all lattice sizes are shown
for better readability. The lines through the data points are
obtained from the multi-histogram analysis. {\it Left:} The order
parameter $D$ as a function of different lattice sizes $L$. {\it
Middle:} The susceptibility $\chi$ of the order parameter. {\it
Right:} The Binder parameter $Q_2$.}
\end{figure*}
Ordinarily, the vast majority of Monte Carlo simulations are performed
using periodic boundary conditions (pbc) which map the lattice onto a
torus topology using the assumption that free-energy contributions
from the surface are thereby minimized. In contrast to this approach,
Mishra {\em et al.}\cite{mishra:207201} argue in their recent
contribution that periodic boundary conditions might not be optimal in
the case of the compass model. Instead, they introduce special, so
called \textit{annealed} boundary condition (abc) to arrive at their
Monte Carlo results. Since a detailed comparison between these two
boundary conditions has to our knowledge not been done, we will
explicitly study and compare their effect on the finite-size scaling
behavior for the classical compass model. This comparison is
especially interesting in view of the fact that we may not easily
apply the annealed case to quantum Monte Carlo since it induces a
minus sign-problem. A characterization and understanding of the
scaling behavior for periodic boundary conditions would therefore be
of advantage before studying the quantum case.
Figure \ref{fig:bc} displays these two types of boundary conditions as
a sketch. The topology of the annealed boundary condition is the same
as for periodic boundary conditions. The annealed case is special
because the sign of couplings on bonds across the border
may fluctuate dynamically according to the Boltzmann distribution.
The bond sign is therefore an additional degree of freedom in the
Monte Carlo update rendering the simulations somewhat more complex.
\section{\label{class2d}The classical compass model in 2D}
In this section we start the presentation of our simulation results.
We consider firstly Monte Carlo simulations of the 2D classical
compass model. The main purpose of this section is to give an explicit
comparison between the different boundary conditions introduced in the
last section. To this end we run simulations for both cases and
compare the observables of Sec.~\ref{sec:methods} and their
finite-size behavior. Figure~\ref{fig:result2d} gives an overview of
our Monte Carlo estimates for $D$, $\chi$ and $Q_2$. There, the top
row contains results for periodic boundary conditions and the bottom
row for annealed boundary conditions. Knowing the different behavior
of reaching the thermodynamic limit is useful in order to appreciate
results of our simulations which follow in subsequent sections.
Simulations are done using lattice sizes {$L
=\{10,12,16,24,32,36,48,64,128\}$} periodic bc and
${L=\{10,12,16,20,24,36,52,64\}}$ for annealed bc, typically taking
about $10^5$ measurements per data point after an equilibration phase
of $10^4$ sweeps. By the behavior of the order parameter in
Fig.~\ref{fig:result2d} (left) it is immediately evident that there is
a phase transition and that directional order with $D>0$ is realized
at low temperatures. We secondly observe that the order parameter for
the pure periodic case has a slow convergence for small lattice sizes
while for larger sizes it suddenly moves considerably. In contrast,
the data for the annealed case show a much smoother movement towards
the infinite-volume limit and it is evident that finite-size effects
are drastically reduced. A difference like this is actually expected
for different boundary conditions. The crucial and interesting
question is whether the two boundary conditions lead to the same
critical temperature in the infinite volume limit where boundary
effects should vanish.
We therefore obtain an estimate of the critical point $T_\mathrm{c}$
in the thermodynamic limit by fitting the pseudocritical temperatures
$T_\mathrm{c}(L)$ taken from the peaks of the susceptibilities in
Fig.~\ref{fig:result2d} (middle) at lattice size $L$ to the
finite-size scaling ansatz
\begin{equation}
T_\mathrm{c}(L)=T_\mathrm{c} + bL^{-1/\nu}(1+cL^{-\omega})\,.
\label{eqn:ffs}
\end{equation}
Here $b,c$ are some constants and $\omega$ is an exponent
describing corrections to scaling. In a first step, we assume nothing
about the value for the correlation length exponent $\nu$ and leave it
as fit parameter.
\begin{figure}
\includegraphics[width=\columnwidth]{pictures/PICffsorder_cmp_color.ps}
\caption{\label{fig:ffsorder_cmp}(Color online) Determination of the
critical temperature $T_\mathrm{c}$ from finite-size scaling of the
pseudocritical temperatures determined from susceptibility peaks.
The two curves correspond to different boundary conditions which
trigger a completely different convergence to the critical point.
The lower curve is obtained for annealed boundary conditions (abc) and
shows the superior scaling compared to periodic boundary conditions (pbc).
Lines are fits to Eq.~\eqref{eqn:ffs} neglecting the correction term
$\omega$.}
\end{figure}
The fitting procedure to the data in Fig.~\ref{fig:ffsorder_cmp}
yields $T_\mathrm{c}=0.144(2)J$ from periodic boundary conditions and
$T_\mathrm{c}=0.1461(8)J$ from annealed boundary conditions and the
estimate for the correlation length critical exponent is $\nu=0.98(4)$
which we take from the straight line fit for the annealed case. Both
results agree within error bars. The annealed value yields a much more
accurate estimate since here the asymptotic scaling regime sets in
much earlier and we have more points available for fitting. These
numerical estimates are in accordance with the value
$T_\mathrm{c}=0.147(1)J$ obtained in Ref.~\onlinecite{mishra:207201}.
With our value for $\nu$ we support the claim that the transition is
of 2D Ising type. To further confirm this conjecture we also determine
the exponent $\gamma$ associated with the susceptibility $\chi$. For
lattice sizes large enough ($L>20$) we obtain $\gamma/\nu=1.73(4)$ from the annealed case
(see Table~\ref{tab:results} and Fig.~\ref{fig:gamma} below) which is
again consistent with 2D Ising universality. In a second step, we can
now assume Ising universality to be given to improve the fit. Using
$\nu=1$ as a fixed parameter the improved value for critical
temperature is $T_\mathrm{c}=0.1464(2)J$.
Let us now turn to a discussion of the Binder parameter $Q_2$
displayed in Fig.~\ref{fig:result2d} (right). For the annealed case a
nice crossing of curves at the critical temperature can be observed
and our estimate for the Binder parameter at the crossing point
(taking the three largest lattice sizes) is $Q_2=0.61(1)$. This is
roughly the known value for the 2D Ising model, which -- however -- is
usually obtained for {\it periodic} boundary conditions.\cite{Kamieniarz} Using the
observed crossing behavior, the Binder parameter supplies a natural
third check of the critical temperature and the critical exponent. We
hence apply our recently developed data collapsing
tool\cite{wenzel:nuclphysB2007} and obtain $T_\mathrm{c}=0.1465(4)J$
and $\nu=1.01(4)$ from the best data collapse. These values are again
fully consistent with our results above and give further confidence to
our analysis.
In contrast, the nice properties of the Binder parameter do not show
up for periodic boundary conditions, where it is hard to judge whether
curves for different lattice sizes cross in a single point at all.
Rather, we see strong finite-size effects and that the crossing points
for large lattice sizes move close to $2/3$, which is totally in
contrast to the expected behavior. It is known that different
boundary conditions cause a discrepancy (see for instance
Refs.~\onlinecite{Kamieniarz} and \onlinecite{selke-2006-51}) in the Binder crossings but such a
drastic behavior was unexpected.
In summary, our investigation for the classical model clearly show
that annealed boundary conditions are favorable because they
drastically reduce finite-size effects and yield good scaling
properties for the finite-size analysis. With periodic boundary
conditions much larger lattice sizes need to be investigated in order
to obtain the critical temperature and to approach the right asymptotic scaling
regime. Additionally, our analysis shows that the Binder parameter for
periodic boundary conditions does not cross at the usual expected
value and that we may not use the crossing point (height) as a good
indication for the critical point, whereas for annealed conditions we
get good properties. These effects are currently not properly
understood. By referring to the typical spin configuration in
Fig.~\ref{fig:disorderorder} it is, however, tempting to argue that
the dominant energy correlations (blue lines) wrap around the torus in
the ordered phase thereby forming some kind of closed loop
excitations. These excitation appear to be more stable against
thermal fluctuations than open excitation. Annealed boundary
conditions seem to prohibit the formation of such loops leading to a
better scaling behavior.
\section{\label{quantum2d}The quantum compass model in 2D}
\begin{figure}
\begin{minipage}{\columnwidth}
\includegraphics[width=\textwidth]{pictures/PICQCM_order_comment_label.eps}
\includegraphics[width=\textwidth]{pictures/PICQCM_susorder_label.eps}
\includegraphics[width=\textwidth]{pictures/PICQCM_Q2_label.eps}
\end{minipage}
\caption{\label{fig:qcm_orderandsus} QMC results for the 2D quantum
compass model with periodic boundary conditions. All lines are a
guide to the eye. (a) The order parameter $D$ for lattice sizes
${L=\{12,14,16,18,20,28,32,40,52,64\}}$ displays a clear signal of a
stable ordered phase at low temperatures. The arrow marks the
transition temperature $T_\mathrm{c}$ from Ref.~\onlinecite{tanaka:256402}.
Our own data indicates a smaller value. (b) The susceptibility
$\chi$ on a logarithmic scale for lattice sizes
$L=\{10,12,14,16,18,20,28,32,40,52,64\}$. (c) the Binder parameter
$Q_2$ in the quantum compass model with periodic boundary
conditions, where steeper slope corresponds to larger $L$
(neglecting $L=28$ and $L=48$ for better clarity). The qualitative
behavior is the same as for the Binder parameter with periodic
boundary conditions in the classical model. No common crossing point
is present for the lattice sizes of this work.}
\end{figure}
Using the knowledge gained from simulations of the classical compass
model we turn to the discussion of the simulation results of the
quantum version. Simulations are done using the stochastic series
expansion as outlined in Sec.~\ref{sec:methods}. The reader is
reminded that annealed boundary conditions, where the sign on boundary
bonds fluctuates, are not possible because such fluctuations induce a
sign problem in the quantum Monte Carlo scheme.
We therefore choose to simulate with periodic boundary conditions and
expect from Fig.~\ref{fig:ffsorder_cmp} that large lattice sizes might
be needed to see the right scaling and to obtain the infinite-volume
critical temperature. Using the parallel tempering scheme and the
reduction of autocorrelation times by two orders of magnitude, we were
finally able to simulate lattice sizes
${L=\{8,10,12,14,16,18,20,24,28,32,40,48,52,64\}}$, where the largest
one is about the limit one can reach in quantum Monte Carlo in
feasible time and resources at the moment.\footnote{In this
temperature regime.} Our largest system size is about three times as
large compared to the simulations of Ref.~\onlinecite{tanaka:256402}.
A detailed check and verification of our
algorithm was done with data from full exact diagonalization (ED) on a
$4\times 4$ lattice. We use our own ED program with some implemented
symmetries\cite{dorier:024448} to reduce the dimension of the Hilbert
space, as well as the ALPS package\cite{ALPS} for smaller system
sizes. During the Monte Carlo runs, a total number of about $4\times
10^5$ measurements are typically taken after each sweep and $2\times
10^4$ sweeps are used for thermalization. Those numbers are, of
course, only meaningful with the additional information that we
construct as many loops in the non-diagonal update such that on
average $2n$ vertices are visited in the SSE configuration.
Figure~\ref{fig:qcm_orderandsus} shows the result for the order
parameter, the susceptibility and the Binder parameter obtained from the simulations in this
manner. The behavior of the order parameter shows a clear signal of a
transition from a disordered to an ordered state at small temperature,
evidently becoming more pronounced with increasing lattice size. This
proves the existence of a directional-ordering transition also in the
quantum case. In the ordered phase, the order parameter seems to take
on a value which is quite different from the classical case and the
order is furthermore less stable against thermal fluctuations as the
temperature regime in the quantum case is evidently much smaller.
Note that the overall estimate for $D$ also agrees roughly with data of
Ref.~\onlinecite{tanaka:256402}.
The dependence of the data on the lattice sizes is, as expected, qualitatively
similar to the classical case, i.e., the order parameter curves and
the susceptibility peaks shift considerably to lower temperatures for
larger lattice sizes. This shift is in fact so large that it is
already obvious from Fig.~\ref{fig:qcm_orderandsus} (a) that the previous estimate of the
critical temperature in the literature is much to large.
\begin{figure}
\includegraphics[width=\columnwidth]{pictures/PICffs_comment.ps}
\caption{\label{fig:scaling}(Color online) Finite-size scaling of pseudocritical temperatures for different
lattice sizes obtained from the susceptibility. For small lattice sizes corrections to scaling are
evident. For large lattice sizes 2D Ising scaling is reached
yielding our estimate for the critical temperature of
$T_\mathrm{c}/J=0.055(1)J$. The curve trough the points represents a fit to
Eq.~\eqref{eqn:ffs}. We also show the crossing points of
the Binder parameters at $L$ and $2L$ for a consistency check. The arrow
indicates the previous result of Ref.~\onlinecite{tanaka:256402}.}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{pictures/PICfitgamma.ps}
\caption{\label{fig:gamma}(Color online) Plot of the susceptibility $\chi$ (at peak locations) versus system size $L$ for
all different simulations in this work (classical annealead bc, classical periodic bc, quantum periodic bc in this order from top to bottom) on a double logarithmic axis.
The straight lines are fits to Eq.~\eqref{eqn:ffsgamma}. All cases are
consistent with a value of $\gamma/\nu=1.75$.}
\end{figure}
Before we quantify this discrepancy for the critical temperature, we
draw our attention to the susceptibility and the Binder parameter in
Figs.~\ref{fig:qcm_orderandsus}(b),(c), both showing a behavior similar
to the classical case with periodic bc. We note especially that the
Binder parameter is again behaving oddly and that there is not a well
defined crossing seen at all at the lattice sizes simulated. A
crossing point might still be achieved for very large lattice lengths
$L$ but is certainly difficult to quantify since the value of $Q$ at
the crossing point is very close to $2/3$. Due to this observation the
Binder parameter is clearly not suited to determine the critical
temperatures by looking at the Binder crossings for small lattice
sizes, where the true behavior is just not seen.
Let us now determine an improved value for the critical temperature
with finite-size scaling from the maxima of the susceptibilities. As
in the classical case, we fit the pseudocritical values to the scaling
ansatz given in Eq.~\eqref{eqn:ffs}. To use as many data points as
possible we include corrections to scaling, described by the exponent
$\omega$, into the fit and leave all fit parameters free. Including
all lattice sizes we obtain $T_\mathrm{c}=0.055(6)J$ and $\nu=0.9(2)$ with a
fit quality of $\chi^2/\mathrm{d.o.f}=0.66$. Those values, however,
are stable also for fitting windows starting at larger lattice sizes.
The precision for the critical exponent $\nu$ is rather low but agrees
with our expectation of 2D Ising universality within the error bar.
Under this assumption, we fix $\nu=1$ and repeat the fit procedure
yielding an improved estimate for the critical temperature as
$T_\mathrm{c}=0.055(1)J$. The relative discrepancy with the previous estimate
of Ref.~\onlinecite{tanaka:256402} is approximately $36\%$. As a
cross check for our analysis we further look at the scaling of the
crossing points of $Q_2$ at lattice sizes $L$ and $2L$, which is also
indicated in Fig.~\ref{fig:scaling}. We observe that this scaling is
consistent with the previous value from the susceptibilities but we do
not attempt a detailed fit by lack of enough data points. It
is then also useful to obtain an independent estimate of the critical
temperature from the maxima in the heat capacity $C$, which again gives
consistent results but does not reach the accuracy of our previous
analysis since $C$ is generally hard to sample in QMC at low temperatures.
To finalize our analysis, we determine the critical exponent $\gamma$
from the susceptibility of the order parameter $D$. For large lattice
sizes we expect a scaling according to Eq.~\ref{eqn:ffsgamma} which
can be tested by plotting $\ln(\chi)$ versus $\ln(L)$. This is done in
Fig.~\ref{fig:gamma} together with the data for the classical cases.
It is evident that asymptotic scaling sets in only for the largest
lattice sizes from which we obtain a value of $\gamma/\nu=1.68(8)$
consistent with 2D Ising universality, but not precise enough to be
absolutely conclusive.
\section{\label{sec:summary}Summary and Conclusions}
\renewcommand{\arraystretch}{1.2}
\begin{table}[t]
\caption{\label{tab:results}Results for the critical temperature and critical exponents as obtained in this work. The upper section contains the results for the classical model taken from annealed bc, which are all pairwise consistent.
The middle section summarizes our estimates for
the quantum compass model for the cases with and without the assumption of 2D Ising behaviour ($\nu=1$). Both cases are consistent with each other.
Lastly the values for $\gamma/\nu$ are summarized as obtained from the largest lattice sizes for the different simulation runs.}
\begin{tabularx}{\columnwidth}{Xllll}
\hline\hline
& $T_\mathrm{c}/J$ & $\nu$ & $\omega$ & $\chi^2/\mathrm{d.o.f}$ \\
\hline
no assumption & $0.1461(8)$ & $0.98(4)$ & $-$ & $1.3$ \\
2D Ising & $0.1464(2)$ & $-$ & $-$ & $1.15$ \\
collapse $Q_2$ & $0.1465(4)$ & $1.01(4)$ & $-$ & $-$ \\
\hline
no assumption & $0.055(6)$ & $0.9(2)$ & $0.5(4)$ & $0.66$ \\
2D Ising & $0.055(1)$ & $-$ & $0.8(2)$ & $0.61$ \\
\hline\hline
& class. (abc) & class. (pbc) & quant. (pbc) & \\
\hline\
$\gamma/\nu$ & $1.73(4)$ & $1.72(5)$ & $1.68(8)$ & \\
\hline\hline
\end{tabularx}
\end{table}
In this paper we reported on comprehensive Monte Carlo simulations of
the classical and quantum compass model. By comparing different
boundary conditions for the classical case, we showed that for
ordinary periodic boundary conditions one needs to go to very large
lattice sizes to see the right scaling and to get good convergence to
the critical point. In order to simulate large lattice sizes, we
implemented a parallel tempering scheme to counteract huge
autocorrelation times. Our results, which are summarized in Table
\ref{tab:results} are perfectly consistent with previous studies in
the literature for the classical model. For the quantum model our
simulations are quantitatively at odds with earlier studies and we
provide here a new estimate for the critical temperature $T_\mathrm{c}$. We
argued that this discrepancy might be explained by the huge
finite-size corrections originating from stable loop excitations
formed by correlation orderings which appear on the torus topology at
periodic boundary conditions. It appears that those excitations even
destroy the usual properties of Binder parameters. Our analysis,
however, shows that one can still arrive at an estimate for
$T_\mathrm{c}$ at periodic boundary conditions provided that one takes
this effect into account. All critical exponents obtained in this
study give further support to the claim that 2D Ising universality
describes the directional-ordering transition in the 2D compass model.
Our findings for the quantum model might have an impact on the
conclusions of Ref.~\onlinecite{tanaka:256402} because a precise
estimate of $T_\mathrm{c}$ enters into the analysis of dilution
effects on the model. It is conceivable that the conclusion obtained
there are still qualitatively valid. For a true quantification
of the dilution effect, however, there is no way around performing a more
detailed investigation of larger lattice sizes. The knowledge gained in this
work should help to start such a study.
The precision of our results for the quantum model are still rather low
compared to many other systems of statistical physics. In this respect
it would be an interesting future project to devise and analyze
special boundary conditions for the quantum model with improved
finite-size scaling behavior compared to periodic boundary
conditions.
\acknowledgments We thank L.-H.~Tang and T.~Platini for discussions, and S.~Wessel for further suggestions.
S.W. acknowledges a PhD fellowship from the Studien\-stiftung des
deutschen Volkes, the kind hospitality of the statistical physics
group at the university Henry Poincare in Nancy and support from the
Deutsch-Franz\"osische Hochschule (DFH) under Contract No.
CDFA-02-07 as well as from the graduate school ``BuildMoNa.'' We also profited from a DAAD PPP programme with China.
Part of the simulations were performed on the JUMP facility of NIC at
Forschungszentrum J\"ulich under Project No. HLZ12.
\bibliographystyle{apsrev}
|
{
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{"url":"https:\/\/what-if.xkcd.com\/127\/","text":"# Tug of War\n\nWould it be possible for two teams in a tug-o-war to overcome the ultimate tensile strength of an iron rod and pull it apart? How big would the teams have to be?\n\n\u2014Markus Andersen\n\nA couple dozen people could pull a half-inch iron rod apart.\n\nTug-of-war, a simple game in which two teams try to pull a rope in opposite directions, has a surprisingly bloody history.\n\nI don't mean that there's some kind of gruesome historical forerunner of modern tug-of-war.[1]Although it's definitely an ancient sport, so I'm sure people have come up with all kinds of horrific variations over the centuries that I don't really want to spend hours reading about. Humans seem to be creative when it comes to that kind of thing. I mean that modern tug-of-war involves a lot more death and mutilation than you might expect\u2014precisely because people underestimate how few people it takes to break \"strong\" things like heavy rope.\n\nAs detailed in a riveting article in Priceonomics, recent games of tug-of-war have resulted in hundreds of serious injuries and numerous deaths\u2014all caused, one way or another, by ropes snapping. In particular, this seems to happen when large groups of students try to set a world record for largest tug-of-war game. When a rope under many tons of tension suddenly snaps, the recoiling ends can\u2014and do\u2014cause a terrifying variety of injuries.\n\nBefore we answer Markus's question, it's worth noting that the physics of tug-of-war can be a little tricky. It seems like common sense that the \"stronger\" team has an advantage, but that's not quite right. To win, you need to resist sliding forward better than the other team. If you can't resist sliding, then increasing your arm strength means you'll just pull yourself forward. Since sliding friction is often proportional to weight, tug-of-war on many surfaces is simply a contest over who's heavier.[2]Champion tug-of-war teams focus on body angle, footwork, digging into the ground, and timing pulls to throw off the other team. The strongest team in the world would lose a tug-of-war with a six-year-old and a sack of bricks, as long as the sack had a firm grip.\n\nSo, how much force can tug-of-war players exert?\n\nA 2011 paper analyzing the immune systems of several \"elite tug-of-war players\"[3]The paper notes that \"Few studies have been done to examine the effects of [the] tug-of-war sport on physiological responses,\" which seems likely enough to me. measured their average pull force (on a school gym floor) to be about 102.5 kilograms-force, or about 1.5x their body weight.\n\nThe ultimate tensile strength of cast iron is about 200 megapascals (MPa), so we can use a simple formula to figure out how many players would be needed to break one.\n\n$\\text{People required}=\\frac{\\pi\\times\\left(\\tfrac{1}{4}\\text{ inch} \\right )^2\\times200\\text{ MPa}}{102.5\\text{ kg}\/\\text{person}}\\approx25\\text{ people}$\n\nTwo teams of 25 people[4]I originally wrote 25 people total, forgetting that two people pulling with 100 units of force each will produce 100 units of tension on the rope, not 200! Thank you for Gordon McDonough for pointing this out. could probably pull a half-inch iron bar apart. An inch-thick iron bar could be torn in half by teams of 101 people,[5]People often play tug-of-war with their dogs. Going by weight alone, 30 humans would probably be about evenly matched against 101 dalmatians. and a 2-inch-diameter bar would need over 400. It's hard to have a tug-of-war with something thicker than about 2 inches. Since you're not allowed to install handles on the rope,[6]Or wrap it around your hand, for reasons which will become clear if you read some of the articles on tug-of-war injuries. it has to be narrow enough to grip easily.\n\nWhile \"400 people\" may be the limit for plain iron bars, there are much stronger substances out there. Common types of steel, for example, have a tensile strength about 10 times that of cast iron. Common half-inch rebar, for example, would in theory take teams of over 200 people to pull apart, compared to 25 for cast iron. Other substances are even stronger; a half-inch shaft made from high-grade steel or a polymer like Kevlar (or, theoretically, a solid silicon crystal) could handle the pulling force from teams of anywhere between 500 and 800 competitive tug-of-war players.\n\nIf we limit ourselves to a two-inch diameter rope, which seems to be about the maximum size for tug-of-wars,[7](William Safire returns from the grave to point out that it should really be tugs-of-war.) then the maximum number of tug-of-war players given a super-strong rope like Kevlar is in the neighborhood of 10,000.[8]Or several times that many, if they're not very athletic.\n\nIf we figured out how to manufacture large ropes out of graphene ribbons, which have tensile strengths over 10 times higher than existing materials, we could theoretically support a tug-of-war between teams of up to 100,000 players each. Such a rope would be over 200 miles long, and could stretch from New York to Washington.\n\nIf our experience with nylon ropes failing is any indication, when the graphene finally snapped, the death toll could be enormous among both players and bystanders. Lengths of graphene would crack across the landscape like bullwhips, slicing down forests and demolishing buildings.\n\nIn the end, trying to develop stronger ropes leads only to greater danger to everyone, both participants and bystanders. In the ultimate game of tug-of-war ...\n\n... the only winning move is not to pull.","date":"2017-05-26 03:40:28","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3125652074813843, \"perplexity\": 1788.1690687383555}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-22\/segments\/1495463608633.44\/warc\/CC-MAIN-20170526032426-20170526052426-00425.warc.gz\"}"}
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El Turó de Can Mallol és una muntanya de 409 metres que es troba al municipi de Barcelona, a la comarca catalana del Barcelonès.
Referències
Can Mallol
Muntanyes de Collserola
Vallvidrera, Tibidabo i les Planes
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Fourth of YouLie 2015 Editorial. "Art is a lie that tells the truth" Pablo Picasso
Read full story | Comments (0)
African-American So-Called Ambassador Noah B. Mamet is Unfit to Represent the United States
Passport Blues Noah Bryson Mamet President, Noah Mamet Associates www.nmapartners.com Noah@nmapartners.com BuenosAires-ACS@state.gov Dear Readers: I have a true story to tell you, so bear with me. I'm an American born in Manhattan whose passport expired…
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Business $2 Billion Welfare Payment to Israel to Buy More Bombs, to Compensate for Iran Nuclear Accord. Americans: When Will We Get Our Country Back?
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American culture There are 1500 newspapers, 1100 magazines, 9000 radios stations, 1500 TV stations, 2400 publishers and all of them are operated by only 6 corporations.
There are 1500 newspapers, 1100 magazines, 9000 radios stations, 1500 TV stations, 2400 publishers and all of them are operated by only 6 corporations. Now here is where the scary part…
Business The Kagans: A Jewish Family Business of Perpetual War at Taxpayer Expense
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Editor's Desk HITLER AND NETANYAHOO: UNCANNY PARALLELS IN CHARACTER. by John Chuckman. Video footage of Jewish Israelis with ISIS-style flags, chanting "niggers go home" at an anti-African rally in Tel Aviv on October 5 has surfaced. Hundreds of protesters could be heard in streets of Tel Aviv chanting anti-African racial slurs, following an Israeli High Court ruling to close down "Holot" detention facility within 90 days. Holot is essentially an internment camp holding about 2,300 of 60,000 African asylum seekers in Tel Aviv.
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Die Ruch Katolicko-Narodowy (, Abkürzung: RKN) ist eine kleine rechtsgerichtete national-klerikale politische Gruppierung in Polen.
Sie wurde nach dem Zerfall der Ruch Odbudowy Polski (ROP) 1997 von Antoni Macierewicz gegründet.
Aufgrund ihrer Nähe zum katholischen Radiosender Radio Maryja wird sie von ihren Anhängern auch "Marienpartei" (Partia Maryja) genannt. Nach Eigendarstellung bestehen ihre grundsätzlichen Ziele im "Schutz der Familie, der katholischen Kirche und der nationalen Interessen".
Bei den Parlamentswahlen 2005 konnte sie über die Liste der Liga Polnischer Familien fünf Abgeordnete in den Sejm entsenden. Seit den Parlamentswahlen 2007 ist sie nicht mehr im Sejm vertreten.
Einzelnachweise
Partei (Polen)
Gegründet 1997
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5 Minutes with… Moos Lamerus
5 minutes with...
MassiveMusic COO and partner on his forged musicianship, insatiable hunger for wisdom and the science of human happiness
By day Moos Lamerus works as global chief operating officer and partner at international creative music agency MassiveMusic in Amsterdam, fusing his love for music with his entrepreneurial spirit, but he's got many more strings to his bow.
He's been involved in the digital tech startup scene for years, has co-founded companies in his spare time and invested in a few others, and when he's not absorbed in the music business he's reading up on the latest developments in science.
Most recently he's been swatting up on the science of happiness. In 2015 he started bringing together findings from the field of positive psychology and by the end of 2018 he'd drawn a lot of it together into a book: 'Gelukkig maar | Een wetenschappelijke zoektocht naar geluk' (In English: 'Fortunately | A scientific search for happiness', although it hasn't yet been translated).
LBB's Alex Reeves caught up with Moos to find out his thoughts on music, business and science.
LBB> Where did you grow up and what kind of kid were you?
Moos> I grew up in Amsterdam. I was a shy kid, doing my own thing but I generally got along very well with all the groups. I didn't particularly fit in with one group. I hung out with the popular kids and the more nerdy, dorky kids.
I remember when I graduated from high school my mentor said: 'Like he improvised on his piano, he improvised his way through education.' It was a good summary because I always found my own path. Sometimes I worked hard and got good grades. Sometimes I was just being creative.
LBB> So music was a big part of your life from early on?
Moos> Yeah. That was all to do with my mum. I grew up with her in Amsterdam and she'd picked up the violin quite late, in her 30s, but she was motivated to learn and played a lot and rehearsed for hours a day. At least four nights a week it was live chamber music with a trio, quartet or quintet. I'd go to bed with chamber music live from the living room. As long as I can remember I've been infused with mainly classical and chamber music. But she was also a big fan of The Beatles.
She always regretted the fact that she started so late, so I had to make up for that and she forced me. I had to pick out an instrument and rehearse every day. At four I tried out the violin. It didn't work, so I switched to play the drums but it was difficult with the neighbours. Then I found the piano when I was six: that's how my formal training started for 12 years of classical music followed by other styles. My youth was piano rehearsal. Every day I had to sit behind it for 30 or 45 minutes. That could be a fight because I didn't want to do anything with music, thought it was stupid and I just wanted to play outside. But my mother had good resolve so she stuck with it and now I'm very grateful.
LBB> When did you first consider going into working in music as a career?
Moos> When I was 15 I started picking up the guitar to play in bands. So trying to be a music professional started then. I was serious with my best friend about starting a band and trying to conquer the world and be world famous artists. We did that for years and years. We obviously didn't succeed but we had a lot of fun.
LBB> What sort of music did you play?
Moos> This was '70s southern rock, some blues-infused rock-and-roll - all basically guitar-heavy.
When it became clear that the world famous artist thing wasn't happening because of a lack of talent or discovering that you've romanticised what it's like, then I started to look towards the industry as a professional, to work in it rather than being an artist. I was in my mid 20s when I actually started a career. Before that it was just college, touring and working behind a bar to make a living.
LBB> What did that early career look like?
Moos> I started as a project manager in the cultural sector, doing projects that tried to bridge between music and other creative disciplines like games and film. That's when I got in contact with MassiveMusic as one of the leading parties being involved in both music and film.
I organised an event where we brought some interesting film composers and directors together in a cinema at the height of summer. It was hot outside and we were sitting in this dark little room. Not everything was as sparkling as it could be so I was kind of bored. Then MassiveMusic came on last. They'd just finished up Nike 'Write the Future' and they blew the whole thing off. It was such an amazing story how this film developed with all the artists they'd reached out to and tried, going all over the world in one epic adventure and they were selling it in a juicy way. Everybody was on the edge of their seat listening. I thought it was just epic.
I started chatting with the guys and the rest is history. It was serendipitous.
The first time I walked into the office I knew this is it. This is my future.
LBB> So fast forward to now. You're global COO. What do your days look like in that role?
Moos> Every day is different. But what I try to focus on is highlighting all the talent we have worldwide and making sure that we keep improving our creative output. We can do that in many ways. Sometimes it's about the tools we have; sometimes it's about how we inspire each other. What I enjoy the most is being part of a big global family of music fanatics and trying to contribute to how this family can flourish.
LBB> Which recent projects are you most proud of at MassiveMusic and why?
Moos> We're particularly proud of a short film that we're just wrapping up with Tom Schlagkamp, a great director. It's called 'Disconnect'. I can't say much more about it. But it's so cool because we're mainly involved with very short films - often 30 seconds - and it's great to be part of a mini feature film. It's also something that we did purely for the art. It wasn't a commercial project. I'm really proud that we carved out time and resources to be able to work on something like this. Also, the end result is great. If it wins a few awards at film festivals that would be great, but that's not why I'm proud of it. It's the fact that we make a conscious choice to do this kind of project. For me, that's what it's all about.
LBB> Tell us about the book you've been writing about the science of happiness. Where did the idea come from?
Moos> I have an insatiable hunger for wisdom. So I devour a lot of books and online courses. Everything I can find. It's mostly centred around what makes us tick as humans, what drives us and why we do the things we do. Sometimes it can be more general about the universe.
While I was devouring all these interesting books, mainly on science, I stumbled across a course 'The Science of Happiness' on edX at UC Berkeley. That really struck a chord because it's a fundamental question about what makes us tick. I was also surprised that the scientific field of positive psychology was relatively young. It's a really curious development in psychology and neuroscience. There's a lot of pathology in there, so you study where things go wrong and then try to cure illnesses. It's interesting that this field focuses on what makes someone flourish, how you can increase wellbeing. What is wellbeing? What is meaning?
This is difficult for science to study. You need the rigour etc. But for me it was very interesting that the scientific lens was now being put on this fundamental question. I felt there was more to discover and I wanted to dive in and do the research, to find out what the scientific literature has to say about this.
LBB> So you basically did a lot of reading?
Moos> Exactly. I did a bit of a preview of the scientific literature, so it's not a self-improvement book. There are many out there that already cover that extensively. This is more about what science has to tell us about happy life and being more objective and nuanced in this. It's not about making a point: 'These are the three simple steps to lead a happier life.' It's more: 'These are some topics that science has studied in depth around this subject. And here are some of the conclusions.' With science it's always very nuanced so they are not strong conclusions, just tendencies here and there. I want the reader to draw their own conclusions from it. My goal is to spark an interest in the topic and reflect on the material and on their lives.
LBB> Who are your creative heroes and why?
Moos> Because our topic is now a bit more intellectual I'll talk about heroes in science. One really nice development is that there are now rising intellectual rock stars. There's a group of individuals somebody gave an umbrella name that sound very ominous and stupid, but the name is the 'Intellectual Dark Web'. There's no formal connection between them. You can maybe see them as a movement about the search for truth and freedom of speech. These are guys like Sam Harris, Jordan Peterson, Bret Weinstein, Eric Weinstein, etc. These are all professors who advocate for freedom of speech. For some people when they hear these names like Jordan Peterson, they say, 'isn't he that guy that's alt right?' Not at all, but they are wildly misportrayed in a lot of the traditional media. These are such creative, clear thinkers in our day and age that they really deserve a platform and a large audience. They have found a large audience within their niche, so in that sense they are celebrities, stars, heroes. But they are not known to a larger audience. For me these are really inspiring figures how they try to dissect and analyse everything that's going on, to make a stand. They are about something. But also how they do it, in a dialogue in the search for truth, using the tools of science in a very appropriate way. It's about the body of knowledge, the weight of the evidence, many studies, meta analysis - only then can you say that a certain signal might arise from all those studies. They are very careful and clear thinkers and they talk with the science in mind about certain topics. From an intellectual point of view they are my heroes.
LBB> And when you're not researching happiness, what do you like to do in your spare time? Any other obsessions?
Moos> Next is music, that's a lot of my time. And like I said I devour anything that explains the world around us. And to give you a sense of how diverse it is (and it might sound a bit pretentious), I'm now doing a course on the science of information with a mathematical strand. I'm also reading about the science of people in the workplace, how people can best work together and collaborate and make beautiful stuff. I'm also putting together a book which is a bundle of essays from leading neuroscientists on the latest in brain research. All that, with a cup of coffee, I'm a happy man.
'Gelukkig Maar' will be launched March 12th in the A'DAM Toren in Amsterdam and will be available online through De Kleine Pauw.
view more - 5 minutes with...
MassiveMusic Amsterdam, Fri, 15 Feb 2019 17:16:21 GMT
MassiveMusic Amsterdam
Fri, 15 Feb 2019 17:16:21 GMT
News from MassiveMusic Amsterdam
MassiveMusic Expands to Dubai
Music and Sound and Its Added Value to the Moving Picture
MassiveMusic and Media.Monks Invite Creative Industry to unMMMMute Themselves at Cannes Lions
MassiveMusic Develops an 'Out There' Sonic Logo for Kathmandu
Work from MassiveMusic Amsterdam
Torres Blancas
NEOM THE LINE - New Wonders for the World
NEOM | THE LINE - New Wonders for the World
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{"url":"http:\/\/skywired.net\/blog\/","text":"## Worst case design and the BITX 40 mic amp\n\nThe BITX 40\u2019s microphone amp has a flaw that may shorten the lifespan of microphones by overstressing their components. Here\u2019s what I found and how I fixed it so far.\n\nHere\u2019s the BITX 40 mic amp schematic:\n\nThe mic bias is supplied by the TX line\u00a0through 4.9K of resistance. TX is the DC power supply, 12-15V.\n\nMost electret microphones have a maximum voltage of 10 V, though I\u2019ve seen a few datasheets with a 9 V limit. The data sheets also say that the mics draw 0.5 mA maximum. No minimum is given.\n\nNow, applying Ohm\u2019s law,\n\n$(0.5 \\mathrm{mA}) * (4.9 \\mathrm{k\\Omega}) = 2.45 \\mathrm{V}$\n\nSubtracting that from 12 V gives us 9.55 V, so one might think that everything is great.\n\nNot really.\n\nFrom the books of Bob Pease I learned the importance of worst case design. In worst case design, one looks at the datasheet minimum and maximum\u00a0specifications and designs the circuit to work under the worst combination of specifications. Ignore the datasheet typical ratings, because those describe the best case. No one wants a circuit that only works in the best possible circumstances.\n\nThe BITX 40 docs specify a maximum power supply of\u00a015 V, so that is the worst-case TX. The worst case microphone current is\u00a0the minimum current, but I haven\u2019t seen a mic datasheet that lists a minimum. Therefore, I have to make a guess. I need the guess to be\u00a0on the safe side without being ridiculous. Knowing that the current is set by a JFET and knowing about both the variability of JFETs and how data sheet maximums are chosen, I could guess that the typical current is maybe half of the maximum and the minimum half of that, so 0.125 mA. If I want to make a truly robust circuit I\u00a0could assume a minimum current of zero. I know a JFET can\u2019t create current out of nothing, so the lowest possible current it draws is zero.\n\nLet\u2019s look at what voltages that creates:\n\n$(15 V) - (0.125 \\mathrm{mA})(4.9 \\mathrm{k\\Omega}) = 14.388$\n\nUh-oh. That\u2019s well above the 10 V maximum for the\u00a0mic.\n\nThe zero current case is easier\u00a0to work out. With zero current through the resistor, the mic sees the full 15 V.\n\nA simple fix is to put a resistor in parallel with\u00a0MIC1, making\u00a0a voltage divider. I use\u00a0a computer headset with my BITX, and I know that computer mics are typically fed with 5 V through a 2.2 k\u03a9 resistor. Knowing that, I picked a 4.3 k\u03a9 resistor across MIC1 to form a voltage divider equivalent to 2.3 k\u03a9 fed by 7 V. That\u2019s close enough, and 7 V is well below the maximum for most electret elements.\n\nUnfortunately, the mic\u00a0now drives an impedance about 30% lower than before, reducing\u00a0its output. The RF drive, R136, should be adjusted to compensate.\n\nOne could\u00a0fix the issue a different way, of course. A Zener diode in series with R121 could drop the voltage just as effectively without lowering the amplifier\u2019s input impedance. An LM780x regulator could do the job, too, albeit with more components. I happened to have 1% resistors within reach and Zener diodes on a different floor of the house. I chose the resistor.\n\n### Is this a\u00a0real problem?\n\nNo one else seems to have noticed this issue, so it\u2019s worth asking\u00a0whether it matters. One answer is that it is worthwhile, and pleasing, to do things right. There is a place for bodging a circuit together that works as a one-off, but I enjoy the n-dimensional puzzle of getting the details right.\n\nThe other answer is that it might be killing microphones, but not often enough that\u00a0anyone has\u00a0noticed. The BITX 20 mailing list, home to BITX 40 discussion, has seen a few comments from hams whose microphone capsules worked for a while, then failed. Some have been able to trace the problems to bad solder joints or physical damage, but I have to wonder if any of the\u00a0remaining unsolved cases were\u00a0caused by overvoltage.\n\nManufacturer maximum specifications are based on an assumption about the device\u2019s expected lifetime. Operating beyond that specification can be expected to shorten the device\u2019s lifetime. Sometimes that shortened lifetime is dramatic, with a flash of light, a puff of smoke, or an outpouring of heat. Other times it is subtle and takes longer. Perhaps mic elements will fail faster than usual. Perhaps their average lifespan will be 2 years instead of 20.\n\nThe difference\u00a0might not be enough to notice, but we can fix it anyway. Worst case design will save the day.\n\n## Replacing the BITX 40 BFO\n\nThe BITX 40 is all about modifications. The design itself is \u201ccheap and cheerful\u201d, and the circuit board is laid out to invite changes and experiments. I have a list of quirks I intend to fix in mine, and with that the mods began.\n\nFirst up was a misalignment of the BFO. Each BITX 40 uses a set of 5 matched crystals, four for the IF filter and the fifth for the BFO. The BFO is set up to pull the crystal frequency slightly to put the audio passband in the right place, or at least it is supposed to. On mine, the BFO frequency was just inside the IF passband.\n\nHaving the BFO in the wrong place had several consequences. First, receive audio was bassy, running from about 0 Hz to 1800 Hz. This transmits audio frequencies that are not useful for communications and omits the ones around 2 kHz that are especially important. Worse, the passband\u00a0actually stretch\u00a0below 0 Hz, into the upper sideband, which meant that my transmitter was not suppressing the carrier. It was sending VSB, vestigial sideband, not SSB. That may be\u00a0fine if you\u2019re a TV transmitter, but\u00a0it\u2019s not the kind of clean SSB signal hams expect.\n\nOne solution, which Wayne NB6M used\u00a0on his BITX 40, is to change the \u201cpulling\u201d capacitor in the BFO to put the frequency where it belongs. Instead, I\u00a0replaced the BFO entirely with a spare channel on the Si5351 frequency synthesizer.\n\nI started off by attaching wires to pins 8 and 9 of the Raduino board. These are Si5351 channel 0 and ground, respectively. For quick progress, I used a twisted pair. When I box it up, I will switch to coax and use a connector to make maintenance easier.\n\nAt the other end, I attached the wires to pins 1 and 6 of T4. This supplies the BFO to the second mixer.\n\nFinally, I unsoldered\u00a0R101 and C106 to remove power from the analog BFO and disconnect it from the second mixer.\n\nWith the hardware work done, I turned to the Arduino code. I downloaded Ashhar Farhan\u2019s original bitx40 sketch and added one line of code near the bottom of setup().\n\nsi5351.set_freq(bfo_freq * 100ULL, SI5351_CLK0);\n\nThen I turned it on, saw that the BFO was now too far above\u00a0the IF passband, and with a couple of experiments, came up with the following edit near the top of the sketch:\n\n#define INIT_BFO_FREQ (11997000L)\nunsigned long baseTune = 7100000L;\nunsigned long bfo_freq = INIT_BFO_FREQ;\n\nWith that, I was done. The rig sounds better and works better. I still haven\u2019t transmitted, though. That will take one more mod which I will write about next.\n\n## BITX 40!\n\nI\u2019ve decided to build a BITX 40. This petite SSB transceiver sells for a mere \\$59, some assembly required. As it comes, it puts out approx. 7W, and with some straightforward upgrades\u00a0it can produce 25W. It comes as a fully-assembled PCB plus most of the parts needed to hook it up. A case, speaker, and a few other incidentals are not included. The board and radio are designed to invite hacking and customization. It is also designed to serve as an introduction to homebrewing for hams who may not be ready to build a radio from scratch. For more information on the BITX-40, see its supplier\u00a0http:\/\/www.hfsigs.com\/\u00a0and the active and helpful support community at groups.io.\n\nMy BITX 40 is operable \u201cal fresco\u201d on my workbench. The included Arduino\/Si5351 VFO works fine and tunes the full 40m band. It needs a little bit of work before I can transmit with it. It\u2019s important to understand that the BITX 40 is not a turnkey rig. Many of them ship with small flaws, and figuring out and fixing the flaws is part of the fun. (This is why that helpful community at groups.io is so important.) Mine shares a flaw with Wayne, N6BM\u2019s BITX 40 \u2014 the BFO frequency is inside the IF passband, instead of about 300 Hz below the passband like it should. This means that stations I tune sound bassy, I can hear part of the opposite sideband, and when I transmit, the carrier is not suppressed.\n\n(I suppose I could call it \u201cvestigial sideband\u201d, like what NTSC TV uses, but it\u2019s supposed to be SSB\u2026)\n\nWayne fixed the problem by changing a capacitor in the BFO circuit to pull the frequency where it needed to be. I plan to fix it by disabling the analog BFO and instead\u00a0use a spare\u00a0Si5351 output. Having a tunable VFO will let me put the passband exactly where I want. With a little more code, it will also give me\u00a0passband tuning.\n\nI think I found a bug in the microphone amp that I\u2019m going to take a closer look at, and I found a perfect case at the Mike and Key hamfest in Puyallup last month. I didn\u2019t even try to negotiate the price.\u00a0It was free!\n\nI\u2019ll have more on my BITX 40 project in upcoming posts.\n\n## Antenna in the air\n\nI finally have an antenna in the air. After a lot of thought about the shape of our lot, the placement of trees, and how to avoid obstructing my children\u2019s use of the yard, I settled on an end-fed halfwave, the LNR Precision EF-10\/20\/40 Mk II.\n\nThe main challenge of the installation was minimizing the ground area obstructed. I have two small children who like to run and play, so guy lines spanning the yard is out of the question. Similarly, I don\u2019t want to set up an inverted-V anchored at the ground. I solved the problem by sloping it between two trees. The high end hangs about 40 feet up from a cedar, and the low end is 8 feet up in a pine. (It\u2019s not for nothing that Washington is the Evergreen State.) Both are right by our property lines, so the entire assembly, including the feed line, are out of the way.\n\nI tuned it with a signal generator and my trusty Oak Hills Research QRP power meter. The SWR bandwidth is broader than I expected. I\u2019ll see how it holds up with higher power output \u2014 it\u2019s always possible my generator\u2019s output is too low for an accurate reading.\n\nI can\u2019t wait to try it on the air.\n\n## The Sound-Following Robot\n\n\u201cWhat do you want to get Nick for Christmas?\u201d \u201cA robot!\u201d My daughter was adamant that her big brother needed a robot for Christmas. I hit the \u2018net, and found many robots. I wanted to give him something that actually worked, that was educational, and that was affordable. Because watching Dad solder is no fun, I also ruled out anything that required soldering. That narrowed it down to two choices. \u201cWhich one do you want to give to Nick, Sally?\u201d I asked, and started to explain their relative merits. \u201cTHE BLUE ONE!\u201d And so it was decided that Nick and I would build an Elenco Sound-Following Robot. The kit is marketed for kids 13 and up. Nick is still in kindergarten, but loves anything motorized or electronic, so we did it together. The manual used exploded assembly diagrams to show how the pieces went together. Few words are used, and some skill in interpreting drawings is helpful in figuring out the steps. In other words, this was very much a collaborative effort for us. I figured out what went where, he found the parts, and we jointly did the assembly. Though one could probably do the project alone, there certainly were times when it helped to have four hands. The robot is built around a drive train, the pieces for which are separately bagged. This suggests it\u2019s common to other robot kits. Though the assembly instructions were clear, the gears were fiddly to assemble. At one point, we managed to undo almost all of our previous work when we tried to slip the last gear onto its shaft. No matter; we were able to put it back together. At this point in the assembly, with a drivetrain and a bottom plate, the possibilities of hacking this robot became clear. The controller for the robot comes preassembled. The IC on top of the board is a microcontroller. It is house-labelled for Elenco, but I suspect it\u2019s a standard part with custom firmware. The manual has a schematic, so I compared the pinout to PICs and AVRs without finding a match. Maybe it\u2019s a 68HC05 or another small micro. The assembly drawing pictured a socket under the microcontroller, but no such luck on this production unit. (By this time, I was started to get intrigued by the modification possibilities for this little robot.)\n\nWe ran into our first glitch when a lead broke off of the speaker. A trip to Radio Shack turned up another speaker that could be trimmed to fit.\n\nThe \u201chead\u201d of the robot has its own PCB ringed by four microphones to detect sound. Two LEDs serve as \u201ceyes\u201d and give feedback on the direction of sound detected by the robot.\n\nThat\u2019s one happy boy playing with his new robot. His mother is a little less excited about it. A sound-following robot likes it noisy. Clap! Clap! Clap! Clap!\u00a0Clap! Clap! Clap! Clap!\u00a0Clap! Clap! Clap! Clap!\u00a0Clap! Clap! Clap! Clap!\u00a0Clap! Clap! Clap! Clap!\u00a0Clap! Clap! Clap! Clap!\u00a0Clap! Clap! Clap! Clap!\u00a0Clap! Clap! Clap! Clap!\u00a0Clap! Clap! Clap! Clap!\u00a0Clap! Clap! Clap! Clap!\u00a0Clap! Clap! Clap! Clap!\u00a0Clap! Clap! Clap! Clap!\u00a0Clap! Clap! Clap! Clap!\u00a0Clap! Clap! Clap! Clap!\u00a0Clap! Clap! Clap! Clap!\u00a0Clap! Clap! Clap! Clap!\u00a0Clap! Clap! Clap! Clap!\u00a0Clap! Clap! Clap! Clap! \u2026.\n\n## pQRP\n\nLast night I attended a very enjoyable gathering of the Pacific Northwest QRP (pQRP) group. I\u2019ve been wishing for some time to find a group of like-minded hams. Like-minded, that is, in the sense that they are interested in design and building as well as operation. Imagine my pleasure at hearing from not one, but four hams involved in design projects as well as others enjoying the experimental side of the hobby, even down to one gentleman who is testing an add-on heat sink for his Elecraft KX-3.\n\nSeveral members of the club are involved in designing and building a variant of Ashhar Farhan\u2019s \u201cMinima\u201d transceiver. They are equipping it with Si570 oscillators and a comfortable user interface, while trying to stay close to the original spirit of the radio. A\u00a0potentiometer is still the interface for frequency control, for example, instead of a rotary encoder.\n\nOther members of the club are active in operating, including SOTA, RTTY contests, and other pursuits.\n\nAs show and tell, I brought along my NorCal 40A, which neither tunes as widely as it should nor outputs full power. So far it has stumped me. The consensus of the group was to post about it to the club mailing list and see who has some ideas to try.\n\nI\u2019m already looking forward to next month\u2019s meeting.\n\n## Digging up a Dinosaur \u2013 a Boatanchor at the Children\u2019s Museum\n\nLast weekend, I took my children to the Imagine Children\u2019s Museum in Everett, WA. On the roof, my daughter found this National NC-173 as part of a \u201cdinosaur dig\u201d exhibit. It may have been gutted. At the very least, the controls are certainly not original! The colorful, kid-resistant buttons cause a speaker to play a variety of recorded messages, supposedly radio communications with a base camp.\n\nIt has been painted so many times, I could hardly identify it. Through the paint, I found an engraved \u201c80\u201d, \u201c40\u201d, and \u201c10-20\u201d by what may have been a band switch. Peering harder, I could just make out the model number.\n\nI\u2019m not a boatanchor guy, and some might bemoan the loss of a historic radio. It made me smile to find it, though, and my daughter enjoyed playing with it. That\u2019s enough for me.\n\nWhen I brought my children to the Imagine Children\u2019s Museum (Everett, WA) last weekend, they showed me the \u201cdinosaur dig\u201d exhibit on the roof. There we found this National NC-173 receiver, of Kon-Tiki fame. It looks like it has been gutted. The controls certainly aren\u2019t original. The brightly-colored, kid-resistant buttons cause it to play recorded messages from a \u201cbase camp\u201d to the pretend-paleontologists at the dig.\n\nI had to peer through layers of paint to read the engravings and identify it. The best remaining marking is \u201c80\u201d, \u201c40\u201d, and \u201c10-20\u201d by what may have been a band switch. I could barely find the model number under all the paint.\n\nI\u2019m not a boatanchor guy, and I understand that some might bemoan the loss of a classic receiver. My daughter enjoyed playing with it, though, and it made me smile to find it, so it\u2019s all right with me.","date":"2018-09-25 22:00:56","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 2, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4018210172653198, \"perplexity\": 2495.1633364133645}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-39\/segments\/1537267162385.84\/warc\/CC-MAIN-20180925202648-20180925223048-00364.warc.gz\"}"}
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{"url":"http:\/\/cie.co.at\/eilvterm\/17-27-010","text":"# 17-27-010\n\nlighting system\n\nsystem designed to provide lighting\n\nNote 1 to entry: A lighting system can be dedicated to\na. the support of one or more specified visual tasks under specified conditions considering other requirements such as human comfort, safety, the appearance of the surrounding environment and energy consumption;\nb. the support of other than human tasks.\n\nNote 2 to entry: A lighting system can include a set of light sources, other physical components, communication protocols, user interfaces, software and networks to provide control and monitoring functions.\n\nNote 3 to entry: The light source(s) and the related equipment can be integrated in a single item, e.g. an LED module, a lamp or a luminaire.\n\nNote 4 to entry: A lighting system can be networked to provide central or remote control and monitoring functions.\n\nNote 5 to entry: A lighting system can be connected to or integrated with other systems or devices.\n\nPublication date: 2020-12","date":"2021-04-19 00:31:38","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9017994999885559, \"perplexity\": 3383.2174968719855}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618038862159.64\/warc\/CC-MAIN-20210418224306-20210419014306-00179.warc.gz\"}"}
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Andrena helenica är en biart som beskrevs av Warncke 1965. Andrena helenica ingår i släktet sandbin, och familjen grävbin. Inga underarter finns listade i Catalogue of Life.
Källor
Sandbin
helenica
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
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