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\section{Introduction}
\label{sec:introduction}
Two-sample hypothesis testing of sparse spatial data is a fundamental problem in a wide variety of scientific applications. It manifests in numerous forms. One example is to compare the cerebral white matter tracts between multiple sclerosis (MS) patients and healthy controls \citep{Goldsmith2011}. The data records the fractional anisotropy measure along the right corticospinal tract, and takes the form of one-dimensional (1D) function. The scientific interest is to compare two sets of fractional anisotropy profiles and locate the tract regions that distinguish cases from controls. Another example is to compare the brain grey matter cortical thickness between subjects diagnosed with attention deficit hyperactivity disorder (ADHD) and typically developing controls \citep{ADHD200}. The data records the volume of grey matter at different brain locations in a three-dimensional (3D) space. The scientific interest is to compare two sets of brain structural images and identify differentiating brain regions. In addition to the above examples, similar problems arise in many other applications, for instance, astronomical surveys \citep{Czakon2009}, disease mapping \citep{Sun2000}, ecology \citep{MauricioBini2009}, and genomics \citep{SunWei2011}.
All these examples can be formulated as a two-sample testing problem, where the data reside in some spatial domain. More specifically, let $\mathbb{S} \subset \mathbb{R}^b$ denote a $b$-dimensional spatial domain, where $b=1,2,3,\ldots$. Let $\mathcal{S} \subset \mathbb{S}$ denote a finite, regular lattice in $\mathbb{S}$, and $\boldsymbol{s} \in \mathcal{S}$ the coordinate of the location. For the 1D MS example, $b=1$ and $\boldsymbol{s}$ is a scalar, whereas for the 3D ADHD example, $b=3$ and $\boldsymbol{s}$ is a three-variate coordinate. Later in our theoretical analysis, we consider the infill-asymptotic framework \citep{stein1999} and assume $\mathcal{S} \to \mathbb{S}$. Suppose the data $Y_d(\boldsymbol{s}) \in \mathbb{R}$ is observed at every location $\boldsymbol{s} \in \mathcal{S}$ for two groups $d=1,2$, and write $\boldsymbol{Y}_d = \{Y_d(\boldsymbol{s}) : \boldsymbol{s} \in \mathcal{S}\}$. Suppose $\boldsymbol{Y}_d$ follows a probability distribution $\mathcal{P}_{\boldsymbol{\beta}_d,\boldsymbol{\eta}_d}$, where $\boldsymbol{\beta}_d=\{ \beta_{d}(\boldsymbol{s}) : \boldsymbol{s} \in \mathcal{S} \}$ denotes the parameters of interest, and $\boldsymbol{\eta}_d$ collects all the nuisance parameters. Suppose we observe two sets of independent samples, $\{\boldsymbol{Y}_{i,d} \}_{i=1}^{n_d}$, where $n_d$ is the sample size for group $d$, $d=1,2$. Our goal is to carry out multiple hypothesis testing given the observed data,
\vspace{-0.01in}
\begin{equation} \label{eqn:hypothesis}
H_0(\boldsymbol{s}): \; \beta_{1}(\boldsymbol{s}) = \beta_{2}(\boldsymbol{s}) \quad \mbox{versus} \quad H_1(\boldsymbol{s}): \; \beta_{1}(\boldsymbol{s}) \neq \beta_{2}(\boldsymbol{s}), \quad \boldsymbol{s} \in \mathcal{S}.
\end{equation}
{We call the location $\boldsymbol{s}$ a signal location or a non-null location if $H_1(\boldsymbol{s})$ holds, and call it a null location otherwise.} We comment that \eqref{eqn:hypothesis} covers a range of testing problems. In this article, we mostly illustrate with the problem of comparing two multivariate means, where $\boldsymbol{\beta}_d$ represents the mean of $\boldsymbol{Y}_d$. Meanwhile, our proposal is equally applicable to the problems of comparing differential networks \citep{ChenKang2015}, or detecting gene-environment interactions \citep{Caspi2006}. Next, we recognize that, since the data resides in a spatial domain, there exists naturally some form of spatial smoothness in the data. Additionally, there is likely sparsity in \eqref{eqn:hypothesis}, in that the alternative hypothesis holds at only a small subset of locations $\boldsymbol{s}$ in the entire $\mathcal{S}$. Sparsity is a common phenomenon in scientific applications and is frequently encountered in multiple testing. The goal of this article is to effectively incorporate both spatial smoothness and sparsity information into the multiple testing problem \eqref{eqn:hypothesis}, while controlling the false discovery rate (FDR) and improving the power of the test.
Both smoothness and sparsity can be viewed as some forms of side information, and there have been a large number of proposals to incorporate side information in multiple testing; see \citet{CaiSun2017} for a review and references therein. In this article, we focus on the strategy of $p$-value weighting, which has been widely used for FDR control and power enhancement \citep[among many others]{BenHoc97, Storey2002, Genovese2006, Roeder2009}. In particular, \cite{Hu2010} adopted the prior knowledge that the hypotheses belong to a known number of groups, and weighed the $p$-values for the hypotheses in each group by $\pi_g / (1-\pi_g)$, where $\pi_g$ is the non-null proportion for the group that needs to be estimated. \cite{Zhang2011} proposed to smooth and aggregate the $p$-values in a local neighborhood to accommodate the spatial information of the neighboring $p$-values. \citet{Liu2014} utilized the sparsity information in the mean vectors and developed an uncorrelated screening-based FDR control procedure. \cite{Li2019} weighed the $p$-values by a heterogeneous weight $1/\{1-\hat\pi(s)\}$, where $\hat\pi(s)$ is the estimated probability of a hypothsis being a non-null. \cite{Ignatiadis2016,LeiFithian2018,lei2021general} incorporated generic side information through secondary data or external covariates, and constructed the $p$-value-based thresholding procedures adaptively. \citet{GAP} proposed the grouping, adjusting, and pooling (GAP) method that exploits the sparsity information, where they adaptively constructed a set of auxiliary statistics, based on which they identified clusters of hypotheses, then weighed the $p$-values by some discrete group-wise weights. \citet{LAWS} proposed the locally adaptive weighting and screening (LAWS) method that utilizes the smoothness information, where they constructed a set of robust and structure-adaptive weights based on the estimated local sparsity levels, then weighed the $p$-values by these continuous weights.
In this article, we propose a \textbf{n}eighborhood-\textbf{a}ssisted and \textbf{p}osterior-\textbf{a}djusted (NAPA) test procedure for the two-sample multiple testing problem \eqref{eqn:hypothesis}, whereas we aim to control the FDR and improve the power by incorporating both spatial and sparsity information. Our key idea is to translate the spatial and sparsity information to construct a set of weights to adjust the $p$-values, where the spatial pattern is encoded by the ordering of the locations, and the sparsity structure is encoded by a set of auxiliary statistics. Specifically, we first construct a set of test statistics $\{T(\boldsymbol{s}) : \boldsymbol{s} \in \mathcal{S}\}$, which contain useful information about the true signal locations $\theta(\boldsymbol{s}) = \id \{\beta_{1}(\boldsymbol{s})\neq \beta_{2}(\boldsymbol{s})\}$, and $\id(\cdot)$ is the indicator function. We then construct a set of auxiliary statistics $\{U(\boldsymbol{s}) : \boldsymbol{s} \in \mathcal{S}\}$, which take the form of $\beta_1(\boldsymbol{s}) + \kappa(\boldsymbol{s}) \beta_2(\boldsymbol{s})$ for some weight function $\kappa(\boldsymbol{s})$, and contain useful information about $\gamma(\boldsymbol{s}) = \id\left\{ \beta_{1}(\boldsymbol{s})\neq 0 \ \mbox{or}\ \beta_{2}(\boldsymbol{s})\neq 0 \right\}$. We note that $\gamma(\boldsymbol{s})=0$ implies $\theta(\boldsymbol{s})=0$. This implication means that, if both $\beta_1(\boldsymbol{s})$ and $\beta_2(\boldsymbol{s})$ are zero, then the null hypothesis must be true, and at least one of $\beta_1(\boldsymbol{s})$ and $\beta_2(\boldsymbol{s})$ must be nonzero for the alternative hypothesis to hold. In other words, the locations where $\gamma(\boldsymbol{s})=1$ capture useful information about the locations of the true signals. Moreover, the location $\boldsymbol{s}$ itself contains useful information about the smoothness embedded in the data. Therefore, we propose the {posterior non-null probability}, $\pi(\boldsymbol{s}, u) = \Pr\{\theta(\boldsymbol{s})=1|U(\boldsymbol{s})=u\}$, and weigh the $p$-value by the weight, $w(\boldsymbol{s},u) = \pi(\boldsymbol{s},u) / \{1 - \pi(\boldsymbol{s},u) \}$, which integrates both the spatial information encoded in $\boldsymbol{s}$ and the sparsity information encoded in $U(\boldsymbol{s})$. Recognizing that $\pi(\boldsymbol{s}, u)$ is a continuous function of $\boldsymbol{s}$ and $u$, we propose to estimate this posterior weight by pooling the information from its neighbors using a smoothing kernel approach. The neighborhood is defined through both $\boldsymbol{s}$ and $U(\boldsymbol{s})$, in that a close spatial location in $\boldsymbol{s}$ and a similar value of $U(\boldsymbol{s})$ both indicate a similar likelihood for the hypothesis at $\boldsymbol{s}$ to be null or alternative. Finally, given the estimated posterior weights, we choose a proper threshold to adjust the multiplicity for FDR control, and further show that the new test enjoys some guaranteed power improvement.
Our proposed NAPA test is built on the recent proposals of multiple testing utilizing the side information. Particularly, our test combines the ideas of the GAP method of \citet{GAP} and the LAWS method of \citet{LAWS}. On the other hand, it is far from a straightforward extension, and is substantially different from both GAP and LAWS, as well as a simple combination of the two. More specifically, GAP translates the sparsity information embedded in the auxiliary statistic into several discrete groups, then constructs a weight for each group. However, the number of groups is generally unknown, and searching among all possible groupings is computationally expensive. Besides, such a discretization may lead to potential power loss. By contrast, our method does not weigh the $p$-values by groups, but instead weigh in a continuous fashion, which is computationally more efficient and can further improve the power. This new weighting strategy, nevertheless, induces new challenges. A key result in GAP, such that the grouping using auxiliary statistic does not distort the null distribution of the $p$-values, is no longer sufficient in our setting. To employ the auxiliary sequence continuously and to ensure the statistical properties, we derive a new conditional normal approximation, a result that is not available in the literature. Next, LAWS focuses on the one-sample testing problem and assumes that $\theta(\boldsymbol{s})$ is fully determined by the location $\boldsymbol{s}$, but ignores additional sparsity information. By contrast, we introduce an auxiliary variable $U(\boldsymbol{s})$ that is constructed adaptively from the data, and explore the posterior binomial variable $\theta(\boldsymbol{s}) | U(\boldsymbol{s})$. The new approach, nevertheless, leads to a more involved theoretical development. This is because we have to tackle the correlations among the spatial locations and the covariates, as well as the dependencies among the primary and auxiliary statistics, and we derive new and sophisticated technical tools to address those challenges. Finally, our proposed test is far from a simple combination of GAP and LAWS. In Section \ref{sec:simulations}, we carry out a simulation experiment and show that our new test is much more powerful than naively combining GAP and LAWS. In summary, we believe our proposal fills an important gap in two-sample inference that utilizes both spatial and sparsity information, and thus helps address a range of scientific questions in areas such as neuroimaging analyses that involve spatial data. Moreover, we develop new technical tools that are potentially useful for general inference problems of complex dependent data.
The rest of the article is organized as follows. Our proposed weight $w(\boldsymbol{s},u)$ and the posterior probability $\pi(\boldsymbol{s}, u)$ hinge on both the location $\boldsymbol{s}$ and the auxiliary statistic $U(\boldsymbol{s})$. In Section \ref{sec:oracle}, we first study this weighting scheme in the oracle setting, where $\pi(\boldsymbol{s}, u)$ is known, the $p$-value obtained from the test statistic $T(\boldsymbol{s})$ is uniformly distributed, and $U(\boldsymbol{s})$ is independent of $T(\boldsymbol{s})$ under the null. We establish the guaranteed power gain under this setting. In Section \ref{sec:napa}, we estimate the weight using smoothing kernel, and develop a multiple testing procedure based on the weighted $p$-values. We further illustrate the construction of the test and auxiliary statistics $\{T(\boldsymbol{s}), U(\boldsymbol{s})\}$ in a multivariate mean comparison problem. In Section \ref{sec:theory}, we show that our weight estimator is consistent, and the proposed test controls the FDR asymptotically. We further use the mean comparison as an example and show that the $p$-value from $T(\boldsymbol{s})$ is asymptotical uniform, and $T(\boldsymbol{s})$ and $U(\boldsymbol{s})$ are asymptotically independent under the null. In Section \ref{sec:simulations}, we study the finite-sample performance, and in Section \ref{sec:realdata}, we illustrate with two neuroimaging data examples. We relegate all technical proofs and additional simulation results to the Supplementary Appendix.
\section{Oracle Weighting and Power Gain}
\label{sec:oracle}
In this section, we first derive and motivate our proposed weight function. We then study the power gain under the oracle setting.
\subsection{Posterior-adjusted weighting}
\label{subsec:weight}
Define the local non-null probability $\pi(\boldsymbol{s})$ at location $\boldsymbol{s}$ as $\pi(\boldsymbol{s}) = \Pr\{\theta(\boldsymbol{s})=1\}$. Define the local posterior non-null probability $\pi(\boldsymbol{s},u)$ at $\boldsymbol{s}$ and $U(\boldsymbol{s})=u$, and the weight function as,
\begin{equation} \label{eqn:pi}
\begin{split}
\pi(\boldsymbol{s},u) & = \Pr\{ \theta(\boldsymbol{s})=1 | U(\boldsymbol{s})=u \} = \frac{\pi(\boldsymbol{s}) \, q_{1}(u|\boldsymbol{s})}{\{ 1-\pi(\boldsymbol{s}) \} \, q_{0}(u|\boldsymbol{s}) + \pi(\boldsymbol{s}) \; q_{1}(u|\boldsymbol{s})}, \\
w(\boldsymbol{s},u) & = \frac{\pi(\boldsymbol{s},u)}{1 - \pi(\boldsymbol{s},u)},
\end{split}
\end{equation}
where {$q_{0}(\cdot | \boldsymbol{s})$ and $q_{1}(\cdot | \boldsymbol{s})$ are respectively the null $(\theta(\boldsymbol{s}) = 0)$ and non-null $(\theta(\boldsymbol{s}) = 1)$ density functions of $U(\boldsymbol{s})$}.
We call $\pi(\boldsymbol{s},u)$ a posterior probability as it is conditioning on the auxiliary variable $U(\boldsymbol{s})$. We then propose to weigh the $p$-value as,
\begin{equation} \label{eqn:weighted-p-value}
p_w(\boldsymbol{s}) = \min\left\{\frac{p(\boldsymbol{s})}{w(\boldsymbol{s},u)}, \; 1\right\} = \min\left\{\frac{1-\pi(\boldsymbol{s},u)}{\pi(\boldsymbol{s},u)}p(\boldsymbol{s}), \; 1\right\}.
\end{equation}
To motivate our weighting scheme, note that
\begin{equation*}
T(\boldsymbol{s})|U(\boldsymbol{s}) \ {\sim} \ f(t|\boldsymbol{s},u)=\{1-\pi(\boldsymbol{s},u)\}f_0(t|\boldsymbol{s},u)+\pi(\boldsymbol{s},u)f_1(t|\boldsymbol{s},u),
\end{equation*}
where $f_0$ and $f_1$ are respectively the {null and non-null conditional density functions} of $T(\boldsymbol{s})$ given $U(\boldsymbol{s})$. Under the oracle setting, we have $f_0(t|\boldsymbol{s},u) = f_0(t|\boldsymbol{s})$. Following \cite{CARS}, when the tests are independent, the optimal test threshold for the above model is based on the ranking of the conditonal local false discovery rate,
\begin{equation*}
\mathrm{CLfdr}(t|\boldsymbol{s},u) = \Pr\{\theta(\boldsymbol{s})=0|t,\boldsymbol{s},u\}=\frac{\{1-\pi(\boldsymbol{s},u)\}f_{0}(t|\boldsymbol{s})}{f(t|\boldsymbol{s},u)}.
\end{equation*}
To extend to the setting of dependent tests, we consider to weigh the $p$-value to approximate CLfdr, which equals $\varphi/ (\varphi +1)$ and is monotonic in $\varphi$, where
\begin{equation*}
\varphi = \frac{1-\pi(\boldsymbol{s},u)}{\pi(\boldsymbol{s},u)} \times \frac{f_0(t|\boldsymbol{s})}{f_1(t|\boldsymbol{s},u)}.
\end{equation*}
Note that the first term in $\varphi$ is the inverse of the weight $w(\boldsymbol{s},u)$, whereas the second term in $\varphi$ reflects the strength of the evidence against the null. On the other hand, it is difficult to directly estimate $f_1(t|\boldsymbol{s},u)$ simultaneously for all locations. Instead, we turn to the $p$-value that is obtained from the test statistic $T(\boldsymbol{s})$, which carries similar information of the null versus alternative hypothesis, to replace the second term in $\varphi$. This leads to our proposed weighted $p$-value in \eqref{eqn:weighted-p-value}.
\subsection{Theoretical improvement}
\label{subsec:ranking}
We next compare our testing method with LAWS of \cite{LAWS} under the oracle setting, and the comparison is illustrated in a simple setting where $U(\boldsymbol{s})$ is independent of $T(\boldsymbol{s})$ under the alternative, i.e., $f_1(t|\boldsymbol{s},u) = f_1(t|\boldsymbol{s})$. The two methods mainly differ in the weight used for the $p$-value.
We use $w(\boldsymbol{s}, u)$ in \eqref{eqn:pi}, whereas LAWS uses $w(\boldsymbol{s}) = \pi(\boldsymbol{s}) / \{1 - \pi(\boldsymbol{s}) \}$. Under some mild conditions, we show our method that incorporates both spatial and sparsity information is guaranteed to improve the power over LAWS that only utilizes the spatial information. We do not analytically compare with GAP of \cite{GAP} here, because GAP uses a discrete weighting scheme and is not directly comparable. Nevertheless, we numerically compare with both LAWS and GAP in Section \ref{subsec:fdr-power}.
We first formally define the evaluation criteria in terms of false discovery and power. Consider a sequence of weighted $p$-values $\{ p_w(\boldsymbol{s}): s \in \mathcal{S}\}$ and a given threshold $t$. When there is no weighting, we set all weights equal to one. Let $\delta^w(\boldsymbol{s}, t) = \id \left\{ p_w(\boldsymbol{s}) \leq t \right\}$ denote the decision rule for the hypotheses in \eqref{eqn:hypothesis}, in that $\delta^w(\boldsymbol{s}, t) = 1$ if we reject the null, and $\delta^w(\boldsymbol{s}, t) = 0$ otherwise. Let $\boldsymbol{\delta}^w_t = \{\delta^w(\boldsymbol{s}, t) : \boldsymbol{s} \in \mathcal{S}\}$ be the collection of all decision rules for $\boldsymbol{s} \in \mathcal{S}$ under the threshold $t$. We define the FDR and the marginal FDR (mFDR) of the test $\boldsymbol{\delta}^w_t$ as,
\begin{equation*}
\mathrm{FDR}(\boldsymbol{\delta}^w_t) = \ep\left(\frac{\sum_{\boldsymbol{s} \in \mathcal{S}}\left[\{1-\theta(\boldsymbol{s})\} \delta^w(\boldsymbol{s}, t)\right]}{\max \left\{\sum_{\boldsymbol{s} \in \mathcal{S}} \delta^w(\boldsymbol{s}, t), 1\right\}}\right), \;
\mathrm{mFDR}(\boldsymbol{\delta}^w_t) = \frac{\ep\left(\sum_{\boldsymbol{s} \in \mathcal{S}}\left[\{1-\theta(\boldsymbol{s})\} \delta^w(\boldsymbol{s}, t)\right]\right)}{\ep\left\{\sum_{\boldsymbol{s} \in \mathcal{S}} \delta^w(\boldsymbol{s}, t)\right\}}.
\end{equation*}
\cite{LAWS} showed that $\mathrm{FDR}(\boldsymbol{\delta}^w_t) = \mathrm{mFDR}(\boldsymbol{\delta}^w_t) + o(1)$ under some mild conditions. Therefore, we can use the leading term $\mathrm{mFDR}(\boldsymbol{\delta}^w_t)$ to approximate $\mathrm{FDR}(\boldsymbol{\delta}^w_t)$ asymptotically. In addition, we define the power of the test as,
\begin{equation*}
\Psi(\boldsymbol{\delta}^w_t) = \ep\left[\sum_{\boldsymbol{s} \in \mathcal{S}} \left\{\theta(\boldsymbol{s}) \delta^w(\boldsymbol{s}, t)\right\}\right].
\end{equation*}
We define the oracle threshold values of the two methods as,
\begin{eqnarray*}
&&t_{\text{LAWS}}=\sup_t \left\{ t: \textrm{mFDR}(\boldsymbol{\delta}^{\text{LAWS}}_t) \leq \alpha \right\},\\
&&t_{\text{NAPA}}=\sup_t \left\{ t: \textrm{mFDR}(\boldsymbol{\delta}^{\text{NAPA}}_t) \leq \alpha \right\},
\end{eqnarray*}
where $\boldsymbol{\delta}^{\text{LAWS}}_t$ and $\boldsymbol{\delta}^{\text{NAPA}}_t$ respectively represent the decision rules based on the LAWS and NAPA weights, and $\alpha$ is a pre-specified significance level.
{Let $F_1$ denote the non-null conditional cumulative distribution function (CDF) of the unweighed $p$-value, and $F_1'$ denote its first derivative.}
The next theorem characterizes the theoretical gain of our NAPA method compared to LAWS.
\begin{thm} \label{thm1}
For each $\boldsymbol{s} \in \mathcal{S}$, suppose $F_1(t|\boldsymbol{s})$ is concave in $t$, and $h(x)=xF_1\{xt / (1-x) | \boldsymbol{s} \}$ is convex in $x$, for $tx/(1-x)\leq 1$. Suppose $\pi(\boldsymbol{s})\in[\zeta,1-\zeta]$, and $F_1\left\{\frac{\pi(\boldsymbol{s})}{1-\pi(\boldsymbol{s})} t_{\text{LAWS}}|\boldsymbol{s}\right\}$ $\leq 1-\varrho$ for some small constants $\zeta,\varrho>0$. If $yF_1'\left(y|\boldsymbol{s}\right)\leq \varrho\zeta$ for any $y\in(0,1)$, then,
\begin{align*}
\begin{split}
& \mathrm{mFDR}\left( \boldsymbol{\delta}^{\text{NAPA}}_{t_{\text{LAWS}}} \right) \leq \mathrm{mFDR}\left( \boldsymbol{\delta}^{\text{LAWS}}_{t_{\text{LAWS}}} \right) \leq \alpha, \\
& \Psi\left( \boldsymbol{\delta}^{\text{NAPA}}_{t_{\text{NAPA}}} \right) \geq \Psi\left( \boldsymbol{\delta}^{\text{NAPA}}_{t_{\text{LAWS}}} \right) \geq \Psi\left( \boldsymbol{\delta}^{\text{LAWS}}_{t_{\text{LAWS}}} \right).
\end{split}
\end{align*}
\end{thm}
\noindent
We make a few remarks. Theorem \ref{thm1} shows that, when using the oracle threshold $t_{\text{LAWS}}$, our NAPA method achieves an mFDR that is no greater than that of LAWS and a power that is no smaller than that of LAWS. Based on this result and the construction of the oracle threshold, $t_{\text{NAPA}}$ is thus no smaller than $t_{\text{LAWS}}$. Therefore, using the threshold $t_{\text{NAPA}}$ leads to an additional power gain of NAPA, and thus it establishes the guaranteed power gain of NAPA over LAWS. Second, \cite{LAWS} showed that LAWS dominates the classical Benjamini and Hochberg (BH) method \citep{Benjamini1995}, and therefore, NAPA dominates BH too. Finally, the concavity and convexity conditions in Theorem \ref{thm1} are easily satisfied by the $p$-value CDF derived from numerous distributions, e.g., a normal distribution, or a $t$-distribution. Similar conditions have been commonly imposed in the FDR literature \citep[e.g.,][]{Storey2002, Genovese2006, Hu2010, GAP, LAWS}. {Besides, the rest of conditions can also be easily verified numerically for the aforementioned distributions. }
\section{Two-sample Testing Procedure}
\label{sec:napa}
In this section, we first discuss how to estimate the posterior non-null probability $\pi(\boldsymbol{s},u)$ given the data. We then develop a general multiple testing procedure based on the weighted $p$-values. Finally, we illustrate the testing procedure with the problem of comparing two multivariate means, with a concrete construction of the test and auxiliary statistics $\{T(\boldsymbol{s}), U(\boldsymbol{s})\}$.
\subsection{Neighborhood-assisted weight estimation}
\label{sec2}
Recognizing that it is rather difficult to directly estimate the posterior non-null probability $\pi(\boldsymbol{s},u)$ in \eqref{eqn:pi}, we first propose an intermediate quantity $\pi_{\tau}(\boldsymbol{s},u)$. We show that $\pi_{\tau}(\boldsymbol{s},u)$ provides a good approximation of $\pi(\boldsymbol{s},u)$, and the weight constructed based on $\pi_{\tau}(\boldsymbol{s},u)$ has the desired theoretical guarantees. A similar approximation has also been used in \cite{Schweder1982, Storey2002, LAWS}. Specifically, define
\begin{equation} \label{eqn:intermediate}
\pi_{\tau}(\boldsymbol{s},u) = 1-\frac{\Pr\big\{ p(\boldsymbol{s})>\tau|U(\boldsymbol{s})=u\big\}}{1-\tau}, \quad \textrm{ for some } \;\; 0<\tau<1.
\end{equation}
To justify the use of $\pi_{\tau}(\boldsymbol{s},u)$, consider the conditional CDF of the $p$-value,
\begin{align*}
\Pr\{p(\boldsymbol{s}) \leq t | U(\boldsymbol{s})=u\} & = \{1-\pi(\boldsymbol{s},u)\} F_0(t|\boldsymbol{s},u)+\pi(\boldsymbol{s},u) F_1(t | \boldsymbol{s},u) \\
& \approx \{1-\pi(\boldsymbol{s},u)\}t+\pi(\boldsymbol{s},u) F_1(t |\boldsymbol{s},u),
\end{align*}
where $t\in[0,1]$, $F_0, F_1$ are respectively the {null and non-null conditional CDFs} of the $p$-value, and the approximation comes from the fact that the null $p$-value is asymptotically uniform and it is asymptotically independent of the auxiliary statistic $U(\boldsymbol{s})$. Then the difference between $\pi_{\tau}(\boldsymbol{s},u)$ and $\pi(\boldsymbol{s},u)$ can be approximated by
\begin{align*}
\frac{\pi_{\tau}(\boldsymbol{s},u)-\pi(\boldsymbol{s},u)}{\pi(\boldsymbol{s},u)}&\approx -\frac{1-F_1(\tau|\boldsymbol{s},u)}{1-\tau}.
\end{align*}
Therefore, the difference between $\pi_{\tau}(\boldsymbol{s},u)$ and $\pi(\boldsymbol{s},u)$ is small with a properly chosen $\tau$, and it is asymptotically negative, which in turn would yield an asymptotically conservative FDR control. We discuss the choice of $\tau$ in Section \ref{subsec:estimation}.
Next, we develop a neighborhood-assisted approach to estimate $\pi_\tau(\boldsymbol{s},u)$. Intuitively, the estimator can be obtained by counting the proportion of $p$-values that are greater than $\tau$ among all $p$-values at the location $\boldsymbol{s}$ and with the same auxiliary covariate value $u$. However, there is only one $p$-value at each $(\boldsymbol{s},u)$ pair. This prompts us to use a smoothing kernel approach to borrow information from the neighborhood of $(\boldsymbol{s},u)$. More specifically, consider a positive, bounded, unimodal kernel function $K(\boldsymbol{x}, y) : \mathbb{R}^{b+1} \to \mathbb{R}$ that is symmetric about zero in each dimension. Let $\boldsymbol{H} \in \mathbb{R}^{(b+1)\times(b+1)}$ be a positive definite bandwidth matrix, and write {$K_{\boldsymbol{H}}(\boldsymbol{s}, u) = |\boldsymbol{H}|^{-1/2}$ $K\{\boldsymbol{H}^{-1/2}(\boldsymbol{s}^{\scriptscriptstyle \sf T}, u)^{\scriptscriptstyle \sf T}\}$,} where $|\cdot|$ is the determinant. We briefly remark that, the bandwidth matrix $\boldsymbol{H}$ is not diagonal, because of the dependency between $\boldsymbol{s}$ and $u$ and possible correlations among the entries of $\boldsymbol{s}$. If we set $\boldsymbol{H}$ a diagonal matrix with the same magnitude along the diagonal and ignore $u$, then our estimator reduces to that in \cite{LAWS}.
For a given $(\boldsymbol{s},u)$, we assign the following weight to the ``pseudo-observation" of the $p$-value at $(\boldsymbol{s}',u')$ {with $u'=U(\boldsymbol{s}')$},
\begin{equation*}
v_{\boldsymbol{H}}\{(\boldsymbol{s},u),(\boldsymbol{s}',u')\}=\frac{K_{\boldsymbol{H}}(\boldsymbol{s}-\boldsymbol{s}',u-u')}{K_{\boldsymbol{H}}(\boldsymbol{0},0)}.
\end{equation*}
Then, the number of ``pseudo-observations" that are greater than $\tau$ at each $(\boldsymbol{s},u)$ can be approximated by $\sum\nolimits_{\boldsymbol{s}'\in\mathcal{I}(\tau)}v_{\boldsymbol{H}}\{(\boldsymbol{s},u),(\boldsymbol{s}',u')\}$, where $\mathcal{I}(\tau) = \{\boldsymbol{s}'\in\mathcal{S}:p(\boldsymbol{s}')>\tau\}$. Meanwhile, the expectation of the $p$-values greater than $\tau$ can be calculated by $\left[\sum\nolimits_{\boldsymbol{s}'\in\mathcal{S}} v_{\boldsymbol{H}}\{(\boldsymbol{s},u),(\boldsymbol{s}',u')\}\right]$ $\left\{1-\pi_\tau(\boldsymbol{s},u)\right\}(1-\tau)$. Setting the two equal, we obtain an estimator of $\pi_\tau(\boldsymbol{s},u)$ as
\begin{equation} \label{eqn:pi-tau}
\hat{\pi}_\tau(\boldsymbol{s},u) = 1 - \frac{\sum_{\boldsymbol{s}'\in \mathcal{I}(\tau)}v_{\boldsymbol{H}}\{(\boldsymbol{s},u),(\boldsymbol{s}',u')\}}{(1-\tau)\sum_{\boldsymbol{s}' \in \mathcal{S}}v_{\boldsymbol{H}}\{(\boldsymbol{s},u),(\boldsymbol{s}',u')\}}.
\end{equation}
We now obtain the neighborhood-assisted and posterior-adjusted weight estimator as
\begin{equation} \label{eqn:weights}
\hat w(\boldsymbol{s},u) = \frac{\hat{\pi}_\tau(\boldsymbol{s},u)}{1-\hat{\pi}_\tau(\boldsymbol{s},u)}.
\end{equation}
\subsection{Multiple testing procedure}
\label{subsec:testproc}
We next develop a general multiple testing procedure. Throughout, we use $\ep_{p|u}$, $\ep_{\theta|u}$ and $\ep_{p,\theta|u}$ to denote the expectations taken over $p(\boldsymbol{s})$, $\theta(\boldsymbol{s})$ and $\{p(\boldsymbol{s}),\theta(\boldsymbol{s})\}$ conditional on $U(\boldsymbol{s})$ for each $\boldsymbol{s}\in\mathcal{S}$, respectively, and denote the corresponding variance terms similarly. We first observe that the expected number of false rejections with a known $\pi(\boldsymbol{s}, u)$ at a given threshold $t$ can be computed as
\begin{align*}
& \ep_{p,\theta|u}\left[\sum_{\boldsymbol{s}\in\mathcal{S}}\id \left\{p_w(\boldsymbol{s})<t,\theta(\boldsymbol{s})=0\right\}\right] \\
= & \sum_{\boldsymbol{s}\in\mathcal{S}}\Pr\{\theta(\boldsymbol{s})=0|U(\boldsymbol{s})=u\}\Pr\{p_w(\boldsymbol{s})\leq t|\theta(\boldsymbol{s})=0,U(\boldsymbol{s})=u\} \\
= & \sum_{\boldsymbol{s}\in\mathcal{S}}\{1-\pi(\boldsymbol{s},u)\}w(\boldsymbol{s},u)t=\sum_{\boldsymbol{s}\in\mathcal{S}}\pi(\boldsymbol{s},u)t,
\end{align*}
in the oracle setting. Given the data, if the $p$-value is uniformly distributed asymptotically, and $U(\boldsymbol{s})$ is asymptotically independent of $T(\boldsymbol{s})$ under the null, then for a given estimate $\hat{\pi}_\tau(\boldsymbol{s},u)$ and the decision rule $\id\{p_{\hat{w}}(\boldsymbol{s})\leq t\}$, we can approximate the number of false rejection by $\sum_{\boldsymbol{s}\in\mathcal{S}}\hat{\pi}_\tau(\boldsymbol{s},u)t$. We aim to reject as many hypotheses as possible, while controlling the estimated false discovery proportion (FDP) not to exceed the pre-specified significance level. This leads to the proposed testing procedure as summarized in Algorithm \ref{alg1}.
\begin{algorithm}[t!]
\caption{The multiple testing procedure of NAPA.}
\label{alg1}{
\begin{description}
\item[\textnormal{Step 1.}] Calculate the weights $\hat w(\boldsymbol{s},u)$ as in \eqref{eqn:weights}, and then adjust $p$-values by $p_{\hat{w}}(\boldsymbol{s})=\min\{p(\boldsymbol{s})/\hat w(\boldsymbol{s},u), 1\}$ for $ \boldsymbol{s} \in \mathcal{S} $.
\item[\textnormal{Step 2.}] Obtain the data-driven threshold
\begin{align*}
{t}_{\hat{w}}=\sup_{t}\left\{t: \frac{\sum_{\boldsymbol{s} \in \mathcal{S} } \hat{\pi}_{\tau}(\boldsymbol{s},u) t}{\max \left\{\sum_{\boldsymbol{s} \in \mathcal{S} } \id\left\{p_{\hat{w}}(\boldsymbol{s}) \leq t\right\}, 1\right\}} \leq \alpha\right\}.
\end{align*}
\item[\textnormal{Step 3.}] Reject $H_{0}(\boldsymbol{s})$ if $p_{\hat{w}}(\boldsymbol{s}) \leq {t}_{\hat{w}}$, $\boldsymbol{s}\in\mathcal{S} $.
\end{description} }
\end{algorithm}
\subsection{Comparison of multivariate means}
\label{subsec:compare-means}
We illustrate the above general testing procedure with a specific testing problem, i.e., comparing two multivariate means, and give a concrete construction of the test statistic $T(\boldsymbol{s})$ and the auxiliary statistic $U(\boldsymbol{s})$. Our method also applies to numerous other testing problems as well.
Given the observed data $\{Y_{i,d}(\boldsymbol{s}): \boldsymbol{s} \in \mathcal{S} \}_{i=1}^{n_d}$, we construct the primary test statistic as
\begin{equation*}
T(\boldsymbol{s}) = \frac{\bar{Y}_1(\boldsymbol{s})-\bar{Y}_2(\boldsymbol{s})}{\left( \hat{\sigma}_{\boldsymbol{s},1}^2 / n_1 + \hat{\sigma}_{\boldsymbol{s},2}^2 / n_2 \right)^{1/2}}, \quad \boldsymbol{s} \in \mathcal{S},
\end{equation*}
where $\bar{Y}_d(\boldsymbol{s}) = n_d^{-1} \sum_{i=1}^{n_d} Y_{i,d}(\boldsymbol{s})$ is the group sample mean, and $\hat{\sigma}_{\boldsymbol{s},d}^2 = n_d^{-1}\sum_{i=1}^{n_d}\{ Y_{i,d}(\boldsymbol{s})-\bar{Y}_d(\boldsymbol{s}) \}^2$ is the sample variance, $d=1,2$. Next, we construct the auxiliary statistic in the form of $\beta_1(\boldsymbol{s}) + \kappa(\boldsymbol{s}) \beta_2(\boldsymbol{s})$, and for the multivariate mean comparison problem, we consider,
\vspace{-0.01in}
\begin{equation*}
U(\boldsymbol{s}) = \frac{\bar{Y}_1(\boldsymbol{s})+\hat{\kappa}(\boldsymbol{s})\bar{Y}_2(\boldsymbol{s})}{\left\{\hat{\sigma}_{\boldsymbol{s},1}^2/n_1+\hat{\kappa}^2(\boldsymbol{s})\hat{\sigma}_{\boldsymbol{s},2}^2/n_2\right\}^{1/2}}, \quad \boldsymbol{s} \in \mathcal{S},
\end{equation*}
where $\hat{\kappa}(\boldsymbol{s})=(n_2\hat{\sigma}_{\boldsymbol{s},1}^2)/(n_1\hat{\sigma}_{\boldsymbol{s},2}^2)$.
\section{Theoretical Properties}
\label{sec:theory}
In this section, we establish the theoretical properties of the NAPA testing procedure. We first show the estimated posterior non-null probability $\hat{\pi}_\tau(\boldsymbol{s}, u)$ is a consistent estimator of $\pi_\tau(\boldsymbol{s}, u)$. We then establish the asymptotic error rate control of NAPA under some conditions. Finally, we illustrate with the mean comparison problem again, and show the required conditions of NAPA are satisfied. Specifically, under the null, the test statistic $T(\boldsymbol{s})$ is asymptotically normally distributed, and thus the corresponding $p$-value is asymptotically uniformly distributed, and $T(\boldsymbol{s})$ is independent of the auxiliary statistic $U(\boldsymbol{s})$ asymptotically. Throughout our asymptotic analysis, we consider the infill-asymptotic framework \citep{stein1999} that $\mathcal{S} \to \mathbb{S}$.
\subsection{Estimation consistency of the posterior probability}
\label{subsec:consistency}
We begin with some notations. Let $\mathcal{S}_0=\{\boldsymbol{s} \in \mathcal{S} : \theta(\boldsymbol{s})=0\}$ and $\mathcal{S}_1=\{\boldsymbol{s} \in \mathcal{S} : \theta(\boldsymbol{s})=1\}$ denote the set of null locations and non-null locations, respectively, and let $\mathcal{S} = \mathcal{S}_0 \cup \mathcal{S}_1$. Let $m = | \mathcal{S} |$, $m_0 = | \mathcal{S}_0 |, m_1 = | \mathcal{S}_1 |$, and $n = n_1 + n_2$, where $|\cdot|$ denotes the cardinality. For two sequences of real numbers $\{a_{n}\}$ and $\{b_{n}\}$, write $a_{n} = O(b_{n})$ if there exists a constant $C$ such that $|a_{n}| \leq C|b_{n}|$ for any sufficiently large $n$, write $a_{n} = o(b_{n})$ if $\lim_{n\rightarrow\infty}a_{n}/b_{n} = 0$, and write $a_{n}\asymp b_{n}$ if there exists constants $C>c>0$ such that $c|b_{n}| \leq |a_{n}| \leq C|b_{n}|$ for any sufficiently large $n$. Let $\lambda_{i}(\cdot)$ and $\tr(\cdot)$ denote the $i$th eigenvalue and the trace of a matrix, respectively.
Next, we show that the estimator $\hat{\pi}_\tau(\boldsymbol{s},u)$ in (\ref{eqn:pi-tau}) converges to the truth $\pi_{\tau}(\boldsymbol{s},u)$ for all $\boldsymbol{s} \in \mathcal{S}$ as $\mathcal{S} \rightarrow \mathbb{S}$. Let $\boldsymbol{A} \in \mathbb{R}^{(b+1)\times (b+1)}$ denote the Hessian matrix of $\Pr\{p(\boldsymbol{s})>\tau|U(\boldsymbol{s})=u\}$ with respect to $(\boldsymbol{s}^{\scriptscriptstyle \sf T},u)^{\scriptscriptstyle \sf T}$. Partition the bandwidth matrix into $\boldsymbol{H}=\begin{pmatrix} \boldsymbol{H}_S & \boldsymbol{a} \\ \boldsymbol{a}^{\scriptscriptstyle \sf T} & h_u^2 \end{pmatrix}$, where $\boldsymbol{H}_S \in \mathbb{R}^{b\times b}$, $\boldsymbol{a} \in \mathbb{R}^{b\times 1}$, and $h_u^2 \in \mathbb{R}$. We introduce the following regularity conditions.
\begin{enumerate}[label=(C\arabic*), series=C]
\item \label{A0}
Suppose the kernel function $K(\boldsymbol{x},y):\mathbb{R}^{b+1}\to \mathbb{R}$ satisfies
\begin{equation*}
\int_{\mathbb{R}^{b}} \boldsymbol{x}^{\scriptscriptstyle \sf T}\boldsymbol{x} K(\boldsymbol{x},0)d\boldsymbol{x} < \infty,\int_{\mathbb{R}^{b}} g^2(\boldsymbol{x})K\{\boldsymbol{Q}\left[\boldsymbol{x}^{\scriptscriptstyle \sf T},g(\boldsymbol{x})\right]^{\scriptscriptstyle \sf T}\}d\boldsymbol{x} <\infty,
\end{equation*}
for any orthogonal matrix $\boldsymbol{Q}$ and any function $g:\mathbb{R}^{b}\to \mathbb{R}$.
\item \label{A1} Suppose $\Pr\{p(\boldsymbol{s})>\tau|U(\boldsymbol{s})=u\}$ has continuous first and second partial derivatives with respect to $(\boldsymbol{s}^{\scriptscriptstyle \sf T},u)^{\scriptscriptstyle \sf T}$, and {$\lambda_i(\boldsymbol{A})=O(1)$} holds uniformly for all $(\boldsymbol{s}^{\scriptscriptstyle \sf T},u)^{\scriptscriptstyle \sf T}$, where $\boldsymbol{s}\in\mathcal{S},u=U(\boldsymbol{s})$, $i=1,\cdots,b+1$.
\item \label{A2} Suppose for any $\boldsymbol{s}\in\mathcal{S}$,
\begin{align*}
&\Var_{p|u}\left(\sum_{\boldsymbol{s}' \in \mathcal{S} }\left[K_{\boldsymbol{H}}(\boldsymbol{s}-\boldsymbol{s}',u-u') \id\left\{p(\boldsymbol{s}')>\tau\right\}\right]\right) \\
=& O\left(\sum_{\boldsymbol{s}'\in \mathcal{S}}\left[ K^2_{\boldsymbol{H}}(\boldsymbol{s}-\boldsymbol{s}',u-u') \Var_{p|u}\id\{p(\boldsymbol{s}')>\tau\}\right] \right).
\end{align*}
\item \label{A3} Suppose $\boldsymbol{H}_{\boldsymbol{S}}$ and $\tilde{h}=h_u^{2}-\boldsymbol{a}^{\scriptscriptstyle \sf T}\boldsymbol{H}_{\boldsymbol{S}}^{-1}\boldsymbol{a}$ are both nonsingular, and $\sum_{\boldsymbol{s}' \in \mathcal{S} } K_{\boldsymbol{H}}(\boldsymbol{s}-\boldsymbol{s}',u-u')\gg m\tilde{h}^{-3/4}\{\tr(\boldsymbol{H})\}^{1/2}$. Furthermore, suppose $\tilde{h}=O\{\tr(\boldsymbol{H})\}, \tr(\boldsymbol{H})=o(1)$ and $m^{-1}|{\boldsymbol{H}}|^{-1/2} \to 0$.
\end{enumerate}
\noindent
We make a few remarks about these conditions. Condition \ref{A0} holds for commonly used multivariate kernels, e.g., the standard normal kernel, the uniform kernel, among others. Condition \ref{A1} regulates the first and second derivatives of the conditional CDF of the $p$-values, and is mild. Condition \ref{A2} assumes that most of the $p$-values are weakly correlated with each other. It can be relaxed with a larger bandwidth selection. For instance, if we set the bandwidth matrix so that it satisfies $|{\boldsymbol{H}}|\gg m^{-1}$, then the right-hand-side of \ref{A2} can be relaxed to $O\left(m^{1/2}\sum_{\boldsymbol{s}'\in \mathcal{S}} \left[K^2_{\boldsymbol{H}}(\boldsymbol{s}-\boldsymbol{s}',u-u') \Var_{p|u}\id\{p(\boldsymbol{s}')>\tau\}\right] \right)$. Condition \ref{A3} is mild too, and can be verified numerically. Besides, the requirement that $m^{-1}|{\boldsymbol{H}}|^{-1/2} \to 0$ in \ref{A3} is the same as Condition (A2) in Lemma 1 of \cite{Duong2005}.
\begin{thm}\label{pro2}
Suppose Conditions \ref{A0} to \ref{A3} hold. Then,
\begin{align*}
\ep_{p|u}\left[\left\{\hat{\pi}_{\tau}\left(\boldsymbol{s},u\right)-\pi_{\tau}\left(\boldsymbol{s},u\right)\right\}^2\right]\to 0, \; \textrm { as } \; \mathcal{S} \rightarrow \mathbb{S},
\end{align*}
uniformly for all $\boldsymbol{s} \in \mathcal{S}$.
\end{thm}
\noindent
Note that, LAWS \citep{LAWS} only considers the spatial information and applies a diagonal smoothing kernel with a homogenous bandwidth to estimate the weight. In comparison, to integrate the neighborhood information encoded by both $\boldsymbol{s}$ and $U(\boldsymbol{s})$, we have developed a more sophisticated kernel estimation procedure that allows both non-orthogonal kernel components and heterogeneous bandwidth magnitudes.
\subsection{Asymptotic error rate control of NAPA}
\label{subsec:asymp-napa}
Next, we show the NAPA procedure controls both the FDR and the FDP asymptotically, where we define the FDP of the test $\boldsymbol{\delta}_t^w = \{\delta^w(\boldsymbol{s}, t) : \boldsymbol{s} \in \mathcal{S}\}$ as,
\begin{equation*}
\mathrm{FDP}\left(\boldsymbol{\delta}_t^{w}\right)=\frac{\sum_{\boldsymbol{s} \in \mathcal{S}}\left[\{1-\theta(\boldsymbol{s})\} \delta^{w}(\boldsymbol{s}, t)\right]}{\max \left\{\sum_{\boldsymbol{s} \in \mathcal{S}} \delta^{w}(\boldsymbol{s}, t), 1\right\}}.
\end{equation*}
We again begin with some regularity conditions.
\begin{enumerate}[resume*=C]
\item \label{A4} Suppose that $n_1\asymp n_2$, $\log m=o(n^{1/8})$, and there exist two independent sets of i.i.d.\ random variables $\{Z_{k}(\boldsymbol{s}), k = 1, \ldots, n_1\}$ and $\{Z_{k}(\boldsymbol{s}), k = n_1 + 1, \ldots, n_1+n_2\}$ satisfying that $\ep\{ Z_{k}(\boldsymbol{s}) \} = 0$ and {$\ep\left\{\exp\left(C_1|Z_{k}(\boldsymbol{s})|/[\Var\{Z_{k}(\boldsymbol{s})\}]^{1/2}\right)\right\} < \infty$} for some $C_1 > 0$, such that
\begin{align*}
&\Pr_{H_0(\boldsymbol{s})}\left\{ \left|T(\boldsymbol{s})-\frac{\sum_{k=1}^{n}Z_{k}(\boldsymbol{s})}{\Var\{\sum_{k=1}^{n}Z_{k}(\boldsymbol{s})\}^{1/2}}\right|\geq b_m \right\} = O(m^{-C_2}),\\
&\Pr_{H_0(\boldsymbol{s})}\left\{\left|\left[U(\boldsymbol{s})-\ep\{U(\boldsymbol{s})\}\right]-\frac{\sum_{k=1}^{n_1}Z_{k}(\boldsymbol{s})-\vartheta(\boldsymbol{s})\sum_{k=n_1+1}^{n}Z_{k}(\boldsymbol{s})}{\Var\{\sum_{k=1}^{n_1}Z_{k}(\boldsymbol{s})-\vartheta(\boldsymbol{s})\sum_{k=n_1+1}^{n}Z_{k}(\boldsymbol{s})\}^{1/2}}\right|\geq b_m\right\} = O(m^{-C_2}),
\end{align*}
where $\vartheta(\boldsymbol{s}) = \left[ n_1\Var\left\{Z_1(\boldsymbol{s})\right\} \right] / \left[ n_2\Var\left\{Z_n(\boldsymbol{s})\right\} \right]$, for some constant $C_2 > 5$ and $b_m = o\{(\log m)^{-1/2}\}$.
\item \label{A6} For $\boldsymbol{Z}_k=\left\{Z_{k}(\boldsymbol{s}):\boldsymbol{s}\in\mathcal{S}\right\}$ as defined in \ref{A4}, let $\boldsymbol{R}_1=\mathsf{Corr}(\boldsymbol{Z}_k)=\left(r_{\boldsymbol{s},\boldsymbol{l};1}\right)_{m \times m}$ for $1\leq k\leq n_1$, and $\boldsymbol{R}_2=\mathsf{Corr}(\boldsymbol{Z}_k)=\left(r_{\boldsymbol{s},\boldsymbol{l};2}\right)_{m \times m}$ for $n_1+1< k\leq n_1+n_2$. Suppose there exists $\gamma>0$ such that $\max_{\boldsymbol{s}\in\mathcal{S}_0}\left|\Gamma_{\boldsymbol{s}}(\gamma)\right| \asymp 1$, where $\Gamma_{\boldsymbol{s}}(\gamma)=\left\{\boldsymbol{l}: \boldsymbol{l}\in\mathcal{S},\left|r_{\boldsymbol{s}, \boldsymbol{l};d}\right| \geq(\log m)^{-2-\gamma}, \ d=1\ \mbox{or}\ 2\right\}$.
\item \label{A7} Suppose, with probability tending to $1$, uniformly for all $\boldsymbol{s}\in \mathcal{S}$, $\pi_{\tau}(\boldsymbol{s},u) \in[\xi, 1-\xi]$ for some sufficiently small constant $\xi>0$ and $\Var_{\theta|u}\left[\sum_{\boldsymbol{s} \in \mathcal{S}} \id\{\theta(\boldsymbol{s})=0\}\right]=o_\pr\left(m^2\right)$.
\item \label{A8} Let $\mathcal{S}_{\nu}=\left\{\boldsymbol{s}\in\mathcal{S}:\frac{\beta_1(\boldsymbol{s})-\beta_2(\boldsymbol{s})}{\Var\{\sum_{k=1}^{n}Z_{k}(\boldsymbol{s})\}^{1/2}/n_2}\geq (\log m)^{1/2+\nu}\right\}$ {for $\boldsymbol{Z}_k=\left\{Z_{k}(\boldsymbol{s}):\boldsymbol{s}\in\mathcal{S}\right\}$ as defined in \ref{A4}.} Suppose $\left|\mathcal{S}_{\nu}\right| \geq \{1 / (c_\pi^{1 / 2} \alpha )+\varepsilon \} (\log m)^{1 / 2}$ for some $\varepsilon, \nu>0$, and $c_\pi$ is the ratio of a circle's circumference to its diameter.
\end{enumerate}
\noindent
Condition \ref{A4} assumes the asymptotic normality of $T(\boldsymbol{s})$ and $U(\boldsymbol{s})$ under the null, which is easily attainable, as we illustrate in Section \ref{subsec:asymp-mean} with the problem of comparing two multivariate means. It also implies the asymptotic independence between $T(\boldsymbol{s})$ and $U(\boldsymbol{s})$ under the null, as we show in Lemma 1 of the appendix.
Condition \ref{A6} requires that not too many variables have strong correlations that exceed $(\log m)^{-2-\gamma}$. Condition \ref{A7} requires $\pi_{\tau}(\boldsymbol{s},u)$ not to be exactly 0 or 1 to ensure the theoretical stability. It also requires the latent variables {$\theta(\boldsymbol{s})|U(\boldsymbol{s})$} are not perfectly correlated, which ensures that $m_0 \asymp m$ has the probability tending to one. Condition \ref{A8} requires a few spatial locations to have the standardized signal magnitude exceeding $(\log m)^{1/2+\nu }$, which avoids an overly conservative FDR. In general, these conditions are mild, and similar conditions of \ref{A6} to \ref{A8} have been imposed in \cite{LAWS}.
The next two theorems establish the asymptotical control of FDR and FDP at the nominal level, first for a known $\pi_{\tau}(\boldsymbol{s},u)$ in Theorem \ref{thm2}, then for the estimated $\hat{\pi}_{\tau}(\boldsymbol{s},u)$ in Theorem \ref{thm3}.
\begin{thm}\label{thm2}
Suppose Conditions \ref{A4} to \ref{A8} hold. Then,
\begin{align*}
\varlimsup_{\mathcal{S} \rightarrow \mathbb{S}} \operatorname{FDR}\left(\boldsymbol{\delta}^{\text{NAPA}}_{t_w}\right) \leq \alpha, \;\; \text { and } \;\;
\lim _{\mathcal{S} \rightarrow \mathbb{S}} \Pr\left\{\operatorname{FDP}\left(\boldsymbol{\delta}^{\text{NAPA}}_{t_w}\right) \leq \alpha+\epsilon\right\} = 1, \text{ for any } \; \epsilon > 0,
\end{align*}
where $t_{w}=\sup _{t}\left\{t: \frac{\sum_{\boldsymbol{s} \in \mathcal{S} } \pi_{\tau}(\boldsymbol{s},u) t}{\max \left\{\sum_{\boldsymbol{s} \in \mathcal{S} } \id\left\{p_{w}(\boldsymbol{s}) \leq t\right\}, 1\right\}} \leq \alpha\right\}$, and $w(\boldsymbol{s},u) = \pi_\tau(\boldsymbol{s},u)/\{1-\pi_\tau(\boldsymbol{s},u)\}$.
\end{thm}
\begin{thm}\label{thm3}
Suppose Conditions \ref{A0} to \ref{A8} hold. Then,
\begin{align*}
\varlimsup_{\mathcal{S} \rightarrow \mathbb{S}} \operatorname{FDR}\left(\boldsymbol{\delta}^{\text{NAPA}}_{t_{\hat w}}\right) \leq \alpha, \;\; \text { and } \;\;
\lim _{\mathcal{S} \rightarrow \mathbb{S}} \Pr\left\{\operatorname{FDP}\left(\boldsymbol{\delta}^{\text{NAPA}}_{t_{\hat w}}\right) \leq \alpha+\epsilon\right\} = 1, \text{ for any } \; \epsilon > 0.
\end{align*}
\end{thm}
\subsection{Asymptotic properties of mean comparison}
\label{subsec:asymp-mean}
Finally, we revisit the example of comparing multivariate means in Section \ref{subsec:compare-means}, and show that the required asymptotic normality and independence both hold. For other testing problems such as comparing the networks and detecting interactions, similar properties can be established accordingly; see also \citet{GAP}. We introduce two additional regularity conditions. {Let $\sigma_{\boldsymbol{s},d}^2 = {\Var\{Y_d(\boldsymbol{s})\}}$ for $d=1,2$, and $\kappa(\boldsymbol{s}) = (n_2\sigma_{\boldsymbol{s},1}^2)/(n_1\sigma_{\boldsymbol{s},2}^2)$, $\boldsymbol{s} \in \mathcal{S}$.}
\begin{enumerate}[resume*=C]
\item \label{C1} Suppose $\log m = o(n^{1/5})$, $n_1\asymp n_2$, and $\sigma_{\boldsymbol{s},1}^2\asymp \sigma_{\boldsymbol{s},2}^2$ for all $\boldsymbol{s} \in \mathcal{S}$.
\item \label{C2} There exists {some constant $C_1>0$,} such that $\ep[\exp \{C_1|Y_{k,d}(\boldsymbol{s})-\beta_d(\boldsymbol{s})|/\sigma_{\boldsymbol{s},d}\}] < \infty$, for $d=1,2$ and all $\boldsymbol{s} \in \mathcal{S}$.
\end{enumerate}
\noindent
Condition \ref{C1} allows the total number of hypotheses to test $m$ to grow exponentially with the total sample size $n$, while requiring the sample size and the variance of each group to be of the same order. Condition \ref{C2} holds for a broad family of distributions with an exponential tail. Both conditions are mild.
\begin{pro}\label{pro1}
Suppose Conditions \ref{C1} and \ref{C2} hold. Then, under the null hypothesis, for any constant $C_3>0$, there exists some ${b_m} = o\left\{(\log m)^{-1/2}\right\}$, such that
\begin{eqnarray*}
&& \Pr_{H_0(\boldsymbol{s})}\left\{\left|{T}(\boldsymbol{s})-\frac{\bar{Y}_1(\boldsymbol{s})-\bar{Y}_2(\boldsymbol{s})}{\left(\sigma_{\boldsymbol{s},1}^2/n_1+\sigma_{\boldsymbol{s},2}^2/n_2\right)^{1/2}}\right|\geq b_m\right\} = O\left( m^{-C_3} \right), \\
&&\Pr_{H_0(\boldsymbol{s})}\left\{\left|\left[U(\boldsymbol{s})-\ep\{U(\boldsymbol{s})\}\right]-\frac{\bar{Y}_1(\boldsymbol{s})-\beta_1(\boldsymbol{s})+\kappa(\boldsymbol{s})\{\bar{Y}_2(\boldsymbol{s})-\beta_2(\boldsymbol{s})\}}{\left\{\sigma_{\boldsymbol{s},1}^2/n_1+\kappa^2(\boldsymbol{s})\sigma_{\boldsymbol{s},2}^2/n_2\right\}^{1/2}}\right|\geq b_m\right\} = O(m^{-C_3}),
\end{eqnarray*}
which implies that, for any constant $C_4>0$,
\begin{align*}
\Pr_{H_0(\boldsymbol{s})}\left\{|T(\boldsymbol{s})|\geq t|U(\boldsymbol{s})= u\right\}=\{1+o(1)\}G(t)+O(m^{-C_4}),
\end{align*}
uniformly in $t = O\{(\log m)^{1/2}\}$, $|u-\ep\{U(\boldsymbol{s})\}| = O \{(\log m)^{1/2}\}$, and all $\boldsymbol{s} \in \mathcal{S}$, where $G(t)=2\{1-\Phi(t)\}$, and $\Phi(\cdot)$ is the CDF of a standard normal random variable.
\end{pro}
\noindent
We make some remarks. First, \cite{GAP} illustrated their GAP test with the problem of comparing two multivariate means too, but they only studied the multivariate normal distribution, while Proposition \ref{pro1} extends to the family of distributions with an exponential tail. Second, compared to the asymptotic independence result in GAP, Proposition \ref{pro1} establishes the exact conditional probability tail of $T(\boldsymbol{s})$ given $U(\boldsymbol{s})$. As a result, the proof of Proposition \ref{pro1} is technically much more involved. Toward our goal, we obtain a conditional normal approximation result in the proof of Proposition \ref{pro1}, which to our knowledge is not available in the literature. Finally, because we incorporate the auxiliary statistic in the construction of the weight, Proposition \ref{pro1} ensures that the null distribution of $T(\boldsymbol{s})$ is not to be affected by the observed $U(\boldsymbol{s})$. By contrast, the auxiliary statistic is only used for the grouping purpose in GAP, and hence their asymptotic independence result can be viewed as a simpler discretized version of Proposition \ref{pro1}.
\section{Simulations}
\label{sec:simulations}
In this section, we first study the capability of NAPA in recovering the posterior non-null probability $\pi(\boldsymbol{s}, u)$. We then investigate the finite-sample performance of NAPA in terms of FDR and power, and compare with BH \citep{Benjamini1995}, GAP \citep{GAP}, LAWS \citep{LAWS}, and a simple combination of GAP and LAWS. This last method first applies GAP with three groups to obtain the group-wise reweighted $p$-values, then feeds into the LAWS method for the second-stage reweighting.
\subsection{Posterior probability estimation}
\label{subsec:estimation}
Given the key role the posterior non-null probability plays in our testing method, we first evaluate the capability of NAPA in recovering $\pi(\boldsymbol{s}, u)$ through the estimator $\hat{\pi}_\tau(\boldsymbol{s},u)$ in \eqref{eqn:pi-tau}.
We simulate two groups of independent samples $\{Y_{i,1}(\boldsymbol{s})\}_{i=1}^{n_1}$ and $\{Y_{i,2}(\boldsymbol{s})\}_{i=1}^{n_2}$ from:
\begin{align} \label{eqn:sim}
\begin{split}
Y_{i,1}(\boldsymbol{s}) \ | \ \theta(\boldsymbol{s}) & \sim \{ 1-\theta(\boldsymbol{s}) \} \ \text{Normal}(0,1) \ + \ \theta(\boldsymbol{s}) \ \text{Normal}\big( \beta_1(\boldsymbol{s}),1 \big),\\
Y_{i,2}(\boldsymbol{s}) \ | \ \theta(\boldsymbol{s}) & \sim \{ 1-\theta(\boldsymbol{s}) \} \ \text{Normal}(0,4) \ + \ \theta(\boldsymbol{s}) \ \text{Normal}\big( \mu+\beta_1(\boldsymbol{s}),4 \big),
\end{split}
\end{align}
where $\theta({\boldsymbol{s}}) \sim \text{Bernoulli}(1, \pi({\boldsymbol{s}}))$, $\beta_1(\boldsymbol{s}) = 1 / \sqrt{20}$, and $\mu = 3/\sqrt{20}$. Note that $\pi(\boldsymbol{s})$ specifies the likelihood of possible signal locations. We consider three examples of generating the signal regions: a 1D example of a piecewise constant-shaped signal, a 2D example of two rectangular-shaped signals, and a 3D example of a cubic-shaped signal. For the 1D case, we consider $s = 1, 2, \ldots, 5000$, and we set $\pi(s)=0.8$ for $s \in [1001, 1200] \cup [2001, 2200]$, and $\pi(s)=0.6$ for $s \in [3001, 3200] \cup [4001, 4200]$. For the 2D case, we consider $\boldsymbol{s} = (s_1, s_2)$, with $s_1 = 1, 2, \ldots, 100, s_2 = 1, 2, \ldots, 50$, and we set $\pi(\boldsymbol{s})=0.8$ for the left signal rectangle when $s_1 \in [20, 40], s_2 \in [10, 30]$, and $\pi(\boldsymbol{s})=0.6$ for the right signal rectangle when $s_1 \in [60, 80], s_2 \in [10, 30]$. For the 3D case, we consider $\boldsymbol{s} = (s_1, s_2, s_3)$, with $s_1 = 1, 2, \ldots, 20, s_2 = 1, 2, \ldots, 25, s_3 = 1, 2, \ldots, 15$, and we set $\pi(\boldsymbol{s})=0.7$ for the signal cube when $s_1 \in [5, 15], s_2 \in [5, 15], s_3 \in [1, 10]$. We set $\pi(\boldsymbol{s})=0.05$ for all the rest of locations. We set the sample size at $n_1 = n_2 = 100$. Given the generative model \eqref{eqn:sim}, we can derive the explicit distribution of $U(\boldsymbol{s}) | \theta(\boldsymbol{s})$, and plugging it into \eqref{eqn:pi} yields the true posterior non-null probability $\pi(\boldsymbol{s}, u)$.
Next, we estimate $\pi(\boldsymbol{s},u)$ using a bivariate Gaussian kernel function with the two-dimensional bandwidth matrix,
\vspace{-0.1in}
\begin{align*}
\boldsymbol{H}=\begin{pmatrix}
h_s^2 & \rho h_sh_u\\
\rho h_sh_u & h_u^2
\end{pmatrix},
\end{align*}
where $h_s$ and $h_u$ are the bandwidths for $\| \boldsymbol{s} - \boldsymbol{s}' \|$ and $\| u - u' \|$, respectively, $\rho$ is the correlation between $\| \boldsymbol{s} - \boldsymbol{s}' \|$ and $\| u - u' \|$, and $\| \cdot \|$ denotes the Euclidean norm. We select the bandwidths $h_s$ and $h_u$ using the cross-validation approach of \citet{Jones1991} and use sample correlation to estimate $\rho$. Moreover, to select $\tau$ in \eqref{eqn:pi-tau}, we follow \cite{LAWS} and choose $\tau$ as the cutoff $p$-value when applying the BH procedure to the sequence of all the unweighted $p$-values at the significance level $0.9$. This ensures that the null cases are dominant in the set $\mathcal{I}(\tau) = \{\boldsymbol{s}'\in\mathcal{S}:p(\boldsymbol{s}')>\tau\}$. Finally, to stabilize the probability estimation, we truncate $\hat{\pi}_{\tau}(\boldsymbol{s},u) = \xi$ if $\hat{\pi}_\tau(\boldsymbol{s},u)<\xi$, and $\hat{\pi}_{\tau}(\boldsymbol{s},u) = 1-\xi$ if $\hat{\pi}_\tau(\boldsymbol{s},u)>1-\xi$, where we set $\xi=10^{-5}$.
\begin{figure}[t!]
\centering
\includegraphics[width=0.95\linewidth, height=3.45in]{figs/fig-pi-1D-const.png}
\caption{Estimation of the posterior non-null probability $\pi(\boldsymbol{s}, u)$ for the 1D example. From top to bottom: the true probability, the estimated probability by NAPA, and the estimated probability by LAWS.}
\label{fig:1D-pi}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[width=0.95\linewidth, height=1.85in]{figs/fig-pi-2D-rectangle.png}
\caption{Estimation of the posterior non-null probability $\pi(\boldsymbol{s}, u)$ for the 2D example. From left to right: the true probability, the estimated probability by NAPA, and the estimated probability by LAWS.}
\label{fig:2D-pi}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[width=0.95\linewidth, height=2.95in]{figs/fig-pi-3D-cubic.png}
\caption{Estimation of the posterior non-null probability $\pi(\boldsymbol{s}, u)$ for the 3D example. From left to right: five selected slices with $s_3 = 3, 6, 9, 12, 15$. From top to bottom: the true probability, the estimated probability by NAPA, and the estimated probability by LAWS.}
\label{fig:3D-pi}
\end{figure}
We compare our posterior probability estimator $\hat{\pi}_\tau(\boldsymbol{s},u)$ in \eqref{eqn:pi-tau} that utilizes both the smoothness and sparsity information with the truth ${\pi}(\boldsymbol{s},u)$. We also compare with the corresponding estimator $\hat{\pi}_\tau(\boldsymbol{s})$ used in LAWS that only utilizes the smoothness information alone. Figures \ref{fig:1D-pi}, \ref{fig:2D-pi} and \ref{fig:3D-pi} report the results based on a single data replication for the 1D, 2D and 3D examples, respectively. It is clearly seen that our posterior probability estimator $\hat{\pi}_\tau(\boldsymbol{s},u)$ is much closer to the truth than the LAWS estimator $\hat{\pi}_\tau(\boldsymbol{s})$.
\subsection{FDR and power comparison}
\label{subsec:fdr-power}
Next, we evaluate the empirical FDR and power of the proposed NAPA method, and compare it with BH, GAP, LAWS, and a simple combination of GAP and LAWS.
We continue to simulate the data from model \eqref{eqn:sim} with the 1D, 2D and 3D examples. Recall that $\mu$ controls the strength of the signal, whereas $\pi(\boldsymbol{s})$ specifies the likelihood of possible signal locations. We consider two scenarios: we vary $\mu$ from $1/\sqrt{5}$ to $3/\sqrt{20}$, while fixing $\pi(\boldsymbol{s})$ in the same way as in Section \ref{subsec:estimation}, and we vary $\pi(\boldsymbol{s})$ in signal regions from $0.3$ to $0.8$, while fixing $\mu=3/\sqrt{20}$. Let $\beta_1(\boldsymbol{s}) \sim \text{Uniform}(-1,1)/\sqrt{5}$ in both scenarios. We set the sample size at $n_1 = n_2 = 100$, and set the nominal level at $\alpha = 0.05$.
\begin{figure}[t!]
\centering
\includegraphics[width=0.95\linewidth, height=3.75in]{figs/fig-1D-const.png}
\caption{Empirical FDR and power for the 1D example. Top panels: varying $\mu$ in scenario 1, and bottom panels: varying $\pi(\boldsymbol{s})$ in scenario 2. Five methods are compared: the proposed method (NAPA), the GAP method \citep{GAP}, the LAWS method \citep{LAWS}, the simple combination of GAP and LAWS, and the BH method \citep{Benjamini1995}.}
\label{fig:1D}
\end{figure}
Figures \ref{fig:1D}, \ref{fig:2D}, and \ref{fig:3D} report the empirical FDR and power of various testing methods based on 200 data replications. It is clearly seen that, in all three examples, while all methods can control the FDR, our proposed NAPA method achieves the most power gain compared to all the alternative methods. These results agree with our theory as well as our intuition that the NAPA method that utilizes both the spatial and sparsity information outperforms the GAP and LAWS methods that utilize only one type of side information alone. Moreover, our method clearly outperforms the simple combination of GAP and LAWS. This simple combination has no theoretical guarantee. Besides, it loses information when applying the GAP method that reweighs the $p$-values in a discrete fashion. Through these examples, we see that our proposed NAPA test is more than just a simple combination of GAP and LAWS.
\begin{figure}[t!]
\centering
\includegraphics[width=0.95\linewidth, height=3.75in]{figs/fig-2D-rectangle.png}
\caption{Empirical FDR and power for the 2D example. The rest is the same as Figure \ref{fig:1D}.}
\label{fig:2D}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[width=0.95\linewidth, height=3.75in]{figs/fig-3D-cubic.png}
\caption{Empirical FDR and power for the 3D example. The rest is the same as Figure \ref{fig:1D}.}
\label{fig:3D}
\end{figure}
\section{Real Data Applications}
\label{sec:realdata}
In this section, we illustrate our proposed test with two neuroimaging applications.
\subsection{Multiple sclerosis study}
The first study is to compare the cerebral white matter tracts between multiple sclerosis (MS) patients and healthy controls \citep{Goldsmith2011}. MS is a demyelinating autoimmune disease that causes lesions in the white matter tracts of a patient and results in severe disability. Diffusion tensor imaging (DTI) is a magnetic resonance imaging (MRI) technique that studies white matter tractography by measuring the diffusivity of water in the brain. The data records the fractional anisotropy measure, which describes the degree of diffusion anisotropy, along the right corticospinal tract for $n_1 = 340$ multiple sclerosis patients and $n_2 = 42$ healthy controls. The tract data are generally modeled as 1D functions, and there are in total $|\mathcal{S}| = 43$ locations for each tract. The dataset is available in the \texttt{R} library \texttt{refund}, and the data processing information can be found in \citet{Luo2017}. The scientific interest is to compare the means of the two sets of functional profiles of diffusivity, and to locate the tract locations that distinguish cases from controls.
We apply the proposed NAPA test to this dataset, and also compare it with the alternative tests, all under the nominal level $\alpha = 0.05$. The number of identified differential locations by NAPA, BH, GAP, LAWS, and their simple combination is 27, 0, 7, 15, and 18, respectively. Besides, the set of locations found by NAPA is a superset of those found by GAP and LAWS, and is also a superset of those found by their simple combination except for one location. Together with our simulation studies, it seems to suggest that our proposed NAPA test manages to achieve the best power. It is also interesting to note that the majority of the locations identified by NAPA concentrate on the regions with distance 13 to 31, and from 37 to 46 along the tract. Such a finding suggests that additional scientific validation is warranted.
\subsection{Attention deficit hyperactivity disorder study}
The second study is to compare the brain grey matter cortical thickness between subjects diagnosed with attention deficit hyperactivity disorder (ADHD) and typically developing controls \citep{ADHD200}. ADHD is one of the most common child-onset neurodevelopmental disorders. Anatomical MRI is an imaging technique that studies brain anatomical structures. The data records the volume of grey matter at different brain locations in a 3D space for $n_1 = 356$ ADHD subjects and $n_2 = 575$ normal controls. The dataset is available at \url{http://neurobureau.projects.nitrc.org/ADHD200/Data.html}. The MRI images were preprocessed by the Neuro Bereau using the burner pipeline \citep{ADHD200}. To reduce the dimensionality of the problem, we further downsize the image resolution from $256 \times 198 \times 256$ to $30 \times 36 \times 30$, following the same data reduction strategy as in \citet{Li2017} and \citet{LAWS}. The scientific interest is to compare the means of the two sets of brain structural images, and to identify the brain regions that differentiate between cases and controls.
We apply the proposed NAPA test to this dataset, and also compare it with the alternative tests, all under the nominal level $\alpha = 0.05$. The number of identified differential locations by NAPA, BH, GAP, LAWS, and their simple combination is 1152, 349, 641, 538, and 947, respectively. Besides, the set of locations found by NAPA contains the majority of those found by BH, GAP, LAWS, and their simple combination, with the overlapping percentage equal to 96.8\% of BH, 90.5\% of GAP, 95.2\% of LAWS, and 82.2\% of the simple combination. Again, together with our simulation studies, it seems to suggest that our proposed NAPA test manages to achieve the best power. Comparing the identified locations with the Desikan-Killiany brain atlas \citep{Desikan2006}, a number of brain regions stand out, including the left and right entorhinal cortex, the left and right posterior cingulate cortex, left precuneus, among others. These findings generally agree with the current literature on ADHD. Particularly, the posterior cingulate cortex forms a central node in the default mode network of the brain, and has been shown to communicate with various brain networks. ADHD has been suggested as a disorder of the default mode network, and there has been evidence showing that abnormalities in the posterior cingulate cortex may disrupt the default mode network that leads to attentional lapses \citep{Nakao2011}.
\bibliographystyle{apa}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,900 |
Q: Sleep inner thread without sleeping outer thread - Java I have a main Mina handler thread is processing and in that thread i made another thread and set it to sleep for specified time. Now i want that this inner thread sleep independently without blocking Handler thread.
following is sample code.
public void messageReceived(IoSession session, Object message) throws Exception {
Integer tts = 5000;
Thread sleepThread = new Thread(obj);
sleepThread.sleep(tts);
}
currently it is blocking main Handler thread.
A: Thread.sleep() is a static method, so calling sleepThread.sleep(tts) is the same as Thread.sleep(tts). Hence your current thread is just sleeping.
You can't cause another thread to sleep by calling a method on its Thread object. At a push, you could set a flag on the object and your thread could check for the presence of that flag and behave accordingly.
A: try
final int tts = 5000;
Thread sleepThread = new Thread() {
public void run() {
try {
Thread.sleep(tts);
} catch (InterruptedException e) {
throw new RuntimeException(e);
}
}
};
sleepThread.start();
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,596 |
\section{Introduction}%
Infrared shells and bubbles are ubiquitous in the Galaxy and can often be associated with H{\sc ii} regions around young, massive stars \citep{Deharveng2010, Simpson2012}, which emit ionising radiation having $E_\gamma > 13.6\; {\rm eV}$. The ionised gas is heated to $\,\sim\! 10^4\,{\rm K}$, and the resulting pressure increase causes the H{\sc ii} region to expand, sweeping up and compressing the much colder ($\sim 10\;{\rm to}\;30\,{\rm K}$) surrounding molecular cloud. Since molecular clouds typically have a complicated, clumpy internal structure, the ionising radiation penetrates to different depths in different directions, producing highly irregular ionisation fronts. Thus, evolved H{\sc ii} regions have diverse morphologies, sometimes appearing as perfectly round shells like RCW 120 \citep{Deharveng2009}, sometimes filamentary and/or clumpy with large holes through which the ionising radiation can escape, like Carina \citep{Smith2010}.
In \citet[][hereafter W12]{Walch2012} we show that all of these morphological features can be reproduced by invoking different values for the fractal dimension, $\mathcal{D}$, of the molecular cloud into which the H{\sc ii} region expands. Low $\mathcal{D}$ (i.e. $\mathcal{D} \le 2.2$) corresponds to clumpy clouds in which the density sub-structure is dominated by large-scale fluctuations. As $\mathcal{D}$ is increased, small-scale fluctuations become increasingly more important. As a result, W12 report a morphological transition from {\it shell-dominated} H{\sc ii} regions for low $\mathcal{D}$, to {\it pillar-dominated} H{\sc ii} regions for large $\mathcal{D}$. In this paper we investigate the formation of cold clumps at the boundaries of H{\sc ii} regions, and the triggering of star formation in them. We show that the statistical properties of the cold clumps, and of the stars that they spawn, are both correlated with the fractal dimension of the initial molecular cloud. In this context, two distinct modes of triggered star formation have traditionally been defined and contrasted: {\it Collect-and-Collapse} and {\it Radiation-Driven Implosion}.
The {\it Collect-and-Collapse} mode \citep[hereafter C\&C; ][]{Elmegreen1977, Whitworth1994b, Dale2007, Dale2009, Wunsch2010} presupposes a rather homogeneous ambient medium; the expanding H{\sc ii} region then sweeps up this medium into a dense shell, which eventually becomes sufficiently massive to fragment gravitationally and form a new generation of stars \citep[e.g.][]{Wunsch2010}. One interesting feature of the C\&C mode is that it is expected to produce quite massive fragments, and therefore the possibility exists that these will spawn a new generation of massive stars, so that the process can repeat itself \citep[e.g.][]{Whitworth1994a}. In several regions (e.g. Sh2-212 \citep{Deharveng2008} and Sh-104 \citep{Deharveng2003}), there is observational evidence for massive star formation in shell-like structures very close to ionisation fronts. However, it has not yet been shown unequivocally that the formation of these massive stars has been triggered by C\&C.
\citet{Wunsch2012} analyse the dense clumps in the Carina flare and find evidence for shell fragmentation, which they explain with the {\bf P}ressure {\bf A}ssisted {\bf G}ravitational {\bf I}nstability \citep[PAGI; ][]{Wunsch2010} formalism, a model based on the C\&C scenario. With PAGI, the mass spectrum of the clumps forming in a swept-up, pressure-confined shell can be derived if the shell surface density and the confining pressure are known. However, \citet{Dale2011b} note that the clump mass spectrum is quickly changed due to oligarchic growth.
The {\it Radiation-Driven Implosion} mode \citep[hereafter RDI; ][]{Sandford1982, Bertoldi1989, Kessel2003} presupposes a rather structured ambient medium; the expanding H{\sc ii} region then advances most rapidly in those directions where the density is relatively low, thereby overtaking and compressing regions where the density is relatively high, and causing them to implode. Existing simulations of RDI have tended to focus on the onset and efficiency of triggered star formation in isolated pre-existing clumps \citep{Bisbas2011, Haworth2012}, and on the driving of turbulence \citep{Peters2011}. Three-dimensional simulations have been used to study the formation of bright rims and pillars \citep{Miao2006, Gritschneder2009, Gritschneder2010, Bisbas2011, Ercolano2011, Mackey2011} as well as the formation of ultra-compact \citep{MacLow2007,Peters2010, Peters2011} and large-scale H{\sc ii} regions \citep{Mellema2006, Krumholz2007, Arthur2011, Walch2011}.
\citet{Walch2011, Walch2012} have shown that in a realistic situation these two modes may operate in tandem: the expansion of the ionised gas acts {\it both} to organise the neutral gas into a range of dense structures, the most massive of which may be extended and shell-like, {\it and} to overrun these structures and compress them, so that they collapse, or collapse faster than they would otherwise have done. Consequently high- and low-mass fragments are formed and collapse, due to the {\bf E}nhancement of pre-existing {\bf D}ensity substructure and subsequent {\bf G}lobal {\bf I}mplosion' \citep[EDGI; ][]{Walch2011}. These conclusions are based on SPH simulations and observables derived by post-processing individual frames with radiative transfer. The initial structure of the cloud is assumed to be fractal. It is important to note that, although the ionising star may accelerate star formation in its vicinity (in the sense that star formation occurs sooner than it would otherwise have done), and organises stars into distinct structures (partial shells, arcs and clusters), it actually reduces the net amount of star formation (as compared with the mass of stars that would eventually have formed in the absence of an ionising star). This is because the ionising star is very effective in dispersing a large fraction of the cloud gas (W12). Similar results have been found by \citet{Daleetal2012}.
Additional feedback processes like the protostellar jet and stellar wind of the central source, as well as radiation pressure, also add momentum and energy and could support the sweeping up of cold gas and the triggering of star formation. Protostellar jets are dynamically not important for triggering star formation as they are highly collimated, slower, $v_{_{\mathrm JET}} \sim 300\;{\rm km/s}$, and overall less energetic than the radiative feedback of the massive star and its stellar wind.
For the type of star considered here, the stellar wind is also likely to have a negligible effect. However, this estimate is more uncertain. Adopting the mass-luminosity scaling relation for main sequence stars, the central star has a mass of $\sim 25 \;{\rm M_\odot}$. A star of this mass has a wind velocity of $\sim 3000\;{\rm km/s}$ and a mass loss rate of $\sim 10^{-7}\;{\rm M_\odot/yr}$ \citep{Ekstroem2012}. Therefore, the kinetic energy input of the wind is $\sim 9\times 10^{48}\;{\rm erg/Myr}$, which is low compared to the radiative energy input of $10^{52}$ erg/Myr. However, the momentum input of the wind may still be significant, as the conversion of radiative to kinetic energy is quite inefficient \citep{Walch2012}.
In paper I \citep{Walch2012}, we show that the cold shell material is accelerated to $\sim 7 {\rm km/s}$. Assuming a typical shell mass of $2000\; M_\odot$, the amount of radiative energy that has been converted to kinetic energy is $\sim 10^{48} {\rm erg/Myr}$. The wind bubble is usually confined within the HII region \citep{Weaver1977}, and thus the hot wind material shocks with the ionised gas that fills the bubble interior. It is unclear how much wind energy is radiated away during this process. In simulations of wind-bubble expansion caused by a $40\;{\rm M_\odot}$ star in uniform media, \citet{Toala2011} show that the fraction of the time-integrated mechanical wind energy, which is retained by the surrounding ISM in form of kinetic energy, is $\sim$ 10\%. Therefore the relative importance of the wind momentum input could be important. We will investigate this in a future paper.
Finally, radiation pressure on dust could add to the expansion of the cloud. In particular, radiation pressure might be the dominant feedback mechanism in the case of massive star clusters forming in giant molecular clouds with $> 10^6\;{\rm M}_\odot$ \citep{Murray2010}. However, \citet{KM2009} estimate that the impact of radiation pressure is small in HII regions driven by single or small N-clusters of massive stars. Therefore, it may be safely neglected in this study.
The plan of this paper is as follows. In Section \ref{SEC:METHOD} we describe how we construct fractal molecular clouds and the numerical scheme that we use. In Section \ref{SEC:CLUMPS} we discuss the properties of the cold clumps swept up and/or compressed by an H{\sc ii} region expanding into a molecular cloud, as a function of its fractal dimension, ${\cal D}$. In Section \ref{SEC:STARS} we focus on the triggering of star formation, and characterise the distribution and the statistical properties of the stars formed, as a function of ${\cal D}$. We summarise our main conclusions in Section \ref{SEC:CONC}.
\section{Numerical method \& Initial conditions}\label{SEC:METHOD}%
\subsection{Generation of fractal molecular clouds}\label{fractal}%
It appears that the internal structure of molecular clouds is broadly self-similar over four orders of magnitude, from $\sim 0.1\,{\rm pc}$ to $500\,{\rm pc}$ \citep[e.g.][]{Bergin2007, Sanchez2010}, and that it is approximately fractal, with dimension in the range $2.0\!\la\!{\cal D}\!\la\!2.8$ \citep[e.g.][]{Falgarone1991, Elmegreen1996, Stutzki1998, Vogelaar1994, Lee2004, Sanchez2005, Sanchez2007, Schneider2011, Miville2010}.
To construct a fractal cloud we consider a $\,2\!\times\!2\!\times\!2\,$ cubic computational domain, and specify three parameters: the fractal dimension, ${\cal D}\!=\!2.0,\,2.2,\,2.4,\,2.6,\,2.8$ (or equivalently \citep{Stutzki1998} the spectral index, $n=8-2{\cal D}=4.0,\,3.6,\,3.2,\,2.8,\,2.4$, where $n$ relates to the 3D density power spectrum, $P_k\propto k^{-n}$, $k$ is wavenumber, and $k\!=\!1$ corresponds to the linear size of the cubic domain, i.e. $\lambda(k)=2/k$); a random seed, ${\cal R}$, which allows us to generate multiple realisations; and a density-scaling parameter, $\rho_{_{\rm O}}$. We populate all modes having integer $k_x$, $k_y$ and $k_z$ in $(1,128)$, with random phases and amplitudes drawn from the power-spectrum. Next we perform an FFT to evaluate the function $\rho_{_{\rm FFT}}(x,y,z)$ on a $128^3$ Cartesian grid spanning the computational domain. Finally we compute the density in the computational domain, according to
\begin{equation}
\rho(x,y,z)=\exp{\left(\frac{\rho_{_{\rm FFT}}(x,y,z)}{\rho_0} \right)}\,,\label{EqScale}
\end{equation}
where $\rho_{_{\rm O}}$ is a dimensionless scaling parameter \citep{Shadmehri2011}. SPH particles are then distributed randomly in each cell of the grid according to its density, and SPH particles that fall outside a unit-radius sphere are culled. The resulting sphere can then be re-scaled to arbitrary total mass, $M_{_{\rm C}}$ and arbitrary radius, $R_{_{\rm C}}$.
With the above procedure, the random seed, ${\cal R}$ completely determines the pattern of the density field. The fractal dimension determines the distribution of power: for small ${\cal D}$ (large $n$), most of the power is on large scales, so the density field is dominated by extended structures; conversely, for large ${\cal D}$, (small $n$), there is more power on small scales, and so the density field is dominated by smaller structures. The scaling parameter, $\rho_{_{\rm O}}$, determines the density contrast, and hence the width of the (approximately log-normal) density PDF; increasing $\rho_{_{\rm O}}$ decreases the density contrast and therefore reduces the width of the PDF.
\subsection{Numerical method}
We use the SPH code \textsc{seren} \citep{Hubber2011}, which is well-tested and has already been applied to many problems in star formation \citep[e.g.][]{Walch2011, Bisbas2011, Stamatellos2011}. We employ the SPH algorithm of \citet{Monaghan1992} with a fixed number of neighbours, $N_{_{\rm NEIGH}}=50$. The SPH equations of motion are solved with a second-order Leapfrog integrator, in conjunction with an hierarchical block time-stepping scheme. Gravitational forces are calculated using an octal spatial decomposition tree \citep{Barnes1986}, with monopole and quadrupole terms and a Gadget-style opening-angle criterion \citep{Springel2001}. We use the standard artificial viscosity prescription \citep{Monaghan1983}, moderated with a Balsara switch \citep{Balsara1995}.
The ionizing radiation is treated with an HEALPix-based, adaptive ray-splitting algorithm, which allows for optimal resolution of the ionization front in high resolution simulations \citep[see ][]{Bisbas2009}. Along each HEALPix ray, the radiative transfer is evaluated at discrete points, $j$. These points are separated by $f_1h_j$, where $f_1=0.6$ is an accuracy parameter and $h_j$ is the local smoothing length (adjusted to enclose $\sim$50 SPH neighbours). A ray is split if the linear separation of neighbouring rays is greater than $f_2h_j$, where $f_2$ is the angular resolution parameter. Typically, good results are achieved for $1.0\leq f_2\leq 1.3$; here we use $f_2=0.8$ to further increase the angular resolution of the ray tracing scheme. We allow for a maximum number of $l_{_{\rm MAX}}=11$ HEALPix levels corresponding to $12 \times 4^{l_{_{\rm MAX}}} \approx 5 \times 10^7$ rays if the whole sphere were refined to $l_{_{\rm MAX}}$. In the simulations presented here, most directions only require $\leq 8$ levels of refinement, and there are typically $\sim 10^5$ rays in total.
The gas is assumed to be either fully molecular with a mean molecular weight of $\mu_{_{\rm NEUT}}=2.38$, or fully ionized with $\mu_{_{\rm ION}}=0.7$. The temperature of ionized gas is set to $T_{_{\rm ION}}=$10,000 K. The temperature of neutral gas is given by a barotropic equation of state,
\begin{equation}
T(\rho)=T_{_{\rm NEUT}} \left[ 1+(\rho/\rho_{_{\rm CRIT}})^{(\gamma-1)}\right]\,,
\end{equation}
where $T_{_{\rm NEUT}}=30\,\rm{K}$, $\rho_{_{\rm CRIT}}=10^{-13}\,{\rm g}\,{\rm cm}^{-3}$, and $\gamma=5/3$. The use of $T_{_{\rm NEUT}}=30\,{\rm K}$ may influence the fragmentation properties of the forming shell. Since the shell becomes very dense and should therefore be allowed to cool further, it might fragment more efficiently than currently seen in our simulations; for this reason we will explore a more complicated cooling function in a future paper.
We introduce sinks at density peaks above $\rho_{_{\rm SINK}}=10^{-11}\,\rm{g}\,\rm{cm}^{-3}$, using the new algorithm developed by \citet{Hubber2013}. Since $\rho_{_{\rm SINK}}\!\gg\!\rho_{_{\rm CRIT}}$, a condensation that is transformed into a sink is normally already well into its Kelvin-Helmholtz contraction phase. Once formed, a sink is able to accrete gas smoothly from its surroundings and thereby grow in mass.
\begin{table}
\begin{center}
\begin{tabular}{ccccccc}\hline
ID & ${\cal D}$ & Seed & $\bar{\rho}_{_{\rm MW}}$ & $\sigma_{_{\rm O}}^2$ & $t_{_1}$ & $t_{_{15}}$ \\
& & & $\overline{10^{-21}{\rm g\,cm}^{-3}}$ & & $\overline{\rm Myr}$ & $\overline{\rm Myr}$ \\\hline
${\cal D}$2.0/O7(1) & 2.0 & 1 & 1.23 & 1.08 & 0.46 & 0.66 \\
${\cal D}$2.0/O7(2) & 2.0 & 2 & 1.17 & 1.42 & 0.50 & 0.61 \\
${\cal D}$2.0/O7(3) & 2.0 & 3 & 1.17 & 1.10 & 0.56 & 0.66 \\\\
${\cal D}$2.2/O7(1) & 2.2 & 1 & 1.17 & 1.06 & 0.47 & 0.62 \\
${\cal D}$2.2/O7(2) & 2.2 & 2 & 0.98 & 1.12 & 0.43 & 0.61 \\
${\cal D}$2.2/O7(3) & 2.2 & 3 & 1.17 & 0.90 & 0.57 & 0.66 \\\\
${\cal D}$2.4/O7(1) & 2.4 & 1 & 1.07 & 0.76 & 0.51 & 0.67 \\
${\cal D}$2.4/O7(2) & 2.4 & 2 & 0.93 & 0.79 & 0.35 & 0.66 \\
${\cal D}$2.4/O7(3) & 2.4 & 3 & 0.93 & 0.55 & 0.47 & 0.87 \\\\
${\cal D}$2.6/O7(1) & 2.6 & 1 & 0.89 & 0.53 & 0.45 & 1.00 \\
${\cal D}$2.6/O7(2) & 2.6 & 2 & 0.89 & 0.58 & 0.44 & 0.72 \\
${\cal D}$2.6/O7(3) & 2.6 & 3 & 0.85 & 0.44 & 0.45 & 0.81 \\\\
${\cal D}$2.8/O7(1) & 2.8 & 1 & 0.81 & 0.41 & 0.67 & 0.97 \\
${\cal D}$2.8/O7(2) & 2.8 & 2 & 0.81 & 0.44 & 0.46 & 0.73 \\
${\cal D}$2.8/O7(3) & 2.8 & 3 & 0.85 & 0.36 & 0.71 & 0.86 \\\hline
\end{tabular}
\caption{Simulation parameters. Column 1 gives the simulation ID; column 2, the fractal dimension (${\cal D}$); column 3, the random seed used; column 4, the mass-weighted mean-density ($\bar{\rho}_{_{\rm MW}}$); column 5, the variance of the log-normal mass-weighted density PDF ($\sigma_{_{\rm O}}^2$); column 6 the time at which the first sink ("protostar") forms ($t_1$); and column 7, the time at which the fifteenth sink forms ($t_{15}$).}
\label{TAB:PARAMS}
\end{center}
\end{table}
\subsection{Initial conditions}
We consider a spherical cloud with total mass ${\rm M}_{_{\rm MC}}=10^4\,{\rm M}_\odot$, radius $R_{_{\rm MC}}=6.4\,\rm{pc}$, mean density $\bar{\rho}=6.17 \times 10^{-22}\,{\rm g}\,{\rm cm}^{-3}$, and mean freefall time $t_{_{\rm FF}}=3\,{\rm Myr}$, with an O7 star at its centre emitting Lyman continuum photons at a rate $\dot{\cal N}_{_{\rm LyC}}=10^{49}\,{\rm s}^{-1}$. We treat five fractal dimensions, ${\cal D}=2.0,\,2.2,\,2.4,\,2.6\,{\rm and}\,2.8$, corresponding to spectral indices $n=4.0,\,3.6,\,3.2,\,2.8\,{\rm and}\,2.4$ respectively. For lower ${\cal D}$ (higher $n$), a larger fraction of the power is invested in the extended structures. For each value of ${\cal D}$, we treat three different realisations, by invoking three different random seeds, ${\cal R}$, but we use the same three random seeds. Simulations with the same seed, ${\cal R}$, but different fractal dimension, ${\cal D}$, have the same pattern of density peaks and troughs, and differ only in the sense that for lower ${\cal D}$ the more extended structures exhibit more density contrast, and the more compact structures exhibit less density contrast. We use a single value of the scaling parameter, $\rho_{_{\rm O}}=1.0$. By using three different random seeds, we are able both to evaluate the extent to which the results depend on the particular choice of ${\cal R}$, and to improve the statistics of the results for each individual value of ${\cal D}$. The simulations are given IDs of the form "${\cal D}$2.0/O7(1)", where characters two through four (following ${\cal D}$) give the fractal dimension, characters six and seven (following the oblique stroke) indicate that the cloud is ionised by an O7 star, and character nine (in parentheses) records which seed was used. All simulations are preformed with ${\cal N}_{_{\rm TOT}}\!\sim\!2.5\times 10^6$ particles, and therefore the minimum mass that can be resolved is $M_{_{\rm MIN}}\!=\!50\,M_{_{\rm MC}}/{\cal N}_{_{\rm TOT}}\!\sim\!0.2\,{\rm M}_{_\odot}$.
\begin{figure}
\includegraphics[width=92mm, angle=0]{PLOTS/seed1_rhopdfs.ps}
\caption{The mass-weighted logarithmic density PDFs for the initial conditions generated with different fractal dimensions (${\cal D}=2.0,\,2.2,\,2.4,\,2.6,\,2.8$) but the same random seed (seed 1). The mass-weighted mean densities and logarithmic variances for these distributions (and those obtained with the other two seeds) are given in Table \ref{TAB:PARAMS}.}
\label{FIG:RHOPDF}
\end{figure}
\begin{figure*}
\begin{center}
\begin{tabular}{c}
\includegraphics[width=162mm]{PLOTS/montage7.eps}
\end{tabular}
\caption{False-colour column density images of all simulations at $t_{15}$ (see Table \ref{TAB:PARAMS}). The ionising source is located at the center of each panel of size $14 \times 14$ pc. From top to bottom ${\cal D}$ increases from ${\cal D}=2.0$ (top) to ${\cal D}=2.8$ (bottom). Each column represents clouds generated with the same random seed, and hence an initial density field with the same pattern. Sink particles are marked as turquoise dots; many of the sinks are in close (unresolved) multiple systems.}
\label{FIG_M1}
\end{center}
\end{figure*}
Fig. \ref{FIG:RHOPDF} shows the mass-weighted density PDFs for the clouds created with the first seed. Since they are approximately log-normal, we can compute a standard deviation, $\sigma_{_{\rm O}}$, for each one. Evidently $\sigma_{_{\rm O}}$ increases with decreasing ${\cal D}$, because at lower ${\cal D}$ a larger fraction of the power is concentrated in a few large-scale structures. This in turn means that for lower ${\cal D}$ there is, at the outset, more gas at large densities, and therefore star formation tends to occur sooner. Table \ref{TAB:PARAMS} gives the basic parameters for the complete suite of simulations.
\section{Spatial distribution and intrinsic statistics of clumps}\label{SEC:CLUMPS}%
\subsection{General morphology}%
The morphology of the evolving H{\sc ii} region is strongly dependent on the fractal dimension, ${\cal D}$, of the initial molecular cloud. Fig. \ref{FIG_M1} shows false-colour column-density images of all 15 simulations (i.e. from top to bottom, all five values of ${\cal D}$, and from left to right, all three random seeds), projected onto the $z\!=\!0$ plane. The snapshots are all taken at time $t_{15}\,$, i.e. the time at which the fifteenth sink is created, and values of $t_{15}$ are given in Table \ref{TAB:PARAMS}. Sinks are shown as turquoise dots. Due to the variance in the initial density fields, different realisations with the same fractal dimension exhibit different star formation rates, but there is a tendency for star formation to occur later when ${\cal D}$ is higher. \citet{Walch2012} report a systematic morphological transition, from a shell-dominated H{\sc ii} region structure for low fractal dimension (${\cal D}=2.0\,{\rm or}\,2.2$), to a pillar-dominated structure at high fractal dimension (${\cal D}=2.6\,{\rm or}\,2.8$). Here, we confirm that this morphological transition persists when using different realisations, even though the detailed appearance of an H{\sc ii} region depends strongly on the random seed used, and on the viewing angle. In the following, we utilise the improved statistics afforded by multiple realisations to investigate the characteristics of the cold, swept-up clumps bordering the H{\sc ii} region and the subsequent triggered star formation, as a function of ${\cal D}$.
\begin{figure*}
\begin{center}
\begin{tabular}{c}
\includegraphics[width=180mm]{PLOTS/partdist_fft_all_montage_seed1.ps}
\end{tabular}
\caption{The positions of high-density SPH particles, projected onto the $z\!=\!0$ plane, for all the simulations set up with the first seed (i.e. ${\cal D}2.0/O7(1)$ (left) to ${\cal D}2.8/O7(1)$ (right) at $t_{_{\rm 15}}$. Particles, $p$, with density $6\times 10^{-20}\,{\rm g\,cm}^{-3}<\rho_p<6\times 10^{-19}\,{\rm g\,cm}^{-3}$ are plotted in black, and those with density $\rho_p>6\times 10^{-19}\,{\rm g\,cm}^{-3}$ are plotted in red. The red particles are the 'core' particles used to determine the core statistics.}
\label{FIG:PARTDIST}
\end{center}
\end{figure*}
\begin{figure*}
\begin{tabular}{ll}
\includegraphics[width=92mm]{PLOTS/fftscaled-1d4msun-25d6-df2.0-pdf1_new_clumps2.ps} &
\includegraphics[width=92mm]{PLOTS/fftscaled-1d4msun-25d6-df2.8-pdf1_clumps2.ps} \\
\end{tabular}
\caption{The dynamical evolution of clump masses and positions projected on the $z\!=\!0$ plane. False colour encodes time (see colour bar), and track width encodes clump mass. The left frame shows a low fractal dimension case (${\cal D}$2.0/O7(1)), and the right frame a high fractal dimension case (${\cal D}$2.8/O7(1)).}
\label{FIG:CLUMPS}
\end{figure*}
\subsection{Clump formation in shells and pillars}%
There is strong observational evidence that H{\sc ii} regions are usually surrounded by shell-like structures that contain dense, molecular clumps, and that these clumps are often the sites of new star formation \citep[e.g.][]{Zavagno2010, Deharveng2010}. However, it is still unclear how these clumps form, i.e. what is the relative importance of RDI, C\&C, PAGI, EDGI, etc. (see Section 1). In some regions the clump mass function (CMF) appears to be rather similar to the one found in low-mass star forming regions \citep{Wunsch2012}, whereas in other regions the presence of more massive clumps suggest that the C\&C mechanism may have been at work \citep{Deharveng2003}.
To investigate the statistics of the cold clumps formed in our simulations, we first identify all the SPH particles having density $\rho_p>6\times 10^{-19}\,{\rm g\,cm}^{-3}$ (i.e., for molecular gas, $n_{{\rm H}_2}>1.5\times 10^5\,{\rm cm}^{-3}$). This density is sufficiently high for the gas to couple thermally to the dust, and a significant proportion of it should be destined to form stars. The free-fall time for a lump of gas with a uniform density of $6\times 10^{-19}\,{\rm g\,cm}^{-3}$ is $\sim$0.1 Myr. The positions of the SPH particles selected in this way are plotted in Fig. \ref{FIG:PARTDIST} for all the simulations performed with the first random seed; SPH particles having density $\rho_p>6\times 10^{-19}\,{\rm g\,cm}^{-3}$ are plotted in red, and -- for comparison -- those having density $6\times 10^{-20}\,{\rm g\,cm}^{-3}<\rho_p<6\times 10^{-19}\,{\rm g\,cm}^{-3}$ in black. Individual clumps are clearly picked up more reliably with the higher density threshold.
We identify individual clumps by applying the Friends-of-Friends (hereafter FoF) algorithm to the selected subset of high-density SPH particles, using a linking length of $\ell=0.05\,{\rm pc}$. Thus a clump represents a collection of high-density SPH particles, all of which are no further than $0.05\,{\rm pc}$ from at least one other member of this collection; any value of $\ell$ in the range $(0.01,0.1){\rm pc}$ delivers broadly similar clumps statistics.
Fig. \ref{FIG:CLUMPS} illustrates the dynamical evolution of the clumps in the simulations ${\cal D}$2.0/O7(1) and ${\cal D}$2.8/O7(1); positions are projected on the the $z\!=\!0$ plane, time is colour-coded, and the width of the symbol encodes the mass of the clump. In the case with low fractal dimension (left-hand frame, ${\cal D}\!=\!2.0$), the plot is dominated by a small number of massive clumps, which are distributed very anisotropically with respect to the ionising star. In the case with high fractal dimension (right-hand frame, ${\cal D}\!=\!2.8$), there are many more clumps, but they are much less massive, and they are distributed much more isotropically. All the clumps are being driven outwards by the rocket effect \citep{Kahn1954, Oort1955}, at speeds up to $\sim\!10\,{\rm km\,s}^{-1}$.
Fig. \ref{FIG:MCLUSTER} illustrates the time evolution of clump masses. Different ${\cal D}$ are represented by different colours. For a given ${\cal D}$, the solid line shows the total mass in clumps; the dotted line shows the mass of the most massive clump; and the dashed line shows the mean clump mass. As ${\cal D}$ is increased, clumps start forming earlier, but ultimately the total mass in clumps and the masses of individual clumps are lower. To make the plot easier to read we only present results for ${\cal D}=2.0,\,2.4\,{\rm and}\,2.8$, but the in-between values show the same trends; for each of these ${\cal D}$-values, we have combined the results from all three realisations.
The growth of a clump is not driven by gravity \citep[as, for example, in Bondi accretion;][]{Bondi1952}, and there is also no strictly oligarchic growth \citep[as e.g. inferred by][]{Dale2011b}. Rather, in the first instance, matter is driven into a clump wherever the shock waves that precede the expanding ionisation front converge. Later on the ionisation front will start to erode the clump, but at the same time the clump will be driven outwards by the rocket effect, and sweep up material from further out in the cloud, like a snowplough. The evolution of its mass is then a competition between these two effects.
\begin{figure}
\includegraphics[width=92mm]{PLOTS/mcluster.eps}
\caption{Time evolution of the total mass in clumps (solid line), the maximum clump mass (dotted line), and the mean clump mass (dashed line), for ${\cal D}=2.0$ (black), ${\cal D}=2.4$ (green) and ${\cal D}=2.8$ (red). In each case, the results from three different realisations have been collated. The results for ${\cal D}=2.2$ and ${\cal D}=2.6$ are not included, simply to make the plot easier to read.}
\label{FIG:MCLUSTER}
\end{figure}
\subsection{Clump statistics}%
Fig. \ref{FIG:CMF} displays clump mass functions (CMFs) for the different ${\cal D}$ values, at times $t=0.40,\,0.50,\,0.60\,{\rm and}\,0.66\,{\rm Myr}$; $\;0.66\,{\rm Myr}$ is the last time reached by all simulations. The bin size is $\Delta \log_{10}\left(M/{\rm M}_{_\odot}\right)\!=\!0.2$, and the plot is logarithmic (so that the Salpeter stellar IMF would have slope $m=-1.35$). Typically, the mass function of cold clumps -- which are not necessarily gravitationally bound -- is somewhat flatter than Salpeter, with slopes $m\sim -0.7$ being commonly reported \citep[e.g.][]{Kramer1998, Wunsch2012}. However, a steeply decreasing power law is not recovered in all molecular clouds; for example, in Orion, \citet{Li2007} even report an {\it increasing} power law of $m=+0.15$ over the mass range $0.1 \;{\rm M}_\odot \le M \le 10\;{\rm M_\odot}$.
\begin{figure*}
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=85mm]{PLOTS/cmf_all_t040.eps} &
\includegraphics[width=85mm]{PLOTS/cmf_all_t050.eps} \\
\includegraphics[width=85mm]{PLOTS/cmf_all_t060.eps} &
\includegraphics[width=85mm]{PLOTS/cmf_all_t066_fit.eps}
\end{tabular}
\caption{CMFs at times, $t=0.4,\,0.5,\,0.6\,{\rm and}\,0.66$ Myr (reading from top left to bottom right). For each ${\cal D}$, we have combined the clump masses from the three different realisations. At $t=0.66$ Myr we determine the power law slopes of the CMFs by means of $\chi^2$-minimisation (dashed lines).}
\label{FIG:CMF}
\end{center}
\end{figure*}
At early times all CMFs appear quite similar, but as time advances, there are three main trends. First, more clumps are formed, as more gas is swept up by the expanding H{\sc ii} region. Second, the CMFs extend to higher mass, as a few of them have trajectories that cause them to intercept and sweep up dense material as they plough outwards. Third, the CMFs are shallow, almost flat, for ${\cal D}=2.0$, and become increasingly steep with increasing ${\cal D}$. We fit the CMFs at $t=0.66\,{\rm Myr}$ with a power-law, using $\chi^2$-minimisation; the resulting fits are shown as dashed lines on Fig. \ref{FIG:CMF}, and have the following slopes:
\begin{center}
\begin{tabular}{rccccc}
${\cal D}\;=\;$ & 2.0 & 2.2 & 2.4 & 2.6 & 2.8 \\
$m\;=\;$ & -\,0.18 & -\,0.32 & -\,0.37 & -\,0.47 & -\,0.91 \\
\end{tabular}
\end{center}
The principal cause of this difference is that low ${\cal D}$ delivers coherent, extended density enhancements in the initial cloud, and this promotes the growth of high-mass clumps within the shells bordering the expanding H{\sc ii} region; although these large clumps are eroded by the ionisation front on the side facing the star, they are also pushed outwards sweeping up the large amounts of neutral gas on the side facing away from the ionising star. In contrast, high ${\cal D}$ delivers smaller structures in the initial density field, and these develop into cometary globules and pillars; the ionisation front wraps round them, and so they are eroded from many different directions and having small cross-sections they do not sweep up so much extra mass as they plough outwards.
\subsection{Internal velocity dispersion}%
On Fig. \ref{FIG:DISPSIZE} (top panel) we plot the velocity dispersion inside each clump, $\sigma_{_{\rm CLUMP}}$, against its linear size, $L_{_{\rm CLUMP}}$, at time $t\!=\!0.66\,{\rm Myr}$. The velocity dispersion of a clump is computed by adding in quadrature the contribution from non-thermal motions (i.e. the velocity dispersion of the constituent SPH particles) and the contribution from thermal velocity dispersion. The linear size of a clump is simply its maximum extent. Many of the clumps conform approximately to Larson's Scaling relation \citep{Larson1981},
\begin{equation}\label{EQ:LARSON}
\frac{\sigma_{_{\rm CLUMP}}}{\rm km/s} = 1.1\; \left (\frac{L_{_{\rm CLUMP}}}{\rm pc}\right)^{\;0.38}\,.
\end{equation}
However, the clumps that are forming stars (hereafter "star-forming clumps", represented by symbols containing a filled circle) all lie well above Larson's Scaling Relation. Their velocity dispersion is higher because they are being accelerated and compressed by the H{\sc ii} region, and because they contain high-velocity flows onto forming protostars.
\citet{Heyer2009} have suggested that the velocity dispersions in molecular clouds might be fitted more accurately if one assumes approximate virialisation, i.e. $\sigma_{_{\rm CLUMP}}\simeq(\pi G\Sigma_{_{\rm CLUMP}} R_{_{\rm CLUMP}}/5)^{1/2}$, where $\Sigma_{_{\rm CLUMP}}$ and $R_{_{\rm CLUMP}}$ are, respectively, the surface-density and radius of a clump (the triggered clumps typically have $10^2 \stackrel{<}{\sim}\Sigma_{_{\rm CLUMP}} \stackrel{<}{\sim}10^4\;{\rm M_\odot/pc^2}$). Putting $\Sigma_{_{\rm CLUMP}}\!=\!M_{_{\rm CLUMP}}/\pi R^2_{_{\rm CLUMP}}$ and $R_{_{\rm CLUMP}}\!=\!L_{_{\rm CLUMP}}/2$, this reduces to
\begin{equation}\label{EQ:HEYER}
\sigma_{_{\rm CLUMP}}\simeq \left(\frac{2GM_{_{\rm CLUMP}}}{5L_{_{\rm CLUMP}}}\right)^{1/2}\,.
\end{equation}
\vspace{-0.1cm}
In the lower panel of Fig. \ref{FIG:DISPSIZE} we plot $\sigma_{_{\rm CLUMP}}$ against $(M_{_{\rm CLUMP}}/L_{_{\rm CLUMP}})^{1/2}$. We find that all clumps lie above the Heyer Scaling Relation (Eqn. \ref{EQ:HEYER}), but those that are forming stars have higher $\sigma$-values, more than a factor $\sim\! 10$ higher.
\begin{figure}
\includegraphics[width=90mm]{PLOTS/dispsize_newsym.eps} \\
\includegraphics[width=90mm]{PLOTS/dispsd2.eps}
\caption{{\sc Top:} The internal velocity dispersion $\sigma$ of a clump plotted against its maximum extent $L$, at $t=0.66\,{\rm Myr}$. Star-forming clumps are marked with an additional filled circle in the same colour. We also show Larson's line-width-size relation (dashed line). {\sc Bottom:} Velocity dispersion $\sigma$ as a function of the square-root of the clump mass-to-size ratio, $(M_{_{\rm CLUMP}}/L_{_{\rm CLUMP}})^{1/2}$. The relationship derived for galactic molecular clouds is indicated by the lowest dotted line \citep[][ see Eq. \ref{EQ:HEYER}]{Heyer2009}. The following dotted lines indicate the \citet{Heyer2009} relation multiplied by a factor of 2, 5, and 10. }
\label{FIG:DISPSIZE}
\end{figure}
\begin{figure}
\begin{tabular}{l}
\includegraphics[width=92mm]{PLOTS/tform_rform.ps} \\
\end{tabular}
\caption{The radial distance from the ionising source at sink formation, $R_{_{\rm FORM}}$, as a function of the sink formation time, $t_{_{\rm FORM}}$ for all simulations. Results obtained with different ${\cal D}$ are represented with different colours and different symbols. The best linear fit to this distribution of sinks is obtained using a slope of $7.1\;{\rm km/s}$ (dashed line).}
\label{FIG:MSINK}
\end{figure}
\section{Spatial distribution and intrinsic statistics of stars}\label{SEC:STARS}%
All the star formation in the simulations presented here is triggered. This has been demonstrated unequivocally in \citet{Walch2012} where, for comparison, we evolve the same fractal clouds without a central ionising star, and show that spontaneous star formation does not occur until long after the evolution time considered here. To characterise the consequences of triggered star formation, we collate the statistical properties of the sink particles, as a function of ${\cal D}$. We employ the improved sink particle algorithm of \citet{Hubber2013}, which has been demonstrated to improve the robustness of sink particle properties against numerical effects. All simulations are advanced until at least 15 sink particles have formed, and their properties are evaluated at this time ($t_{15}$).
\subsection{The location of stars, relative to the ionising star}%
Fig. \ref{FIG:MSINK} shows how the formation radius, $R_{_{\rm FORM}}$ (i.e. the distance from the ionising star to a sink when it is first created), varies with the sink formation time, $t_{_{\rm FORM}}$. As ${\cal D}$ increases, the sinks form later and at larger radii. We derive a linear fit to the distribution of sinks in the ($R_{_{\rm FORM}},t_{_{\rm FORM}}$)-plane, using a $\chi^2$-minimization method. The best fit has a slope of $7.1\;{\rm km/s}$ (see dashed line in Fig. \ref{FIG:MSINK}). This velocity is comparable to the radial velocity of the ionisation front, and shows that stars are progressively triggered by the expansion of the H{\sc ii} region.
\subsection{The location of stars, relative to the ionisation front}%
Fig. \ref{FIG:RHIST} shows two histograms of the number of sinks as a function of $R_{_{\rm 15}}/R_{_{\rm IF}}$, where $R_{_{\rm IF}}$ is the three-dimensional distance from the ionising star to the ionisation front along the line from the ionising star to the sink. The number of sinks is divided by the area of the annulus corresponding to each bin, so as to yield a surface density \citep[see][]{Thompson2012}. For simplicity, we are assuming that a 2-dimensional projection of the system along an arbitrary line of sight would on average yield the same ratio $R_{_{\rm 15}}/R_{_{\rm IF}}$. In the top panel, we compile all sinks formed in all simulations into one histogram. There is a clear over-density of triggered stars at, or close to, $R_{_{\rm IF}}$. In the bottom panel, we repeat the same analysis but distinguish the distributions for different $\mathcal{D}$. For low ${\cal D}$, the sinks typically stay ahead of the ionisation front, because the gas in the clumps from which these sinks are formed has been accelerated by the rocket effect. Therefore the sinks that condense out of it have significant radial velocities, whereas the expansion of the ionisation front is slowing down. Conversely, for high ${\cal D}$ the sinks form in the heads of pillars and are frequently left behind in the H{\sc ii} region, thus leading to a flat distribution of $R_{_{\rm 15}}/R_{_{\rm IF}} $.
Overall, the derived distribution of triggered stars compares remarkably well with the observational findings of \citet{Thompson2012}, who study the over-density of young stellar objects around {\it Spitzer} bubbles. However, they find a significantly enhanced source density at small radii ($R_{_{\rm 15}}/R_{_{\rm IF}} < 0.5$), which is not present in our analysis. One reason for this is the fact that they use projected positions of stars, whereas their estimate of the radius of the H{\sc ii} region is determined by its lateral extent. Thus, for a star on the far or near side of the H{\sc ii} region, close to the line of sight through the ionising star(s), $R_{_{\rm 15}}/R_{_{\rm IF}}$ will appear much smaller than the true three-dimensional value.
\begin{figure}
\includegraphics[width=90mm]{PLOTS/if_sinkpos_all.ps}
\includegraphics[width=90mm]{PLOTS/if_sinkpos.ps}
\caption{Histograms of the number of sinks at $t_{_{\rm 15}}$, as a function of the radial distance of the sink from the ionising star, $R_{_{\rm 15}}$, divided by the radial distance of the ionisation front from the ionising star, $R_{_{\rm IF}}$, in the same direction. The number counts are scaled by area of the annulus corresponding to each bin and thus represent a surface density. {\sc Top:} Total number count compiled using all sinks in all simulations. {\sc Bottom:} Number counts compiled for each fractal dimension.}
\label{FIG:RHIST}
\end{figure}
\begin{figure}
\includegraphics[width=92mm]{PLOTS/rfin_mfin.ps} \\
\includegraphics[width=90mm]{PLOTS/msink_dmdt_all.ps}
\caption{{\sc Top panel:} Final sink mass, $M_{_{\rm 15}}$, as a function of final radial distance from the center, $R_{_{\rm 15}}$. Results obtained with different ${\cal D}$ are represented with different colours and different symbols. The star symbols show the mean for each value of ${\cal D}$. {\sc Bottom panel:} Mean mass accretion rate as a function of sink mass at $t_{_{\rm 15}}$. Sinks located above the dotted lines can evolve into a massive star with $M > 8\;{\rm M}_\odot$ if they continue accreting at their previous rate for 0.1 Myr (top line), 0.2 Myr (middle line), or 0.3 Myr (bottom line). }
\label{FIG:MASSES}
\end{figure}
\subsection{The masses of stars}%
Observational results from the Milky Way project \citep{Kendrew2012} and \citet{Deharveng2010} indicate that approximately 20\% of young, massive star formation may have been triggered. This estimate is also in agreement with \citet{Thompson2012}, who derive 14\% - 30\%.
Fig. \ref{FIG:MASSES} (top panel) shows how the sink mass, $M_{_{\rm 15}}$, varies with the three-dimensional radial distance from the center $R_{_{\rm 15}}$ at $t_{_{\rm 15}}$. Massive sinks ($M\!>\!8\,{\rm M}_{_\odot}$) are quite common for low ${\cal D}$ (shell-dominated morphology), and much rarer for high ${\cal D}$ (pillar-dominated morphology). The sinks seem to be aligned in vertical stripes, which are caused by two effects. First, $t_{15}$ and therefore the mean ionisation front radius varies for the three different realisations of every $\mathcal{D}$, which causes preferential triggering at different radii. Second, in some cases multiple arcs form around the H{\sc ii} region and therefore the sinks can be clustered about different radii.
In the bottom panel of Fig. \ref{FIG:MASSES} we plot the mean mass accretion rates onto sinks up to $t_{_{\rm 15}}$, $\dot{M}$, as a function of their masses, $M_{_{\rm 15}}$. Above and to the right of the dotted lines a sink that continues to accrete at the observed rate for $0.1\,,0.2\;{\rm or}\;0.3\,{\rm Myr}$ will exceed $8\,{\rm M}_{_\odot}$, and therefore would be classified as a massive star. We see that for low ${\cal D}$ a significant fraction of sinks either are already, or will soon be, massive in this sense, whereas for higher ${\cal D}$ fewer of them are destined to be massive. The simulations with low-${\cal D}$ also appear to produce more low-mass stars, i.e. a bigger range of masses at both extremes is produced. The mass accretion rates are generally quite high ($\dot{M} \sim 5\times 10^{-5}\;{\rm M}_\odot/{\rm yr}$), which is not unexpected for these early stages of star formation. However, they decline as soon as the surrounding cold material has been accreted onto a sink, or ablated by the ionising radiation.
With respect to triggered star formation, the major limitation of the simulations presented here is that radiative and mechanical feedback from newly-formed stars is not included. Therefore, the quoted mass accretion rates are upper limits and we probably over-estimate the number of massive stars formed. At $\;t_{_{\rm 15}}$ the results are still credible, since the percentage of sinks with $M \ge 8\;{\rm M}_\odot$ is only 6.7\%. If all sinks were to continue accreting at their measured rate after $t_{_{\rm 15}}$, by $t_{_{\rm 15}} +0.1$ Myr the percentage of sinks with $M \ge 8\;{\rm M}_\odot$ would be $\sim$ 25\%. If feedback from newly-formed stars were included, it is not clear whether such a high percentage of massive stars would be able to form.
\subsection{The clustering of stars}%
For each simulation, at $t_{15}$, we perform a Minimum Spanning Tree (MST) analysis. To construct an MST, we project all the star positions onto a plane, and then identify the system of straight lines (``edges") with minimum total length that links all the stars together; for an ensemble of ${\cal N}_{_\star}$ stars, there are ${\cal N}_{_\star}-1$ edges, and no closed loops. Having done this, we analyse the distribution of edge lengths, $\ell$. To improve the statistics, we project the star positions onto each of the fundamental Cartesian planes, and we consider all three realisations, so we end up with ${\cal N}_\ell=9({\cal N}_{_\star}-1)$ edge-lengths. If we define the $k$th moment about the mean, for the ensemble of edge-lengths,
\begin{eqnarray}
m_k&=&\overline{\left(\ell-\bar{\ell}\right)^k}\,,
\end{eqnarray}
the standard deviation of the ensemble is $\sigma_\ell=m_2^{1/2}$, and the skewness of the ensemble is $\gamma_\ell=m_3/\sigma_\ell^3$; the standard deviation is a measure of the width of the distribution, and the skewness is a measure of the asymmetry of the distribution. Fig. \ref{FIG:MST} shows the cumulative distribution of edge-lengths, for the different ${\cal D}$ values (top plot), and the skewness plotted against the mean (bottom plot). These plots demonstrate that with low ${\cal D}$ the stars are strongly clustered, whereas with high ${\cal D}$ they are more uniformly distributed. From the top plot we see that with ${\cal D}\!=\!2.0$, $\,\sim\! 90\%$ of edges are less than $1\,{\rm pc}$, whereas, with ${\cal D}\!=\!2.8$, only $\,\sim\! 62\%$ of edges are less than $1\,{\rm pc}$. From the bottom plot we see that with ${\cal D}\!=\!2.0$, the mean, $\mu_\ell$, is small, because most of the stars are in compact groups connected by small $\ell$, but the skew, $\gamma_\ell$ is large, because there is a significant tail of large $\ell$ that connect up the individual groups. As ${\cal D}$ is increased, $\mu_\ell$ increases because the stars become more homogeneously distributed, and their nearest neighbours are typically further away. The skewness tends to decrease, although not monotonically; it would be interesting to improve the statistics and explore whether this non-monotonicity is simply the result of small-number statistics.
\begin{figure}
\includegraphics[width=90mm]{PLOTS/MST_new.ps}
\includegraphics[width=90mm]{PLOTS/skew_new.ps}
\caption{{\sc Top panel:} The cumulative distribution of the edge lengths derived from a Minimum Spanning Tree analysis of the sinks in each simulation at $t_{15}$. For each simulation, the MST is constructed using three projections, onto the fundamental Cartesian planes, and the results added to improve statistics. {\sc Bottom panel:} The skew of the edge length distributions plotted against the mean separation. }
\label{FIG:MST}
\end{figure}
\section{Conclusions}\label{SEC:CONC}%
In this paper, we use high-resolution 3D SPH simulations to explore the effect of a single O7 star emitting photons at $10^{49}\,{\rm s}^{-1}$ and located at the centre of a molecular cloud with mass $10^4\,{\rm M}_{_\odot}$ and radius $6.4\,{\rm pc}$. We focus on the statistics of dense clumps and triggered star formation, as a function of the initial fractal dimension, ${\cal D}$, of the molecular cloud into which the H{\sc ii} region expands. We find that most properties show a clear correlation with ${\cal D}$.
Cold clumps form due to the sweeping up of gas by the H{\sc ii} region. The clumps are pushed outward by the rocket effect and grow in mass by collecting material in a snowplough manner. Thus, large clumps cover a bigger surface area and may accrete faster, even though the growth is not caused by self-gravity.
\begin{itemize}
\item For low ${\cal D} \le 2.2$ (shell-dominated regime), we find a small number of massive clumps, whereas high ${\cal D} \ge 2.6$ (pillar-dominated regime) results in many low-mass clumps.
\item The clumps have trans- to super-sonic internal velocity dispersions. For non-star-forming clumps the internal velocity dispersion increases with clump size following Larson's Relation. For star-forming clumps the internal velocity dispersion is significantly higher than predicted by Larson's Relation.
\item The resulting CMFs are well fitted by power-laws, with the slope increasing with increasing ${\cal D}$. Typically observed CMF slopes of $-0.7$ are recovered for intermediate ${\cal D}$.
\end{itemize}
The statistical properties of triggered stars are also well correlated with ${\cal D}$. On average, clouds with lower ${\cal D}$
\begin{itemize}
\item form stars earlier in the simulation and at smaller distances from the ionising source (these stars are mostly located within the dense shell-like structures present for lower {\cal D}; for higher {\cal D} most stars sit in the tips of pillar-like structures);
\item are more prone to massive star formation;
\item form mainly small star clusters, whereas for higher ${\cal D}$ star formation occurs in small-N multiple systems spaced at large distances from one another.
\end{itemize}
Stars are strongly concentrated near the ionisation front ($R_{_{\rm 15}}/R_{_{\rm IF}}=1$), but stars that form in pillars (high ${\cal D}$) tend to be left behind within the H{\sc ii} region.
\section*{Acknowledgments}%
We thank the anonymous referee for helpful comments and suggestions, which helped us to improve the paper.
SKW thanks D. Kruijssen for useful and interesting discussions on the manuscript, and the Deutsche Forschungsgemeinschaft (DFG) for funding through the SPP 1573 'The physics of the interstellar medium'. SKW and AW further acknowledge the Marie Curie {\sc rtn constellation}. RW acknowledges the support of the Czech Science Foundation grant 209/12/1795 and by the project RVO:67985815. The simulations have been performed on the Cardiff {\sc arcca} Cluster. T.G.B. acknowledges support from STFC grant ST/J001511/1.
\bibliographystyle{mn2e
| {
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} | 3,963 |
Raising the minimum buying age for tobacco could mean fewer people start to smoke
April 2, 2015 1.12am EDT
Micah Berman, The Ohio State University
Micah Berman
Assistant Professor of Public Health and Law, The Ohio State University
Micah Berman receives funding from the US Food and Drug Administration and the National Institutes of Health. The work discussed in this article were not supported by any funding, and the views expressed are his own.
Too young to smoke. Image of teens smoking via www.shutterstock.com.
In 2005, the Boston suburb of Needham tried a new tactic to reduce youth tobacco use: the town raised the legal age for purchasing tobacco from 18 to 21. The results were dramatic – tobacco use among high school students dropped almost in half, and Needham's decline in high school smoking rates far outpaced the surrounding suburbs.
In the past two years, communities around the country have begun to follow Needham's lead. To date, more than 50 communities in seven states have raised their tobacco sales age to 21, including New York City in 2014. And the momentum keeps growing. And at least 10 state legislatures are now considering Tobacco 21 legislation.
Earlier this month, the Institute of Medicine (IOM), which is part of the National Academy of Sciences, released a 335-page report detailing the benefits of raising the tobacco sales age to 21, which would match the minimum age for purchasing alcohol.
Of all the options for addressing tobacco use, why are Tobacco 21 policies catching on? Why do they work?
Smoking is a pediatric epidemic. Teenager and cigarette via www.shutterstock.com.
Tobacco use is a 'pediatric epidemic'
Think about the people you know who smoke. How many of them smoked their first cigarette before age 21? How many of them wish they had never smoked that first cigarette? The likely answer to both of these questions is: all of them.
The US Surgeon General has referred to tobacco use as a "pediatric epidemic," because it almost always begins in youth. Indeed, despite all we know about the harms of tobacco, it is still the case that one in four high school seniors is a smoker and youth tobacco rates have barely budged over the past decade.
Of those who begin smoking as youth, 80% will smoke into adulthood, and one-half of all adult smokers will die prematurely from tobacco-related diseases.
The flip side of tobacco use being a "pediatric epidemic" is that the likelihood of starting to smoke declines markedly with age. The older you are, the less likely you are to start smoking. Although the tobacco industry has been increasingly targeting college-age students with its marketing, it remains the case that if someone makes it through high school without smoking, it is unlikely that he or she will ever start.
The tobacco industry has recognized this for years. In a 1982 memo, a researcher from the tobacco company R J Reynolds stated: "If a man has never smoked by age 18, the odds are three-to-one he never will. By age 21, the odds are twenty-to-one."
Make it through high school without smoking and it's likely that you will never start. Cigarettes via www.shutterstock.com.
Why do Tobacco 21 policies work?
Tobacco 21 policies are effective because they make it much more difficult for middle and high school students to access tobacco. This is because youth tobacco experimentation and use is driven by legal tobacco sales, not by illegal ones.
Today, at least in most places, 18- and 19-year-olds can legally purchase tobacco products and then supply them to younger kids (who, at least in the early stages of smoking, only use cigarettes occasionally). Raising the minimum age to 21 puts legal purchasers outside the social circle of most high school students.
Of course, raising the tobacco sales age to 21 will not keep all high school students from finding ways to access tobacco products, but the experience in Needham suggests it will significantly reduce the amount of youth tobacco use.
Given the scope of the problem – more than 3,800 kids under the age of 18 start smoking every day – the public health benefits could be enormous. Using conservative assumptions, the IOM study concluded that a nationwide Tobacco 21 policy would avoid nearly 250,000 premature deaths among those born between 2000 and 2019. Other public health benefits, such as a reduction in low birth weight and pre-term births, would be far more immediate.
The desire for nicotine can become wired in the developing brain. Brain via www.shutterstock.com
We also now know that a legal age of 18 for tobacco is out of touch with what the scientific evidence says about adolescent brain development. As discussed in the recent book The Teenage Brain, brains do not fully mature until people reach their early 20s (and possibly later).
For a still-developing brain, exposure to nicotine causes long-term neurological harm; in essence, the addiction to nicotine gets hard-wired into the developing brain. This leads to a stronger nicotine addiction and makes it much more difficult to quit later on. For this reason, the recent explosion in youth e-cigarette use is deeply troubling, and Tobacco 21 policies should also include e-cigarettes, hookahs and other products that deliver nicotine.
18 is not a magic number
Federal law prohibits the FDA from raising the tobacco sales age above 18; only Congress can do that for the nation as a whole – and it's hard for Congress to get anything done these days. But every state and most communities have the legal authority to adopt Tobacco 21 laws, which is exactly what they are starting to do.
The opposition to this emerging movement (primarily tobacco companies and tobacco retailers) chants "old enough to vote, old enough to smoke." But tobacco use is not a right or a privilege; it is an addictive and deadly activity. For the overwhelming majority of smokers, tobacco use is not in fact an "adult choice;" it is the result of an addiction that began when they were in high school or younger, and one that they are trying hard to kick.
There is nothing natural or unchangeable about a minimum age of 18. In traditional British common law, the "age of majority" (adulthood) was 21. In the US, the voting age was not lowered from 21 to 18 until 1971, but soon thereafter states began raising their drinking age from 18 to 21 when they realized that teens were disproportionately responsible for drunk driving accidents.
More recently, states that have sanctioned the legal use of marijuana – a drug far less deadly than tobacco – have set 21 as the minimum age. In short, it has long been the case that there are different minimum ages at different times and for different purposes.
Something we can agree on
Because no one (except for tobacco companies) wants the next generation to smoke, raising the minimum age to 21 is one tobacco control policy that nearly everyone can agree upon. It's no surprise then, that a recently published study found that more than seven in 10 adults favored increasing the tobacco sales age to 21, including strong majorities in every demographic category (including current smokers and 18-20 year olds). This is the rare policy measure that is bipartisan, popular, and effective. What are we waiting for? | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,712 |
iIVBGD0 = "Intel® HD Graphics 4000"
iIVBGM0 = "Intel® HD Graphics 4000"
iIVBGD0SRV = "Intel® HD Graphics"
iIVBGD0GT1 = "Intel® HD Graphics"
iIVBGM0GT1 = "Intel® HD Graphics"
iIVBGD0SRVGT1 = "Intel® HD Graphics"
iVLVGMT0 = "Intel® HD Graphics"
Note: It is recommended to upgrade the below list of applications to avoid any unexpected issues.
This driver causes the display brightness to keep resetting to max after sleeping.
Just a warning I would advise using these drivers. I got a blank screen after installing them but Windows was working as I could hear the sounds. Even after a reboot still blank screen. Only managed to get it to boot ok when I plugged in power cable.
Best thing to do is go to Intel Download page, installed Intel Driver update utility and let it find the latest GPU drivers. I did this and it worked fine with latest 2015 drivers.
varkanoid wrote: Just a warning I would advise using these drivers. I got a blank screen after installing them but Windows was working as I could hear the sounds. Even after a reboot still blank screen. Only managed to get it to boot ok when I plugged in power cable. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,875 |
\section{Introduction}
Mirror symmetry is originally stated as a duality between Calabi--Yau manifolds. Mirror symmetry predicts that the symplectic geometry (or the complex geometry) of a Calabi--Yau manifold is equivalent to the complex geometry (or the symplectic geometry, respectively) of the mirror Calabi--Yau manifold. The mirror duality has been generalized to Fano varieties, more generally, to log Calabi--Yau pairs. The mirror of a smooth log Calabi--Yau pair $(X,D)$ is a Landau--Ginzburg model, which is a variety $X^\vee$ with a proper map, called the superpotential, $W:X^\vee \rightarrow \mathbb C$. One further expects that the generic fiber of the superpotential $W$ is mirror to a smooth anticanonical divisor $D$ of $X$. We call the mirror duality between a smooth log Calabi--Yau pair $(X,D)$ and a proper Landau--Ginzburg model \emph{relative mirror symmetry}. To construct the mirror of the smooth log Calabi--Yau pair $(X,D)$, one needs to construct the variety $X^\vee$ and the proper Landau--Ginzburg potential $W$.
The variety $X^\vee$ is considered as the mirror of the complement $X\setminus D$. A general construction of the variety $X^\vee$ is through intrinsic mirror symmetry \cite{GS19} in the Gross--Siebert program. One considers a maximally unipotent degeneration $g: Y\rightarrow S$, where $S$ is an affine curve, of the pair $(X,D)$. The mirror is constructed as the projective spectrum of the degree zero part of the relative quantum cohomology $QH^0_{\log}(Y,D^\prime)$ of $(Y,D^\prime)$, where $D^\prime$ is a certain divisor that contains $g^{-1}(0)$.
It remains to compute the proper Landau--Ginzburg potential $W$. Following the Gross--Siebert program, the Landau--Ginzburg potentials are given by the theta functions. The theta functions are usually difficult to compute.
Recently, \cite{GRZ} computed the proper Landau--Ginzburg potentials for toric del Pezzo surfaces. They considered a toric degeneration of the smooth pair $(X,D)$, then applied the tropical view of the Landau--Ginzburg models \cite{CPS}. The theta function in \cite{GRZ} was defined tropically. By proving a tropical correspondence theorem in \cite{Graefnitz2022}, they showed that the theta function can be written as a generating function of two-point relative invariants. The idea of computing two-point relative invariants in \cite{GRZ} was to relate these two-point relative invariants with one-point relative invariants of a blow-up $\tilde {X}$. Then use the local-relative correspondence of \cite{vGGR} to relate these invariants to local invariants of the Calabi--Yau threefold $K_{\tilde {X}}$. By the open-closed duality of \cite{LLW11}, these local invariants are open invariants of the local Calabi--Yau threefold $K_X$ which form the open mirror map. Therefore, \cite{GRZ} showed that, for toric del Pezzo surfaces, the proper Landau--Ginzburg potentials are the open mirror maps.
In this paper, we study the Landau--Ginzburg model from the intrinsic mirror symmetry construction in \cite{GS19}. The goal of this paper is to generalize the result of \cite{GRZ} to all dimensions via a direct computation of two-point relative Gromov--Witten invariants. The computation is based on the relative mirror theorem of \cite{FTY}, where we only need to assume that $D$ is nef. The variety $X$ is not necessarily toric or Fano.
\subsection{Intrinsic mirror symmetry and theta functions}
Besides the tropical view of the Landau--Ginzburg model \cite{CPS}, the proper Landau--Ginzburg can also be constructed through intrinsic mirror symmetry. We learnt about the following construction from Mark Gross.
Given a smooth log Calabi--Yau pair $(X,D)$. We recall the maximally unipotent degeneration $g:Y\rightarrow S$ and the pair $(Y,D^\prime)$ from the construction of $X^\vee$. The theta functions in $QH^0_{\log}(Y,D^\prime)$ form a graded ring. The degree zero part of the ring agrees with $QH^0_{\log}(X,D)$. The base of the Landau--Ginzburg mirror of $(X,D)$ is $\on{Spec}QH^0_{\log}(X,D)=\mathbb A^1$ and the superpotential is $W=\vartheta_{1}$, the unique primitive theta function of $QH^0_{\log}(X,D)$.
We claim that the theta functions of $QH^0_{\log}(X,D)$ are generating functions of two-point relative Gromov--Witten invariants as follows.
\begin{definition}[=Definition \ref{def-theta-func}]\label{intro-def-theta}
For $p\geq 1$, the theta function is
\begin{align}\label{intro-theta-func-def}
\vartheta_p=x^{-p}+\sum_{n=1}^{\infty}nN_{n,p}t^{n+p}x^n,
\end{align}
where
\[
N_{n,p}=\sum_{\beta} \langle [\on{pt}]_n,[1]_p\rangle_{0,2,\beta}^{(X,D)}
\]
is the sum of two-point relative Gromov--Witten invariants with the first marking having contact order $n$ along with a point constraint and the second marking having contact order $p$.
\end{definition}
By \cite{GS19}, theta functions should satisfy the following product rule
\begin{align}\label{intro-theta-func-multi}
\vartheta_{p_1}\star \vartheta_{p_2}=\sum_{r\geq 0, \beta}N_{p_1,p_2,-r}^{\beta} \vartheta_r,
\end{align}
where the structure constants $N_{p_1,p_2,-r}^{\beta}$ are punctured invariants with two positive contacts and one negative contact.
In Proposition \ref{prop-struc-const}, we show that the structure constants can be written in terms of two-point relative invariants. In other words, we reduce relative invariants with two positive contacts and one negative contact to relative invariants with two positive contacts. Then we show that the theta functions in Definition \ref{intro-theta-func-def} indeed satisfy the product rule (\ref{intro-theta-func-multi}) with the correct structure constants $N_{p_1,p_2,-r}^{\beta}$. In particular, in Proposition \ref{prop-wdvv}, we prove an identity of two-point relative invariants generalizing \cite{GRZ}*{Lemma 5.3} and show that it follows from the WDVV equation.
\begin{remark}
During the preparation of our paper, we learnt that Yu Wang\cite{Wang} also obtained the same formula as in Proposition \ref{prop-struc-const} but using the punctured invariants of \cite{ACGS}. Some formulas for two-point relative invariants are also obtained in \cite{Wang} via a different method.
\end{remark}
\subsection{Relative mirror maps}
The relative mirror theorem of \cite{FTY} states that, under the assumption that $D$ is nef, a genus zero generating function of relative Gromov--Witten invariants (the $J$-function) can be identified with the relative periods (the $I$-function) via a change of variables called the relative mirror map. This provides a powerful tool to compute genus zero relative Gromov--Witten invariants.
Our computation of these two-point relative invariants is straightforward but complicated. It is straightforward to see that these invariants can be extracted from the relative $J$-function after taking derivatives. On the other hand, although such computation for (one-point) absolute invariants is well-known, the computation of two-point relative invariants is much more complicated due to the following reasons.
First of all, we need to compute two-point relative invariants instead of one-point invariants. For one-point relative invariants, one can also use the local-relative correspondence of \cite{vGGR} (see also \cite{TY20b}) to reduce the computation to local invariants when the divisor $D$ is nef. To compute these two-point invariants one need to consider the so-called extended relative $I$-function, instead of the much easier non-extended relative $I$-function.
Secondly, the relative mirror map has never been studied systematically. There have been some explicit computations of relative invariants when the relative mirror maps are trivial, see, for example, \cite{TY20b}. When the relative mirror maps are not trivial, more complicated invariants will appear. One of the important consequences of this paper is to provide a systematic analysis of these invariants and set up the foundation of future applications of the relative mirror theorem.
We would like to point out a related computation in \cite{You20}, where we computed one-point relative invariants of some partial compactifications of toric Calabi--Yau orbifolds. The computation is much easier in \cite{You20} because of the two reasons that we just mentioned. First of all, one can apply the local-relative correspondence to compute these invariants, although we did not use it in \cite{You20}. Secondly, although the mirror map is not trivial, the mirror map is essentially coming from the absolute Gromov--Witten theory of the partial compactifications. The relative theory in \cite{You20} does not contribute to the non-trivial mirror map. Therefore we were able to avoid all these complexities. We would also like to point out that the computation in \cite{You20} is restricted to the toric case. In this paper, we work beyond the toric setting.
In order to apply the relative mirror theorem of \cite{FTY}, we need to study the relative mirror map carefully.
Let $X$ be a smooth projective variety and $D$ be a smooth nef divisor, we recall that the $J$-function for the pair $(X,D)$ is defined as
\[
J_{(X,D)}(\tau,z)=z+\tau+\sum_{\substack{(\beta,l)\neq (0,0), (0,1)\\ \beta\in \on{NE(X)}}}\sum_{\alpha}\frac{q^{\beta}}{l!}\left\langle \frac{\phi_\alpha}{z-\bar{\psi}},\tau,\ldots, \tau\right\rangle_{0,1+l, \beta}^{(X,D)}\phi^{\alpha},
\]
and
the (non-extended) $I$-function of the smooth pair $(X,D)$ is
\[
I_{(X,D)}(y,z)=\sum_{\beta\in \on{NE(X)}}J_{X,\beta}(\tau_{0,2},z)y^\beta\left(\prod_{0<a\leq D\cdot \beta-1}(D+az)\right)[1]_{-D\cdot \beta}.
\]
We refer to Definition \ref{def-relative-J-function} and Definition \ref{def-relative-I-function} for the precise meaning of the notation. The extended $I$-function $I(y,x_1,z)$ takes a more complicated form than the non-extended $I$-fnction $I(y,z)$. We refer to Definition \ref{def-relative-I-function-extended} for the precise definition of the extended $I$-function. We further assume that $-K_X-D$ is nef. The extended relative mirror map is given by the $z^0$-coefficient of the extended $I$-function:
\begin{align*}
\tau(y,x_1)=\sum_{i=1}^r p_i\log y_i+x_1[1]_{1}+\sum_{\substack{\beta\in \on{NE}(X)\\ D\cdot \beta \geq 2}}\langle [\on{pt}]\psi^{D\cdot \beta-2}\rangle_{0,1,\beta}^Xy^\beta (D\cdot \beta-1)![1]_{-D\cdot \beta}.
\end{align*}
The (non-extended) relative mirror map is given by $\tau(y,0)$, denoted by $\tau(y)$. The relative mirror theorem of \cite{FTY} states that
\[
J(\tau(y,x_1),z)=I(y,x_1,z).
\]
Therefore from the expression of $\tau(y,x_1)$, we can see that relative invariants with several negative contact orders will appear when the relative mirror map is not trivial. We obtain the following identity which shows that the negative contact insertion $[1]_{-k}$ is similar to the insertion of a divisor class $[D]_0$.
\begin{proposition}[=Proposition \ref{prop-several-neg-1}]
\begin{align*}
\langle [1]_{-k_1},\cdots, [1]_{-k_l}, [\gamma]_{k_{l+1}} \bar{\psi}^a\rangle_{0,l+1,\beta}^{(X,D)}=\langle [D]_0,\cdots, [D]_0, [\gamma]_{D\cdot \beta} \bar{\psi}^a\rangle_{0,l+1,\beta}^{(X,D)},
\end{align*}
where $\gamma \in H^*(D)$, $k_i$'s are positive integers, and
\[
D\cdot \beta=k_{l+1}-\sum_{i=1}^l k_i\geq 0.
\]
\end{proposition}
Since we need to compute invariants with insertion $[1]_1$, we also prove similar propositions when there are also insertions of $[1]_1$. We refer to Proposition \ref{prop-several-neg-2} and Proposition \ref{prop-several-neg-3} for the precise formula. We also compute degree zero relative invariants with two positive contact orders and several negative contact orders in Proposition \ref{prop-degree-zero}.
\begin{remark}
We would like to point out that a key point of the computation of the proper potential is to observe the subtle (but vital) difference between the above mentioned formulas for invariants with two positive contacts and formulas for invariants with one positive contact.
\end{remark}
\begin{remark}
The proof of Proposition \ref{prop-several-neg-1}, Proposition \ref{prop-several-neg-2}, Proposition \ref{prop-several-neg-3}, and Proposition \ref{prop-degree-zero} make essential use of the (both orbifold and graph sum) definitions of relative Gromov--Witten invariants with negative contact orders in \cite{FWY}. Since these invariants reduce to relative invariants with two positive contact orders, we do not need to assume a general relation between the punctured invariants of \cite{ACGS} and \cite{FWY}. To match with the intrinsic mirror symmetry in the Gross--Siebert program, we only need to assume that punctured invariants of a smooth pair with one negative contact order defined in \cite{ACGS} coincide with the ones in \cite{FWY}. A more general comparison result is an upcoming work of \cite{BNR22}, so we do not attempt to give a proof of this special case.
\end{remark}
With all these preparations, we are able to express invariants in $J(\tau(y,x),z)$ in terms of relative invariants without negative contact orders and the relative mirror map can be written as the following change of variables
\begin{align}\label{intro-relative-mirror-map}
\sum_{i=1}^r p_i\log q_i=\sum_{i=1}^r p_i\log y_i+g(y)D.
\end{align}
\subsection{The proper Landau--Ginzburg potential}
The main result of the paper is to relate the proper Landau--Ginzburg potential with the relative mirror map.
\begin{theorem}[=Theorem \ref{thm-main}]\label{intro-thm-main}
Let $X$ be a smooth projective variety with a smooth nef anticanonical divisor $D$. Let $W:=\vartheta_1$ be the mirror proper Landau--Ginzburg potential. Set $q^\beta=t^{D\cdot \beta}x^{D\cdot\beta}$. Then
\[
W=x^{-1}\exp\left(g(y(q))\right),
\]
where
\[
g(y)=\sum_{\substack{\beta\in \on{NE}(X)\\ D\cdot \beta \geq 2}}\langle [\on{pt}]\psi^{D\cdot \beta-2}\rangle_{0,1,\beta}^Xy^\beta (D\cdot \beta-1)!
\]
and $y=y(q)$ is the inverse of the relative mirror map (\ref{intro-relative-mirror-map}).
\end{theorem}
\begin{remark}
This is a natural expectation from the point of view of relative mirror symmetry. Recall that the proper Landau--Ginzburg model $(X^\vee,W)$ is mirror to the smooth log Calabi--Yau pair $(X,D)$. The proper Landau--Ginzburg potential $W$ should encode the instanton corrections. On the other hand, the relative mirror theorem relates relative Gromov--Witten invariants with relative periods (relative $I$-functions) via the relative mirror map. In order to have a mirror construction with a trivial mirror map, the instanton corrections should be the inverse relative mirror map. This provides an enumerative meaning of the relative mirror map.
\end{remark}
In \cite{GRZ}, the authors conjectured that the proper Landau--Ginzburg potential is the open mirror map. We also have a natural explanation for it. The open mirror map of \cite{GRZ} is given by open Gromov--Witten invariants of the local Calabi--Yau $\mathcal O_X(-D)$. These open invariants encode the instanton corrections and are expected to be the inverse mirror map of the local Gromov--Witten theory of $\mathcal O_X(-D)$. We observe that the relative mirror map (\ref{intro-relative-mirror-map}) and the local mirror map coincide up to a sign. So we claim that the proper Landau--Ginzburg potential is also the open mirror map. When $X$ is a toric variety, we proved the conjecture of \cite{GRZ}.
\begin{theorem}[=Theorem \ref{thm-toric-open}]\label{intro-thm-toric-open}
Let $(X,D)$ be a smooth log Calabi--Yau pair, such that $X$ is toric and $D$ is nef. The proper Landau--Ginzburg potential of $(X,D)$ is the open mirror map of the local Calabi--Yau manifold $\mathcal O_X(-D)$.
\end{theorem}
In general, Theorem \ref{intro-thm-toric-open} is true as long as the open-closed duality (e.g. \cite{CLT} and \cite{CCLT}) between open Gromov--Witten invariants $K_X$ and closed Gromov--Witten invariants $P(\mathcal O(-D)\oplus \mathcal O)$ is true. Therefore, we have the following.
\begin{corollary}
The open-closed duality implies the proper Landau--Ginzburg potential is the open mirror map.
\end{corollary}
\begin{remark}
The reason that the relative mirror map is the same as the local mirror map can also be seen from the local-relative correspondence of \cite{vGGR} and \cite{TY20b}. It has already been observed in \cite{TY20b} that the local and (non-extended) relative $I$-functions can be identified. And this identification has been used to prove the local-relative correspondence for some invariants in \cite{TY20b}.
\end{remark}
Theorem \ref{intro-thm-main} provides explicit formulas for the proper potentials whenever the relevant genus zero absolute Gromov--Witten invariants of $X$ are computable. These absolute invariants can be extracted from the $J$-function of the absolute Gromov--Witten theory of $X$. Therefore, we have explicit formulas of the proper Landau--Ginzburg potentials whenever a Givental style mirror theorem holds. Givental style mirror theorem has been proved for many cases beyond the toric setting (e.g. \cite{CFKS} for non-abelian quotients via the abelian/non-abelian correspondence). Therefore, we have explicit formulas for the proper Landau--Ginzburg potentials for large classes of examples. Note that there may be non-trivial mirror maps for absolute Gromov--Witten theory of $X$. If we replace the absolute invariants in $g(y)$ by the corresponding coefficients of the absolute $I$-function, we also need to plug-in the inverse of the absolute mirror map. This can be seen in the case of toric varieties in Section \ref{sec-toric-semi-Fano}.
For Fano varieties, the invariants in $g(y)$ are usually easier. We observed that $g(y)$ is closely related to the regularized quantum periods in the Fano search program \cite{CCGGK}.
\begin{theorem}\label{intro-thm-quantum-period}
The function $g(y)$ coincides with the anti-derivative of the regularized quantum period.
\end{theorem}
By mirror symmetry, it is expected that regularized quantum periods of Fano varieties coincide with the classical periods of their mirror Laurent polynomials. Therefore, as long as one knows the mirror Laurent polynomials, one can compute the proper Landau--Ginzburg potentials. For example, the proper Landau--Ginzburg potentials for all Fano threefolds can be explicitly computed using \cite{CCGK}. More generally, Theorem \ref{intro-thm-quantum-period} allows one to use the large databases \cite{CK22} of quantum periods for Fano manifolds to compute the proper Landau--Ginzburg potentials.
Interestingly, the Laurent polynomials are considered as the mirror of Fano varieties with maximal boundaries (or as the potential for the weak, non-proper, Landau--Ginzburg models of \cite{Prz07}, \cite{Prz13}). Therefore, we have an explicit relation between the proper and non-proper Landau--Ginzburg potentials.
\subsection{Acknowledgement}
The author would like to thank Mark Gross for explaining the construction of Landau--Ginzburg potentials from intrinsic mirror symmetry. The author would also like to thank Yu Wang for illuminating discussions regarding Section \ref{sec:intrinsic}. This project has received funding from the Research Council of Norway grant no. 202277 and the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska -Curie grant agreement 101025386.
\section{Relative Gromov--Witten invariants with negative contact orders}
\subsection{General theory}\label{sec-rel-general}
We follow the presentation of \cite{FWY} for the definition of genus zero relative Gromov--Witten theory with negative contact orders.
Let $X$ be a smooth projective variety and $D\subset X$ be a smooth divisor. We consider a topological type (also called an admissible graph)
\[
\Gamma=(0,n,\beta,\rho,\vec \mu)
\]
with
\[
\vec \mu=(\mu_1, \ldots, \mu_\rho)\in (\mathbb Z^*)^{\rho}
\]
and
\[
\sum_{i=1}^\rho \mu_i=\int_\beta D.
\]
\begin{defn}[\cite{FWY}, Definition 2.4]
A rubber graph $\Gamma'$ is an admissible graph whose roots have two different types. There are
\begin{enumerate}
\item $0$-roots (whose weights will be denoted by $\mu^0_1,\ldots,\mu^0_{\rho_0}$), and
\item $\infty$-roots (whose weights will be denoted by $\mu^\infty_1,\ldots,\mu^\infty_{\rho_\infty}$).
\end{enumerate}
The map $b$ maps $V(\Gamma)$ to $H_2(D,\mathbb Z)$.
\end{defn}
\begin{defn}[\cite{FWY}, Definition 4.1]\label{def:admgraph0}
\emph{A (connected) graph of type $0$} is a weighted graph $\Gamma^0$ consisting of a
single vertex, no edges, and the following.
\begin{enumerate}
\item {$0$-roots},
\item {$\infty$-roots of node type},
\item {$\infty$-roots of marking type},
\item {Legs}.
\end{enumerate}
$0$-roots are weighted by
positive integers, and $\infty$-roots are weighted by negative integers. The vertex is associated with a tuple $(g,\beta)$ where $g\geq 0$
and $\beta\in H_2(D,\mathbb Z)$.
\end{defn}
A graph $\Gamma^\infty$ of type $\infty$ is an admissible graph such that the roots are distinguished by node type and marking type.
\begin{defn}[\cite{FWY}, Definition 4.8]\label{defn:locgraph}
\emph{An admissible bipartite graph} $\mathfrak G$ is a tuple
$(\mathfrak S_0,\Gamma^\infty,I,E,g,b)$, where
\begin{enumerate}
\item $\mathfrak S_0=\{\Gamma_i^0\}$ is a set of graphs of type $0$;
$\Gamma^\infty$ is a (possibly disconnected) graph of type $\infty$.
\item $E$ is a set of edges.
\item $I$ is the set of markings.
\item $g$ and $b$ represent the genus and the degree respectively.
\end{enumerate}
Moreover, the admissible bipartite graph must satisfy some conditions described in \cite{FWY}*{Definition 4.8}. We refer to \cite{FWY} for more details.
\end{defn}
Let $\mathcal B_\Gamma$ be a connected admissible bipartite graph of topological type $\Gamma$. Given a bipartite graph $\mathfrak G\in \mathcal B_\Gamma$, we consider
\[
\overline{\mathcal M}_{\mathfrak G}=\prod_{\Gamma_i^0\in \mathfrak S_0}\overline{\mathcal M}_{\Gamma_i^0}^\sim (D) \times_{D^{|E|}}\overline{\mathcal M}_{\Gamma^\infty}^{\bullet}(X,D),
\]
where
\begin{itemize}
\item $\overline{\mathcal M}_{\Gamma_i^0}^\sim (D)$ is the moduli space of relative stable maps to rubber target over $D$ of type $\Gamma_i^0$;
\item $\overline{\mathcal M}_{\Gamma^\infty}^{\bullet}(X,D)$ is the moduli space of relative stable maps of type $\Gamma^\infty$;
\item $\times_{D^{|E|}}$ is the fiber product identifying evaluation maps according to edges.
\end{itemize}
We have the following diagram.
\begin{equation}\label{eqn:diag}
\xymatrix{
\overline{\mathcal M}_{\mathfrak G} \ar[r]^{} \ar[d]^{\iota} & D^{|E|} \ar[d]^{\Delta} \\
\prod\limits_{\Gamma^0_i\in \mathfrak S_0}\overline{\mathcal M}^\sim_{\Gamma^0_i}(D) \times \overline{\mathcal M}^{\bullet}_{\Gamma^\infty}(X,D) \ar[r]^{} & D^{|E|}\times D^{|E|}.
}
\end{equation}
There is a natural virtual class
\[
[\overline{\mathcal M}_{\mathfrak G}]^{\on{vir}}=\Delta^![\prod\limits_{\Gamma^0_i\in \mathfrak S_0}\overline{\mathcal M}^\sim_{\Gamma^0_i}(D) \times \overline{\mathcal M}^{\bullet}_{\Gamma^\infty}(X,D)]^{\on{vir}}
\]
where $\Delta^!$ is the Gysin map.
For each $\overline{\mathcal M}^\sim_{\Gamma_i^0}(D)$, we have a stabilization map $\overline{\mathcal M}^\sim_{\Gamma_i^0}(D) \rightarrow \overline{\mathcal M}_{0,n_i+\rho_i}(D,\beta_i)$ where $n_i$ is the number of legs, $\rho_i$ is the number of $0$-roots plus the number of $\infty$-roots of marking type, and $\beta_i$ is the curve class of $\Gamma_i^0$. Hence, we have a map
\[
\overline{\mathcal M}_{\mathfrak G}=\prod\limits_{\Gamma^0_i\in \mathfrak S_0}\overline{\mathcal M}^\sim_{\Gamma^0_i}(D) \times_{D^{|E|}} \overline{\mathcal M}^{\bullet}_{\Gamma^\infty}(X,D)
\rightarrow
\prod\limits_{\Gamma^0_i\in \mathfrak S_0}\overline{\mathcal M}_{0,n_i+\rho_i}(D,\beta_i) \times_{D^{|E|}} \overline{\mathcal M}^{\bullet}_{\Gamma^\infty}(X,D).
\]
Composing with the boundary map
\[
\prod\limits_{\Gamma^0_i\in \mathfrak S_0}\overline{\mathcal M}_{0,n_i+\rho_i}(D,\beta_i) \times_{D^{|E|}} \overline{\mathcal M}^{\bullet}_{\Gamma^\infty}(X,D) \rightarrow \overline{\mathcal M}_{0,n+\rho}(X,\beta)\times_{X^{\rho}} D^{\rho},
\]
we obtain a map
\[
\mathfrak t_{\mathfrak G}:\overline{\mathcal M}_{\mathfrak G}\rightarrow \overline{\mathcal M}_{0,n+\rho}(X,\beta)\times_{X^{\rho}} D^{\rho}.
\]
Following the definition in \cite{FWY}, we need to introduce the relative Gromov--Witten cycle of the pair $(X,D)$ of topological type $\Gamma$.
Let $t$ be a formal parameter. Given a $\Gamma^\infty$, we define
\begin{align}\label{neg-rel-infty}
C_{\Gamma^\infty}(t)=\frac{t}{t+\Psi}\in A^*(\overline{\mathcal M}^\bullet_{\Gamma^\infty}(X,D))[t^{-1}].
\end{align}
Then we consider $\Gamma_i^0$. Define
\[
c(l)=\Psi_\infty^l-\Psi_\infty^{l-1}\sigma_1+\ldots+(-1)^l\sigma_l,
\]
where $\Psi_\infty$ is the divisor corresponding to the cotangent line bundle determined by the relative divisor on the $\infty$ side. We then define
\[
\sigma_k=\sum\limits_{\{e_1,\ldots,e_k\}\subset \on{HE}_{m,n}(\Gamma_i^0)} \prod\limits_{j=1}^{k} (d_{e_j}\bar\psi_{e_j}-\ev_{e_j}^*D),
\]
where $d_{e_j}$ is the absolute value of the weight at the root $e_j$.
For each $\Gamma_i^0$, define
\begin{align}\label{neg-rel-0}
C_{\Gamma_i^0}(t)= \frac{\sum_{l\geq 0}c(l) t^{\rho_\infty(i)-1-l}}{\prod\limits_{e\in \on{HE}_{n}(\Gamma_i^0)} \big(\frac{t+\ev_e^*D}{d_e}-\bar\psi_e\big) } \in A^*(\overline{\mathcal M}^\sim_{\Gamma_i^0}(D))[t,t^{-1}],
\end{align}
where $\rho_\infty(i)$ is the number of $\infty$-roots (of both types) associated with $\Gamma_i^0$.
For each $\mathfrak G$, we write
\begin{equation}\label{eqn:cg}
C_{\mathfrak G}=\left[ p_{\Gamma^\infty}^*C_{\Gamma^\infty}(t)\prod\limits_{\Gamma_i^0\in \mathfrak S_0} p_{\Gamma_i^0}^*C_{\Gamma_i^0}(t) \right]_{t^{0}},
\end{equation}
where
$[\cdot]_{t^{0}}$ means taking
the constant term, and $p_{\Gamma^\infty}, p_{\Gamma_i^0}$ are projections from $\prod\limits_{\Gamma^0_i\in \mathfrak S_0}\overline{\mathcal M}^\sim_{\Gamma^0_i}(D) \times \overline{\mathcal M}^{\bullet}_{\Gamma^\infty}(X,D)$ to the corresponding factors. Recall that
\[
\iota: \overline{\mathcal M}_{\mathfrak G}\rightarrow \prod\limits_{\Gamma^0_i\in \mathfrak S_0}\overline{\mathcal M}^\sim_{\Gamma^0_i}(D) \times \overline{\mathcal M}^{\bullet}_{\Gamma^\infty}(X,D)
\]
is the closed immersion from diagram \eqref{eqn:diag}.
\begin{defn}[\cite{FWY}*{Definition 5.3}]\label{def-rel-cycle}
The relative Gromov--Witten cycle of the pair $(X,D)$ of topological type $\Gamma$ is defined to be
\[
\mathfrak c_\Gamma(X/D) = \sum\limits_{\mathfrak G \in \mathcal B_\Gamma} \dfrac{1}{|\on{Aut}(\mathfrak G)|}(\mathfrak t_{\mathfrak G})_* ({\iota}^* C_{\mathfrak G} \cap [\overline{\mathcal M}_{\mathfrak G}]^{\on{vir}}) \in A_*(\overline{\mathcal M}_{0,n+\rho}(X,\beta)\times_{X^{\rho}} D^{\rho}),
\]
where $\iota$ is the vertical arrow in Diagram \eqref{eqn:diag}.
\end{defn}
\begin{proposition}[=\cite{FWY}, Proposition 3.4]
\[\mathfrak c_\Gamma(X/D) \in A_{d}(\overline{M}_{0,n+\rho}(X,\beta)\times_{X^{\rho}} D^{\rho}),\] where
\[d=\mathrm{dim}_{\mathbb C}X-3+\int_{\beta} c_1(T_X(-\mathrm{log} D)) + n + \rho_+, \]
where $\rho_+$ is the number of relative markings with positive contact.
\end{proposition}
Let
\begin{align*}
\alpha_i\in H^*(X), \text{ and } a_i\in \mathbb Z_{\geq 0} \text{ for } i\in \{1,\ldots, n\};
\end{align*}
\begin{align*}
\epsilon_j\in H^*(D), \text{ and } b_j\in \mathbb Z_{\geq 0} \text{ for } j\in \{1,\ldots, \rho\}.
\end{align*}
We have evaluation maps
\begin{align*}
\ev_{X,i}:\overline{\mathcal M}_{0,n+\rho}(X,\beta)\times_{X^{\rho}} D^{\rho}&\rightarrow X, \text{ for } i\in\{1,\ldots,n\};\\
\ev_{D,j}:\overline{\mathcal M}_{0,n+\rho}(X,\beta)\times_{X^{\rho}} D^{\rho}&\rightarrow D \text{ for } j\in \{1,\ldots, \rho\}.
\end{align*}
\begin{defn}[\cite{FWY}, Definition 5.7]\label{rel-inv-neg}
The relative Gromov--Witten invariant of topological type $\Gamma$ is
\[
\langle \prod_{i=1}^n \tau_{a_i}(\alpha_i) \mid \prod_{j=1}^\rho\tau_{b_j}(\epsilon_j) \rangle_{\Gamma}^{(X,D)} = \displaystyle\int_{\mathfrak c_\Gamma(X/D)} \prod\limits_{j=1}^{\rho} \bar{\psi}_{D,j}^{b_j}\ev_{D,j}^*\epsilon_j\prod\limits_{i=1}^n \bar{\psi}_{X,i}^{a_i}\ev_{X,i}^*\alpha_i,
\]
where $\bar\psi_{D,j}, \bar\psi_{X,i}$ are pullback of psi-classes from $\overline{\mathcal M}_{0,n+\rho}(X,\beta)$ to $\overline{\mathcal M}_{0,n+\rho}(X,\beta)\times_{X^{\rho}} D^{\rho}$ corresponding to markings.
\end{defn}
\begin{remark}
In \cite{FWY}, relative Gromov--Witten invariants of $(X,D)$ with negative contact order are also defined as a limit of the corresponding orbifold Gromov--Witten invariants of the $r$-th root stack $X_{D,r}$ with large ages
\[
\langle \prod_{i=1}^{n+\rho} \tau_{a_i}(\gamma_i)\rangle_{0,n+\rho,\beta}^{(X,D)}:=r^{\rho_-}\langle \prod_{i=1}^{n+\rho} \tau_{a_i}(\gamma_i)\rangle_{0,n+\rho,\beta}^{X_{D,r}},
\]
where $r$ is sufficiently large; $\rho_-$ is the number of orbifold markings with large ages (=relative markings with negative contact orders); $\gamma_i$ are cohomology classes of $X$ or $D$ depending on the markings being interior or orbifold/relative.
We refer to \cite{FWY}*{Section 3} for more details.
\end{remark}
We recall the topological recursion relation and the WDVV equation which will be used later in our paper.
\begin{prop}[\cite{FWY}*{Proposition 7.4}]\label{prop:TRR}
Relative Gromov--Witten theory satisfies the topological recursion relation:
\begin{align*}
&\langle\bar\psi^{a_1+1}[\alpha_1]_{i_1}, \ldots, \bar\psi^{a_n}[\alpha_n]_{i_n}\rangle_{0,\beta,n}^{(X,D)} \\
=&\sum \langle\bar\psi^{a_1}[\alpha_1]_{i_1}, \prod\limits_{j\in S_1} \bar\psi^{a_j}[\alpha_j]_{i_j}, \widetilde T_{i,k}\rangle_{0,\beta_1,1+|S_1|}^{(X,D)} \langle\widetilde T_{-i}^k, \bar\psi^{a_2}[\alpha_2]_{i_2}, \bar\psi^{a_3}[\alpha_3]_{i_3}, \prod\limits_{j\in S_2} \bar\psi^{a_j}[\alpha_j]_{i_j}\rangle_{0,\beta_2,2+|S_2|}^{(X,D)},
\end{align*}
where the sum is over all $\beta_1+\beta_2=\beta$, all indices $i,k$ of basis, and $S_1, S_2$ disjoint sets with $S_1\cup S_2=\{4,\ldots,n\}$.
\end{prop}
\begin{prop}[\cite{FWY}*{Proposition 7.5}]\label{prop:WDVV}
Relative Gromov--Witten theory satisfies the WDVV equation:
\begin{align*}
&\sum \langle\bar\psi^{a_1}[\alpha_1]_{i_1}, \bar\psi^{a_2}[\alpha_2]_{i_2}, \prod\limits_{j\in S_1} \bar\psi^{a_j}[\alpha_j]_{i_j}, \widetilde T_{i,k}\rangle_{0,\beta_1,2+|S_1|}^{(X,D)} \\
& \quad \cdot \langle\widetilde T_{-i}^k, \bar\psi^{a_3}[\alpha_3]_{i_3}, \bar\psi^{a_4}[\alpha_4]_{i_4}, \prod\limits_{j\in S_2} \bar\psi^{a_j}[\alpha_j]_{i_j}\rangle_{0,\beta_2,2+|S_2|}^{(X,D)} \\
=&\sum \langle\bar\psi^{a_1}[\alpha_1]_{i_1}, \bar\psi^{a_3}[\alpha_3]_{i_3}, \prod\limits_{j\in S_1} \bar\psi^{a_j}[\alpha_j]_{i_j}, \widetilde T_{i,k}\rangle_{0,\beta_1,2+|S_1|}^{(X,D)} \\
& \quad \cdot \langle\widetilde T_{-i}^k, \bar\psi^{a_2}[\alpha_2]_{i_2}, \bar\psi^{a_4}[\alpha_4]_{i_4}, \prod\limits_{j\in S_2} \bar\psi^{a_j}[\alpha_j]_{i_j}\rangle_{0,\beta_2,2+|S_2|}^{(X,D)},
\end{align*}
where each sum is over all $\beta_1+\beta_2=\beta$, all indices $i,k$ of basis, and $S_1, S_2$ disjoint sets with $S_1\cup S_2=\{5,\ldots,n\}$.
\end{prop}
\subsection{A special case}
As explained in \cite{FWY}*{Example 5.5}, relative Gromov--Witten invariants with one negative contact order can be written down in a simpler form. In this case, we only have the graphs $\mathfrak G$ such that $\{\Gamma_i^0\}$ consists of only one element (denoted by $\Gamma^0$). Denote such a set of graphs by $\mathcal B_\Gamma'$. The relative Gromov--Witten cycle of topological type $\Gamma$ is simply
\[
\mathfrak c_\Gamma(X/D) = \sum\limits_{\mathfrak G \in \mathcal B_\Gamma'} \dfrac{\prod_{e\in \on{HE}_{n}(\Gamma^0)}d_e}{|\on{Aut}(\mathfrak G)|}(\mathfrak t_{\mathfrak G})_* \big([\overline{\mathcal M}_{\mathfrak G}]^{\on{vir}}\big).
\]
Note that
\[
[\overline{\mathcal M}_{\mathfrak G}]^{\on{vir}}=\Delta^![\overline{\mathcal M}^\sim_{\Gamma^0}(D)\times \overline{\mathcal M}^\bullet_{\Gamma^\infty}(X,D)]^{\on{vir}}.
\]
Let
\begin{align*}
\alpha_i\in H^*(X), \text{ and } a_i\in \mathbb Z_{\geq 0} \text{ for } i\in \{1,\ldots, n\};
\end{align*}
\begin{align*}
\epsilon_j\in H^*(D), \text{ and } b_j\in \mathbb Z_{\geq 0} \text{ for } j\in \{1,\ldots, \rho\}.
\end{align*}
Without loss of generality, we assume that $\epsilon_1$ is the insertion that corresponds to the unique negative contact marking. Then the relative invariant with one negative contact order can be written as
\begin{align*}
&\langle \prod_{i=1}^n \tau_{a_i}(\alpha_i) \mid \prod_{j=1}^\rho\tau_{b_j}(\epsilon_j) \rangle_{\Gamma}^{(X,D)}\\
=& \sum_{\mathfrak G\in \mathcal B_\Gamma'} \frac{\prod_{e\in E}d_e}{|\on{Aut}(E)|}\sum \langle \prod_{j\in S_{\epsilon,1}}\tau_{b_j}(\epsilon_j) | \prod_{i\in S_{\alpha,1}}\tau_{a_i}(\alpha_i) | \eta, \tau_{b_1}(\epsilon_1)\rangle^\sim_{\Gamma^0}\langle \check{\eta}, \prod_{j\in S_{\epsilon,2}}\tau_{b_j}(\epsilon_j) | \prod_{i\in S_{\alpha,2}}\tau_{a_i}(\alpha_i) \rangle_{\Gamma^\infty}^{\bullet, (X,D)},
\end{align*}
where $\on{Aut}(E)$ is the permutation group of the set $\{d_1,\ldots, d_{|E|}\}$; $\check{\eta}$ is defined by taking the Poincar\'e dual of the cohomology weights of the cohomology weighted partition $\eta$; the second sum is over all splittings of
\[
\{1,\ldots, n\}=S_{\alpha,1}\sqcup S_{\alpha,2}, \quad \{2,\ldots, \rho\}=S_{\epsilon,1}\sqcup S_{\epsilon,2}
\]
and all intermediate cohomology weighted partitions $\eta$.
The following comparison theorem between punctured invariants of \cite{ACGS} and relative invariants with one negative contact order of \cite{FWY} for smooth pairs is an upcoming work of \cite{BNR22b}.
\begin{theorem}\label{theorem-puncture-relative}
Given a smooth projective variety $X$ and a smooth divisor $D\subset X$, the punctured Gromov--Witten invariants of $(X,D)$ and the relative Gromov--Witten invariants of $(X,D)$ with one negative contact order coincide.
\end{theorem}
\begin{remark}
\cite{BNR22b} studies the comparison between punctured invariants of \cite{ACGS} and relative invariants with several negative contact orders. For the purpose of this paper, we only need the case with one negative contact order. In this case, the comparison is significantly simpler because we have simple graphs as described above and the class $\mathfrak c_\Gamma(X/D)$ is trivial. Since the general comparison is obtained in \cite{BNR22b}, we do not attempt to give a proof for this special case.
Theorem \ref{theorem-puncture-relative} is sufficient for us to fit our result into the Gross--Siebert program as theta functions and structure constants in \cite{GS19} and \cite{GS21} only involve punctured invariants with one punctured marking. Note that relative invariants with several negative contact orders will also appear in this paper. However, the general comparison theorem is not necessary because we will reduce relevant invariants with several negative contact orders to invariants without negative contact order.
\end{remark}
\section{The proper Landau--Ginzburg potential from intrinsic mirror symmetry}\label{sec:intrinsic}
\subsection{The Landau--Ginzburg potential}
A tropical view of the Landau--Ginzburg potential is given in \cite{CPS} using the toric degeneration approach to mirror symmetry. We consider the intrinsic mirror symmetry construction instead and focus on the case when the Landau--Ginzburg potential is proper.
Following intrinsic mirror symmetry \cite{GS19}, one considers a maximally unipotent degeneration $g:Y\rightarrow S$ of the smooth pair $(X,D)$. The mirror of $X\setminus D$ is constructed as the projective spectrum of the degree zero part of the relative quantum cohomology of $(Y,D^\prime)$, where $D^\prime$ is certain divisor of $Y$ that includes $g^{-1}(0)$.
Let $(B^\prime,\mathscr P, \varphi)$ be the dual intersection complex or the fan picture of the degeneration. Recall that $B^\prime$ is an integral affine manifold with finite polyhedral decomposition $\mathscr P$ and a multi-valued strictly convex piecewise linear function $\varphi$.
An \emph{asymptotic direction} is an integral tangent vector of a one-dimensional unbounded cell in $(B^\prime,\mathscr P, \varphi)$ that points in the unbounded direction.
\begin{definition}
The dual intersection complex $(B^\prime, \mathscr P)$ is asymptotically cylindrical if
\begin{itemize}
\item $B^\prime$ is non-compact.
\item For every polyhedron $\sigma$ in $\mathscr P$, all of the unbounded one-faces of $\sigma$ are parallel with respect to the affine structure on $\sigma$.
\end{itemize}
\end{definition}
We consider the case when $D$ is smooth. Hence, $(B^\prime, \mathscr P)$ is asymptotically cylindrical and $B^\prime$ has one unbounded direction $m_{\on{out}}$. We choose $\phi$ such that $\phi(m_{\on{out}})=1$ on all unbounded cells.
For the smooth pair $(X,D)$, one can also consider its relative quantum cohomology. Let $QH^0_{\on{log}}(X,D)$ be the degree zero subalgebra of the relative quantum cohomology ring $\on{QH}^*_{\on{log}}(X,D)$ of a pair $(X,D)$. Let $S$ be the dual intersection complex of $D$. Let $B$ be the cone over $S$ and $B(\mathbb Z)$ be the set of integer points of $B$. Since $D$ is smooth, $B(\mathbb Z)$ is the set of nonnegative integers. The set
\[
\{\vartheta_p\}, p\in B(\mathbb Z)
\]
of theta functions form a canonical basis of $\on{QH}^0_{\on{log}}(X,D)$. Moreover, theta functions satisfy the following multiplication rule
\begin{align}\label{theta-func-multi}
\vartheta_{p_1}\star \vartheta_{p_2}=\sum_{r\geq 0, \beta}N_{p_1,p_2,-r}^{\beta} \vartheta_r.
\end{align}
\if{
Given a point $P$ on the dual intersection complex, the theta function can be defined by
\[
\vartheta_p(P)=\sum_{\mathfrak b}a_{\mathfrak b} m^{\mathfrak b},
\]
where the sum is over all broken lines $\mathbf b$ with asymptotic monomial $p$ and $\mathbf b(0)=P$. Then we have the following relation
\[
N_{p_1,p_2,-r}^{\beta}=\sum_{(\mathfrak a, \mathfrak b)} a_{\mathfrak a}a_{\mathfrak b}.
\]
}\fi
Recall that the structure constants $N^{\beta}_{p_1,p_2,-r}$ are defined as the invariants of $(X,D)$ with two ``inputs'' with positive contact orders given by $p_1, p_2\in B(\mathbb Z)$, one ``output'' with negative contact order given by $-r$ such that $r\in B(\mathbb Z)$, and a point constraint for the punctured point. Namely,
\begin{align}\label{def-stru-const}
N^{\beta}_{p_1,p_2,-r}=\langle [1]_{p_1},[1]_{p_2},[\on{pt}]_{-r}\rangle_{0,3,\beta}^{(X,D)}.
\end{align}
We learnt about the following intrinsic mirror symmetry construction of the proper Landau--Ginzburg potential from Mark Gross.
\begin{construction}
We recall the maximally unipotent degeneration $g:Y\rightarrow S$ and the pair $(Y,D^\prime)$ from the intrinsic mirror construction of the mirror $X^\vee$. The degree zero part of the graded ring of theta functions in $QH^0_{\log}(Y,D^\prime)$ agrees with $QH^0_{\log}(X,D)$. The base of the Landau--Ginzburg mirror of $(X,D)$ is $\on{Spec}QH^0_{\log}(X,D)=\mathbb A^1$ and the superpotential is $W=\vartheta_{1}$, the unique primitive theta function of $QH^0_{\log}(X,D)$.
\end{construction}
Under this construction, to compute the proper Landau--Ginzburg potential, we just need to compute the theta function $\vartheta_1$. We would like to compute the structure constants $N^{\beta}_{p_1,p_2,-r}$ and then provide a definition of the theta functions, in terms of two-point relative invariants of $(X,D)$, which satisfy the multiplication rule (\ref{theta-func-multi}). The notion of broken lines will not be mentioned here.
\subsection{Structure constants}
We first express the structure constants in terms of two-point relative invariants.
\begin{prop}\label{prop-struc-const}
Let $(X,D)$ be a smooth log Calabi--Yau pair. Without loss of generality, we assume that $p_1\leq p_2$. Then the structure constants $N^{\beta}_{p_1,p_2,-r}$ can be written as two-point relative invariants (without negative contact):
\begin{align}\label{equ-punctured-2}
N^{\beta}_{p_1,p_2,-r}=\left\{
\begin{array}{cc}
(p_1-r)\langle [\on{pt}]_{p_1-r}, [1]_{p_2}\rangle_{0,2,\beta}^{(X,D)}+ (p_2-r)\langle [\on{pt}]_{p_2-r}, [1]_{p_1}\rangle_{0,2,\beta}^{(X,D)} & \text{ if }0\leq r<p_1;
\\
(p_2-r)\langle [\on{pt}]_{p_2-r}, [1]_{p_1}\rangle_{0,2,\beta}^{(X,D)} & \text{ if }p_1\leq r<p_2;\\
0 & \text{ if } r\geq p_2, r\neq p_1+p_2;\\
1 & \text{ if } r=p_1+p_2.
\end{array}
\right.
\end{align}
\end{prop}
\begin{proof}
We divide the proof into different cases.
\begin{enumerate}
\item $0<r<p_1:$
We use the definition of relative Gromov--Witten invariants with negative contact orders in \cite{FWY}. Recall that $N^{\beta}_{p_1,p_2,-r}$ (\ref{def-stru-const}) is a relative invariant with one negative contact order. It can be written as
\begin{align}\label{def-one-negative}
\sum_{\mathfrak G\in \mathcal B_\Gamma}\frac{\prod_{e\in E}d_e}{|\on{Aut}(E)|}\sum \langle \prod_{j\in S_1}\epsilon_j, \, |\,|[\on{pt}]_r,\eta\rangle^{\sim}_{\Gamma^0} \langle \check{\eta},\prod_{j\in S_2}\epsilon_j\rangle^{\bullet, (X,D)}_{\Gamma_\infty},
\end{align}
where $S_1\sqcup S_2=\{1,2\}$, $\{\epsilon_j\}_{j=1,2}=\{[1]_{p_1},[1]_{p_2}\}$; the sum is over the cohomology weighted partition $\eta$ and the splitting $S_1\sqcup S_2=\{1,2\}$.
\begin{itemize}
\item [(I) $|S_1|=\emptyset$:]
Then by the virtual dimension constraint on $\langle \, |\,|[\on{pt}]_r,\eta\rangle^{\sim}_{\Gamma^0}$, the insertions in $\eta$ must contain at least one element with insertion $[1]_k$, for some integer $k>0$.
Let $\pi:P:=\mathbb P_D(\mathcal O_D\oplus N_D)\rightarrow D$ be the projection map and $D_0$ and $D_\infty$ be the zero and infinity divisors of $P$. Let
\[
p: \overline{\mathcal M}_{\Gamma^0}^\sim(P,D_0\cup D_\infty)\rightarrow \overline{\mathcal M}_{0,m}(D,\pi_*(\beta_1))
\]
be the natural projection of the rubber map to $X$ and contracting the resulting unstable components. By \cite{JPPZ18}*{Theorem 2}, we have
\[
p_*[\overline{\mathcal M}_{\Gamma^0}^\sim(P,D_0\cup D_\infty)]^{\on{vir}}=[\overline{\mathcal M}_{0,m}(D,\pi_*(\beta_1))]^{\on{vir}}.
\]
The marking $[1]_k$ becomes the identity class $ 1\in H^*(D)$. Applying the string equation implies that the rubber invariant $\langle \, |\,|[\on{pt}]_r,\eta\rangle^{\sim}_{\Gamma^0}$ vanishes unless $\pi_*(\beta_1)=0$ and $m=3$. However, $\pi_*(\beta_1)=0$ implies that $D_0\cdot \beta_1=D_\infty\cdot \beta_1$. On the other hand, there is no relative marking at $D_0$. Therefore $D_0\cdot \beta_1=0$. We also know that $D_\infty\cdot \beta_1>0$ because $\eta$ is not empty. This is a contradiction. Therefore, we can not have $|S_1|=\emptyset$.
\item [(II) $|S_1|\neq \emptyset$:]
In this case, for $\langle \prod_{j\in S_1}\epsilon_j, \, |\,|[\on{pt}]_r,\eta\rangle^{\sim}_{\Gamma^0} $, the relative insertion $\prod_{j\in S_1}\epsilon_j$ at $D_0$ is not empty. That is, it must contain at least one of $[1]_{p_1},[1]_{p_2}$.
Again, we consider the natural projection
\[
p: \overline{\mathcal M}_{\Gamma^0}^\sim(P,D_0\cup D_\infty)\rightarrow \overline{\mathcal M}_{0,m}(D,\pi_*(\beta_1)).
\]
Since $\prod_{j\in S_1}\epsilon_j$ must contain at least one of $[1]_{p_1},[1]_{p_2}$, by the projection formula and the string equation,
\[
\langle \prod_{j\in S_1}\epsilon_j, \, |\,|[\on{pt}]_r,\eta\rangle^{\sim}_{\Gamma^0}
\]
vanishes unless $\pi_*(\beta_1)=0$ and $m=3$. Note that $\pi_*(\beta_1)=0$ implies $D_0\cdot \beta=D_\infty\cdot \beta$, for any effective curve class $\beta$ of $P$. We recall that we assume that $r<p_1\leq p_2$, therefore $\eta$ must contain at least one markings with positive contact order. Then $\prod_{j\in S_1}\epsilon_j$ must contain exactly one of $[1]_{p_1},[1]_{p_2}$ when $r<p_1$. Therefore, $\eta$ contains exactly one element $[1]_{p_1-r}$ or $[1]_{p_2-r}$ respectively. Hence, (\ref{def-one-negative}) is the sum of the following two invariants
\[
(p_1-r)\langle [\on{pt}]_{p_1-r}, [1]_{p_2}\rangle_{0,2,\beta}^{(X,D)}, \quad \text{and} \quad (p_2-r)\langle [\on{pt}]_{p_2-r},[1]_{p_1}\rangle_{0,2,\beta}^{(X,D)},
\]
which are exactly the invariants that appear on the RHS of (\ref{equ-punctured-2}) when $r<p_1$.
\end{itemize}
\item $r=0$:
In this case, there are no negative contacts. We can require the marking with the point insertion $[\on{pt}]_0$ maps to $D$. Consider the degeneration to the normal cone of $D$ and apply the degeneration formula. After applying the rigidification lemma \cite{MP}*{Lemma 2}, we also obtain the formula (\ref{def-one-negative}) with $r=0$. Then the rest of the proof is the same as the case when $0<r<p_1$.
\item $p_1\leq r< p_2$:
We again have the formula (\ref{def-one-negative}) as in the first case. The difference is that we can not have $\prod_{j\in S_1}\epsilon_j=[1]_{p_1}$ because this will imply that $\eta$ contains the non-positive contact order element $[1]_{p_1-r}$. Therefore, we have
\[
(p_2-r)\langle [\on{pt}]_{p_2-r}, [1]_{p_1}\rangle_{0,2,\beta}^{(X,D)}.
\]
\item $r\geq p_2$ and $r\neq p_1+p_2$:
Similar to the previous case, we can not have $\prod_{j\in S_1}\epsilon_j=[1]_{p_1}$ or $\prod_{j\in S_1}\epsilon_j=[1]_{p_2}$. The invariant is $0$.
\item $p_1+p_2=r$:
In this case, we can have $\eta$ to be empty. Then there is no $\Gamma^\infty$ and the curves entirely lie in $D$. Therefore, there is only one rubber integral. The invariant is just $1$.
\end{enumerate}
\end{proof}
\begin{remark}
A special case of Proposition \ref{prop-struc-const} also appears in \cite{Graefnitz2022}*{Theorem 2} for del Pezzo surfaces via tropical correspondence. Our result here uses the definition of \cite{FWY} for punctured invariants and it works for all dimensions and $X$ is not necessarily Fano.
\end{remark}
Later, we will also need to consider invariants of the following form:
\[
\langle [1]_{p_1},[\on{pt}]_{p_2},[1]_{-r}\rangle_{0,3,\beta}^{(X,D)}.
\]
The proof of the following identity is similar to the proof of Proposition \ref{prop-struc-const}.
\begin{prop}\label{prop-theta-2}
Let $(X,D)$ be a smooth log Calabi--Yau pair. If $p_1\leq p_2$, then
\begin{align*}
&\langle [1]_{p_1},[\on{pt}]_{p_2},[1]_{-r}\rangle_{0,3,\beta}^{(X,D)}\\
=&\left\{
\begin{array}{cc}
(p_1-r)\langle [\on{pt}]_{p_2}, [1]_{p_1-r}\rangle_{0,2,\beta}^{(X,D)}+ (p_2-r)\langle [\on{pt}]_{p_2-r}, [1]_{p_1}\rangle_{0,2,\beta}^{(X,D)} & \text{ if }0\leq r<p_1;
\\
(p_2-r)\langle [\on{pt}]_{p_2-r}, [1]_{p_1}\rangle_{0,2,\beta}^{(X,D)} & \text{ if }p_1\leq r<p_2;\\
0 & \text{ if } r\geq p_2, r\neq p_1+p_2;\\
1 &\text{ if } r=p_1+p_2.
\end{array}
\right.
\end{align*}
If $p_2\leq p_1$, then
\begin{align*}
&\langle [1]_{p_1},[\on{pt}]_{p_2},[1]_{-r}\rangle_{0,3,\beta}^{(X,D)}\\
=&\left\{
\begin{array}{cc}
(p_1-r)\langle [\on{pt}]_{p_2}, [1]_{p_1-r}\rangle_{0,2,\beta}^{(X,D)}+ (p_2-r)\langle [\on{pt}]_{p_2-r}, [1]_{p_1}\rangle_{0,2,\beta}^{(X,D)} & \text{ if }0\leq r<p_2;
\\
(p_1-r)\langle [\on{pt}]_{p_2}, [1]_{p_1-r}\rangle_{0,2,\beta}^{(X,D)} & \text{ if }p_2\leq r<p_1;\\
0 & \text{ if } r\geq p_1, r\neq p_1+p_2;\\
1 & \text{ if } r=p_1+p_2.
\end{array}
\right.
\end{align*}
\end{prop}
\subsection{Theta functions}
Now we define the theta function in terms of two-point relative Gromov--Witten invariants of $(X,D)$.
\begin{definition}\label{def-theta-func}
Write $x=z^{(-m_{\on{out}},-1)}$ and $t=z^{(0,1)}$. For $p\geq 1$, the theta function is
\begin{align}\label{theta-func-def}
\vartheta_p:=x^{-p}+\sum_{n=1}^{\infty}nN_{n,p}t^{n+p}x^n,
\end{align}
where
\[
N_{n,p}=\sum_{\beta} \langle [\on{pt}]_n,[1]_p\rangle_{0,2,\beta}^{(X,D)}.
\]
\end{definition}
We also write
\[
N_{p_1,p_2,-r}:=\sum_\beta N_{p_1,p_2,-r}^{\beta}.
\]
To justify the definition of the theta function, we need to show that this definition satisfies the multiplication rule (\ref{theta-func-multi}). Plug-in $(\ref{theta-func-def})$ to $\vartheta_{p_1}\star \vartheta_{p_2}$, we have
\begin{align}\label{theta-p-q}
\notag \vartheta_{p_1}\star \vartheta_{p_2}=&(x^{-p_1}+\sum_{m=1}^{\infty}mN_{m,p_1}t^{m+p_1}x^m)(x^{-p_2}+\sum_{n=1}^{\infty}nN_{n,p_2}t^{n+p_2}x^n)\\
=&x^{-(p_1+p_2)}+\sum_{n=1}^{\infty}nN_{n,p_2}t^{n+p_2}x^{n-p_1}+\sum_{m=1}^{\infty}mN_{m,p_1}t^{m+p_1}x^{m-p_2}\\
\notag &+\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}mnN_{m,p_1}N_{n,p_2}t^{m+p_1+n+p_2}x^{m+n}.
\end{align}
On the other hand, we have
\begin{align}\label{theta-r}
\notag \sum_{r\geq 0, \beta}N_{p_1,p_2,-r}^{\beta}t^\beta \vartheta_r&=\sum_{r\geq 0}N_{p_1,p_2,-r}t^{p_1+p_2-r}\vartheta_r\\
&=\sum_{r\geq 0}N_{p_1,p_2,-r}t^{p_1+p_2-r}(x^{-r}+\sum_{k=1}^{\infty}kN_{k,r}t^{k+r}x^k),
\end{align}
where the second line follows from (\ref{theta-func-def}). Note that $N_{k,r}=0$ when $r=0$ by the string equation.
By Proposition \ref{prop-struc-const}, it is straightforward that the coefficients of $x^k$, for $k\leq 0$, of (\ref{theta-p-q}) and (\ref{theta-r}) are the same: without loss of generality, we assume that $p_1\leq p_2$. Then, we have the following cases.
\begin{itemize}
\item $k\leq -p_2$, and $k\neq -p_1-p_2$: we see that the coefficient of $\vartheta_{p_1}\star \vartheta_{p_2}$ in (\ref{theta-p-q}) is zero. The corresponding coefficient $N_{p_1,p_2,k}$ in (\ref{theta-r}) is also zero because of Proposition \ref{prop-struc-const}.
\item $k=-p_1-p_2$: we see that the coefficient of $\vartheta_{p_1}\star \vartheta_{p_2}$ in (\ref{theta-p-q}) is $1$. The corresponding coefficient $N_{p_1,p_2,k}$ in (\ref{theta-r}) is also $1$ because of Proposition \ref{prop-struc-const}.
\item $-p_2<k\leq -p_1$: the coefficient of $\vartheta_{p_1}\star \vartheta_{p_2}$ in (\ref{theta-p-q}) is $(p_2+k)N_{p_2+k,p_1}$. By Proposition \ref{prop-struc-const}, the corresponding coefficient $N_{p_1,p_2,k}$ in (\ref{theta-r}) is:
\[
N_{p_1,p_2,k}=(p_2+k)N_{p_2+k,p_1}.
\]
\item $-p_1<k\leq 0$: the coefficient of $\vartheta_{p_1}\star \vartheta_{p_2}$ in (\ref{theta-p-q}) is
\[
(p_1+k)N_{p_1+k,p_2}+(p_2+k)N_{p_2+k,p_1}.
\]
This coincides with the corresponding coefficient $N_{p_1,p_2,k}$ by Proposition \ref{prop-struc-const}.
\end{itemize}
For the coefficients of $x^k$, for $k>0$, the coefficients also match because of the following result.
\begin{prop}\label{prop-wdvv}
Let $(X,D)$ be a smooth log Calabi--Yau pair. We have
\begin{align}\label{identity-wdvv}
(k+p_1)N_{k+p_1,p_2}+(k+p_2)N_{k+p_2,p_1}+\sum_{m,n>0, m+n=k}mnN_{m,p_1}N_{n,p_2}=\sum_{r>0}N_{p_1,p_2,-r}kN_{k,r}.
\end{align}
\end{prop}
\begin{proof}
It can be proved using the WDVV equation of relative Gromov--Witten theory in \cite{FWY}*{Proposition 7.5}. Set
\[
[\alpha_1]_{i_1}=[1]_{p_1}, [\alpha_2]_{i_2}=[1]_{p_2}, [\alpha_3]_{i_3}=[\on{pt}]_{k+p_1+p_2}, [\alpha_4]_{i_4}=[1]_{-p_1-p_2}.
\]
Then the WDVV equation states that
\begin{align}\label{wdvv}
\sum\langle [1]_{p_2},[\on{pt}]_{k+p_1+p_2},[\gamma]_{-i}\rangle_{0,\beta_1,3}^{(X,D)} \langle [\gamma^\vee]_{i}, [1]_{p_1},[1]_{-p_1-p_2} \rangle_{0,\beta_2,3}^{(X,D)} \\
\notag =
\sum\langle [1]_{p_1},[1]_{p_2},[\gamma]_{-i}\rangle_{0,\beta_1,3}^{(X,D)} \langle [\gamma^\vee]_{i}, [\on{pt}]_{k+p_1+p_2},[1]_{-p_1-p_2} \rangle_{0,\beta_2,3}^{(X,D)} ,
\end{align}
where each sum is over the curve class $\beta$ such that $D\cdot \beta=p_1+p_2+k$, all splittings of $\beta_1+\beta_2=\beta$ and the dual bases $\{[\gamma]_{-i}\}$ and $\{[\gamma^\vee]_{i}\}$ of $\mathfrak H$.
\begin{enumerate}
\item[\textbf{(I)}] We first consider the LHS of the WDVV equation (\ref{wdvv}):
\begin{align}\label{wdvv-lhs}
\sum\langle [1]_{p_2},[\on{pt}]_{k+p_1+p_2},[\gamma]_{-i}\rangle_{0,\beta_1,3}^{(X,D)} \langle [\gamma^\vee]_{i}, [1]_{p_1},[1]_{-p_1-p_2} \rangle_{0,\beta_2,3}^{(X,D)}.
\end{align}
We analyze the invariant
\[
\langle [\gamma^\vee]_{i}, [1]_{p_1},[1]_{-p_1-p_2} \rangle_{0,\beta_2,3}^{(X,D)}
\]
in (\ref{wdvv-lhs}).
\begin{itemize}
\item[(i) $i<0$:] we claim that the invariant vanishes.
By the virtual dimension constraint, $\deg(\gamma^\vee)=\dim_{\mathbb C}X-2$.
We apply the definition of relative Gromov—Witten invariants with negative contact orders in Section \ref{sec-rel-general}. The marking with negative contact order $[1]_{-p_1-p_2}$ is distributed to a rubber space. The marking becomes a relative marking at $D_\infty$ with insertion $[1]_{p_1+p_2}$.
We further divide it into two cases.
\begin{itemize}
\item [(Case 1):] the first marking and the third marking are distributed to different rubber spaces. Then the class $\mathfrak c_\Gamma$ is trivial. We consider the rubber moduli space $\overline{\mathcal M}_{\Gamma^0_v}^\sim(P,D_0\cup D_\infty)$ where the third marking is distributed to and pushforward this rubber moduli space to the moduli space $\overline{M}_{0,m}(D,\pi_*\beta_v)$ of stable maps to $D$. The marking with $[1]_{p_1+p_2}$ becomes the identity class $1\in H^*(D)$. Apply the string equation, we see that the rubber invariant vanishes unless $\pi_*(\beta_v)=0$ and $m=3$. However $\pi_*(\beta_v)=0$ implies that $D_0\cdot \beta_v=D_\infty\cdot \beta_v$. This is not possible because, based on the insertions of the markings, we must have $D_\infty\cdot \beta_v\geq p_1+p_2>D_0\cdot \beta_v$.
\item [(Case 2):] the first marking and the third marking are distributed to the same rubber space. The class $\mathfrak c_\Gamma$ is a sum of descendant classes of degree one. By the virtual dimension constraint, $\eta$ must contain at least one element with insertion $[1]_k$ for some positive integer $k$. Pushing forward to the moduli space of stable maps to $D$ and applying the string equation twice, we again conclude that the invariant vanishes as in (Case 1).
\end{itemize}
\item [(ii) $i\geq 0$:]
The invariants in (\ref{wdvv-lhs}) are genus zero $3$-point relative invariants of $(X,D)$ with one negative contact order. Therefore, the virtual dimensions of the moduli spaces are $(\dim_{\mathbb C} X-1)$. By the virtual dimension constraint, we must have
\[
[\gamma]_{-i}=[1]_{-i}, \quad [\gamma^\vee]_{i}=[\on{pt}]_{i}.
\]
By Proposition \ref{prop-theta-2}, $ \langle [\on{pt}]_{i}, [1]_{p_1},[1]_{-p_1-p_2} \rangle_{0,\beta_2,3}^{(X,D)} $ vanishes unless $i>p_1+p_2$ or $i=p_2$.
\begin{itemize}
\item When $i=p_2$, we have the term
\[
\langle [1]_{p_2},[\on{pt}]_{k+p_1+p_2},[1]_{-i}\rangle_{0,\beta_1,3}^{(X,D)} \langle [\on{pt}]_{i}, [1]_{p_1},[1]_{-p_1-p_2} \rangle_{0,\beta_2,3}^{(X,D)}=(k+p_1)\langle [\on{pt}]_{k+p_1}, [1]_{p_2}\rangle_{0,\beta,2}^{(X,D)}
\]
in (\ref{wdvv-lhs}), by Proposition \ref{prop-theta-2}.
\item When $i>p_1+p_2$, we have
\[
\langle [\on{pt}]_{i}, [1]_{p_1},[1]_{-p_1-p_2} \rangle_{0,\beta_2,3}^{(X,D)}=(i-p_1-p_2)\langle [\on{pt}]_{i-p_1-p_2}, [1]_{p_1}\rangle_{0,\beta_2,2}^{(X,D)},
\]
by Proposition \ref{prop-theta-2}.
\end{itemize}
\end{itemize}
Similarly, by Proposition \ref{prop-theta-2}, $\langle [1]_{p_2},[\on{pt}]_{k+p_1+p_2},[1]_{-i}\rangle_{0,\beta_1,3}^{(X,D)}$ vanishes unless $i<k+p_1+p_2$ or $i=k+p_1+2p_2$.
\begin{itemize}
\item When $i=k+p_1+2p_2$ we have the term
\[
\langle [1]_{p_2},[\on{pt}]_{k+p_1+p_2},[1]_{-i}\rangle_{0,\beta_1,3}^{(X,D)} \langle [\on{pt}]_{i}, [1]_{p_1},[1]_{-p_1-p_2} \rangle_{0,\beta_2,3}^{(X,D)}=(k+p_2)\langle [\on{pt}]_{k+p_2}, [1]_{p_1}\rangle_{0,\beta,2}^{(X,D)},
\]
in (\ref{wdvv-lhs}), by Proposition \ref{prop-theta-2}.
\item When $i<k+p_1+p_2$, we have
\[
\langle [1]_{p_2},[\on{pt}]_{k+p_1+p_2},[1]_{-i}\rangle_{0,\beta_1,3}^{(X,D)} =(k+p_1+p_2-i)\langle [\on{pt}]_{k+p_1+p_2-i}, [1]_{p_2}\rangle_{0,\beta_2,2}^{(X,D)},
\]
by Proposition \ref{prop-theta-2}.
\end{itemize}
We summarize the above analysis of (\ref{wdvv-lhs}) in terms of $i$:
\begin{itemize}
\item $i<0$, the summand is $0$.
\item $i=p_2$, the summand is
\[
(k+p_1)\langle [\on{pt}]_{k+p_1}, [1]_{p_2}\rangle_{0,\beta,2}^{(X,D)}.
\]
\item $i=k+p_1+2p_2$, the summand is
\[
(k+p_2)\langle [\on{pt}]_{k+p_2}, [1]_{p_1}\rangle_{0,\beta,2}^{(X,D)}.
\]
\item $p_1+p_2 <i<k+p_1+p_2$, the summand is
\[
(k+p_1+p_2-i)\langle [\on{pt}]_{k+p_1+p_2-i}, [1]_{p_2}\rangle_{0,\beta_2,2}^{(X,D)}(i-p_1-p_2)\langle [\on{pt}]_{i-p_1-p_2}, [1]_{p_1}\rangle_{0,\beta_2,2}^{(X,D)}.
\]
\end{itemize}
Set $m:=i-p_1-p_2$ and $n:=k+p_1+p_2-i$. Then (\ref{wdvv-lhs}) becomes
\[
(k+p_1)N_{k+p_1,p_2}+(k+p_2)N_{k+p_2,p_1}+\sum_{m,n>0, m+n=k}mnN_{m,p_1}N_{n,p_2}.
\]
\item[\textbf{(II)}] Now we look at the RHS of the WDVV equation (\ref{wdvv}):
\begin{align}\label{wdvv-rhs}
\sum\langle [1]_{p_1},[1]_{p_2},[\gamma]_{-i}\rangle_{0,\beta_1,3}^{(X,D)} \langle [\gamma^\vee]_{i}, [\on{pt}]_{k+p_1+p_2},[1]_{-p_1-p_2} \rangle_{0,\beta_2,3}^{(X,D)}.
\end{align}
\begin{itemize}
\item If $i<0$, then the invariant $\langle [\gamma^\vee]_{i}, [\on{pt}]_{k+p_1+p_2},[1]_{-p_1-p_2} \rangle_{0,\beta_2,3}^{(X,D)}$ is a genus zero three-point relative invariant with two negative contact orders. The virtual dimension of the moduli space is $(n-2)$. On the other hand, the second marking has a point insertion:
\[
\deg([\on{pt}]_{k+p_1+p_2})=n-1>n-2.
\]
It is a contradiction.
\item If $i=0$, by the virtual dimension constraint, we must have
\[
[\gamma^\vee]_{i}=[1]_0.
\]
The string equation implies the invariant is zero.
\item If $i>0$, we must have
\[
[\gamma^\vee]_{i}=[1]_{r}, \quad \text{and } [\gamma]_{-i}=[\on{pt}]_{-r} \quad \text{for } r:=i>0.
\]
We have
\[
\langle [1]_{p_1},[1]_{p_2},[\gamma]_{-i}\rangle_{0,\beta_1,3}^{(X,D)}=\langle [1]_{p_1},[1]_{p_2},[\on{pt}]_{-r}\rangle_{0,\beta_1,3}^{(X,D)}.
\]
By Proposition \ref{prop-theta-2},
\[
\langle [1]_{r}, [\on{pt}]_{k+p_1+p_2},[1]_{-p_1-p_2} \rangle_{0,\beta_2,3}^{(X,D)}=k\langle [1]_{r}, [\on{pt}]_{k} \rangle_{0,\beta_2,2}^{(X,D)}.
\]
\end{itemize}
Therefore, (\ref{wdvv-rhs}) becomes
\[
\sum_{r>0}N_{p_1,p_2,-r}kN_{k,r}.
\]
\end{enumerate}
This completes the proof.
\end{proof}
\begin{remark}
We can consider the special case when $p_1=1$, then the LHS of (\ref{identity-wdvv}) is
\[
(k+1)N_{k+1,p_2}+(k+p_2)N_{k+p_2,1}+\sum_{m,n>0, m+n=k}mnN_{m,1}N_{n,p_2}
\]
and the RHS of (\ref{identity-wdvv}) is
\begin{align*}
&\sum_{r>0}N_{1,p_2,-r}kN_{k,r}\\
=& kN_{k,p_2+1}+ \sum_{r=1}^{p_2-1}(p_2-r)N_{p_2-r,1}kN_{k,r},
\end{align*}
by Proposition \ref{prop-struc-const}. Identity (\ref{identity-wdvv}) becomes
\[
(k+1)N_{k+1,p_2}+(k+p_2)N_{k+p_2,1}+\sum_{m,n>0, m+n=k}mnN_{m,1}N_{n,p_2}
=kN_{k,p_2+1}+ \sum_{r=1}^{p_2-1}(p_2-r)N_{p_2-r,1}kN_{k,r}.
\]
If we further specialize to the case of toric del Pezzo surfaces with smooth divisors, we recover \cite{GRZ}*{Lemma 5.3}. Here, we give a direct explanation of Identity (\ref{identity-wdvv}) in terms of the WDVV equation of the relative Gromov--Witten theory in \cite{FWY}.
\end{remark}
\section{A mirror theorem for smooth pairs}
\subsection{Relative mirror theorem}
Let $X$ be a smooth projective variety. Let $\{p_i\}_{i=1}^r$ be an integral, nef basis of $H^2(X)$. For the rest of the paper, we assume that $D$ is nef.
Recall that the $J$-function for absolute Gromov--Witten theory of $X$ is
\[
J_{X}(\tau,z)=z+\tau+\sum_{\substack{(\beta,l)\neq (0,0), (0,1)\\ \beta\in \on{NE(X)}}}\sum_{\alpha}\frac{q^{\beta}}{l!}\left\langle \frac{\phi_\alpha}{z-\psi},\tau,\ldots, \tau\right\rangle_{0,1+l, \beta}^{X}\phi^{\alpha},
\]
where $\tau=\tau_{0,2}+\tau^\prime\in H^*(X)$; $\tau_{0,2}=\sum_{i=1}^r p_i \log q_i\in H^2(X)$; $\tau^\prime\in H^*(X)\setminus H^2(X)$; $\on{NE(X)}$ is the cone of effective curve classes in $X$; $\{\phi_\alpha\}$ is a basis of $H^*(X)$; $\{\phi^\alpha\}$ is the dual basis under the Poincar\'e pairing. We can decompose the $J$-function as follows
\[
J_{X}(\tau,z)=\sum_{\beta\in \on{NE(X)}}J_{X,\beta}(\tau,z)q^\beta.
\]
The $J$-function of the smooth pair $(X,D)$ is defined similarly.
We first define
\[
\mathfrak H_0:=H^*(X) \text{ and }\mathfrak H_i:=H^*(D) \text{ if }i\in \mathbb Z \setminus \{0\}.
\]
The ring of insertions (state space) of relative Gromov--Witten theory is defined as
\[
\mathfrak H:=\bigoplus\limits_{i\in\mathbb Z}\mathfrak H_i.
\]
Each $\mathfrak H_i$ naturally embeds into $\mathfrak H$. For an element $\gamma\in \mathfrak H_i$, we denote its image in $\mathfrak H$ by $[\gamma]_i$. Define a pairing on $\mathfrak H$ by the following.
\begin{equation}\label{eqn:pairing}
\begin{split}
([\gamma]_i,[\delta]_j) =
\begin{cases}
0, &\text{if } i+j\neq 0,\\
\int_X \gamma\cup\delta, &\text{if } i=j=0, \\
\int_D \gamma\cup\delta, &\text{if } i+j=0, i,j\neq 0.
\end{cases}
\end{split}
\end{equation}
The pairing on the rest of the classes is generated by linearity.
\begin{defn}\label{def-relative-J-function}
Let $X$ be a smooth projective variety and $D$ be a smooth nef divisor, the $J$-function for the pair $(X,D)$ is defined as
\[
J_{(X,D)}(\tau,z)=z+\tau+\sum_{\substack{(\beta,l)\neq (0,0), (0,1)\\ \beta\in \on{NE(X)}}}\sum_{\alpha}\frac{q^{\beta}}{l!}\left\langle \frac{\phi_\alpha}{z-\bar{\psi}},\tau,\ldots, \tau\right\rangle_{0,1+l, \beta}^{(X,D)}\phi^{\alpha},
\]
where $\tau=\tau_{0,2}+\tau^\prime\in H^*(X)$; $\tau_{0,2}=\sum_{i=1}^r p_i \log q_i\in H^2(X)$; $\tau^\prime\in \mathfrak H\setminus H^2(X)$; $\{\phi_\alpha\}$ is a basis of the ambient part of $\mathfrak H$; $\{\phi^\alpha\}$ is the dual basis under the Poincar\'e pairing.
\end{defn}
\begin{defn}\label{def-relative-I-function}
The (non-extended) $I$-function of the smooth pair $(X,D)$ is
\[
I_{(X,D)}(y,z)=\sum_{\beta\in \on{NE(X)}}J_{X,\beta}(\tau_{0,2},z)y^\beta\left(\prod_{0<a\leq D\cdot \beta-1}(D+az)\right)[1]_{-D\cdot \beta},
\]
where $\tau_{0,2}\in H^2(X)$.
\end{defn}
\begin{theorem}[\cite{FTY}, Theorem 1.4]\label{thm:rel-mirror}
Let $X$ be a smooth projective variety and $D$ be a smooth nef divisor such that $-K_X-D$ is nef. Then the $I$-function $I_{(X,D)}(y,\tau,z)$ coincides with the $J$-function $J_{(X,D)}(q,z)$ via change of variables, called the relative mirror map.
\end{theorem}
The relative mirror theorem also holds for the extended $I$-function of the smooth pair $(X,D)$. For the purpose of this paper, we only write down the simplest case when the extended data $S$ is the following:
\[
S:=\{1\}.
\]
\begin{defn}\label{def-relative-I-function-extended}
The $S$-extended $I$-function of $(X,D)$ is defined as follows.
\[
I_{(X,D)}^{S}(y,x_1,z)=I_++I_-,
\]
where
\begin{align*}
I_+:=&\sum_{\substack{\beta\in \on{NE}(X),k\in \mathbb Z_{\geq 0}\\ k<D\cdot \beta} }J_{X, \beta}(\tau_{0,2},z)y^{\beta}\frac{ x_1^{k}}{z^{k}k!}\frac{\prod_{0<a\leq D\cdot \beta}(D+az)}{D+(D\cdot \beta-k)z}[{1}]_{-D\cdot \beta+k},
\end{align*}
and
\begin{align*}
I_-:=&\sum_{\substack{\beta\in \on{NE}(X),k\in \mathbb Z_{\geq 0}\\ k\geq D\cdot \beta} }J_{X, \beta}(\tau_{0,2},z)y^{\beta}\frac{ x_1^{k}}{z^{k}k!}\left(\prod_{0<a\leq D\cdot \beta}(D+az)\right)[{ 1}]_{-D\cdot \beta+k}.
\end{align*}
\end{defn}
\begin{theorem}[\cite{FTY}, Theorem 1.5]\label{thm:rel-mirror-extended}
Let $X$ be a smooth projective variety and $D$ be a smooth nef divisor such that $-K_X-D$ is nef. Then the $S$-extended $I$-function $I^S_{(X,D)}(y,x_1,z)$ coincides with the $J$-function $J_{(X,D)}(q,z)$ via change of variables, called the relative mirror map.
\end{theorem}
Although the relative mirror theorem of \cite{FTY} has been used in the literature several times, the relative mirror map has never been studied in detail. We would like to provide a detailed description of the relative mirror map here.
We consider the extended $I$-function in Definition \ref{def-relative-I-function-extended} under the assumption that $D$ is nef and $-K_X-D$ is nef. The extended $I$-function can be expanded as follows
\begin{align*}
&I^S_{(X,D)}(y,x_1,z)\\
=&z+\sum_{i=1}^r p_i\log y_i+x_1[1]_{1}+\sum_{\substack{\beta\in \on{NE}(X)\\ D\cdot \beta \geq 2}}\langle [\on{pt}]\psi^{D\cdot \beta-2}\rangle_{0,1,\beta}^Xy^\beta (D\cdot \beta-1)![1]_{-D\cdot \beta}+\sum_{k=1}^{\infty}I_{-k}z^{-k},
\end{align*}
where the coefficient of $z^0$, denoted by $\tau(y,x_1)$, is the relative mirror map:
\begin{align}\label{rel-mirror-map}
\tau(y,x_1)=\sum_{i=1}^r p_i\log y_i+x_1[1]_{1}+\sum_{\substack{\beta\in \on{NE}(X)\\ D\cdot \beta \geq 2}}\langle [\on{pt}]\psi^{D\cdot \beta-2}\rangle_{0,1,\beta}^Xy^\beta (D\cdot \beta-1)![1]_{-D\cdot \beta}.
\end{align}
The relative mirror theorem of \cite{FTY} states that
\[
J(\tau(y,x_1),z)=I(y,x_1,z).
\]
The function $\tau(y,x_1)$ is the mirror map computed from the extended $I$-function. We will refer to $\tau(y,x_1)$ as the \emph{extended relative mirror map}. The relative mirror map for the non-extended $I$-function is $\tau(y,0)$. We will refer to it as the \emph{relative mirror map} and denote it by $\tau(y)$.
To be able to compute invariants from the relative mirror theorem, we need to understand the invariants that appear in $J(\tau(y,x_1),z)$. In particular, we need to understand the following invariants in order to compute the theta function $\vartheta_1$.
\begin{itemize}
\item Relative invariants with one positive contact order and several negative contact orders:
\[
\langle [1]_{-k_1},\cdots, [1]_{-k_l}, [\gamma]_{k_{l+1}} \bar{\psi}^a\rangle_{0,l+1,\beta}^{(X,D)},
\]
where
\[
D\cdot \beta=k_{l+1}-\sum_{i=1}^l k_i\geq 0, \text{ and } k_i>0.
\]
This is needed to understand the relative mirror map.
\item Relative invariants with two positive contact orders and several negative contact orders of the following form:
\[
\langle [1]_1, [1]_{-k_1},\cdots, [1]_{-k_l}, [\gamma]_{k_{l+1}} \rangle_{0,l+2,\beta}^{(X,D)},
\]
\[
D\cdot \beta-1=k_{l+1}-\sum_{i=1}^l k_i\geq 0, \text{ and } k_i>0.
\]
\item Degree zero relative invariants with two positive contact orders and several negative contact orders of the following form:
\[
\langle [1]_1,[1]_{-k_1},\cdots, [1]_{-k_l},[\on{pt}]_{k_{l+1}}\rangle_{0,l+2,0}^{(X,D)}.
\]
\end{itemize}
We will compute these invariants in the following sections.
\subsection{Relative invariants with several negative contact orders}
Based on the expression of relative mirror map in (\ref{rel-mirror-map}), to be able to compute relative invariants from the relative mirror theorem, we first need to study relative invariants with several insertions of $[1]_{-i}$ for $i\in \mathbb Z_{>0}$. We start with the case when $x=0$. That is, there is only one marking with positive contact order and no marking with insertion $[1]_1$. We would like to claim that the insertion $[1]_{-i}$ behaves like the divisor class $D$ in the sense that there is an analogous of the divisor equation as follows.
\begin{proposition}\label{prop-several-neg-1}
Given a curve class $\beta$, Let $k_i\in \mathbb Z_{>0}$ for $i\in \{1,\ldots, l+1\}$ such that
\[
D\cdot \beta=k_{l+1}-\sum_{i=1}^l k_i\geq 0.
\]
Then we have the following relation.
\begin{align}\label{identity-several-neg-1}
\langle [1]_{-k_1},\cdots, [1]_{-k_l}, [\gamma]_{k_{l+1}} \bar{\psi}^a\rangle_{0,l+1,\beta}^{(X,D)}=\langle [D]_0,\cdots, [D]_0, [\gamma]_{D\cdot \beta} \bar{\psi}^a\rangle_{0,l+1,\beta}^{(X,D)},
\end{align}
where $\gamma \in H^*(D)$.
\end{proposition}
\begin{proof}
\textbf{The base case I: $a=0$.}
In this case, there are no descendant classes. Then the identity becomes
\begin{align}\label{identity-several-neg-1-0}
\langle [1]_{-k_1},\cdots, [1]_{-k_l}, [\gamma]_{k_{l+1}} \rangle_{0,l+1,\beta}^{(X,D)}=\langle [D]_0,\cdots, [D]_0, [\gamma]_{D\cdot \beta} \rangle_{0,l+1,\beta}^{(X,D)}.
\end{align}
By Section \ref{sec-rel-general}, relative Gromov--Witten theory is defined as graph sums by gluing moduli spaces of relative stable maps with moduli space of rubber maps using fiber products. When there are more than one negative contact orders, the invariants are usually complicated and involve summation over different graphs as described in Section \ref{sec-rel-general}. But for invariants on the LHS of (\ref{identity-several-neg-1-0}), the situation is significantly simplified.
Every negative contact marking must be distributed to a rubber moduli $\overline{\mathcal M}^\sim_{\Gamma_i^0}(D)$ labelled by $\Gamma_i^0$. Since $D$ is a nef divisor in $X$, we have
\[
\int_{\beta_D}c_1(N_{D/X})\geq 0,
\]
for every effective curve class $\beta_D$ in $D$. Let $\beta_v$ be a curve class associated to a vertex $v$ in $\Gamma^0$, we must have
\[
D_0\cdot \beta_v-D_\infty\cdot \beta_v=\int_{\pi_*(\beta_v)}c_1(N_{D/X})\geq 0,
\]
where $\pi:P\rightarrow D$ is the projection map.
Therefore, the nefness of $D$ implies that, for each $\overline{\mathcal M}^\sim_{\Gamma_i^0}(D)$, the relative insertion at $D_0$ can not be empty. Hence, at least one of the positive contact markings on the LHS of (\ref{identity-several-neg-1-0}) must be distributed to $\overline{\mathcal M}^\sim_{\Gamma_i^0}(D)$. Since there is only one positive contact marking, there can only be one rubber moduli, denoted by $\overline{\mathcal M}^\sim_{\Gamma^0}(D)$. Therefore, all the negative contact markings, as well as the positive contact marking, are distributed to $\overline{\mathcal M}^\sim_{\Gamma^0}(D)$. The invariant can be written as
\[
\sum_{\mathfrak G\in \mathcal B_\Gamma}\frac{\prod_{e\in E}d_e}{|\on{Aut}(E)|}\sum_\eta \langle \check{\eta}\rangle^{\bullet,\mathfrak c_\Gamma, (X,D)}_{\Gamma^\infty}\langle \eta,[1]_{k_1},\cdots, [1]_{k_l},| \, | [\gamma]_{k_{l+1}} \rangle^{\sim,\mathfrak c_{\Gamma}}_{\Gamma^0},
\]
where the superscript $\mathfrak c_\Gamma$ means capping with the class $\mathfrak c_\Gamma(X/D)$ in Definition \ref{def-rel-cycle}.
Let
\[
p: \overline{\mathcal M}_{\Gamma^0}^\sim(P,D_0\cup D_\infty)\rightarrow \overline{\mathcal M}_{0,m}(D,\pi_*(\beta_1))
\]
be the natural projection of the rubber map to $X$ and contracting the resulting unstable components. By \cite{JPPZ18}*{Theorem 2}, we have
\[
p_*[\overline{\mathcal M}_{\Gamma^0}^\sim(P,D_0\cup D_\infty)]^{\on{vir}}=[\overline{\mathcal M}_{0,m}(D,\pi_*(\beta_1))]^{\on{vir}},
\]
where $\pi:P\rightarrow D$ is the projection map.
Note that there are $l$ identity classes $[1]$ and the degree of the class $\mathfrak c_\Gamma(X/D)$ is less or equal to $l-1$. We can apply the string equation $l$-times. Then the invariant
\[
\langle \eta,[1]_{k_1},\cdots, [1]_{k_l},| \, | [\gamma]_{k_{l+1}} \rangle^{\sim,\mathfrak c_{\Gamma}}_{\Gamma^0}
\]
vanishes unless $\pi_*(\beta)=0$ and $\eta$ contains exactly one element. Moreover, $\eta$ needs to be $[\check{\gamma}]_{D\cdot \beta}$ and $\check{\eta}$ needs to be $[\gamma]_{D\cdot \beta}$. Therefore,
\[
\langle \check{\eta}\rangle^{\bullet,\mathfrak c_\Gamma, (X,D)}_{\Gamma^\infty}=\langle [\gamma]_{D\cdot \beta}\rangle_{0,1,\beta}^{(X,D)}.
\]
There is only one edge, hence
\[
\frac{\prod_{e\in E}d_e}{|\on{Aut}(E)|}=D\cdot \beta.
\]
It remains to compute
\[
\langle [\check{\gamma}]_{D\cdot \beta},[1]_{k_1},\cdots, [1]_{k_l},| \, | [\gamma]_{k_{l+1}} \rangle^{\sim,\mathfrak c_{\Gamma}}_{\Gamma^0}
\]
with $\pi_*(\beta)=0$. This is the same as the rubber invariant with the base being a point. Set $d=D\cdot \beta$. We claim that it coincides with the following relative Gromov--Witten invariants of $(\mathbb P^1,0\cup\infty)$ with negative contact orders
\begin{align}\label{rel-inv-P-1}
\langle [1]_{d}, [1]_{-k_1},\cdots, [1]_{-k_l}, | \, | [1]_{k_{l+1}}\rangle_{0,l+2,d}^{(\mathbb P^1,0\cup\infty)}.
\end{align}
This is because one can run the above computation of the LHS of (\ref{identity-several-neg-1-0}) to the invariant (\ref{rel-inv-P-1}), we see that (\ref{rel-inv-P-1}) equals to
\[
d\langle [1]_d | \, | [1]_d \rangle_{0,l+2,d}^{(\mathbb P^1,0\cup\infty)} \langle [\check{\gamma}]_{D\cdot \beta},[1]_{k_1},\cdots, [1]_{k_l},| \, | [\gamma]_{k_{l+1}} \rangle^{\sim,\mathfrak c_{\Gamma}}_{\Gamma^0}.
\]
It is straightforward to compute that
\[
\langle [1]_d | \, | [1]_d \rangle_{0,l+2,d}^{(\mathbb P^1,0\cup\infty)}=\frac{1}{d}.
\]
This proves the claim that
\[
\langle [\check{\gamma}]_{D\cdot \beta},[1]_{k_1},\cdots, [1]_{k_l},| \, | [\gamma]_{k_{l+1}} \rangle^{\sim,\mathfrak c_{\Gamma}}_{\Gamma^0}
\]
with $\pi_*(\beta)=0$ equals to (\ref{rel-inv-P-1}). The invariant (\ref{rel-inv-P-1}) has already been computed in \cite{KW}*{Proposition B.2}:
\[
\langle [1]_{d}, [1]_{-k_1},\cdots, [1]_{-k_l}, | \, | [1]_{k_{l+1}}\rangle_{0,l+2,d}^{(\mathbb P^1,0\cup\infty)}=d^{l-1}.
\]
Therefore the LHS of (\ref{identity-several-neg-1-0}) is
\[
(D\cdot \beta)^l \langle [\gamma]_{D\cdot \beta}\rangle_{0,1,\beta}^{(X,D)}=\langle [D]_0,\cdots, [D]_0, [\gamma]_{D\cdot \beta} \rangle_{0,l+1,\beta}^{(X,D)}
\]
by the divisor equation.
\textbf{ The base case II: $l=1$.}
Then the LHS of (\ref{identity-several-neg-1-0}) is a relative invariant with one negative contact order. Similar to the proof of Proposition \ref{prop-struc-const}, the invariant is of the form,
\begin{align*}
\sum_{\mathfrak G\in \mathcal B_\Gamma}\frac{\prod_{e\in E}d_e}{|\on{Aut}(E)|}\sum \langle [\gamma]_{k_2}\psi^a |\,|[1]_{k_1},\eta\rangle^{\sim}_{\Gamma^0} \langle \check{\eta}\rangle^{\bullet, (X,D)}_{\Gamma_\infty}.
\end{align*}
For the RHS of (\ref{identity-several-neg-1-0}), we consider the degeneration to the normal cone of $D$ and apply the degeneration formula. After applying the rigidification lemma \cite{MP}*{Lemma 2}, the invariant is of the form
\begin{align*}
\sum_{\mathfrak G\in \mathcal B_\Gamma}\frac{\prod_{e\in E}d_e}{|\on{Aut}(E)|}\sum \langle [\gamma]_{k_2-k_1}\psi^a |\,|[1]_{0},\eta\rangle^{\sim}_{\Gamma^0} \langle \check{\eta}\rangle^{\bullet, (X,D)}_{\Gamma_\infty}.
\end{align*}
The only difference between the LHS of (\ref{identity-several-neg-1-0}) and the RHS of (\ref{identity-several-neg-1-0}) is the contact orders of two markings (contact orders $k_2$ and $k_1$ for the LHS and contact orders $k_2-k_1$ and $0$ for the RHS) for the rubber invariants. We pushforward the rubber moduli spaces to the moduli space $\overline{M}(D)$ of stable maps to $D$. Since the genus zero double ramification cycle is trivial, it does not depend on the contact orders. Therefore, the LHS of (\ref{identity-several-neg-1-0}) and the RHS of (\ref{identity-several-neg-1-0}) are the same.
\textbf{Induction:}
Now we use the induction to prove the case when $a>0$ and $l>1$. Suppose Identity (\ref{identity-several-neg-1}) is true when $a=N\geq 0$. When $a=N+1$, we apply the topological recursion relation
\begin{align*}
&\langle [1]_{-k_1},\cdots, [1]_{-k_l}, [\gamma]_{k_{l+1}} \bar{\psi}^{N+1}\rangle_{0,l+1,\beta}^{(X,D)}\\
=&\sum \langle [\gamma]_{k_{l+1}} \bar{\psi}^{N}, \prod_{j\in S_1}[1]_{-k_j},\tilde {T}_{i,k}\rangle_{0,\beta_1,2+|S_1|}^{(X,D)} \langle \tilde {T}_{-i}^k, \prod_{j\in S_2}[1]_{-k_j},[1]_{-k_1},[1]_{-k_2} \rangle_{0,\beta_2,3+|S_2|}^{(X,D)},
\end{align*}
where the sum is over all $\beta_1+\beta_2=\beta$, all indices $i,k$ of basis and $S_1, S_2$ disjoint sets with $S_1\cup S_2=\{3,\ldots,l\}$. The nefness of the divisor $D$ implies that
\[
\tilde {T}_{-i}^k=[\alpha]_{b} \text{ and } \tilde {T}_{i,k}=[\check{\alpha}]_{-b},
\]
for some positive integer $b\geq k_1$.
Note that
\[
\langle [\alpha]_{b}, \prod_{j\in S_2}[1]_{-k_j}, [1]_{-k_1},[1]_{-k_2} \rangle_{0,\beta_2,3+|S_2|}^{(X,D)}=\langle [\alpha]_{b-k_1-k_2},\prod_{j\in S_2}[1]_{-k_j}, [D]_0,[D]_0 \rangle_{0,\beta_2,3+|S_2|}^{(X,D)}
\]
follows from the base case.
On the other hand, we have
\[
\langle [\gamma]_{k_{l+1}} \bar{\psi}^{N}, \prod_{j\in S_1}[1]_{-k_j}, [\check{\alpha}]_{-b}\rangle_{0,\beta_1,2+|S_1|}^{(X,D)}=\langle [\gamma]_{k_{l+1}-k_1} \bar{\psi}^{N}, \prod_{j\in S_1}[1]_{-k_j}, [\check{\alpha}]_{-b+k_1}\rangle_{0,\beta_1,2+|S_1|}^{(X,D)}.
\]
This is because, for these invariants, the graph sum in the definition of the relative invariants only has one rubber space and all the markings are in the rubber space. Moreover, the class $\mathfrak c_\Gamma$ does not depend on the value of $k_{l+1}$ and $b$. Therefore, we have the identity.
Therefore, we have
\begin{align*}
&\sum \langle [\gamma]_{k_{l+1}} \bar{\psi}^{N}, \prod_{j\in S_1}[1]_{-k_j}, \tilde {T}_{i,k}\rangle_{0,\beta_1,2+|S_1|}^{(X,D)} \langle \tilde {T}_{-i}^k, \prod_{j\in S_2}[1]_{-k_j}, [1]_{-k_1},[1]_{-k_2} \rangle_{0,\beta_2,3+|S_2|}^{(X,D)}\\
=&\sum \langle [\gamma]_{k_{l+1}-k_1} \bar{\psi}^{N}, \prod_{j\in S_1}[1]_{-k_j}, [\check{\alpha}]_{-b+k_1}\rangle_{0,\beta_1,2+|S_1|}^{(X,D)} \langle [\alpha]_{b-k_1},\prod_{j\in S_2}[1]_{-k_j}, [D]_0,[D]_0 \rangle_{0,\beta_2,3+|S_2|}^{(X,D)}\\
=&\langle [D]_0, [D]_0, \prod_{j\in\{3,\ldots,l\}}[1]_{-k_j},[\gamma]_{k_{l+1}-k_1} \bar{\psi}^{N+1}\rangle_{0,3,\beta}^{(X,D)},\\
\end{align*}
where the third line is the topological recursion relation. We have the identity
\begin{align*}
\langle [1]_{-k_1},\cdots, [1]_{-k_l}, [\gamma]_{k_{l+1}} \bar{\psi}^{N+1}\rangle_{0,l+1,\beta}^{(X,D)}
=\langle [D]_0, [D]_0,\prod_{j\in\{3,\ldots,l\}}[1]_{-k_j}, [\gamma]_{D\cdot \beta-k_{1}} \bar{\psi}^{N+1}\rangle_{0,l+1,\beta}^{(X,D)}.
\end{align*}
Run the above argument multiple times to trade markings with negative contact orders with markings with insertion $[D]_0$. We end up with either one negative contact order or no negative contact order. The former case is the base case II: $l=1$, the latter case is just Identity (\ref{identity-several-neg-1}).
\if{
When $a>0$ and $l>1$, we use induction (on $a$ and $l$) and apply the topological recursion relation for relative Gromov--Witten theory in \cite{FWY}. Suppose the identity (\ref{identity-several-neg-1}) is true for $a=N$. By the topological recursion, the LHS of (\ref{identity-several-neg-1}) when $a=N+1$ is
\[
\sum \langle [\gamma]_{k_{l+1}} \bar{\psi}^{N}, \prod_{j\in S_1}[1]_{-k_j}, \tilde {T}_{i,k}\rangle_{0,\beta_1,1+|S_1|}^{(X,D)} \langle \tilde {T}_{-i}^k, [1]_{-k_1},[1]_{-k_2},\prod_{j\in S_2}[1]_{-k_j} \rangle_{0,\beta_2,2+|S_2|}^{(X,D)},
\]
where the sum is over all $\beta_1+\beta_2=\beta$, all indices $i,k$ of basis, and $S_1, S_2$ disjoint sets with $S_1\cup S_2=\{3,\ldots,l\}$. We first consider
\[
\langle \tilde {T}_{-i}^k, [1]_{-k_1},[1]_{-k_2},\prod_{j\in S_2}[1]_{-k_j} \rangle_{0,\beta_2,2+|S_2|}^{(X,D)}.
\]
The nefness of the divisor $D$ implies that
\[
\tilde {T}_{-i}^k=[\alpha]_{b}
\]
for some positive integer $b$.
}\fi
\end{proof}
The identity will be slightly different if we add an insertion of $[1]_1$. For the purpose of this paper, we only consider the case when there are no descendant classes. There is also a (more complicated) identity for descendant invariants, but we do not plan to discuss it here.
\begin{proposition}\label{prop-several-neg-2}
Given a curve class $\beta$, Let $k_i\in \mathbb Z_{>0}$ for $i\in \{1,\ldots, l+1\}$ such that
\[
D\cdot \beta-1=k_{l+1}-\sum_{i=1}^l k_i\geq 0.
\]
Then we have the following relation.
\begin{align}\label{identity-several-neg-2}
\langle [1]_1, [1]_{-k_1},\cdots, [1]_{-k_l}, [\gamma]_{k_{l+1}} \rangle_{0,l+2,\beta}^{(X,D)}=(D\cdot \beta-1)^l\langle [1]_1, [\gamma]_{D\cdot \beta} \rangle_{0,2,\beta}^{(X,D)},
\end{align}
where $\gamma \in H^*(D)$.
\end{proposition}
\begin{proof}
The proof is similar to the proof of Proposition \ref{prop-several-neg-1}. We first consider the LHS of (\ref{identity-several-neg-2}). By definition, every negative contact marking must be in a rubber moduli $\overline{\mathcal M}^\sim_{\Gamma_i^0}(D)$ labelled by $\Gamma_i^0$ and each rubber moduli $\overline{\mathcal M}^\sim_{\Gamma_i^0}(D)$ has at least one negative contact marking distributed to it. Similar to the proof of Proposition \ref{prop-several-neg-1}, the nefness of $D$ implies that the last marking (with insertion $[\gamma]_{k_{l+1}}$) has to be distributed to the rubber space.
Now we examine the first marking. Since the contact order of the first marking is $1$, the nefness of $D$ implies that the first marking and the last marking can not be in different rubber space. On the other hand, if the first marking and the last marking (both with positive contact orders) are in the same rubber space, then we claim the invariant vanishes. This is because, after pushing forward to $\overline{M}_{g,n}(D, \pi_*(\beta))$, there are $(l+1)$-identity class $1$ and the degree of the class $\mathfrak c_\Gamma$ is $l-1$. Applying the string equation $(l+1)$-times implies that the invariant is zero.
Therefore, there is only one rubber moduli, labelled by $\Gamma^0$, and the first marking can not be distributed to the rubber moduli. The LHS of (\ref{identity-several-neg-2}) is of the following form
\[
\sum_{\mathfrak G\in \mathcal B_\Gamma}\frac{\prod_{e\in E}d_e}{|\on{Aut}(E)|}\sum_\eta \langle [1]_1,\check{\eta}\rangle^{\bullet,\mathfrak c_\Gamma, (X,D)}_{\Gamma^\infty}\langle \eta,[1]_{k_1},\cdots, [1]_{k_l},| \, | [\gamma]_{k_{l+1}} \rangle^{\sim,\mathfrak c_{\Gamma}}_{\Gamma^0}.
\]
Then the rest of the proof follows from the proof of Proposition \ref{prop-several-neg-1} that the Equation (\ref{identity-several-neg-2}) holds. Note that the contact order of the unique marking in $\eta$ is $D\cdot \beta-1$ instead of $D\cdot \beta$. Therefore, we have the factor $(D\cdot \beta-1)^l$ instead of $(D\cdot \beta)^l$.
\end{proof}
One can add more insertions of $[1]_1$, then we have a similar identity as follows.
\begin{prop}\label{prop-several-neg-3}
Given a curve class $\beta$, Let $k_i\in \mathbb Z_{>0}$ for $i\in \{1,\ldots, l+1\}$ such that
\[
D\cdot \beta-a=k_{l+1}-\sum_{i=1}^l k_i\geq 0.
\]
Then we have the following relation.
\begin{align}\label{identity-several-neg-3}
\langle [1]_1, \ldots, [1]_1, [1]_{-k_1},\cdots, [1]_{-k_l}, [\gamma]_{k_{l+1}} \rangle_{0,a+l+1,\beta}^{(X,D)}=(D\cdot \beta-a)^l\langle [1]_1,\ldots, [1]_1, [\gamma]_{D\cdot \beta} \rangle_{0,a+1,\beta}^{(X,D)},
\end{align}
where $\gamma \in H^*(D)$.
\end{prop}
\begin{proof}
The proof is similar to the proof of Proposition \ref{prop-several-neg-1} and Proposition \ref{prop-several-neg-2}. We do not repeat the details.
\end{proof}
\subsection{Degree zero relative invariants}\label{sec-deg-0}
The following invariants will also appear in the $J$-function when plugging in the mirror map:
\[
\langle [1]_1, [1]_{-k_1},\cdots, [1]_{-k_l}, [\gamma]_{k_{l+1}} \rangle_{0,l+2,\beta}^{(X,D)}, \text{ with } D\cdot \beta=0.
\]
We again apply the definition of relative Gromov--Witten invariants with negative contact orders in Section \ref{sec-rel-general} and then pushforward the rubber moduli to $\overline{M}_{0,l+2}(D,\pi_*(\beta))$. Then applying the string equation $(l+1)$-times, then the invariants vanish unless $\beta=0$.
When $l=1$, the invariants have two markings with positive contact orders and one marking with negative contact order. By direct computation, we have
\[
\langle [1]_1,[1]_{-k_1},[\on{pt}]_{k_{2}}\rangle_{0,3,0}^{(X,D)}=1.
\]
Therefore, We still need to compute degree zero, genus zero relative invariants with two positive contacts and several negative contacts. By the definition of relative invariants with negative contact orders in Section \ref{sec-rel-general}, the bipartite graphs simplifies to a single vertex of type $0$ and the moduli space is simply the product $\overline{M}_{0,n}\times D$.
We will have the following result.
\begin{prop}\label{prop-degree-zero}
\begin{align}\label{identity-zero-several-neg}
\langle [1]_1,[1]_{-k_1},\cdots, [1]_{-k_l},[\on{pt}]_{k_{l+1}}\rangle_{0,l+2,0}^{(X,D)}=(-1)^{l-1},
\end{align}
where $k_1, \ldots, k_l$ are positive integers and
\[
1+k_{l+1}=k_1+\cdots +k_l.
\]
\end{prop}
These invariants can be computed from the $I$-function. The $I$-function is a limit of the $I$-function for the twisted invariants of $B\mathbb Z_r$. The $I$-function is also the restriction of the $I$-function for $(X,D)$ to the degree zero case.
Before computing the corresponding orbifold invariants, we briefly recall the relative-orbifold correspondence of \cite{FWY}:
\[
r^{m_-}\langle \prod_{i=1}^m \tau_{a_i}(\gamma_i)\rangle_{0,m,\beta}^{X_{D,r}}=\langle \prod_{i=1}^m \tau_{a_i}(\gamma_i)\rangle_{0,m,\beta}^{(X,D)},
\]
where there are $m$ markings in total and $m_-$ of them are markings with negative contact orders; $a_i\in \mathbb Z_{\geq 0}$; $\gamma_i$ are cohomology classes of $X$ (or $D$) when the marking is interior (or relative/orbifold, respectively). We would like to point out that it is important to keep in mind the factor $r^{m_-}$.
We need to consider the $S$-extended $I$-function for the twisted Gromov--Witten invariants of $B\mathbb Z_r$ for sufficiently large $r$, with extended data
\[
S=\{1,-k_1,\ldots,-k_l\}.
\]
The $I$-function is
\begin{align}\label{I-func-BZ}
z\sum \frac{x_1^{a}\prod_{i=1}^l x_{-k_i}^{a_{i}}}{z^{a+\sum_{i=1}^l a_i}a!\prod_{i=1}^l a_i!}\frac{\prod_{b\leq 0, \langle b\rangle=\langle -\frac{1}{r}-\sum_{i=1}^l (a_i(1-\frac{k_i}{r}))\rangle} (bz) }{\prod_{b\leq -\frac{1}{r}-\sum_{i=1}^l (a_i(1-\frac{k_i}{r})), \langle b\rangle=\langle -\frac{1}{r}-\sum_{i=1}^l (a_i(1-\frac{k_i}{r}))\rangle} (bz) } [1]_{\langle a-\sum_{i=1}^l \frac{a_ik_i}{r}\rangle},
\end{align}
where $\langle b \rangle$ is the fractional part of the rational number $b$.
The orbifold mirror map, the $z^0$-coefficient of the $I$-function, is
\[
x_1[1]_{\frac{1}{r}}+\sum_{\{i_1,\ldots, i_j\}\subset \{1,\ldots, l\}} x_{-k_{i_1}}\cdots x_{-k_{i_j}} \prod_{b=0}^{j-1}\left(\frac{ k_{i_1}+\cdots+ k_{i_j}}{r}-b\right)[1]_{-\frac{k_{i_1}+\cdots+ k_{i_j}}{r}}.
\]
Since the expression of the $I$-function and the mirror map looks quite complicated for general $l$. We would like to first start with the computation for the case when $l=2$ for a better explanation of the idea.
\subsubsection{Computation for $l=2$}
When $l=2$, the $I$-function becomes
\[
z\sum \frac{x_1^{a} x_{-k_1}^{a_{1}}x_{-k_2}^{a_{2}}}{z^{a+a_1+a_2}a! a_1!a_2!}\frac{\prod_{b\leq 0, \langle b\rangle=\langle -\frac{1}{r}-\sum_{i=1}^2 a_i(1-\frac{k_i}{r}))\rangle} (bz) }{\prod_{b\leq -\frac{1}{r}-\sum_{i=1}^2 (a_i(1-\frac{k_i}{r})), \langle b\rangle=\langle -\frac{1}{r}-\sum_{i=1}^2 (a_i(1-\frac{k_i}{r}))\rangle} (bz) } [1]_{\langle a-\sum_{i=1}^l \frac{a_ik_i}{r}\rangle}.
\]
The orbifold mirror map is
\[
x_1[1]_{\frac{1}{r}}+x_{-k_1}[1]_{1-\frac{k_1}{r}}+x_{-k_2}[1]_{1-\frac{k_2}{r}}+x_{-k_1}x_{-k_2}\left(\frac{k_1+k_2}{r}-1\right)[1]_{-\frac{k_1+k_2}{r}}.
\]
We would like to take the coefficient of $x_1x_{-k_1}x_{-k_2}z^{-1}[1]_{-\frac{k_1+k_2-1}{r}}$ of the $I$-function and the $J$-function. The coefficient of the $I$-function is
\[
\frac{k_1+k_2-1}{r}-1.
\]
By the mirror theorem, this coefficient of the $I$-function coincides with the coefficient of the $J$-function:
\begin{align*}
&\langle [1]_{\frac{1}{r}}, [1]_{1-\frac{k_1}{r}}, [1]_{1-\frac{k_2}{r}}, r[1]_{\frac{k_1+k_2-1)}{r}}\rangle_{0,4}^{B \mathbb Z_r, tw}\\
+&\left(\frac{k_1+k_2}{r}-1\right)\langle [1]_{\frac{1}{r}}, [1]_{1-\frac{k_1+k_2}{r}}, r[1]_{\frac{k_1+k_2-1}{r}}\rangle_{0,3}^{B \mathbb Z_r, tw}.
\end{align*}
Note that the invariant
\[
\langle [1]_{\frac{1}{r}}, [1]_{1-\frac{k_1+k_2}{r}}, r[1]_{\frac{k_1+k_2-1}{r}}\rangle_{0,3}^{B \mathbb Z_r, tw}
\]
coincides with degree zero relative invariant with one negative contact order and the value of the invariants is $1$ by direct computation. Therefore, we have
\[
\frac{k_1+k_2-1}{r}-1=\langle [1]_{\frac{1}{r}}, [1]_{1-\frac{k_1}{r}}, [1]_{1-\frac{k_2}{r}}, r[1]_{\frac{k_1+k_2-1}{r}}\rangle_{0,4}^{B \mathbb Z_r, tw}+\left(\frac{k_1+k_2}{r}-1\right).
\]
Hence, we have
\[
r\langle [1]_{\frac{1}{r}}, [1]_{1-\frac{k_1}{r}}, [1]_{1-\frac{k_2}{r}}, [1]_{\frac{k_1+k_2-1}{r}}\rangle_{0,4}^{B \mathbb Z_r, tw}=-\frac{1}{r}.
\]
We conclude that, the degree zero relative invariant with two negative contact orders is
\begin{align*}
&\langle [1]_1,[1]_{-k_1}, [1]_{-k_2},[\on{pt}]_{k_1+k_2-1}\rangle^{(X,D)}_{0,4,0}\\
=& r^2\langle [1]_{\frac{1}{r}}, [1]_{1-\frac{k_1}{r}}, [1]_{1-\frac{k_2}{r}}, [1]_{\frac{k_1+k_2-1}{r}}\rangle_{0,4}^{B \mathbb Z_r, tw}\\
=&-1.
\end{align*}
\subsubsection{Computation for general $l$}
\begin{proof}[Proof of Proposition \ref{prop-degree-zero}]
We proceed with induction on $l$. Suppose Identity (\ref{identity-zero-several-neg}) is true for $l=N>0$. For $l=N+1$, extracting the coefficient $x_1\prod_{i=1}^{N+1} x_{-k_i}z^{-1}[1]_{-1-\frac{\sum_{i=1}^{N+1} k_i}{r}}$ of the $I$-function (\ref{I-func-BZ}), we have
\[
\prod_{b=1}^{N} \left( \frac{-1+\sum_{i=1}^{N+1} k_i }{r}-b\right).
\]
The corresponding coefficient of the $J$-function is
\begin{align*}
&\langle [1]_{\frac{1}{r}}, \prod_{i=1}^{N+1} [1]_{1-\frac{k_i}{r}}, r[1]_{\frac{-1+\sum_{i=1}^{N+1}}{r}}\rangle_{0,N+3}^{B \mathbb Z_r, tw}\\
+&\sum_{\{i_1,i_2\}\subset \{1,\ldots,N+1\}}\left(\frac{k_{i_1}+k_{i_2}}{r}-1\right)\langle [1]_{\frac 1 r}, \prod_{i\in\{1,\ldots, N+1\}\setminus \{i_1,i_2\} }[1]_{k_i}, [1]_{1-\frac{k_{i_1}+k_{i_2}}{r}}, r[1]_{\frac{-1+\sum_{i=1}^{N+1} k_i}{r}}\rangle_{0,N+2}^{B \mathbb Z_r, tw}\\
+& \cdots\\
+& \prod_{b=1}^{N} \left( \frac{\sum_{i=1}^{N+1} k_i }{r}-b\right)\langle [1]_{\frac{1}{r}}, [1]_{1-\frac{\sum_{i=1}^{N+1}k_i}{r}}, r[1]_{\frac{-1+\sum_{i=1}^{N+1} k_i}{r}}\rangle_{0,3}^{B \mathbb Z_r, tw}.
\end{align*}
We further multiply both coefficients of the $I$-function and the $J$-function by $r^{-N}$, then take the constant coefficient (which is the coefficient of the lowest power of $r$). This coefficient of the $I$-function is
\[
\left( -1+\sum_{i=1}^{N+1} k_i \right)^N.
\]
Apply the induction on $l$, the coefficient of the $J$-function is
\begin{align*}
& r^{N+1}\langle [1]_{\frac{1}{r}}, \prod_{i=1}^{N+1} [1]_{1-\frac{k_i}{r}}, [1]_{\frac{-1+\sum_{i=1}^{N+1}}{r}}\rangle_{0,N+3}^{B \mathbb Z_r, tw}\\
+& \sum_{\{i_1,i_2\}\subset \{1,\ldots,N+1\}}\left(k_{i_1}+k_{i_2}\right)(-1)^{N-1}\\
+& \sum_{\{i_1,i_2,i_3\}\subset \{1,\ldots,N+1\}}\left(k_{i_1}+k_{i_2}+k_{i_3}\right)(-1)^{N-2}\\
+& \cdots\\
+& \left( \sum_{i=1}^{N+1} k_i \right)^N.
\end{align*}
The coefficient of the $J$-function can be simplified to
\begin{align*}
& r^{N+1}\langle [1]_{\frac{1}{r}}, \prod_{i=1}^{N+1} [1]_{1-\frac{k_i}{r}}, [1]_{\frac{-1+\sum_{i=1}^{N+1}}{r}}\rangle_{0,N+3}^{B \mathbb Z_r, tw}\\
+& N\left(\sum_{i=1}^N k_i\right)(-1)^{N-1}+ {N\choose 2} \left(\sum_{i=1}^N k_i\right)(-1)^{N-2}+\cdots
+ \left( \sum_{i=1}^{N+1} k_i \right)^N.
\end{align*}
Therefore, the identity of the (coefficients of the) $I$-function and the $J$-function is
\begin{align*}
\left( -1+\sum_{i=1}^{N+1} k_i \right)^N=& r^{N+1}\langle [1]_{\frac{1}{r}}, \prod_{i=1}^{N+1} [1]_{1-\frac{k_i}{r}}, [1]_{\frac{-1+\sum_{i=1}^{N+1}k_i}{r}}\rangle_{0,N+3}^{B \mathbb Z_r, tw}\\
+& N\left(\sum_{i=1}^N k_i\right)(-1)^{N-1}+ {N\choose 2} \left(\sum_{i=1}^N k_i\right)(-1)^{N-2}+\cdots
+ \left( \sum_{i=1}^{N+1} k_i \right)^N.
\end{align*}
The binomial theorem implies that
\[
(-1)^N=r^{N+1}\langle [1]_{\frac{1}{r}}, \prod_{i=1}^{N+1} [1]_{1-\frac{k_i}{r}}, [1]_{\frac{-1+\sum_{i=1}^{N+1}}{r}}\rangle_{0,N+3}^{B \mathbb Z_r, tw}
\]
The orbifold definition of the relative Gromov--Witten invariants with negative contact orders implies Identity (\ref{identity-zero-several-neg}).
\end{proof}
\subsection{Relative mirror map}\label{sec-rel-mirror-map}
We recall that the extended relative mirror map (\ref{rel-mirror-map}) is
\begin{align*}
\tau(y,x_1)=\sum_{i=1}^r p_i\log y_i+x_1[1]_{1}+\sum_{\substack{\beta\in \on{NE}(X)\\ D\cdot \beta \geq 2}}\langle [\on{pt}]\psi^{D\cdot \beta-2}\rangle_{0,1,\beta}^Xy^\beta (D\cdot \beta-1)![1]_{-D\cdot \beta}.
\end{align*}
Let
\[
g(y)=\sum_{\substack{\beta\in \on{NE}(X)\\ D\cdot \beta \geq 2}}\langle [\on{pt}]\psi^{D\cdot \beta-2}\rangle_{0,1,\beta}^Xy^\beta (D\cdot \beta-1)!.
\]
Let $\iota: D\hookrightarrow X$ be the inclusion map and $\iota_!:H^*(D)\rightarrow H^*(X)$ be the Gysin pushforward. Recall that
\[
\mathfrak H:=\bigoplus_{i\in \mathbb Z}\mathfrak H_i,
\]
where $\mathfrak H_0=H^*(X)$ and $\mathfrak H_i=H^*(D)$ for $i\neq 0$.
We also denote $\iota_!:\mathfrak H\rightarrow H^*(X)$ for the map such that it is the identity map for the identity sector $\mathfrak H_0$ and is the Gysin pushforward for twisted sectors $\mathfrak H_i$. We first let $x_1=0$,
then
\[
\iota!J_{(X,D)}(\tau(y),z)=e^{(\sum_{i=1}^r p_i\log y_i+g(y)D)/z}\left(z+\sum_{ \beta\in \on{NE(X)}}\sum_{\alpha}y^{\beta}e^{g(y)(D\cdot \beta)}\left\langle \frac{\phi_\alpha}{z-\bar{\psi}}\right\rangle_{0,1, \beta}^{(X,D)}D\cup \phi^{\alpha}\right).
\]
This has the same effect with the change of variables
\begin{align}\label{relative-mirror-map}
\sum_{i=1}^r p_i\log q_i=\sum_{i=1}^r p_i\log y_i+g(y)D,
\end{align}
or,
\[
q^\beta=e^{g(y)D\cdot \beta}y^\beta.
\]
We will also refer to the change of variables (\ref{relative-mirror-map}) as the \emph{relative mirror map}. In particular, relative mirror map coincides with the local mirror map of $\mathcal O_X(-D)$ after a change of variables $y\mapsto -y$.
\if{
Then we write the $J$-function $J(\tau(-y),z)$ as
\[
e^{(\sum_{i=1}^r p_i\log q_i)/z}\left(z+\sum_{\substack{(\beta,l)\neq (0,0)\\ \beta\in \on{NE(X)}}}\sum_{\alpha}\frac{(-q)^{\beta}e^{-lg(-y(q))}}{l!}\left\langle \frac{\phi_\alpha}{z-\psi},[1]_1,\ldots, [1]_1\right\rangle_{0,1+l, \beta}^{X}x^l\phi^{\alpha}\right),
\]
where
\[
q^\beta=e^{g(-y)D\cdot \beta}y^\beta.
\]
}\fi
When $x_1\neq 0$, we will be able to compute invariants with more than one positive contact order. We will consider it in the following section.
\section{Theta function computation via relative mirror theorem}
\begin{theorem}\label{thm-main}
Let $X$ be a smooth projective variety with a smooth nef anticanonical divisor $D$. Let $W:=\vartheta_1$ be the mirror proper Landau--Ginzburg potential. Set $q^\beta=t^{D\cdot \beta}x^{D\cdot\beta}$. Then
\[
W=x^{-1}\exp\left(g(y(q))\right),
\]
where
\[
g(y)=\sum_{\substack{\beta\in \on{NE}(X)\\ D\cdot \beta \geq 2}}\langle [\on{pt}]\psi^{D\cdot \beta-2}\rangle_{0,1,\beta}^Xy^\beta (D\cdot \beta-1)!
\]
and $y=y(q)$ is the inverse of the relative mirror map (\ref{relative-mirror-map}).
\end{theorem}
We will prove Theorem \ref{thm-main} in this section through the mirror theorem for the smooth pair $(X,D)$ proved in \cite{FTY}.
For the purpose of the computation of the theta function:
\[
x\vartheta_1=1+\sum_{\beta: D\cdot \beta=n+1}n\langle [1]_1,[\on{pt}]_{n}\rangle_{0,\beta,2}^{(X,D)}q^\beta,
\]
we will only consider the $S$-extended $I$-function with
\[
S=\{1\}.
\]
Recall that the $S$-extended $I$-function of $(X,D)$ is defined as follows:
\[
I_{(X,D)}^{S}(y,x_1,z)=I_++I_-,
\]
where
\begin{align*}
I_+:=&\sum_{\substack{\beta\in \on{NE}(X),k\in \mathbb Z_{\geq 0}\\ k<D\cdot \beta} }J_{X, \beta}(\tau_{0,2},z)y^{\beta}\frac{ x_1^{k}}{z^{k}k!}\frac{\prod_{0<a\leq D\cdot \beta}(D+az)}{D+(D\cdot \beta-k)z}[{1}]_{-D\cdot \beta+k},
\end{align*}
and
\begin{align*}
I_-:=&\sum_{\substack{\beta\in \on{NE}(X),k\in \mathbb Z_{\geq 0}\\ k\geq D\cdot \beta} }J_{X, \beta}(\tau_{0,2},z)y^{\beta}\frac{ x_1^{k}}{z^{k}k!}\left(\prod_{0<a\leq D\cdot \beta}(D+az)\right)[{ 1}]_{-D\cdot \beta+k}.
\end{align*}
\subsection{Extracting the coefficient of the $J$-function}
We consider the $J$-function
\[
J(\tau(y,x_1),z),
\]
where
\[
J_{(X,D)}(\tau,z)=z+\tau+\sum_{\substack{(\beta,l)\neq (0,0), (0,1)\\ \beta\in \on{NE(X)}}}\sum_{\alpha}\frac{q^{\beta}}{l!}\left\langle \frac{\phi_\alpha}{z-\bar{\psi}},\tau,\ldots, \tau\right\rangle_{0,1+l, \beta}^{(X,D)}\phi^{\alpha},
\]
and
\begin{align*}
\tau(y,x_1)=\sum_{i=1}^r p_i\log y_i+x_1[1]_{1}+\sum_{\substack{\beta\in \on{NE}(X)\\ D\cdot \beta \geq 2}}\langle [\on{pt}]\psi^{D\cdot \beta-2}\rangle_{0,1,\beta}^Xy^\beta (D\cdot \beta-1)![1]_{-D\cdot \beta}.
\end{align*}
\if{
\[
e^{(\sum_{i=1}^r p_i\log q_i)/z}\left(z+x[1]_1+\sum_{\substack{(\beta,l)\neq (0,0), (0,1)\\ \beta\in \on{NE(X)}}}\sum_{\alpha}\frac{q^{\beta}e^{-lg(y(q))}}{l!}\left\langle \frac{\phi_\alpha}{z-\psi},[1]_1,\ldots, [1]_1\right\rangle_{0,1+l, \beta}^{X}x^l\phi^{\alpha}\right).
\]
}\fi
The sum over the coefficient of $x_1z^{-1}$ of $J(\tau(y,x_1),z)$ that takes value in $[1]_{-n}$, for $n\geq 1$, is the following
\begin{align}\label{coeff-J}
[J(\tau(y,x_1),z)]_{x_1z^{-1}}=& \sum_{\beta: D\cdot \beta\geq 1,n\geq 1}\langle [1]_1,\tau(y),\cdots, \tau(y),[\on{pt}]_{n}\rangle_{0,\beta,k+2}^{(X,D)}q^\beta \\
\notag & +\sum_{\beta: D\cdot \beta=0}\sum_{n\geq 1, k>0}\langle[1]_1,\tau(y),\cdots,\tau(y),[\on{pt}]_n \rangle_{0,\beta,k+2}^{(X,D)}.
\end{align}
By Proposition \ref{prop-several-neg-2}, we have
\begin{align}\label{identity-deg-geq-1}
& \sum_{\beta: D\cdot \beta \geq 1, n\geq 1}\langle [1]_1,\tau(y),\cdots, \tau(y),[\on{pt}]_{n}\rangle_{0,\beta,k+2}^{(X,D)}q^\beta\\
\notag = &\exp\left(-g(y)\right)\sum_{\beta: D\cdot \beta=n+1,n\geq 1}\langle [1]_1,[\on{pt}]_{n}\rangle_{0,\beta,2}^{(X,D)}q^\beta,
\end{align}
where
\[
g(y)=\sum_{\substack{\beta\in \on{NE}(X)\\ D\cdot \beta \geq 2}}\langle [\on{pt}]\psi^{D\cdot \beta-2}\rangle_{0,1,\beta}^Xy^\beta (D\cdot \beta-1)!
\]
and
\[
q^\beta=e^{g(y)D\cdot \beta}y^\beta.
\]
When $D\cdot \beta=0$, the invariants are studied in Section \ref{sec-deg-0}. As mentioned at the beginning of Section \ref{sec-deg-0}, we need to have $\beta=0$. The degree zero invariants are computed in Proposition \ref{prop-degree-zero}. Therefore, we have
\begin{align}\label{identity-deg-0}
&\sum_{\beta: D\cdot \beta=0}\sum_{n\geq 1, k>0}\langle[1]_1,\tau(y),\cdots,\tau(y),[\on{pt}]_n \rangle_{0,\beta,k+2}^{(X,D)}\\
\notag =& g(y)+\sum_{l\geq 2} g(y)^l(-1)^{l-1}\\
\notag =& -\exp\left(-g(y)\right)+1.
\end{align}
Therefore, (\ref{identity-deg-geq-1}) and (\ref{identity-deg-0}) imply that (\ref{coeff-J}) is
\begin{align}\label{coeff-J-1}
& [J(\tau(y,x_1),z)]_{x_1z^{-1}}\\
\notag =& \exp\left(-g(y)\right)\sum_{\beta: D\cdot \beta=n+1,n\geq 1}\langle [1]_1,[\on{pt}]_{n}\rangle_{0,\beta,2}^{(X,D)}q^\beta -\exp\left(-g(y)\right)+1.
\end{align}
Note that (\ref{coeff-J-1}) is not exactly the generating function of relative invariants in the theta function $\vartheta_1$. We want to compute $\sum_{\beta: D\cdot \beta=n+1,n\geq 1}n\langle [1]_1,[\on{pt}]_{n}\rangle_{0,\beta,2}^{(X,D)}q^\beta$ instead of $\sum_{\beta: D\cdot \beta=n+1,n\geq 1}\langle [1]_1,[\on{pt}]_{n}\rangle_{0,\beta,2}^{(X,D)}q^\beta$.
Write
\[
D=\sum_{i=1}^r m_ip_i
\]
for some $m_i\in \mathbb Z_{\geq 0}$.
In order to compute $\vartheta_1$, we apply the operator $\Delta_D=\sum_{i=1}^r m_iy_i\frac{\partial}{\partial y_i}-1$ to the $J$-function $J(\tau(y,x_1),z)$. Then (\ref{coeff-J-1}) becomes
\begin{align}\label{coeff-J-der}
& \left(-\sum_{i=1}^r m_iy_i\frac{\partial (g(y))}{\partial y_i}\right)\exp\left(-g(y)\right)\sum_{\beta: D\cdot \beta=n+1}\langle [1]_1,[\on{pt}]_{n}\rangle_{0,\beta,2}^{(X,D)}q^\beta \\
\notag +& \exp\left(-g(y)\right)\sum_{i=1}^r m_iy_i\sum_{j=1}^r\frac{\partial q_j}{\partial y_i}\frac{\partial}{\partial q_j}\sum_{\beta: D\cdot \beta=n+1}\langle [1]_1,[\on{pt}]_{n}\rangle_{0,\beta,2}^{(X,D)}q^\beta \\
\notag -& \exp\left(-g(y)\right)\sum_{\beta: D\cdot \beta=n+1}\langle [1]_1,[\on{pt}]_{n}\rangle_{0,\beta,2}^{(X,D)}q^\beta+ \left(1+\sum_{i=1}^r m_iy_i\frac{\partial (g(-y))}{\partial y_i}\right)\exp(-g(y))-1.
\end{align}
We compute the partial derivatives
\[
\frac{\partial q_j}{\partial y_i}=\left\{
\begin{array}{cc}
y_je^{m_jg(y)}m_j\frac{\partial g(y)}{\partial y_i} & j\neq i; \\
e^{m_j g(y)}+y_je^{m_jg(y)}m_j\frac{\partial g(y)}{\partial y_j} & j=i.
\end{array}
\right.
\]
Therefore,
\begin{align*}
&\sum_{i=1}^r m_iy_i\sum_{j=1}^r\frac{\partial q_j}{\partial y_i}\frac{\partial}{\partial q_j}\\
=&\sum_{j=1}^r \left(m_jy_j\left(e^{m_j g(y)}+y_je^{m_jg(y)}m_j\frac{\partial g(y)}{\partial y_j}\right)+m_iy_i\sum_{j\neq i} y_je^{m_jg(y)}m_j\frac{\partial g(y)}{\partial y_i} \right)\frac{\partial}{\partial q_j}\\
=&\sum_{j=1}^r\left(1+\sum_{i=1}^r m_iy_i\frac{\partial g(y)}{\partial y_i}\right)m_jq_j\frac{\partial}{\partial q_j}\\
=&\left(1+\sum_{i=1}^r m_iy_i\frac{\partial g(y)}{\partial y_i}\right)\sum_{j=1}^rm_jq_j\frac{\partial}{\partial q_j}.
\end{align*}
Hence, (\ref{coeff-J-der}) is
\begin{align*}
&\left(-\sum_{i=1}^r m_iy_i\frac{\partial (g(y))}{\partial y_i}\right)\exp\left(-g(y)\right)\sum_{\beta: D\cdot \beta=n+1}\langle [1]_1,[\on{pt}]_{n}\rangle_{0,\beta,2}^{(X,D)}q^\beta \\
& +\exp\left(-g(y)\right)\left(1+\sum_{i=1}^r m_iy_i\frac{\partial g(y)}{\partial y_i}\right)\sum_{j=1}^rm_jq_j\frac{\partial}{\partial q_j}\sum_{\beta: D\cdot \beta=n+1}\langle [1]_1,[\on{pt}]_{n}\rangle_{0,\beta,2}^{(X,D)}q^\beta \\
& -\exp\left(-g(y)\right)\sum_{\beta: D\cdot \beta=n+1}\langle [1]_1,[\on{pt}]_{n}\rangle_{0,\beta,2}^{(X,D)}q^\beta + \left(1+\sum_{i=1}^r m_iy_i\frac{\partial (g(-y))}{\partial y_i}\right)\exp(-g(y))-1\\
= &\left(-1-\sum_{i=1}^r m_iy_i\frac{\partial (g(y))}{\partial y_i}\right)\exp\left(-g(y)\right)\left(\sum_{\beta: D\cdot \beta=n+1}\langle [1]_1,[\on{pt}]_{n}\rangle_{0,\beta,2}^{(X,D)}q^\beta \right)\\
& +\exp\left(-g(y)\right)\left(1+\sum_{i=1}^r m_iy_i\frac{\partial g(y)}{\partial y_i}\right)\left(\sum_{j=1}^r\sum_{\beta: D\cdot \beta=n+1}(n+1)\langle [1]_1,[\on{pt}]_{n}\rangle_{0,\beta,2}^{(X,D)}q^\beta +1\right)-1\\
=& \left(1+\sum_{i=1}^r m_iy_i\frac{\partial (g(y))}{\partial y_i}\right)\exp\left(-g(y)\right)\left(\sum_{\beta: D\cdot \beta=n+1}n\langle [1]_1,[\on{pt}]_{n}\rangle_{0,\beta,2}^{(X,D)}q^\beta +1\right)-1.
\end{align*}
\subsection{Extracting the coefficient of the $I$-function}
Recall that, when $\beta=0$, we have
\[
J_{X, 0}(\tau_{0,2},z)=z.
\]
When $\beta\neq 0$, we have
\[
J_{X, \beta}(\tau_{0,2},z)=e^{\tau_{0,2}/z}\sum_\alpha\left\langle\psi^{m-2}\phi_\alpha\right\rangle_{0,1,\beta}^Xy^\beta\phi^\alpha\left(\frac{1}{z}\right)^{m-1}
\]
and
\[
m=\dim_{\mathbb C} X+D\cdot \beta-\deg (\phi_\alpha)\geq D\cdot\beta.
\]
The $I$-function can be expanded as
\[
I=z+x_1[1]_1 +\tau_{0,2}+\sum_{\substack{\beta\in \on{NE}(X)\\ D\cdot \beta \geq 2}}\langle [\on{pt}]\psi^{D\cdot \beta-2}\rangle_{0,1,\beta}^Xy^\beta (D\cdot \beta-1)![1]_{-D\cdot \beta}+\sum_{k=1}^{\infty}I_{-k}z^{-k}.
\]
We would like to sum over the coefficient of $x_1z^{-1}$ of the $I$-function that takes value in $[1]_{-n}$ for $n\geq 1$. By direct computation, the sum of the coefficients is
\begin{align}\label{coeff-I}
[I(y,z)]_{x_1z^{-1}}=\sum_{\beta: D\cdot \beta =n+1, n\geq 1}\langle [\on{pt}]\psi^{n-1}\rangle_{0,1,\beta}^Xy^\beta \frac{(n+1)!}{n}.
\end{align}
We apply the operator
\[
\Delta_D=\sum_{i=1}^r m_iy_i\frac{\partial}{\partial y_i}-1
\]
to the $I$-function $I(y,z)$. Then (\ref{coeff-I}) becomes
\[
\sum_{\beta: D\cdot \beta =n+1, n\geq 1}\langle [\on{pt}]\psi^{n-1}\rangle_{0,1,\beta}^X(y)^\beta (n+1)!.
\]
\subsection{Matching}
The relative mirror theorem of \cite{FTY} states that the coefficients of the $I$-function and the $J$-function are the same. Therefore, we have
\begin{align}\label{identity-coeff-I-J}
& \sum_{\beta: D\cdot \beta =n+1, n\geq 1}\langle [\on{pt}]\psi^{n-1}\rangle_{0,1,\beta}^X(y)^\beta (n+1)!\\
\notag = & \left(1+\sum_{i=1}^r m_iy_i\frac{\partial (g(y))}{\partial y_i}\right)\exp\left(-g(y)\right)\left(\sum_{\beta: D\cdot \beta=n+1}n\langle [1]_1,[\on{pt}]_{n}\rangle_{0,\beta,2}^{(X,D)}q^\beta +1\right)-1.
\end{align}
Recall that
\[
g(y)=\sum_{\substack{\beta\in \on{NE}(X)\\ D\cdot \beta \geq 2}}\langle [\on{pt}]\psi^{D\cdot \beta-2}\rangle_{0,1,\beta}^Xy^\beta (D\cdot \beta-1)!.
\]
Therefore
\begin{align*}
&\sum_{i=1}^r m_iy_i\frac{\partial (g(y))}{\partial y_i}\\
=&\sum_{\substack{\beta\in \on{NE}(X)\\ D\cdot \beta \geq 2}}\langle [\on{pt}]\psi^{D\cdot \beta-2}\rangle_{0,1,\beta}^Xy^\beta (D\cdot \beta)!\\
=& \sum_{\beta: D\cdot \beta =n+1, n\geq 1}\langle [\on{pt}]\psi^{n-1}\rangle_{0,1,\beta}^X(y)^\beta (n+1)!
\end{align*}
we conclude that (\ref{identity-coeff-I-J}) becomes
\begin{align*}
& 1+ \sum_{\beta: D\cdot \beta =n+1, n\geq 1}\langle [\on{pt}]\psi^{n-1}\rangle_{0,1,\beta}^X(y)^\beta (n+1)!\\
\notag = & \left(1+\sum_{\beta: D\cdot \beta =n+1, n\geq 1}\langle [\on{pt}]\psi^{n-1}\rangle_{0,1,\beta}^X(y)^\beta (n+1)!\right)\exp\left(-g(y)\right)\left(\sum_{\beta: D\cdot \beta=n+1}n\langle [1]_1,[\on{pt}]_{n}\rangle_{0,\beta,2}^{(X,D)}q^\beta +1\right).
\end{align*}
Therefore
\[
1=\exp\left(-g(y)\right)\left(\sum_{\beta: D\cdot \beta=n+1}n\langle [1]_1,[\on{pt}]_{n}\rangle_{0,\beta,2}^{(X,D)}q^\beta +1\right).
\]
We have
\begin{align}
1+\sum_{\beta: D\cdot \beta=n+1}n\langle [1]_1,[\on{pt}]_{n}\rangle_{0,\beta,2}^{(X,D)}q^\beta=\exp\left(g(y(q))\right),
\end{align}
where $y=y(q)$ is the inverse mirror map. This concludes Theorem \ref{thm-main}.
\section{Toric varieties and the open mirror map}
In this section, We will specialize our result to the toric case. The proper Landau--Ginzburg potential can be computed explicitly.
Let $X$ be a toric variety with a smooth, nef anticanonical divisor $D$.
Recall that the small $J$-function for absolute Gromov--Witten theory of $X$ is
\[
J_{X}(z)=e^{\sum_{i=1}^r p_i\log q_i/z}\left(z+\sum_{\substack{(\beta,l)\neq (0,0), (0,1)\\ \beta\in \on{NE(X)}}}\sum_{\alpha}\frac{q^{\beta}}{l!}\left\langle \frac{\phi_\alpha}{z-\psi}\right\rangle_{0,1, \beta}^{X}\phi^{\alpha}\right),
\]
where $\tau_{0,2}=\sum_{i=1}^r p_i \log q_i\in H^2(X)$; $\{\phi_\alpha\}$ is a basis of $H^*(X)$; $\{\phi^\alpha\}$ is the dual basis under the Poincar\'e pairing.
By \cite{Givental98}, the $I$-function for a toric variety $X$ is
\[
I_{X}(y,z)
= ze^{t/z}\sum_{\beta\in \on{NE}(X)}y^{\beta}\left(\prod_{i=1}^{m}\frac{\prod_{a\leq 0}(\bar{D}_i+az)}{\prod_{a \leq D_i\cdot \beta}(\bar{D}_i+az)}\right),
\]
where $t=\sum_{a=1}^r \bar p_a \log y_a$, $y^\beta=y_1^{p_1\cdot \beta}\cdots y_r^{p_r\cdot \beta}$ and $\bar{D}_i$'s are toric divisors.
The $I$-function can be expanded as
\[
z+\sum_{a=1}^r \bar p_a \log y_a+\sum_{j}\bar{D}_j\sum_{\substack{c_1(X)\cdot \beta=0,D_j\cdot \beta<0\\ D_i\cdot \beta\geq 0, \forall i\neq j}} \frac{(D_j\cdot \beta-1)!}{\prod_{i=1, i\neq j}^m (D_i\cdot \beta)!}y^\beta+O(z^{-1}).
\]
The $J$-function and the $I$-function are related by the following change of variables, called the (absolute) mirror map:
\[
\sum_{i=1}^r p_i\log q_i=\sum_{i=1}^r p_i\log y_i+\sum_{j}\bar{D}_j\sum_{\substack{c_1(X)\cdot \beta=0,D_j\cdot \beta<0\\ D_i\cdot \beta\geq 0, \forall i\neq j}} \frac{(D_j\cdot \beta-1)!}{\prod_{i=1, i\neq j}^m (D_i\cdot \beta)!}y^\beta
\]
Set $z=1$, the coefficient of the $1\in H^0(X)$ of the $J$-function is
\[
\sum_{\substack{\beta\in \on{NE}(X)\\ D\cdot \beta \geq 2}}\langle [\on{pt}]\psi^{D\cdot \beta-2}\rangle_{0,1,\beta}^X
\]
The corresponding coefficient of the $I$-function is
\begin{align*}
&\sum_{D_i\cdot \beta\geq 0, \forall i} \frac{1}{\prod_{i=1}^m (D_i\cdot \beta)!}y^\beta\\
=&\sum_{\substack{\beta\in \on{NE}(X)\\ D\cdot \beta \geq 2}} \frac{1}{\prod_{i=1}^m (D_i\cdot \beta)!}y^\beta.
\end{align*}
\subsection{Toric Fano varieties}
When $X$ is Fano, the absolute mirror map is trivial. Then we have
\[
g(y)=\sum_{D_i\cdot \beta\geq 0, \forall i} \frac{ (D\cdot \beta-1)!}{\prod_{i=1}^m (D_i\cdot \beta)!}y^\beta
.
\]
Then Theorem \ref{thm-main} specialize to
\[
W=\exp\left(\sum_{D_i\cdot \beta\geq 0, \forall i} \frac{ (D\cdot \beta-1)!}{\prod_{i=1}^m (D_i\cdot \beta)!}y(q)^\beta
\right),
\]
where $y(q)$ is the inverse to the relative mirror map
\begin{align*}
\sum_{i=1}^r p_i\log q_i=\sum_{i=1}^r p_i\log y_i+g(y)D.
\end{align*}
If we further specialize the result to dimension $2$ case, we recover the main result of \cite{GRZ}.
\subsection{Toric varieties with a smooth, nef anticanonical divisor}\label{sec-toric-semi-Fano}
Theorem \ref{thm-main} specializes to
\[
W=\exp\left(\sum_{D_i\cdot \beta\geq 0, \forall i} \frac{ (D\cdot \beta-1)!}{\prod_{i=1}^m (D_i\cdot \beta)!}y(q)^\beta
\right),
\]
where $y(q)$ is the inverse to the relative mirror map
\begin{align*}
\sum_{i=1}^r p_i\log q_i=\sum_{i=1}^r p_i\log y_i+g(y)D + \sum_{j}\bar{D}_j\sum_{\substack{c_1(X)\cdot \beta=0,D_j\cdot \beta<0\\ D_i\cdot \beta\geq 0, \forall i\neq j}} \frac{(D_j\cdot \beta-1)!}{\prod_{i=1, i\neq j}^m (D_i\cdot \beta)!}y^\beta.
\end{align*}
Let $X$ be a Fano variety with a smooth anticanonical divisor $D$. In \cite{GRZ}, the authors proposed that the mirror proper Landau--Ginzburg potential is the open mirror map of the local Calabi--Yau $\mathcal O_X(-D)$. When $X$ is toric, the open mirror map for the toric Calabi--Yau $\mathcal O_X(-D)$ has been computed in \cite{CCLT} and \cite{CLT} (see also \cite{You20} for the computation in terms of relative Gromov--Witten invariants).
The SYZ mirror construction for a toric Calabi--Yau manifold $Y$ was constructed in \cite{CCLT} and \cite{CLT}. We specialize to the case when $Y=\mathcal O_X(-D)$ where $D$ is a smooth, nef, anticanonical divisor of $X$. Note that, we do not need to assume $X$ is Fano.
The SYZ mirror of $Y$ is modified by the instanton corrections. Following \cite{Auroux07} and \cite{Auroux09}, the SYZ mirror of a toric Calabi--Yau manifold was constructed in \cite{CCLT} and \cite{CLT} where the instanton corrections are given by genus zero open Gromov--Witten invariants. These open Gromov--Witten invariants are virtual counts of holomorphic disks in $Y$ bounded by fibers of the Gross fibration. It was shown in \cite{CCLT} and \cite{CLT} that the generating function of these open invariants is the inverse of the mirror map for $Y$. This generating function of these open invariants is referred to as the open mirror map in \cite{GRZ}.
Comparing the open mirror map of \cite{CCLT} and \cite{CLT} with our relative mirror map, we directly have
\begin{theorem}\label{thm-toric-open}
Let $(X,D)$ be a smooth log Calabi--Yau pair, such that $X$ is toric and $D$ is nef. The proper Landau--Ginzburg potential of $(X,D)$ is the open mirror map of the local Calabi--Yau manifold $\mathcal O_X(-D)$.
\end{theorem}
\subsection{Beyond the toric case}
Beyond the toric setting, the conjecture of \cite{GRZ} is also expected to be true as long as we assume the following principal (open-closed duality) in mirror symmetry.
\begin{conjecture}\label{conj-open-closed}
The instanton corrections of a local Calabi--Yau manifold $\mathcal O_X(-D)$ is the inverse mirror map of the local mirror theorem that relates local Gromov--Witten invariants with periods.
\end{conjecture}
Open Gromov--Witten invariants have not been defined in the general setting. Moreover, open Gromov--Witten invariants are more difficult to compute. On the other hand, the local mirror map can usually be computed. As we have seen in Section \ref{sec-rel-mirror-map}, the local mirror map and the relative mirror map coincide. Therefore, we have the following result.
\begin{corollary}
Assuming Conjecture \ref{conj-open-closed}, the proper Landau--Ginzburg potential is the open mirror map.
\end{corollary}
\section{Fano varieties and quantum periods}
For a Fano variety $X$, the function $g(y)$ is closely related to the quantum period of $X$. In fact, we have
\begin{theorem}\label{thm-quantum-period}
The function $g(y)$ coincides with the anti-derivative of the regularized quantum period.
\end{theorem}
\begin{proof}
We recall that
\[
g(y)=\sum_{\substack{\beta\in \on{NE}(X)\\ D\cdot \beta \geq 2}}\langle [\on{pt}]\psi^{D\cdot \beta-2}\rangle_{0,1,\beta}^Xy^\beta (D\cdot \beta-1)!.
\]
We consider the change of variables
\[
y^\beta=t^{D\cdot \beta}.
\]
Then the derivative $\frac{d}{dt}$ of $g(t)$ is
\[
\sum_{\substack{\beta\in \on{NE}(X)\\ D\cdot \beta \geq 2}}\langle [\on{pt}]\psi^{D\cdot \beta-2}\rangle_{0,1,\beta}^Xt^{D\cdot\beta} (D\cdot \beta)!,
\]
which is precisely the regularized quantum period in \cite{CCGGK}.
\end{proof}
Following the Fanosearch program, the regularized quantum period of a Fano variety coincides with the classical period of its mirror Laurent polynomial. A version of the relation between the quantum period and the classical period was obtained in \cite{TY20b} using the formal orbifold invariants of infinite root stacks \cite{TY20c}. Combining with the Fanosearch program, one can explicitly compute the proper Landau--Ginzburg potential of a Fano variety as long as one knows its mirror Lanurent polynomial. In particular, we have found explicit expressions of the proper Landau--Ginzburg potentials for all Fano threefolds using the expression of the quantum periods in \cite{CCGK}.
\begin{example}
We consider the Fano threefold $V_{10}$ in \cite{CCGK}*{Section 12}. It is a Fano threefold with Picard rank 1, Fano index 1, and degree $10$. It can be considered as a complete intersection in the Grassmannian $\on{Gr}(2,5)$. Following \cite{CCGK}, the quantum period is
\[
G_{V_{10}}(y)=e^{-6y}\sum_{l=0}^\infty\sum_{m=0}^\infty (-1)^{l+m}y^{l+m}\frac{((l+m)!)^2(2l+2m)!}{(l!)^5(m!)^5}(1-5(m-l)H_m),
\]
where $H_m$ is the $m$-th harmonic number.
Therefore,
\[
g_{V_{10}}(y)=e^{-6y}\sum_{l=0}^\infty\sum_{m=0}^\infty (-1)^{l+m}y^{l+m}\frac{((l+m)!)^2(2l+2m)!}{(l!)^5(m!)^5}(1-5(m-l)H_m)(l+m-1)!
\]
The proper Landau--Ginzburg potential is
\[
W=x^{-1}\exp\left(g_{V_{10}}(y(tx))\right),
\]
where $y(tx)$ is the inverse of
\[
tx=y\exp\left(g_{V_{10}}(y)\right).
\]
\end{example}
Similarly, one can compute the proper Landau--Ginzburg potential for all Fano threefold using the quantum period in \cite{CCGK}. Moreover, there are large databases \cite{CK22} of quantum periods for Fano manifolds which can be used to compute the proper Landau--Ginzburg potential.
\begin{remark}
We noticed that H. Ruddat \cite{Ruddat} has been working on the relation between the proper Landau--Ginzburg potential and the classical period. This can also be seen from Theorem \ref{thm-quantum-period} because it is expected from mirror symmetry that the regularized quantum period of a Fano variety equals the classical period of the mirror Laurent polynomial. The Laurent polynomials are considered as the potential for the weak, non-proper, Landau--Ginzburg model of \cite{Prz07}, \cite{Prz13}. Therefore, Theorem \ref{thm-quantum-period} provides an explicit relation between the proper and non-proper Landau--Ginzburg potentials.
\end{remark}
\bibliographystyle{amsxport}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,803 |
Last Friday, January 25th, Mayor McDaniel and I had the opportunity to visit AM General. AM General opened their Research & Development facility in Auburn Hills in October 2017, and they continue to grow in that location.
As a major defense industry company, they represent the diversity of the Auburn Hills' economy in a segment that we certainly welcome and encourage growth. Friday's meeting was a mix of industry people including those who are Officers in the military and government representatives who attended on behalf of their elected Senator or Congressperson, among many others. There was a great deal stressed on the importance of our elected officials, particularly at the federal level, to tout Michigan as a great state to invest in the defense industry.
As we all know, Michigan provides many reasons for companies to locate and expand here, not the least of which is due to its talented and educated workforce. Auburn Hills stands ready to welcome other defense industry businesses who might want to follow the lead of industry leaders such as, AM General. Thank you to AM General's President and CEO Andy Hove, for welcoming us back to your place of business to learn more about the significance of the defense industry. We certainly appreciate you being here in Auburn Hills. | {
"redpajama_set_name": "RedPajamaC4"
} | 131 |
Home / Athearn Genesis: EMD SD70M / HO Scale: EMD SD70M - DCC & Sound - Southern Pacific
HO Scale: EMD SD70M - DCC & Sound - Southern Pacific
Road Number 9803 9811 9820 9817
Athearn Genesis HO Scale: EMD SD70M - DCC & Sound - Southern Pacific
SP FEATURES:
The last new EMD units delivered to the SP, these 25 SD70Ms were built in 1994.
Early Cab with nose door
Whip and can-type antennas
P3 horn
Large snowplow
Spare knuckles on trucks
Appropriately-colored MU cables
4900 gallon reinforced fuel tank
Battery charging receptacle
Illuminated front ditch lights
Continuing upon the successful SD60-series, in 1992 EMD debuted the next step in locomotive evolution with the SD70-series. While outwardly similar at first glance to the SD60M, the D.C.-drive SD70M featured several external design refinements from the predecessor model. Battery boxes were relocated to the left-hand side walkway immediately behind the cab, a large, boxy forward traction motor blower housing replaced the angular version used on SD60s, the raised walkway duct on the left hand walkway was eliminated, and an intake for the rear traction motor blower on the left hand side of the carbody, directly under the rear radiator intake grill, was added.
Internally, the SD70 boasted improvements as well; a 16-710GB prime mover, rated at 4,000hp, was coupled to a new alternator design, the AR20. New D70TR traction motors were standard, and controlling all of this power and locomotive function was EMD's new EM2000 microprocessor, which boasted more memory, twice the processing speed, and improved locomotive self-diagnostic capabilities compared to the processor suite used in the SD60. Even more revolutionary was the inclusion of EMD's patented "Radial" truck design, the HTC-R. This design, which made its debut under EMD Demo SD60 #3, replaced the venerable HT-C truck, and is unique in its ability to shift, or "steer", the wheelsets laterally through curves, resulting in greatly reduced wheelset and track wear, and coupled with the new D70TR traction motors and EM2000 microprocessor, greatly improved adhesion.
Continuing with previous practice, EMD built a set of Demonstrator SD70Ms, EMD 7000-7002, all equipped with the North American safety-cab (hence the "M" in their model designation), and decked out in an attractive gray, silver, and burgundy paint scheme. These units travelled all over North America, showing off the latest technology from EMD. Southern Pacific, Union Pacific and eventually Norfolk Southern would sign up for the SD70M models. Although NS started out with the "Spartan" cab SD70, they eventually returned for the SD70M model. By 2004 the SD70M began to be delivered with the familiar "flared" radiator grilles. Both NS and UP have examples of these locomotives on their roster, and you can still find the entire fleet of SD70Ms in regular service today.
HO Scale: EMD SD70M - DCC Ready - Norfolk Southern 'Grey Ghost'
HO Scale: EMD SD70M - DCC Ready - Norfolk Southern 'Flare w/ PTC'
HO Scale: EMD SD70M - DCC Ready - Union Pacific 'ex SP w/PTC' | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,963 |
FAMILIAR
ALSO BY J. ROBERT LENNON
The Great Zombini (e-book, illustrated by Lou Beach)
Castle
Pieces for the Left Hand: 100 Anecdotes
Happyland
Mailman
On the Night Plain
The Funnies
The Light of Falling Stars
FAMILIAR
A Novel
J. Robert Lennon
Graywolf Press
Copyright © 2012 by J. Robert Lennon
This publication is made possible in part by a grant provided by the Minnesota State Arts Board, through an appropriation by the Minnesota State Legislature from the Minnesota general fund and its arts and cultural heritage fund with money from the vote of the people of Minnesota on November 4, 2008, and a grant from the Wells Fargo Foundation Minnesota. Significant support has also been provided by the National Endowment for the Arts; Target; the McKnight Foundation; and other generous contributions from foundations, corporations, and individuals. To these organizations and individuals we offer our heartfelt thanks.
Published by Graywolf Press
250 Third Avenue North, Suite 600
Minneapolis, Minnesota 55401
All rights reserved.
www.graywolfpress.org
Published in the United States of America
ISBN 978-1-55597-625-5
ISBN 978-1-55597-061-1
2 4 6 8 9 7 5 3 1
First Graywolf Printing, 2012
Library of Congress Control Number: 2012936386
Cover design: Kyle G. Hunter
Cover photo: Lauri Rotko/Folio Images/Getty Images
After the death of his mother, he had spent five years in the house of his brother. It was not from what he said but from the way he said it that his enormous animosity toward the domineering, cold, and unfriendly nature of his brother became evident.
Then, in short, not very pregnant sentences, he related that he had a friend now who very much loved and admired him. Following this communication, there was a prolonged silence. A few days later he reported a dream: he saw himself in a strange city with his friend, except that the face of his friend was different.
—Wilhelm Reich, Character Analysis
PART ONE
1.
She's driving. A Monday morning in July, hot outside, the windows of her Honda are down and the highway air is rushing in. It's the sixth hour of a daylong trip from the town where her dead son is buried to the town where she lives now with her husband and living son.
Her name is Elisa Macalaster Brown. She makes this trip once a year: drives to Wisconsin, stays in a motel, drinks coffee and reads magazines in the places she used to go, when both boys were alive and they all lived there together. She visits the grave, never for long. She watches television in her room. She remembers people from when she lived there, and if she sees them she says hello, but she never seeks them out. They know why she's there and they don't want to talk about it. When she's finished she gets back into her car and goes home, and that's what she's doing now.
Interstate 90 is a dull gray strap laid over brown land. There's a drought here and everything is dead. Somewhere in Ohio. Other cars can be seen far ahead and far behind, but nobody passes anybody else. Soon she'll have to stop for lunch, but for now she is content being hungry.
She likes to drive this car—she's had it for a dozen years. Silas and Sam used to ride in back, on the way to soccer practice or music lessons, in the days when they could be persuaded to care about such things. There was a dog for a while, Derek's dog from before they married, but it died. The car still smells a little like dog, but all traces of Silas are gone. Sam borrows the car sometimes, filling the ashtray with his cigarette butts and leaving them there for Elisa to clean up. No, that's not fair—it has nothing to do with her. He just forgets. He's twenty-five, a year older than his brother would be.
But for now Elisa is alone. Solitude is something she learned how to love, and now she loves it better than almost anything. She loves it at home, in her filthy little studio at the back of the house, with her radio and paints, where she goes on her day off when Derek is at work. But she is most alone in a car, on a trip, this trip, with the windows down and the wind and highway in her ears. Or in a storm, windows shut, defogger roaring, the rain thudding against the metal and glass. She doesn't listen to the radio. It's too easy to get invested in something, only to hear it fade away. Not that there's much on the radio to capture her interest. She doesn't like the news. Politics are meaningless to her. She didn't care who won the 2008 election—this disgusted people she knew. But they forgave her because her son was dead and they figured that had something to do with it.
She believes that Silas's death makes people feel superior to her. They decided at some point that it was probably her fault. In any event, her days of worrying about other people's opinions of her are long over.
Everything's going to change in a couple of minutes.
2.
The Honda has a crack in the windshield that runs from the lower left-hand corner to a spot at eye level on the passenger side. She is looking at the crack, or rather through the crack, to the white line on the roadside that it parallels. This is a habit of hers that she is now indulging, keeping the crack aligned with the line.
When she was a teenager and her father took her out to the suburbs to teach her to drive, he recommended that, in order to stay on the road, she should keep the hood ornament aligned in her vision with the white line. It was one of four stations, as he put it, for the eye to visit: side mirror, rearview mirror, speedometer, hood ornament. Even the shortest journey should consist of a constant cycling among the four. He was methodical that way. When her mother drank or grew depressed, and Elisa ended up in tears in her bedroom with the radio turned up, her father's knock was never far behind; he would sit at the foot of her bed, ostentatiously not touching her, and offer tried and tested methods for "dealing with" her mother: never look directly into her eyes when she is in "a state"; never tell her that she reminds you of her mother; always offer audible feedback to indicate you're listening; frequently use the phrase "of course you're right." As though the woman were a problem to be solved.
Which in some ways she was. Which in some ways Elisa is, as she is painfully aware. Derek, to his credit, doesn't try. He lets her wend her unthinking way through her life, making only the subtlest suggestions, usually coiled up inside the phrase "You're a scientist, you figure it out." A reminder of what she gave up. An unasked question of why.
The problem, of course, with her father's driving method is that it leaves out actually looking where you're going. In truth it is not necessary to align anything with anything, in order to drive. You just focus on a point in the distance and move toward it. Here on the highway, Elisa feels proud to have overcome this limitation in her technique; gazing through the crack is just a tic. But outside the car, in her life, this has long been, and remains, a problem. There is only what lies directly in front of her, and she is afraid to look away from it. She is resourceful; tough, she hopes; able to cope with whatever she finds immediately before her. But she doesn't want to look up and see the future.
A shrink told her that once, about herself. Facile nonsense, she told Derek. Derek nodded in the way he nodded when he thought she was wrong.
She is feeling the feeling of being hungry and thinking about what food she will settle for, when she finally decides it's time to eat. A spring inside the driver's seat is worrying at her back. One cloud is in the sky, its bottom edge just visible in the tinted area running along the top of the windshield. Sunlight has slashed the dashboard in two and she can see her fingerprints in the dust, from where she braced herself several days ago, at a rest stop on the outbound leg of the trip, while she was picking some money up off the passenger-side floor.
Everything is so clear and vivid, she is almost moved.
3.
She married at twenty-one. Derek was in law school—he was twenty-five, wore a tie; he took her out and drank one bottle of beer and they had sex in his room. He was consistent in everything, the things he said, the ways he touched her: it flattered her, that he should care enough to do the things he knew would please her. She took him home to Chicago, and her parents embarrassed her, with their sagging bookcases and thick eyeglasses and Oriental rugs worn through to the boards. Growing up, she had been smuggled into jazz clubs, she read fantasy novels in the back row of Marxist lectures, inhaled pot smoke on her way to bed. Blacks and gays came to the house and she sat on their knees. Cops broke up her parents' parties. Karl Popper once took her to a Cubs game.
But Derek managed to impress them, laughing at their jokes, pretending to sympathize with their politics. When they told him he was welcome to sleep with Elisa in her room he said no thanks, the couch was fine. During the night she went to him.
"Sorry," she whispered.
"They're all right."
"They're not."
"They love you."
"They love themselves."
Derek gazed at her levelly. "We aren't any different," he said, and she said, "Fuck you, Derek!" just loud enough to be sure her parents would hear.
This was a part of him she professed to dislike—the thing they would fight about, when they fought. The way he would challenge her, reflexively, as though, because she was younger, because she was a woman, because she was less advanced in her studies, because her parents were bohemians, she lacked rigor, lacked the ability to see herself clearly. She fought against his patronizing tone, shouted at him, cursed, pushed him hard in the chest with both hands. His amusement at her rage enraged her further still.
But of course this was what attracted them, too. She liked his immovability, his physical solidity and unswerving beliefs, the way she could fling herself against him until she was exhausted. He liked her volatility, her war against the genetic and cultural heritage she ultimately had no hope of overcoming (and perhaps, at least at times, secretly loved).
And so why, when they got married and she got pregnant, did she give in? She'd been working on a master's in plant biology at UW, in Madison. Derek told her she didn't have to quit, she shouldn't, but she surprised herself by wanting to. And as soon as she did, she felt a profound relief. She drank coffee (until her growing body developed an aversion) and read, and decided that she was going to love motherhood. She made a careful study of parenting books and tallied up their shortcomings in her head even before she'd reached her third trimester; they seemed to her like bad science. When both births went smoothly, when she appeared happy, Derek professed to be pleased. If he missed her old intensity, he didn't say so. In any event, the only advice her parents could give her now was parenting advice, and they wouldn't dare. They were proud of their inexpertise. "The shittiest parents in Chicago," Elisa's mother used to boast to her friends. Her parents approved wholeheartedly of her early marriage, her abandonment of her studies, seeming to delight in having raised someone so different from themselves, someone who turned out to be conventional after all. Or perhaps Elisa was happy, and they saw this, and it pleased them? Surely it couldn't be that simple?
No, of course not. But by now it was time to stop caring about what they thought or why they thought it. She had her own life to live. Her own children to confuse.
She raised the boys, talked on the phone with friends, listened to the morning therapist on the radio and laughed at people's problems. Derek always worked late but it was all right. He didn't have anybody on the side. He was just busy. He specialized in maritime law as it applied to the Great Lakes. When she began to feel that perhaps she had thrown her youth away, she got a part-time job as a technician at the lab where she had once interned, before she met Derek. It was all right. She kept doing it. They spent most of a decade like this, turning into Wisconsinites. She was happy enough with Derek, with her lab. This is what she told herself.
Years later, when Derek accepted the job at SUNY Reevesport, Elisa considered not going with him. Keeping Sam and staying in Madison. Or even giving up Sam for a little while, staying there alone. Derek didn't understand. He had actually been offered the job the previous year—it was what they wanted then, a change, maybe a way to break the pattern of Silas's bad behavior. But then Silas died, and he had to turn it down. Now the college had called and offered it to him again—the other guy hadn't worked out. He hounded Elisa until she told him—this came to her without forethought—that it was Silas's grave, she didn't want to leave Silas's grave.
"That's it?" he said.
It wasn't, but she said, "Yes."
"But that's ridiculous."
"Leaving our son behind?"
"But you're willing to leave our living son behind."
She said nothing.
"A grave can be moved," he said.
He was right—it embarrassed her not to have thought of that.
"That's not what I want," she said.
"Then what do you want?"
This was the question Derek could always use to end an argument, because it was the question she could never seem to answer. Even when she did know what she wanted, she often found it difficult to articulate. Where to go, what to do. Should we spend this money on the house, or on a vacation? Do you want to go back to school? Do you want to go out to eat? Should Silas's friends, the ones who led him wrong, be invited to the funeral, or barred from it? Does Sam need counseling, do we all need counseling? What do you want to do for the rest of your life? What have you ever wanted to do?
The only thing she always knew she wanted was love, was Derek, was her boys, but then even that went wrong, and she didn't want anything at all. And so she began to feel as though there was no want, there was no you.
What do you want?
They moved to Reevesport together, and the grave stayed behind, an unresolved conflict between them. An artifact. It is still there, and that's why she's on the road today, five years later, July 23, 2012. That's why she has her job waiting for her in Reevesport, in another lab, managing it this time. That's why she has her art studio and her love of solitude, and an affair she's been having with a man in the town, the man who runs the frame shop. That's why she's where she is now, poised to apprehend the inexplicable thing that's about to happen.
4.
Her initial method of coping was to ignore everything. To obliterate her usual mode of attention. She didn't let her eyes or mind rest. She kept moving, doing things of little or no importance. This was in Madison. She didn't answer the phone or return messages. She threw most mail away. Derek paid the bills, talked to Elisa's mother on the phone, took Sam to the shrink. She got thinner and no longer drank for pleasure, or for any reason at all. She stopped having sex, while Derek carried on, semi-openly and without evident pleasure, with a colleague. She rarely spoke and didn't look people in the eye when she did. She did not feel alive.
Then, about a year in, she made a discovery. If she managed to focus on something very small, she could enter into a state of deep concentration that her body registered as satisfaction. Embroidery. She had a kit: a long-forgotten gift from Derek's mother that she had almost thrown out. Now she spent three months doing almost nothing else. Her eyes were red and tired from incessant close focus. Her fingertips were callused from pricking. She would finish a project—flowers, ducks, it didn't matter—set it aside, pick up another. This is what she did while they prepared for the move, while Derek packed boxes, marked boxes, hauled boxes.
As she worked she would talk. In her head, she thought. The first time was with the memory of a college roommate, a sullen, lachrymose girl named Naomi with flat damp eyes and a long chin, who had stayed in Elisa's dorm for only a semester before transferring. She had worn flannel pajamas printed with a pattern of ivy not unlike the one Elisa was embroidering now. The two of them were not friends. They argued over petty things—water in the soap dish, moldy food. These were the kinds of fights they had now, in Elisa's head, but with a wild and violent intensity: the two of them facing off in the narrow vestibule of their dorm room, shoving each other against the walls, screaming and spitting in each other's faces. There was something deeply satisfying and fascinating about these fantasies; daily, Elisa took up her needlework with excitement and dread, like a junkie.
At some point the arguments with Naomi left the hallway and moved to the bedroom, where, in predawn light, Elisa could make out the ivy pajamas discarded on the floor, and in the bed, her roommate's thick stubbled ankles crossed over a man's tight and clenching ass: Derek, her husband, fucking the horsey Naomi!
The women shoved and shouted while Derek cowered, half-covered by the bedclothes. Elisa could feel the rough synthetic carpet underneath her feet, she could smell deodorant and sweaty bras and the funk of sex. You don't go to bed without washing your fucking dinner dishes! And your fucking hair in the shower drain! And that's my husband, get your hands off my husband! Before dismissing Naomi, poor shy unhappy Naomi, from her thoughts, Elisa got her libido back, made Derek screw her, told him to give up the woman from work, You're mine, she told him, you're mine.
In the days that followed, in her head, she found herself in a pseudointellectual standoff with an old philosophy professor; embroiled in a religious debate with an Orthodox rabbi, the father of an old friend; in the checkout aisle of the supermarket, demanding to see the manager.
And then Silas, the nonexistent adult Silas, in the hospital waiting area, the two of them separated by twenty feet of linoleum, shouting at one another, their fingers gripping the soft and soiled arms of their chairs. She could smell the stale coffee in the little café beyond the reception desk; she could see nurses hurrying by, white moths at the edges of her vision. This was the hospital where they were told to go after the accident; this is where the doctor told them he was dead. Indeed, in this fantasy, the doctor was waiting just offstage, invisible but present; Elisa warded him off with an outstretched hand.
Here, Silas was tall—he was still growing when he died—and stooped, around twenty-five, though he seemed to have aged further. Sullen, as in life, but sadder now. Angry, but a new kind of anger. Calculated, precise. She was afraid of him, not afraid of what he might do, but that whatever he said, it might be right.
You always thought you were so smart! You thought you were smarter than everyone else! But look at you now!
He raised his face to reply, and his mouth moved and sounds came out of it. They frightened her, though they had not yet coalesced into sense. Any second now, they might.
What are you doing here? she shouted, Why did you make us come here? (Though Derek wasn't there in this fantasy, only Elisa; she needed Silas to think his father was present to back her up, though it wasn't clear why this was important.)
He was pointing at her now, as he spoke, pointing first at Elisa and then down the hall, where his own dead body lay, and his eyes were blazing and his lips were white with spit. He stood up, accusing her of something, she didn't know what. The twenty feet between them seemed to collapse into ten, then five.
There is nothing else I could have done, she was saying, I would have had to become someone else.
And now he was right there, she could feel the heat from his enraged face and she understood him now, That's what I wanted! he shouted, That's what I wanted!
(And all the while she poked and pulled at the embroidery hoop, a toad seated on a mushroom, blades of grass springing up on either side, I HEAR IT'S SPRING! was the caption, and her lips moved, muttered sounds came out, and later Sam would say he paused outside the open door calling Mom, Mom! and she didn't answer.)
She stops for gas and food at the first place she sees when she can no longer stand not eating. It's a sprawling rest stop, with separate parking lots for cars and big rigs, and multiple pavilions under which gas can be pumped. She chooses one, fills her tank with the wind blowing her hair into her face, then goes inside and eats something so bland and generic she isn't even aware of what it is while she's eating it. She is reminded of every other highway journey she has undertaken, every other undistinguished meal. She thinks they stopped here when they moved from Wisconsin—here or someplace exactly like it. They sat in a silent cluster, chewing their food and staring into space.
They never did find a rhythm, the three of them. A way for them to fit together without Silas. They would get along, life would be peaceful in their new home, but they had no sense of purpose. The house in Reevesport was quiet; it had more rooms, for fewer people. One of them was a studio for Elisa in which, presumably, she would find something to do. She loaded all her embroidery into it, sat there for a few days, then threw it all away.
A few weeks later, Sam said to her, "That stuff was fucking you up. You should paint." This was a habit that Sam had begun, gradually and then with increasing confidence, to indulge: treating his mother like a casual acquaintance, another teenager. He swore. He let her see his cigarettes, spilling out of the military-surplus canvas satchel on which he had Sharpied an anarchy symbol. This is something his brother might once have mocked. Sam was seventeen then, but at times seemed much younger.
Elisa said, "I don't paint."
"So?" he said. He himself was into drawing. He was trying to start a band. "It doesn't matter," he said to her. "Just paint."
She did what he said. A few months later, it was almost all she did. For a long time she tried to paint particular subjects—she bought some watercolors and an easel and went out in nature and made terrible pictures of things. But she preferred to be in the studio. So she switched to still lifes. Then she took up acrylics because the watercolors were so thin as to seem to consist of nothing at all. She liked the acrylics so much, the rich pastes in their metal tubes, that she stopped caring about what she was painting and just put the paint onto the canvas right out of the tube. Then she gave up on canvas and just slathered the paint around using a palette knife on squares of plywood she cut to size in the shed. When she broke the palette knife trying to open a frozen door lock, she switched to a heavy-duty paint scraper from the hardware store.
She did not talk to herself or to anyone else, real or imaginary, while she was painting. At first it was an effort not to. This was a thing her mind craved. But she silenced the voices and tried to appreciate the integrity of the materials before her, and as the voices went away so did her subject matter, until she was left with pure form. She tried to make her mind mirror the paintings, to render the slurry of memory and impulse as colored fields, complementary blanknesses connected by line and hue. This became the new project, doomed to failure. Which is part of why she liked it.
All of this took a year. She was working again, at the new lab. She worked, and made these paint-covered squares, in perfect contentment. The paintings weren't art. They weren't for other people. They were just a thing she was doing.
But Derek told her she should go out and find a gallery to hang them in. She believed that he was growing weary of her cadaverous presence in the house, her sexual appetite (which had not been quelled by its regular satisfaction), her newly rediscovered intensity and sobriety that appeared, once and for all, to have nothing of substance behind it. So she took a couple of the most obscure-looking paintings out to get framed, and the man at the frame shop looked at them and said, "They don't need much. They don't really need frames at all, do they?" A few weeks after that, she went to bed with him.
She gets back on the road, and again she is driving, and the windows are down and the air is rushing in. This stretch of highway is both different and exactly the same. They made it to be consistent. So that wherever you go it looks like the American highway. And the highways of Ohio, she has often thought, are the precise average of all the other highways in America. When people say "the open highway" they are thinking of Wyoming, Colorado, northern California, but if you are driving a car in America, chances are you're someplace that looks a lot like this. Elisa is surprised that more people don't fall asleep and crash.
Two cars and a pickup are approaching from the east. They are still a great distance away. Behind her, equally distant, is something that looks like a minivan. Far out in front of her is a white sports car that entered the highway ten minutes ago and accelerated quickly away. She expected it would continue to outrace her, to disappear, but it slowed down and is still visible.
The guardrail beyond the white line is gently bowed, as though a giant paused there for a rest. Underneath it grows an unfamiliar weed, some kind of fern, and beside the fern lies a perfect undented aluminum can.
5.
The crack in the windshield disappears. She tries to blink it back into place because at first she thinks that her vision has blurred, but blinking doesn't bring it back, and now she is noticing other things. The sound inside the car has changed. It's quiet. The window is closed. The window's closed and the air-conditioning is on, the dashboard isn't dusty anymore, and the taste of mint gum is in her mouth. In fact the gum is there, she has gum in her mouth right now. She pushes it out with her tongue and it falls into her lap.
The gum lands not on her cutoff jeans, but on a gray cotton skirt draped over a pair of stockings. These aren't her clothes—she doesn't have clothes like these. She's wearing an ivory silk blouse and there's a sticker on the blouse that reads HELLO! MY NAME IS, then in her own block printing, ELISA MACALASTER BROWN.
She notices that the spring in the seat is no longer bothering her, and that she is wearing an uncomfortable bra.
Elisa looks up at the road. Only a second, less than a second, has passed, and the road has grown. It's wider, the sky is taller. And it's cloudy now, partly cloudy, many small clouds, as though the single cloud has spawned. No—it isn't the road that's wider, it's the windshield, the windshield is larger.
She glances around her, at the interior of the car, and it isn't her car.
She signals and pulls over. The shoulder here is wide, and she comes to a stop as close to the guardrail as she dares. Another car passes her from behind, startling her, because it wasn't there before. She shifts into park and leaves it running: there are her hands on the wheel, her familiar hands. One of them reaches into her lap and picks up the gum; the other reaches for the window crank. But this car has power windows. Okay. The switch, then. The window rolls down and out goes the gum.
After a moment, she opens the door and gets out herself. When she tries to stand up she nearly falls over. It's her shoes. She's wearing pumps with low heels. Not sneakers. Right. She slips off the shoes and stands on the hot pavement in her stockings. Stockings! In the summer! But of course the car is air-conditioned, why not?
She turns and gets a good look at this car. It's American. A Dodge Intrepid, sort of copper-colored.
An eighteen-wheeler passes, roiling the air, and she squints. Maybe I should find a doctor, she thinks.
Instead she gets back into the idling car, closes the window, and sits very still for a few minutes. She peels the name tag off her blouse and folds it over, onto itself, and sets it in the drink holder. (Drink holder!) Beside her, on the passenger seat, is a thick plastic three-ring binder. There's a label stuck to it, bearing her name—laser-printed, this time—under the title 6TH ANNUAL CONFERENCE OF ACADEMIC ADMINISTRATORS. She opens it. A schedule, presentation agendas, handouts. New networked software applications offer statistics-driven space management models for research and pedagogical purposes. Has she attended this? It doesn't seem possible. She thumbs through the pages until she arrives at the plastic pocket in the back of the binder. There are several printed-out e-mails, hotel reservations, maps. An envelope marked RECEIPTS in her own handwriting. She chooses an e-mail at random. It consists of an exchange between, apparently, herself and a conference organizer. Supported by a small forest of carets is a boilerplate sig line: Elisa Macalaster Brown, Graduate Studies Coordinator, Levinson Biotech Center, SUNY Reevesport. And a phone number.
This is not her job.
There's a handbag on the passenger side floor, and she reaches over and picks it up. The movement is familiar to her, she performed it just the other day, picking up that change. But it feels more effortful now, for reasons that are not yet clear. The bag is familiar too—it's her bag. Inside it is a cell phone, not one she recognizes, but no matter—the bag interior looks right, it contains the same mass of scribbled notes and receipts and ticket stubs and dead ballpoint pens she is accustomed to. Okay then. She turns on the phone and there's a photo of Derek on the screen. It looks recent—he is groomed and relaxed, smiling in the shade of a tree.
She hesitates for a moment, and in that moment feels the world trembling, as though it might implode. An involuntary gasp. Have to do something. She casts her gaze around the interior of the car, settles on the printed e-mail, picks it up. She opens the phone and calls the number in her own sig line.
She expects voicemail. But: "Levinson Center."
"Ah... hello?"
There's a pause. Then, "Lisa?"
She laughs. She's got to admit, it's funny! She laughs and the voice on the other end laughs and says "Are you at the conference?" and Elisa says "Yes, no, I mean I'm on the road, sorry, I meant to call Derek" and the voice laughs again, says "See you tomorrow."
Elisa ends the call.
If there's a time to panic, this is it, while she's alone. She takes several deep breaths and rests her head on the wheel. All right, she thinks, this is very unusual, this is frightening. Or it should be frightening. But she isn't afraid, not really. Instead, she is intensely aware.
She is reminded of times in her life when everything felt different, all the time. When the small changes in her social circles, her patterns of thought, the texture of her emotions, would register as tectonic shifts, altering utterly the landscape of her life. College, grad school. The early days with Derek. Often she would stop what she was doing, close her eyes, take stock, and it would feel as though her life of just a week before, even of the previous day, was thoroughly, inalienably over, and that everything was starting again, beginning with now. And there were times when she would apprehend the impending extinction of the present moment. Out walking around the lake, or in bio lab, the sweat beading on her forehead and trickling down into her safety glasses. She would think, this moment was just born, and soon it will be gone. She would meet it, fall in love with it, mourn it, all at once. She cried more in those days. Almost daily, and often in public: silently, unobtrusively. The brutal immediacy of thoughts and emotions. This is what she has forgotten. It is here now. Her throat is tight and her jaw trembles.
Something must be wrong with her. Yet she doesn't feel dizzy or light-headed. She only feels different. Her body is different. Jesus, this bra.
She lets out breath. Leans back in the seat and unbuttons her blouse. Quickly strips it off, unclasps the bra and removes that too. She tosses the bra onto the open binder and puts the blouse back on. Then, after a moment's thought, she lifts the blouse and looks at her stomach. It is definitely different. Fatter. She's what, ten pounds heavier? Fifteen? Suddenly she can feel her thighs chafing.
But I'm thin, she thinks. I walk to work.
To her lab, that is. Maybe she doesn't walk to the Levinson Biotech Center at SUNY Reevesport. Maybe she drives this car, not her Honda. Maybe she picks up a box of doughnuts on the way.
This is fucked up.
From inside her bag comes the sound of electronically rendered mariachi music.
She reaches in and takes out the phone. There's the picture of Derek again, and the screen tells her it's him who's calling. She isn't sure if she wants to hear his voice right now. But perhaps something is wrong. Maybe it's Sam, something's wrong with Sam. Why would he call her when he knew she was driving? He doesn't ever call her. If he needs her, he just waits for her to appear.
The music is awful. She answers.
"Hi, it's me! You got on the road okay?"
She doesn't speak.
"Lisa?"
"I'm here. Yes, I'm fine, I'm on the road."
"How far are you?"
"I don't know. I'm in Ohio."
After a silence, he says, "Are you all right?"
"Sure." He sounds different. So must she.
"You're sure?"
"I'll be home soon."
"Should I have dinner ready?"
Dinner. This isn't something he does.
"Yes, okay."
"Will do," he says.
"Okay."
"Love you."
"Okay," she says. She draws another breath, then hangs up. She sits and stares at nothing for a moment then figures out how to turn the phone off entirely. Hands trembling, she drops it back into the bag.
Now she is frightened. That voice that both is and is not Derek's. The presence of love where there is supposed to be none, or at least a different kind: habitual, practical, inert. A bulwark. Whereas the voice—even in the early days, when he whispered to her in the narrow apartment bed, when their love was a force field around them, buzzing like a short circuit in the energy of the world, it was not like that. Derek was never sweet. This wasn't a thing she thought she ever wanted. Her father was sweet. No, Elisa wanted a serious man.
The Derek who just called her was not anyone she recognized. He didn't sound like her husband impersonating somebody else. He sounded like somebody else.
There is something reassuring, isn't there, about the absence of love. This is what she has often told herself. The only real marriage is the marriage of the body and the mind. Until death do us part: a romantic lie. People can indeed be parted. Love can end, and the body and mind soldier on. To pick up the phone and find that love is gone, that's something a person can understand. That's a thing that happens. To pick up the phone and find that love is here, where it doesn't belong: well.
She has felt this before. The imminence of something enormous and terrible, bearing down. Not knowing. The last time, she was not ready. She should have been, of course; all the signs were there, she should have seen it coming. She might have turned and faced it, stood in its path and stared it down. Instead she let it crush her.
Another big rig roars past. The car shakes. She grips the wheel with both hands and lowers her head to it. All the air seems to be leaving the car. She thinks of the last thing she saw before everything changed, the soda can at the side of the road, and she sees it imploding, some invisible force crumpling it, it folds and twists in on itself, a death rattle escapes it. The steering wheel presses its fake leather texture into her forehead and blood rushes to the spot, and it is the heat from the blood and from the friction of her skin against the plastic that keeps her steady as the moment passes, the car inhales, the air around her cools.
Elisa sits very still, opening and closing her fingers on the wheel, breathing deeply. She feels a cloud pass over the sun, then disappear; her vision behind her closed eyes goes black and then red. After an interval she sits up.
She could look through her bag right now. It would tell her things, no doubt. Maybe it would prepare her better for her arrival at home. She considers, then decides to check one thing only. She feels around in the bag for her wallet, which is as she remembers, and takes a look at her driver's license. It isn't the same: her photo is unfamiliar. But the address is the same. She closes the wallet, shoves it into the bag, and pushes the bag onto the floor.
A minute later she's back on the highway, again heading home. At least she knows where it is.
6.
There was a time, back in Madison, after the accident, when she thought she was going to get a job, and she decided that she ought to get a picture of the boys to hang in her office.
This wasn't like her. She wasn't sentimental. But now she needed to be reminded of what she had lost and how she had failed. And the people she worked with, at this imaginary job, they would pity and coddle her, and she wanted to show them that she didn't need their pity, that she could take it, having this reminder there in the office with her.
As it happened, she couldn't take it after all. And she wouldn't work again until Reevesport. But she didn't know that yet.
Derek's mother was staying with them—she had come out from North Carolina for the funeral and lingered to help out around the house. As though Elisa, incapacitated by grief, would no longer be able to clean up after herself. This, anyway, was how Elisa chose to see it, Lorraine as an interloper, in collusion with Derek against her. Even from the depths of her misery, she understood that this wasn't the case, that Silas's death had changed her, that order had collapsed in their small house on Gorham Street and Derek feared for their marriage and for their life together: but still, he shouldn't have invited Lorraine to stay. They had lodged her in Derek's study, and she emerged mornings and walked slowly from room to room, coffee mug in hand, alert for signs of disorderliness. She often sat on the sofa with her arm around Sam's shoulders, as though protecting him. At times she gazed into Elisa's eyes and shook her head sadly, demonstrating her sympathy.
Elisa disliked Lorraine. She had always considered this dislike to be a reasonable response to Lorraine's dislike of her, but Derek had long maintained that Lorraine liked her just fine, or as much as it was possible for her to like any lover of Derek's. Derek's father had died young, Lorraine never remarried, and their attachment seemed unhealthy to Elisa. Crypto-sexual. For Derek's part, he shrugged and smiled when Elisa criticized his mother, as though the very thought of the woman charmed him. Of course this was exasperating, but on the other hand, what mother wouldn't welcome this kind of devotion?
Well: Elisa, for one. She had not been a cold mother, or an unaffectionate mother, but she had taken pains to keep a distance, a slight detachment. She had not allowed the boys to climb on her, to sit on her lap, to lie with her when she lounged reading on the sofa or on her bed. She stopped breastfeeding each after one year (Sam, of course, lost out to his brother, who had arrived so soon, so unexpectedly; she might have nursed them both, but somehow this had seemed a kind of taboo, or perhaps a visceral reaction to Silas's fussiness at the breast). What she feared, she would one day realize, was the overwhelming power of love: that if she ever loved her boys with the same level of intensity, the same flavor of intensity, that had once defined her devotion to Derek, she might never again take pleasure in solitude, in the fact of herself, in her self-containment, the intricacies of her mind. She feared she might give herself over to them, to the boys and men, and exist for them alone.
Lorraine blamed Elisa for Silas's troubles. She was the type of person for whom someone always had to be to blame: there were no innocent mistakes, no accidents. And if Silas was troubled, and Derek infallible, then who else's fault could it possibly be? To her credit, Lorraine never spoke her mind, not to Elisa anyway. But in her posture, her demeanor, the pitying way she looked at the boys, at her own boy: it was clear how she felt. And now that Silas was dead, now that the worst that could happen had happened, Elisa had begun to agree with her. It was her fault. It was. Cutting off the nursing. Pushing him away when she was trying to read. Failing to give the second helping of dessert. Letting him cry it out in the crib. Since the funeral Elisa had become obsessed with the past, with all the wrong turns their lives together had taken. If only she hadn't shouted. If only she hadn't smacked. Wandering through the house, in those days of Lorraine's tenancy, she would slow, then stop, mouth hanging open, staring at a patch of wall, thinking. Seeing herself doing this, understanding how it must look to Lorraine. And Lorraine was pleased to have been right about Elisa; her every gesture seemed to communicate triumph.
Elisa would have liked to discuss this phenomenon with Derek, but he didn't believe it to be real. He believed that his mother meant well and harbored no ill will toward Elisa at all.
In any event, and in spite of everything, she thought she would soon be going back to work, and she thought she would like to have a photo of the boys there. They rarely took photographs, they weren't that kind of family. They owned a small digital camera that now had a year's worth of pictures on it, on the memory card, that they had never transferred to a computer, or gotten printed. The camera was always lying around somewhere, in the basket of pencils in the kitchen drawer, or on the coffee table, or in somebody's bag, and every now and then one of them would pick it up and gaze at the photos on the tiny screen.
Elisa was looking for the camera. She couldn't find it. Sam and Derek didn't know where it was. She searched for twenty minutes before discovering it, at last, in Lorraine's room. Later, when Lorraine came home—she had taken Sam and Derek to the movies—and saw the camera in Elisa's hands, she said, "I see you found the camera."
"What did you do?"
"I'm sorry, dear," Lorraine said, "I don't know what you mean."
Derek was standing beside his mother, keys still in hand. They were in the kitchen, Derek staring at Elisa, who sat, still gripping the camera, at the table. Where she had been waiting for them. Sam had disappeared. Lorraine was looking at something on the floor, or perhaps just at the floor.
"They're gone," Elisa said. "The pictures."
No one spoke for a moment. Then Derek said, "What do you mean?"
"I'm talking to your mother."
Now Lorraine looked up, her face scrubbed of emotion. "I don't think so, dear. I was just looking at them."
"Not all of them," Elisa said. "The pictures of Silas. I took one of the boys together. It's not in here."
"Well, whatever is on there now, that's what was on there when I picked it up." She swept her hand over the countertop, as though wiping away dust, though there was no dust.
Elisa looked at Derek, who was staring at her with curiosity. Elisa held out the camera to him.
He sat at the table and clicked through the photos with his thumb. He said, "I'm not sure anything's missing."
"All the pictures of Silas."
"He hated having his picture taken." His eyes were still on the screen. "I don't think there were any."
"Derek," she said. "You remember the one where they're in the booth at that diner. And at the state park. And the one where he's throwing rocks into the creek."
"Did he come to the state park with us? Didn't he stay with what's-his-name, that fat kid?"
"He came to the park. You took his picture. I took his picture."
"Well," he said, and handed the camera back. "Maybe something happened to them. Some digital thing."
This, of course, was asinine. Derek wasn't an idiot. He was a lawyer, an academic. He was the most organized person she had ever met. (Later, it would be his uncharacteristic unconcern that would tell her their marriage was falling apart: his unwillingness to solve their problems, any problems, however insignificant.) He did not believe that "some digital thing" had happened—and even if he did, he would know which digital thing he was referring to. This was an act he was putting on for his mother. To signal his sympathy for her position, her reason for deleting the photos, whatever it might have been.
"Things are always malfunctioning," Lorraine said from across the room. Her back was turned, and she was making tea at the counter.
Now Elisa stared at the floor. The room was silent. She wasn't going to raise her head to look at him, not just yet. There would be an apology in his eyes. Indulge me. She didn't want to give him the satisfaction.
But so what? There was nothing here to satisfy anyone, even Lorraine. There were no points for anyone to score. Later that night she would feel this even more powerfully, as she imagined the conditions under which Lorraine would have deleted the photos: alone in the study, knowing her dead grandson was there, on the memory card, a poison pill. Maybe she was drinking. What did Elisa know? Lorraine picked up the camera in a fit of anger and grief, and deleted them. The seeds of misery, obliterated. It wasn't about Elisa at all. It was what Lorraine's body had done to quell the sadness she did not otherwise know how to express.
The photos were still on there, of course. Elisa herself understood how it worked, the subterfuge of digital storage. The data wasn't overwritten, it remained in place, there on the memory card. All that was removed was the camera's willingness to acknowledge it. Where the camera once saw the photos, it now saw holes: neat blank spaces, like graves.
Flawlessly implemented denial: this is what pressing the DELETE key accomplished, on the family camera. There were places in town where Elisa could take the card to have the data recovered, to have the photos exhumed. But, with much the same variety of inertia that would later prevent them from moving Silas's real grave to Reevesport, she held back. Something had been done, and that was that. A part of Elisa had moved on. She did not want to go back. Indeed, she was grateful, in a way, to Lorraine for eliminating the temptation to look, to prolong her agony. Or she would be grateful, lying in bed that night, once she had more time to think it over.
But not yet. Now, in the kitchen, Elisa gave in: she looked up at her husband. But he wasn't looking at her. He was watching his mother with apparent love and concern, and never did turn to face his wife.
7.
The trouble with driving is that there is nothing for the body to do. The mind sees the body moving; the landscape is rushing by outside. It believes that the body must be occupied. But the body is motionless, and the mind does not accept the experience as real.
Several hours of this state lay ahead of her: she should use them to prepare.
But how can you prepare for the unknown? For the impossible? She wants to know what to do, how to behave, but there are no precedents in her life, or any other life she has heard of, to follow. She can only think of movies. A spy picture: the agent going undercover, pretending to be somebody else, ferreting out secrets. Of course there's always a moment when the spy's cover is blown and he blasts his way out of the alternate life. Or he gets the information he needs and his mission ends. He returns home. He resumes his real life.
Elisa is, in fact, returning home right now. Home is the mission. This is real life.
Or it's science fiction. Someone was doing an experiment, and the fabric of space and time was torn. She is the unwitting victim of a top-secret military project, code-named Omega or Vanguard. Somewhere an angry man in a uniform is shouting, demanding to know what went wrong. In her quest to discover the truth, she'll go all the way to the top. And meanwhile the president picks up a red phone and, jaw tight, says Get... me... that... woman.
Or it's a psychological thriller. The heroine has amnesia. That's a real phenomenon, not just a movie trope; it happens to regular people. You forget who you are and what you're doing, and your past disappears. The people you love, the house you live in, they're familiar to you. But you're lost in your own life.
Of course, her past has not disappeared. With effort, she could tell you where she was during any week of her adult life. She remembers every moment that led up to this thing, this state of being, whatever it is. But maybe this past life she believes is hers, is part of the amnesia—a kind of dream she was having. This is her real life, the one where she was at the conference. And the other one, the one where she was skinny and wearing cutoffs and driving her old Honda, that one was imaginary.
But she knows that isn't true. Or rather, if it is, she has no intention of accepting it, so it might as well not be true.
In spite of the air-conditioning, she's hot, and feels fat. She wants her old body back, the one she woke up in this morning. It's her father's body, lean, stringy, a little stooped. It embarrassed her in high school and college, but even then it felt good to live in, like the sparse apartments she favored, the casual relationships with boys she preferred, until Derek. She was surprised and grateful that this body came back after her boys quit nursing—she looked just like them.
Now she feels like her mother, with boobs and a gut. Small boobs, small gut, but still. She's reached some critical mass—hot enough on the inside to sweat, and the cold air chilling her. Fuck it! She turns off the AC and lowers the driver's side window. The pressure inside the car changes and her ears pop. She has to open the passenger window too, to even it out. The car isn't built to be driven with the windows open. It's supposed to seal its driver off from the world.
She doesn't want to be sealed off. She would never buy a car like this, not in a million years.
In the rearview mirror her eyes are different, softer somehow, the cheeks puffier, but the face tired. Has this body been up late, worrying? Drinking? Working? Having sex? Or did it lie in bed and watch TV, just like she remembers her real body doing?
Driving, she is surprised at how calm she feels. Her hands on the wheel are still and her right foot maintains a constant speed. A moment of unease when a police siren sounds and flashing lights appear, but the cruiser is moving fast and passes before she even has a chance to react. Still, for the next ten minutes her neck and shoulders are tight and she keeps glancing in the rearview, alert for another. A part of her, ridiculously, always believes that she's the one the cops are after; she is forever in a state of readiness for them to arrive on the scene, point, seize and detain her.
Eight years ago, when the police phoned to give her the news, she could feel some primitive device taking control, creaking and groaning in her mind. There was no breakdown, no fit of grief. She just bore the extra weight. She thought she was doing this for Sam and Derek, but Derek spent all of his time smoking in the kitchen or finding reasons to go to campus, and Sam stayed in his room, and Elisa didn't care to see either one of them. Probably they didn't want to see her, either, or each other. Along with everything else they were feeling, they also felt relief, guilt. Guilt at the relief.
They worried about Sam, especially when he retreated to his room or disappeared for hours on long walks by himself. A therapist prescribed drugs he declined to take. Only after weeks had passed did Elisa realize that he had what he wanted now, he had privacy, solitude. He could live for hours, days, without anyone judging him. His life was simpler and quieter without Silas in it. He put on muscle weight and his skin took on a healthier color. He spoke less, and rarely in complaint. He wasn't depressed—he was learning how to be Sam. And Derek changed as well. He became more serious, more dedicated to his habits, if that is even possible. The world returned to him.
But Elisa wasn't sure, still isn't sure, if she ever got the world back. There have been times, over the past ten years, when she has wondered whether some essential part of herself was missing. She seemed to have lost her capacity for delight—what happened to that version of her that was moved by everything? Whose emotions were so overwhelming, debilitating?
There was a morning not so long ago when she woke up in the middle of the night, sat up in bed, and with absolute certainty knew that she was dead. She actually laughed at the force of this epiphany: it was summertime, the window was open, the sounds of insects and traffic carried through the room on the wind, and she understood that she could rise up out of the bed and fly out that window and disappear into the clouds. And then what? Dissolve into nothing: first her physical form, and then her mind, and then her... soul? This isn't a thing she believes in, the soul, but she believed in it that night. Oblivion! It was hers if she wanted it; there was no need to haunt this house, this marriage. She tried to remember how she died; all she knew was that she fell from a great height and never hit the ground. She was still falling.
Derek shifted, groaned, pulled his arms out from under his pillow. "You're dreaming, go back to sleep." His voice had the strange quality it had when he talked in his sleep: the crisp consonants, the slurred vowels. "Oh sure," she said, and she lay down and closed her eyes. Two adults, half-married, half-awake, talking to one another from their respective dreams.
Indeed, that's what it feels like to her most of the time. Their marriage is like sleepwalking: each afraid to rouse the other, for fear of what they'll see when they come to. Elisa does know some things about herself, though. She has hardened, sharpened, intensified. But also calmed. She gained a new steadiness, she likes to think, a new strength. She is proud of what she has become.
But now she seems to have become something else.
8.
As she exits the highway onto county route 31, she begins to think that perhaps all is, that all will be, well. That an explanation for what has happened is just ahead. Everything looks the same as when she left. The same dilapidated barns and rusted road signs and the abandoned farm fields overtaken by woody shrubs and goldenrod. Reevesport is a large town on the southern tip of Kineota Lake, home to the college and a hospital and a factory that makes chains. It's unassuming, and it assumes nothing now, as Elisa crosses over the Reeve Avenue bridge and into the city limits.
Here is the farm supply store, the burger joint, the Asian food market that is still open in spite of the always-empty parking lot and a nearly Asian-free local population. Here is the Walmart, the Hiway Motel, the Italian place the senator ate at once during the campaign. They live in a neighborhood overlooking the lake, but from a long way away; if you go out on the roof through the bedroom window and climb up to the top you can see it, hazy in the distance, green and rippling. She signals, turns, climbs the hill that leads her past the college, past the supermarket and horse pasture and onto their winding suburban street. It is all the same. She actually takes a quick look around the car, to make sure that it's all really there, the binder, the blouse, the power windows, to make sure she wasn't deceiving herself, back on the interstate.
The mailbox is the same and the driveway she pulls into is the same. But the house is not.
It is white, for one thing. It's supposed to be a pale yellow-gray. It had been white when they bought it, but they changed it. The rhododendrons are gone, replaced by a row of sculpted yews. Or rather the yews they tore out a few years ago are still there. The grass, to which she had always been indifferent, is healthy and trim, and the pink dogwood, the one that had seemed certain to die but then rallied and came back to life, that dogwood is gone and in its place stands a Japanese maple.
The house could be described as landscaped, well cared for. It is not an aesthetic that she particularly appreciates or feels capable of achieving through her own actions.
Only when she begins to sweat again does she realize she has been sitting in the car, with the engine turned off, for several minutes. It would seem that she has entered into a minor state of shock. It is one thing to be driving an unfamiliar car, but her house? A motion catches her eye—it's Derek, waving from the front window. She raises her hand to him. A moment later the front door opens and he stands in the frame, smiling at her, at the car.
Derek in the doorway, smiling. Elisa closes her eyes, gropes for the calm, the steel. Her mind conjures a memory, not a specific one but a palimpsest of nearly identical experiences: hauling herself up out of the lake she and her parents used to go to in the summer, somewhere in Michigan: an aluminum ladder, bolted to the dock, the rank water sluicing off her body and out of her swimsuit, and Elisa at last standing steady on the bare wood in her bare feet—goggles on her head, the rubber strap tugging at her wet hair—gazing out over the choppy surface roiled by a coming storm. Up and out of the lake, under her own power, looking back at what she has emerged from: that's the image that will give her strength. She draws, lets out, breath, takes up her purse and conference binder and holds them in her lap.
Then, with deep reluctance, she gets out. She makes her way up the walk and Derek comes out to meet her. He's wearing jeans, a clean white tee shirt, bare feet. He seems to be approaching too fast, and she flinches—but he only pauses to kiss her cheek, squeeze her elbow, and then he is past.
She watches him pop open the trunk of the car and pull from it a small suitcase, hers apparently. Which he knew would be there. She turns, climbs the front step, enters the house.
It's tidy. There's a new carpet. For years they talked about tearing up the old carpet, refinishing the pine floor, but here they have gone and gotten new carpet instead. A chair has been reupholstered. There are different things on the walls. She is startled by the sight of a studio portrait of the four of them, from when the boys were five and six, that she took down soon after Silas died. And here Derek has put it back.
"The chicken's almost done," Derek says from the doorway. "You want a glass of wine?"
"Please." And he nods and disappears.
She goes to the bedroom. It's similar, aside from a new comforter and, again, the carpet. She slips her shoes off, then the stockings, then she sits on the bed for several minutes, taking deep breaths. She says "Okay" and goes down to the kitchen. A glass is waiting for her on the table. It's the same table. But the cabinets are new, the linoleum is gone and replaced by synthetic hardwood, the refrigerator and stove are new. The glass is full of white wine. She doesn't drink white wine, but she drinks it now. Derek is busying himself in front of the stove, sawing away at a roasted chicken in a pan. He turns, smiles again, comes to her and takes her face in his hands. They kiss.
"I missed you," says her husband.
9.
She is impressed with herself, at her ability to pretend. She lets him do the talking. He talks about a man in his department; she's supposed to know who this is. He talks about world events. He looks good—if she has let herself go here, Derek has become more disciplined. He's leaner, his skin has some color. She suspects that he has gotten a gym membership—it's something he used to talk about doing but in the old life never did.
"The old life." It's only been a few hours. Look how she's adjusting! Derek is cheerful, cheerful, cheerful. The food is good, no wonder she's overweight. The same magnets are on the refrigerator and she is wondering how to tell him what has happened to her.
She gets up to load the dishwasher and he laughs at her. Come on, he says, and motions her into the living room, their wine glasses in his hand, bottle in the other. She expects a crisis—something—a confession—a discussion. Instead they sit down and he keeps talking, and she realizes that he wants to fuck her.
So this is something they do, in this life. She remembers it now, this mood, the barely suppressed laughter, the ridiculousness of their desire. Here, in this life, it has returned. It isn't that sex between them has ended in the other life, it's just that it isn't very funny, or very fun. It's more like a reminder to them that they are married. It's pleasurable and necessary and serious.
They're on the sofa and he's flirting with her. He's stroking her shoulder and her arms and she finds herself saying "Well!" She's playing along, but of course it's easy, this is her, this is Derek, even if it isn't her, it isn't Derek. This Derek is certainly attractive to her, she'll give him that. She feels bad about the extra weight. But then doesn't, not really, as he kisses her, unbuttons her blouse, takes her breasts into his hands. All of her feels a little more... luxurious. She thinks of Larry, her lover, the man from the frame shop, and she feels guilty—not as if she's betrayed Derek, but as if she's betraying Larry now. She doesn't feel that way when she has sex with Derek in the other life. She wonders if this Elisa is sleeping with the Larry of this life. Everything is becoming confused in her mind, the Elisas and Dereks and Larrys, but this Elisa is turned on by this Derek. His hand is under her skirt. They're undressing. They're making little sounds.
Suddenly Derek stands up, his shirt off, his pants unbuttoned, and holds out his hand. It's time to go to the bedroom. She follows him, letting her skirt fall onto the carpeted floor. It's quiet—the carpet makes everything quiet. Suddenly she likes the idea of carpet very much. She climbs the stairs, passing the photo of the boys as toddlers, then moves down the hallway, watching the muscles in Derek's back.
Something catches her eye then, something she saw without seeing when she came up here to take off her shoes, and her mind tells her, Don't look, just follow him down the hall. But she slows, looks, stops. There's another photo of the boys here, with Derek this time, and something is the matter with it. The photo has been taken someplace she doesn't immediately recognize, someplace outdoors. Hills and water in the background. They are all wearing windbreakers. Derek is the only one smiling, but you can tell it isn't genuine, he is under strain, setting a good example. He is looking not at the camera but at the photographer, presumably Elisa but she doesn't, still doesn't, remember this. Sam is pale, drawn; he looks like this is his first time outdoors in a while. His face is riddled with acne and his hair arcs over his head in a cowlick. His most awkward time, when most boys had begun to look like men. Eighteen? He must be eighteen or nineteen. But that doesn't seem right. Why not?
It's Silas, though, whom she is staring at now, Silas who is the problem. He's looking at the camera, directly at the camera, as if he is thinking as he does it that it's the future he's looking at, future versions of his brother, his parents, himself, people he doesn't know yet, people who might not even be born. In his eyes is the expression of calm calculation she remembers, of a sneakiness so subtle that he could not be accused of harboring it, not without the accuser looking like a paranoid, a fool. He's smirking, that at least is how she reads this expression. She can't remember where this was, she can't remember, and it is suddenly very important to her that she remember, and she reaches out a hand to steady herself against the wall as she tells herself to remember, remember, you have to remember.
"Lisa?"
Derek, I want to come down the hall, I want to make love in this strange body, to your strange body, but I have to remember now, I have to remember this impossible thing or I am going to scream, because Silas didn't live to seventeen, or even sixteen, he was dead two months after his fifteenth birthday, and this photograph cannot exist. And so I need to be wrong now, before we go to bed, I need to remember this moment.
But it's hopeless, because the moment never happened, not anywhere in her memory. And so she doesn't go to Derek, doesn't go to the bedroom, instead she stumbles into the bathroom, collapses onto the toilet seat, and bunches a bath towel into her hands and covers her face with it and screams and screams.
10.
When Derek knocks and enters she is still holding the towel in her hands. It's unclear how long she's been sitting here. She doesn't look up because she doesn't want to see him.
"Lisa?"
He's kneeling in front of her now, inserting his head into her line of sight. He says, "Tell me what's wrong."
"I'm confused," she says.
"Okay."
"I think I had a stroke."
His forehead creases and she can tell that he is more doubtful than concerned. She can't blame him. He opens his mouth but most of a second passes before he says, "Do you want to go to the hospital?"
The hospital, at this moment, sounds wonderful to her. Clean and sensible. She nods. He nods. He extracts the bunched-up towel, drops it on the floor, helps her to her feet. A few minutes later she has somehow gotten her clothes back on and so has Derek and they're in his truck, on their way to the opposite side of town. It's a long drive. This is the wrong side of town to have had a stroke on, she thinks. Downtown traffic is backed up because of a passing freight train. People are honking for no reason. Someone up ahead has a stereo cranked and the bass makes her feel nauseated. She shuts her eyes. The coins in the ashtray buzz: she thinks of Brownian motion, atoms vibrating too fast to see. Everything around her is vibrating. Her forehead is pressed against the passenger side window and Derek is holding her hand.
"Is anything tingling? Is anything numb?"
"I don't think so."
There is irritation in his silence. Surely she should know if she is tingling or numb.
She says, "I'm confused."
"About what?"
"I don't remember things."
"What don't you remember?"
She doesn't answer. She wants to tell him everything. Traffic begins to move again.
They wait in the emergency room while Derek fills out forms. It's strangely silent, among the ferns and rows of airport chairs, though the room is filled with people. There's no music, shouldn't there be music? Elisa uses one hand to massage the other. Her breathing is shallow; her body feels insubstantial. Somehow the act of coming here—the act of agreeing to come here—has lightened her. She is no less terrified, but the terror is like the wings of some small bird, beating ineffectually against her. Where is her steely resolve? The image of the lake, of pulling herself up out of the lake, is of no use; she thinks of it and she can feel the dock swaying and lurching under her feet, and the boards are slimed with algae.
They are called in to the examination area and led to one of a dozen rooms surrounding an open central zone of cubicles and workstations. It's more like an office than it should be. She is given a thin gown and asked to undress and sit on a small bed. Then they wait again. Eventually a doctor introduces himself, Dr. Mayles. He's bespectacled, nerdy, and friendly. The word socialized springs to mind. It's the learned friendliness of a man who is not naturally friendly. He is around the same age as Elisa and Derek, and she suddenly imagines the three of them in a play group at the age of four—it's as if they all grew up together, making their slow way toward this moment, this little tableau. She almost laughs.
Dr. Mayles asks about her medical history, though Derek has already written it on the clipboard the doctor now holds. He shines a penlight in her eyes, holds her chin gently and studies her face. He takes her pulse and blood pressure, taps her knee with a tiny hammer, all the while talking slowly, quietly, asking her questions. What year is it? What's five times seven? Who's the president?
2012. Thirty-five. Obama.
She tenses. What if the president's somebody else? But it seems to be the right answer.
"You have two sons," the doctor says, reading from the clipboard. "What are their names?"
And as she says the names she realizes that she doesn't want to be here, doesn't want to have had a stroke, has made a terrible mistake. She says, "I'm sorry."
"Why are you sorry?" the doctor asks.
"I'm fine. I think I'm fine actually."
Derek is standing beside the bed, both hands gripping the side rail. He is studying her. "You said you couldn't remember things," he says.
"What can't you remember, Elisa?" says the doctor.
The fluttering wings again. She feels her heartbeat growing faster and weaker. "My... job. I was on a trip—I didn't know what it was for. I don't... I saw a photo in my house. Of our sons. And I couldn't remember where it was taken. It was unfamiliar."
The doctor seems to have adopted Derek's reckoning gaze. It's contagious. He says, "When was the last time you had a CT scan?"
"Never?" she says.
"Okay. Let's make sure everything looks normal up there." Meaning, presumably, her head. "The nurse will take some blood. And then we'll get you down to radiology." He sounds disappointed. A look passes between him and Derek. But then Derek turns back to her and there's nothing on his face but sympathy and worry. He takes her hand again.
"Your job? What about it don't you remember?"
She squeezes her eyes shut.
As promised, a nurse arrives and draws blood from her arm. She doesn't look. Later she's asked to sit in a wheelchair and is taken to the room where they keep the CT scanner. It looks like a giant toilet seat. Derek is asked to remain outside. She lies down on a table and a nurse, a different one, injects a dye into her arm, not far from where the blood was taken out. They are replacing my blood with dye, she thinks. A technician instructs her to remain still, the room clears, and the machine knocks and hums as the table she lies on makes abrupt gradual movements.
This is the most relaxing part of the entire experience. She is alone here with the machine. She isn't thinking of anything at all. The machine is rocking her gently. She falls asleep.
In the end, there's nothing. Her brain looks normal. Her blood is normal—she's maybe a little anemic. The doctor prescribes eating. "And B vitamins. You could maybe pick some of those up at the supermarket." He tells Derek to keep an eye on her.
"Are you beginning to remember?" he asks her, almost as an afterthought.
She manages a smile. "It's all coming back to me now."
11.
It went this way:
He wasn't driving. The driver lived but ended up paralyzed and in a coma. The boy in the passenger seat was killed, as was Silas, who was in back with a fourth passenger.
The fourth passenger was a boy named Kevin Framus. His injuries were not serious. He was the one who told the police what happened. There was an alley behind the brick factory, on the southern shore of Lake Monona, with a stone retaining wall that ran along one side and a chain-link fence along the other. The alley was long and lightless, and the driver, a twenty-seven-year-old man named Richard Samuelson, liked to drive down it at high speed with the headlights off.
"Did he ever say why he liked this?" the prosecuting attorney asked Kevin Framus, many months later, during Samuelson's manslaughter trial, which Elisa and Derek watched, over several days, from the hard wooden benches behind the attorney's table, their backs sweating and aching, their minds exhausted, blank.
"He liked getting rushed up."
"And this meant what?"
"That's what he used to say. Like, all excited. For the adrenaline rush. We were going out looking for girls."
This man Samuelson was a high school dropout. He played the drums in local bands. Silas had learned to play bass and was in one of the bands. The other boys in the van were band members. Samuelson, it emerged, hung around with schoolboys in order to attract underage girls. He and his companions would pair up with the girls and have sex with them in the van. The van was full of candles and incense and drugs. Apparently Samuelson liked to watch the younger boys having sex with their girls, while he was having sex with his.
On this night, they never made it to the girls. A police cruiser was patrolling the factory grounds and pulled into the end of the alley.
"And why didn't Mr. Samuelson stop when he saw the police car?"
"Objection," said the defense attorney.
"Did you notice anything about Mr. Samuelson that would indicate why he didn't stop?"
"He had his eyes closed," said Kevin Framus.
"Did he always have his eyes closed when he drove down the alley?"
"Every time I was with him."
"And you never objected?"
"I figured what was the harm. I never saw anybody back there."
"Did you see the police car on the night in question?"
"No, I was in back."
"And so you didn't try to stop Mr. Samuelson."
"No. Silas did. He never came with us before."
"And so he didn't know about Mr. Samuelson's habit of driving behind the brick factory with his eyes closed."
"No. This was the first time he came. Silas wasn't... he didn't like this kind of thing. He didn't like to, like, hang out. He liked to do stuff. He was pissed at Ricky that we were just, you know, fucking around." To the judge he said, "Sorry."
"All right," the prosecutor said. "And how did Silas react when he saw the police car?"
"He started shouting at Ricky to stop."
"But Mr. Samuelson didn't stop."
"No. Silas was grabbing for Ricky when we hit. He got up and he was between the seats." He paused. "Silas didn't belong there. He was... we were idiots. Silas was all right."
"And when the van struck the police cruiser, he was thrown through the windshield."
"I guess. I didn't open my eyes until after."
"Did Silas say anything about the police cruiser? Did he specifically tell Mr. Samuelson that the cruiser was there?"
"I really don't remember. It was just screaming. I don't remember."
"Do you have any idea why Mr. Samuelson didn't listen? Why he didn't open his eyes and look?"
The defense attorney didn't object. He was staring at Kevin Framus. He seemed to want to know the answer himself.
Framus said, "I don't know. I just..." He paused, seemed to consider a moment. "Honestly? I always figured he didn't really have his eyes closed all the way. Otherwise how could he keep it straight? He never scraped the wall or whatnot. It was always straight down the middle."
"So he was just pretending to close his eyes."
"Yeah. He'd make little slits, you know. He didn't really close them."
"So you think he knew the police car was there."
The defense attorney seemed to wake up. He objected, flailing his arms in the air.
This Samuelson was the kind of person nobody knew but would pretend to have done so later, after it was all over. That he was bad news, a notorious town character. In truth he was more or less anonymous, a cipher. The police didn't have him on their radar. It was the trial that taught Madison about him, and a few weeks after it was all over they would forget entirely. The trial revealed his insistence that nothing more than white cotton briefs be worn by band members during rehearsal, the suspicious fire that claimed his childhood home and the lives of his mother and sister, his dishonorable discharge from the Army. They learned about his bust for cocaine possession.
For Elisa and Derek's part, they had met the man once. They hadn't thought well of him but agreed that the band was a good thing for Silas and that they should not interfere with his private life as long as it was legal and safe. It had been neither, but they didn't know, and now he was dead.
A few days after his trial began, Samuelson stopped breathing in the middle of the night, and then he was dead, too. The two police officers who were in the cruiser were treated for their injuries and recovered. Kevin Framus hadn't been injured. He wasn't charged with anything. Six months later his family moved away, and the family of the other dead boy moved away, then Elisa and Derek and Sam moved away, too, and that was the end of that.
It was not the kind of thing that people remembered later. In Reevesport, maybe it would have been, but Madison is a big town, and more important things soon pushed it from memory. Indeed, except for the handful of acquaintances Elisa might run into on her visits, who could be forgiven for regarding her arrival as more of a burden than a pleasure, nobody thought about it much at all. It was just a thing that happened, and it was over now.
12.
But if Silas is alive, then it never did happen.
She is lying on the bed, half-naked. Derek has covered her with a duvet and brought her a cup of tea before retreating back down the stairs. The tea is now cold and her head throbs without actually hurting. It's growing dark outside.
"Tell me when you're ready to talk," he said when he left the tea. She can't imagine that moment arriving.
The darkness deepens. She's cold. She turns a lamp on long enough to confirm that none of her usual clothes are in the closet. Of course not. Some digging does reveal her favorite nightshirt, long and blue and covered with clouds, wadded into a ball in a far corner of the shelf. She undresses, allows herself a glance at her body in the mirror, hauls the musty shirt over it. What on earth could have made her stop liking it? Who is this woman?
She goes to the door, opens it quietly, steps into the hall. She hears Derek shifting downstairs but he doesn't call out. In the bathroom she rubs her face, examines it in the mirror. She is prettier here. This Lisa takes care to pretty herself. She strips off the nightshirt and then showers, careful to wash off all the makeup, not that there was much.
Back in the hallway she pauses beside the incriminating photo. There are things she needs to see and do, to figure out, and she doesn't know how much, if any, to explain to Derek. She recalls his earlier ardent affection and feels a kind of longing. But she doesn't go downstairs.
Instead she makes her way to the door of the room they called the box room. In the old life. When they moved in, they chose their bedrooms, she and Derek and Sam, and there was one left, the one that would have been Silas's. Or, rather, it would have been Sam's, as Silas, though the younger, would have taken the larger one, if he was alive.
Or maybe not—if they gave Sam the choice, Silas could make him feel, somehow, that the larger was inferior. He could find a way to make Sam uncomfortable there, make him wish he had chosen differently. And so perhaps this room, the old life's box room, is now, in fact, Silas's.
Of course the boys are grown now. They don't live here. Or maybe they do and simply aren't home? She is alarmed not to know this. In the old life, Sam has maintained his old room more or less as it was when he was in high school. He has left his posters up, kept his old mix CDs, the model airplanes he was obsessed with, belatedly, during the years he spent puzzling over his sexuality and his uneasy relationship with her and Derek. Even today, with a job and (she suspects) a boyfriend, he still hangs out there, in evident satisfaction, when he comes over; he still smokes cigarettes sitting cross-legged on the bed beside the open window, with earbuds in.
Maybe he's there now, in a world she evidently no longer inhabits. She misses him, that reality, suddenly, painfully. She hasn't seen Sam for days—or, alternately, for a lifetime.
The box room was where they put Silas's things. Derek had packed them, quite neatly, into plain cardboard boxes, and when they arrived here they chose a room for these things, and told the movers that anything unmarked should go in this room. Over time, the boxes were opened, rummaged through, crushed into corners or piled into stacks. Sam took what he wanted, music mostly, a stereo receiver and turntable and speakers, and took them to college and then later brought them back home. Derek removed some impersonal things—a radio, a lamp, a rug—and sold them at a yard sale. But the rest is still there, in the other life, along with other unwanted items, behind the always-closed door. Whenever one of them needs something from the box room, it is like entering a tomb that has been excavated from the floor of a desert. The stillness is so complete it seems to have a physical manifestation, like a liquid form of air.
Hand on the doorknob, Elisa has a sudden memory of Silas at fourteen, in his room in Madison, music emanating from the walls, filling the house: ludicrously repetitive heavy metal songs (progressive metal, Silas would correct her), some of them more than ten minutes long, interrupted only by, alternately, the mumbling and shouting of a crazed lead singer. This is the music he did his homework to. It drove her nuts. How could anyone concentrate to that? (Derek: "Let us be glad he is doing homework.") He listened at high volume, on vinyl records, sometimes for hours on end, transferred the records to recordable CDs and listened to those in the car or on headphones while he was walking around town or, presumably, the halls of his high school.
Elisa had learned to tune it out, mostly anyway, while she was preparing dinner or trying to read, but one afternoon something about it was bothering her: the song that was playing had been playing too long, the words and chords repeating with uncanny precision. She stepped into the hall and concentrated. There: a hiccup in the progression, a missed beat. The record was skipping. It had been skipping for twenty minutes.
"Silas!" She was banging on the door, but either he couldn't hear or was ignoring her. Two singers, engaged in an incomprehensible conversation: Ah mutter-mutter-mutter. Ah mutter-mutter-mutter? Ah mutter. Ah mutter. Ah screaaaam! Ah screaaa— Ah mutter-mutter-mutter. Ah mutter-mutter-mutter? Ah mutter. Ah mutter. Ah screaaaam! Ah screaaa—
"Silas!!"
The door was flung open, the music belched out its full spectrum of loud, she could feel her teeth itch. He got right up in her face. "What!!"
"Your! Record! Is skipping!" Elisa screamed now, to match his screaming, to overwhelm the singer's.
He appeared for a moment as though he might explode in anger at her, for criticizing, for daring to interrupt. But then his face went slack, his head tilted, his eyes unfocused, and he blinked. The record skipped.
Silas almost smiled. It was funny, it really was. But something held him back. The eyes found her face, locked on, and his mouth tightened and he said, "Maybe I like it that way!" before slamming the door and, with a little scrape, lifting the music out of its closed loop.
Something there that she missed. Some opportunity. She doesn't know how she might have played it differently. Surely there was a way: they might have ended laughing.
She opens the door and turns on the light.
It's a home office. There is a computer desk and a laptop and a small radio. Derek's diploma is on the wall. The only boxes are sitting open on the floor, and they're filled with files. A bookcase is lined with law books, Derek's.
In the old life, Derek doesn't work at home. He stays late at work.
Elisa backs out, turns off the light, closes the door. She goes to the other room, Sam's room, and it isn't Sam's room, it's a guest room. It is even tidier here than the rest of the house. There's a brass bed and a caramel-colored bureau and signed prints of sentimental rural clichés: a basket, a barn, an owl. A large vase of dried flowers occupies a corner. The room bears the marks of Lorraine's taste. So—in this life, Elisa has allowed Lorraine to decorate a room in her house. She can't decide what this says about her—that she has no integrity, or that she has so much that she can allow such things without feeling insulted.
Or maybe this Elisa likes Lorraine. Maybe they're friends.
She stands in the hallway for some time, just breathing and feeling her body against the nightshirt. Her feet are cold but the rest of her is warm from the shower and from the agitation of her mind. She has to face Derek, has to explain what just happened—the bathroom, the interruption of their lovemaking, the hospital, the non-stroke—and she is trying to determine how to do it.
Eventually she walks down the stairs. Derek is there, on the sofa, and he smiles at her. He must have sneaked into their bedroom to change while she showered, because he's wearing his pajama pants and bedroom slippers. There's a beer on the table beside him and a law journal is open on his lap.
"Are you all right?"
She can hear, in his voice, the old irritation and doubt. He thinks she is hysterical, that nothing is wrong with her, that they went to the hospital for no reason. All these things may be more or less true.
"I think so."
"Haven't seen that in a while," he says. He means the nightshirt.
She sits down on the other end of the sofa and Derek closes the book and sets it on the table. He folds his hands together and faces her squarely, his eyes locked with hers. This gesture—this readiness to listen—is unfamiliar. And it occurs to her that he has to have learned it somewhere. Therapy? He has had therapy, or they have had it together.
She says, "I need you to tell me our story."
13.
He looks at her. He says, "Something happened at the conference."
"No."
"You're not the same."
"I'll try to explain. Maybe not right away. But first I need to hear everything up until now. From you."
He appears confused. "Everything since you came home?"
"No. No. Everything. Our story. Since we met."
A kind of fear ripples across his face. He sighs, looks out the window, and it appears to be an effort for him to turn his head back to her. "'Need.'"
"What?"
"It's disturbing that you are coming back here and saying you need things from me. That you're just asking for this and expect to get it."
She is eager to respond, to defend the request, but of course she can't. She gazes at her own tightly folded hands in her lap and waits.
"It's your tone, Lisa. Making this demand. This weird demand."
"I'm sorry."
He grunts, shifts his body. He doesn't like the apology, either.
But he says, "Okay. Our story. From when we met, that's what you want?"
She nods, puts her hand on his knee, withdraws it.
He begins talking. Grudgingly at first, and not without sarcasm. But then he warms to it. It's good to hear his voice. It's like it was back when they met, hearing him late at night, talking about his family, trying to tell her everything. To fill her in, so that it would seem they'd always been together. Those first months, they barely slept. Friends mocked their tired eyes, made sexual jokes, but it was mostly talking, talking, talking that kept them awake. He's telling her this now, and of course he is talking now the way he talked then, methodically, linearly, with the kind of confidence that renders editorializing as incontrovertible fact. He's good. She doesn't understand why he didn't become a trial lawyer; he could convince anyone of anything. When they were young, this was bracing for her—the antidote to her parents' equivocation and moral relativism. He said things he believed were true, as though they were true, and she accepted them. Then, in bed with him, listening, she laughed, thinking of her mother's recurring admonishment: "Accept what is offered." The trouble with that world was that nothing offered was ever any good: a ratty paperback book, marred by some anonymous fool's notations; a flavorless stew; a heavily qualified compliment ("Good work, but don't get a big head about it"). Well, she accepted what Derek had to offer. Strong opinions, a strong body, a strong will. She opened herself up completely and let him pour in whatever he liked.
He is enjoying himself now. "You were into me," he is saying. "We were into each other." And though he is deep in the past, and is not, she thinks, referring to the way things have changed, she understands that it is no longer so, that they are not into each other now. It isn't merely the inevitable passing of desperate infatuation, the settling and hardening of love, it's that they no longer dominate one another's frame of reference. They are not the most interesting thing in each other's lives.
Derek says, "You were the only woman I'd ever known who actually listened to what I was saying. The way you concentrated. Even if you didn't know what the hell I was talking about—you figured it out. You asked questions. I liked answering your questions."
He pauses, and his face flushes. Maybe because he has realized that's what he's doing now: answering her question. Or maybe because he is contemplating what he says next:
"You fucked me. Not like other girls. You wanted it and you came and got it. You didn't give a shit how that might look to me."
"It looked good to you," she says.
"It looked good to me."
Maybe too good, she thinks. He drew out the narcissist in her. The self she saw reflected in him, she came to mistake for her real self. She forgot how to desire things that weren't him. She would blame him for this, later. Unfairly. Now, recalling that state, listening to his voice, feeling bad that she ever found fault with anything, she takes his hand in both of hers. He lets her do this. He tells her about his proposal ("You will marry me, won't you?"), their wedding (family only, no church, which ought to have pleased everyone but pleased only the two of them), their first rental house, a crumbling bungalow two blocks from the Willy Street Co-op, with crooked floors, squirrels living under the roof, and mushrooms sprouting in between the bathtub and wall. Eventually that house was condemned, then collapsed on itself, folding spontaneously into the ground, a briefly famous neighborhood event.
The time his older brother Nate, drunk, groped her. She did not take it personally—it was something between Derek and him, a rivalry with deep and mysterious roots. Maybe it was about Lorraine, or about their dead father. At any rate Derek simply grabbed his brother by the back of the shirt, dragged him to the door, pushed him outside, and locked it behind him. He left Elisa to drop Nate's duffel bag and car keys out a second-story window. She had known of many such rivalries in her life—there was a time, in college, when she thrived on other people's stories of familial disharmony—but she had never heard of one that ended so suddenly or endured so completely. She has never seen Nate again, rarely even heard his name uttered—even now, retelling the story, Derek refers only to "my brother." You're dead to me. Derek would never say anything so dramatic, but he could mean it, live by it. He is a man of regulations, absolutes.
He is telling her, now, about the time he broke up a fight on the street and ended up in the hospital. (She remembers she was angry at him, and proud; frustrated that he should insert himself into someone else's life in this typically masculine fashion. But why is he reminiscing about this, of all things? Why is it important to him, to their life story?) And now he begins to tell her about the boys. (His gaze leaves her face and body and drifts to the window; his voice quiets. He does not want to discuss this. But he will do as she has asked, what he has agreed to.) Sam: the pregnancy test in the convenience store bathroom. Her craving for peanut butter cups. Lorraine's absurd objection to certain baby names: "If you call it that, I will never speak to you again." Elisa waking in the night, eight months pregnant, telling him the baby didn't feel right. The drive to the hospital, the midwife's vexation; Sam had shifted; he wasn't where he belonged. The doctor's efforts to shove him into place, the sudden gush of fluid onto the table, the emergency cesarean. And Silas, eleven months and three weeks later: unexpected, uncanny. Her obstetrician had retired in the interim and the new one, a young, nervous man, advised against a natural birth. She insisted. She knew it would be easy. It was easy, waking to labor, arriving at the hospital just after eight, birth by noon.
It is all the same so far, the story is as she remembers. But his voice is growing increasingly strained, as if he has been forced into a lie. Then he stops, and says, as if exasperated, "Does this have something to do with them? With the boys?"
She doesn't respond.
"Is something going on that you haven't told me?"
"No, no. They're—when was the last time you talked to them? The last time we talked to them. Remind me."
His eyes widen, then narrow into a scowl.
"I don't remember," he says. "Why are you asking me this?"
She doesn't know what to say. She folds her legs up under her nightshirt.
"Is it Sam?" he asks. "Have you heard something from Sam?" As if this is a plausible circumstance, as if he's been waiting for it to happen.
"No. I don't know. I don't know, that's why I'm asking."
He stares at her a long time. She says, "Please, just go on with the story."
He closes his eyes, slowly, draws breath. There is a very fine sheen on his forehead. He tells her about their place on Gorham, Elisa's abandonment of her studies. He says this carefully, as though it might hurt her to hear it. The boys' early childhood. The Montessori preschool where they ate brown rice and seaweed for snack and made toys out of sticks and leaves and feathers. Or was it Waldorf? And shouldn't she have cared enough to remember? She hated the place at first, until she met the aide, a kindly old German woman who told the children violent folktales in a barely penetrable accent while waving her fat arms over her head. Sam's fall from his high chair, his stitches, the scar he still has (why does so much of this narrative consist of injuries, accidents, fights?). The time Silas went missing: they spent an hour and a half stalking through the neighborhood, calling his name, searching the park, knocking on doors. Panic, terror.
She remembers a thought that came to her then that she has repeated to herself many times over the years and that she still doesn't fully understand: I have finally gotten what I deserve. (In the end Silas had crawled under their bed and fallen asleep.)
And now the stories focus on Silas. He begins to change. His unwillingness to take naps. His boredom. At four, he is no longer content to hear a story or to play with the toys he has. He doesn't listen to what they tell him, or rather he seems not to hear. He is impassive in the face of punishment: where his brother wails with frustration and regret, Silas tends to endure, quietly, with evident puzzlement. If Sam is absorbed in something, Silas will disrupt it. It isn't the behavior that disturbs them, but the evident lack of malice. He isn't being mean. It's as if he is conducting a social experiment. He will tear a magazine out of Elisa's hands and throw it across the room. Or he will spill his dinner on the floor, then quickly turn to Derek to see his reaction.
They learn not to react emotionally. Or to react at all. They leave the spilled dinners where they lie. They finish their own meals deliberately, silently, while Silas pounds the table in a monotonous rhythm and Sam cries and cries.
Derek hesitates now, in the telling. He says, "And of course the time..." He pauses, his expression sour, and glares at her.
"Derek, don't, you don't have to," she says, because she knows where this has been going. It all leads to this, for him. She feels, suddenly, as though she has made a major tactical error—that this, on the heels of the fake stroke, the hospital visit, has led her into a cul-de-sac she can't get out of. Surely she seems completely insane to him.
"There's the time I hit him," Derek says.
"Derek, stop, it's okay, I didn't mean—"
She squeezes his hand but he pulls it away. He says, "That night at dinner. He threw the sippy cups on the floor. And then while we were trying to clean them up he grabbed your glass and then mine—"
"Stop, stop. I know this, you don't need to do this."
"—and he smashed them on the table until one broke. And he cut himself, cut his hand," Derek says, gazing levelly at her, "and I hit him, didn't I. Open-handed across the cheek. I knocked him over."
"It's all right," she says, "stop."
"And raised bruises on his face."
"Derek," she says. "Stop, please." He is staring at her, slack-jawed. Elisa feels a deep sympathy for him: he had been looking forward to this day, the dinner, the wine, going to bed. She has fucked everything up for no clear reason. Now he shrugs. He's finished with the story.
Of course she remembers it, too. When Silas smashed the glasses, he flinched—Derek flinched, but he didn't cry out, and he didn't shout. Instead he stood up, leaned over Silas, drew his arm back. Elisa thought, Do it. And he did. A moment later Silas lay on the floor, broken glass around and beneath him, the first expression of genuine surprise they had seen on his face in months. Silas cried this time, for sure. Then Sam, then Elisa. Then Derek.
He says, "Why are you doing this."
"I'm so sorry."
"There had better be a goddamn good reason." He leans back, covers his face with his hands. "A stroke, Lisa? You didn't have a stroke."
"I thought..."
"Something happened. You met somebody."
"No, no."
"Then what." His voice muffled by his fingers.
She says, "I can't explain, not yet. There's no one else." Though there is, there's Larry.
He's waiting.
"Just tell me about one more thing. One more. From later."
It takes a moment for him to react. To understand, evidently, that she needs him to agree to hear her question. His hands slide off his face and lie limp at his sides. "All right," he says.
She has grown cold now, even in the flannel nightshirt, and she bunches her hands into fists and shoves them together into her lap. "When the boys were fifteen and sixteen," she says.
"Okay."
She hesitates a moment before saying, quietly, "Was there a crash? In a van?"
She expects him to gape, shake his head in disappointment, walk out of the room. Or, Of course there was, he nearly died, why are you doing this? But instead he only blinks. He is bewildered.
"A crash?"
"Yes."
"I don't... no. No? I don't remember a crash."
"Silas was in a band, a rock band with a guy named Ricky. He was older, he was in his twenties. And that boy Kevin."
"Uh huh," he says, but he doesn't remember. And then he does. "Oh," he says. "They were in a crash. They were playing chicken or something?"
"Yes!"
"And they died, some people died."
"Yes, yes. And Silas?"
"Silas? There was a funeral, we tried to get him to go. He wouldn't go."
She says, "He had nothing to do with the crash? He wasn't in the van?"
He coughs out a little laugh. "No!"
"Nothing happened then. To Silas. Around that time."
"Well," Derek says after a moment, "there was his lost weekend. Was that before or after the van thing?"
She wants to say "Lost weekend?" but bites it back. Derek goes on.
"I barely remember the van thing, I can't tell you when that was."
She says, "Tell me about the lost weekend. I want to know how you remember it."
The eyes again narrow. When he speaks, she can tell that he's had nearly enough of this game: he's going to quit soon. He says, "All right, well, he left school that Thursday afternoon and didn't come back. Somebody saw him out in the parking lot, smoking, as though he was waiting for somebody—I remember the school secretary told you that, on the phone. And then we didn't see him again until Monday night." He is staring at her. "Monday night? Is that right?"
"I think so," she says, quietly.
"The police were looking for him, we took turns driving all over town—I think we were out all night two nights running, then we just couldn't stay awake anymore. I canceled my appointments Monday, and we just sat at home, waiting. You said, 'He's never coming back.'"
"I did?"
There is a tightness to his voice when he says, "Yes, you did." He pauses. "I thought so too. As you well know. As we have discussed many times, alone and with Amos."
She does not ask who Amos is, though she wants to. Then she wonders if this is a test, if there is no Amos, and she is supposed to ask who he is. But no, why would he do that? In any event, Derek has moved on.
"He never said where he'd been. But he'd lost weight—he looked terrible and reeked of cigarettes and body odor. He appeared, in every respect, homeless. His school attendance was poor after that. He rarely spoke. Then I got my job offer and we moved here, and entered into the next phase of our strange life together, Lisa."
He's angry. She turns away from him, looks out the window, into the darkness; superimposed over the glass, her face looks heavy and old.
Derek gets up, delivers his empty beer bottle into the kitchen. When he returns he passes by the sofa and climbs the stairs to bed. Halfway up, he turns. "I still think there's somebody else," he says. "I think this was all about the third rule."
She doesn't know what this means.
He waits.
"I'm sorry. The third... I'm forgetting..."
His mouth turns down in a way that she recognizes from the real Derek, the one she knows. Polite displeasure. He thinks he is being mocked.
"Refusing. You refused."
She says, "Refused..."
Angry now: "Intimacy."
"I don't—" But now she is just making it worse. His shoulders and jaw are tensed.
"You fell in love with somebody, and you thought you could come home and pretend it never happened, but you couldn't. You panicked." He shakes his head. "A stroke."
"It's not like that. That's not what's happening."
"So what's happening?"
Elisa's throat is half-closed, and her voice is strangled when she says, "I don't know!"
He twitches, as if he's about to reply. And then she watches him master himself—the eyes close, the muscles relax. He lets out breath. "I love you," he says, quietly and with resignation, then turns and continues on his way. A few moments later she hears the mattress groan underneath him.
It's an hour before she is able to join him, and another before she sleeps. The first day is over.
14.
She's awake at four thirty. She is lying in the bed beside Derek, not touching him; his chest rises and falls in the gloom but he makes no sound. She permits herself a moment of hoping that yesterday never happened, or was a dream, but this is folly, even lying still she can feel herself occupying the wrong body.
Out of bed, down the stairs. The carpet bothers her now, it's like she's still in bed. The place smells different here—synthetic, unlived-in.
In this life, there's a laptop computer sitting on the kitchen island. Maybe it's just where Derek does his work. But she doesn't think so—he has his study. The computer must be hers. Beside it lies the binder she brought back from the conference that she has no memory of. As the computer boots she opens the binder, stares at the pages, at the printed-out e-mails. Her palms are sweating, her feet are cold, her behind hurts where it meets the hard wooden kitchen stool. She feels hung over.
She has to make a decision.
When the computer is ready, she brings up the SUNY Reevesport website, finds the biology department, reads about it. (They have wireless internet in this life, like a normal professional family.) It's fairly well regarded, particularly for plant biology. She recognizes some of the names of faculty members—people whom she has encountered through her lab in the old life. There's a guy who did some important work on fungi, another who is an expert on fatty acids in seeds, and won an award for it.
There's a list of support staff. And there she is. Graduate studies coordinator. No picture, no job description.
She could call in sick. It would be easy—she's been out of town, she could have picked up a virus. Nobody would find this unusual—it would buy her some time.
But then what? She will have to go to work eventually. She will have to do something.
The way she sees it, there are two choices. Fake it, or not. If she chooses not to fake it, to quit her job, she'll be starting over. She'd have to depend upon Derek's love and patience, which, however devoted (resigned might be the better word) he might seem in his present incarnation, she knows have their limits. In this life, she will be the woman who suddenly dropped out, who had a nervous breakdown—though she is not dropping out, is not having a breakdown. Or, at least, she doesn't feel as though she's having one. Is a breakdown a thing you feel, or a thing that changes your relationship to other people? In any event, friendships will end—though what does she care? She didn't have many in her real life, and she is not invested in this one.
It occurs to her to wonder what this means. She thinks, I expect that this is temporary, and that I will soon return to my real life. But, if this is my real life, then I am a woman whose only emotional investment is in an imaginary life. Thus, I am insane. And so I'd better hedge my bets—I'd better be invested in this life. Just in case.
She realizes now that she will never be able to explain to anyone what has happened. No one she knows now, at least not in her real life, would understand. Elisa doesn't understand, for that matter. How could she explain?
No—she will have to fake it. How hard, really, could this job be? It's summer—there are no graduate classes in the summer, right? It isn't academic application season, she doesn't think. From Derek, she knows that summer is the time for overhauls, for long-term projects. There will be time and space to figure it out. And once she has done so, she can figure out the rest. Whatever it is that has happened to her.
It's decided, then. She's going to give it a try. She's going to go to work. Of course the implications of that decision won't begin to reveal themselves for several hours, but there's nothing she can do about it now.
There is something she can do, however, and with Derek still asleep and the house quiet, this seems the time to do it. She moves the cursor to the search bar and types Silas's name into it. She reaches for the ENTER key, lets her finger hover over it a moment. Pulls it away.
It isn't clear when this happened to her; perhaps it happened to everybody at once. But at some point the internet became more real than the physical world. There was a time when it seemed like a dream—an impossible thing with uncertain implications. And then suddenly it was everything. There are people, she knows, who don't use it, who have no presence on it, who can't be searched for, who can only be accessed by going to their house and knocking on their door. But those people are the dream now. They're like ghosts.
There was a time, she thinks, as her hand moves back toward the keyboard, when a physical artifact—a letter, a piece of clothing, a room full of still-unopened boxes in another world—was the conduit to what could be known about a person. Touch that thing, hold it, smell it. Inhabit it. Close your eyes and remember.
Now, you search first, remember later. We don't need memory anymore—the internet has replaced it. And it's a good thing for Elisa, because it is all she has. She lets her hand fall. She hits ENTER.
There are, it turns out, many Silas Browns. A blind computer scientist, an audio recording engineer. But it doesn't take long to find her Silas, her living son. He is a programmer for a video game company. There he is, photographed in a parking lot in front of a low buff-colored cinder-block building, standing unsmiling in a small crowd of other unsmiling young men. She leans close to the screen. Her throat catches: he's an adult, he's really there. He's wearing sunglasses.
The company is called Infinite Games; they make violent first-person fantasies for game consoles. Titles like Berserker 4 and Ultimate Warlock. She searches a little more. They're popular, these games, but not all that popular. Popular enough, though, so that Silas appears to be quite successful. He is quoted in magazines and on gaming websites. There are a lot of pictures of him, almost always in direct sun. He wears his hair slicked back, gelled, and his arms are crossed. It's tempting to think that this is how she imagined he would be, if he'd lived, but the truth is she'd never imagined such a thing, never allowed herself that indulgence. Here, he looks like a third brother, another person entirely.
California, that's where he is. She feels both relief and longing. He isn't near. And it would seem that he is not in frequent touch.
But he's alive. Something has kept him alive, has given him a viable life. Something that was lacking in the other world. The real world.
Because that's what this is, isn't it? Unreal? Or, an alternate to the real? She has been put here for a reason, surely. To do something. To find out what she did wrong in her real life, to find out how she could have saved her son. This is a dream she'll wake up from, once she learns what she is supposed to learn.
But even as she thinks this, Elisa has her doubts. Because she doesn't believe in God. And what else could have put her here?
15.
A few hours later, she is standing in the driveway, ready for work. Derek will drive her there in his pickup—that seems to be their routine. The notion of Derek driving her anywhere seems absurd. The other car is obviously hers, she could drive herself. But this must be something they decided to do. For their marriage?
Everything so far today has been excruciating. She wanted to make coffee—but does she make coffee? Or does Derek make coffee? Their coffee machine was the same, but the can of ground coffee was not in its familiar cave in the freezer, enfolded in frozen years-old hamburger rolls. Instead there were whole beans, and a grinder. How many to put in? How fine to grind them? She made choices, proceeded. The grinder was startlingly loud. Would it wake Derek up? It didn't; he didn't get out of bed, anyway. The coffee brewed. There was half-and-half in the refrigerator—hers? It didn't matter, she drank it black. She was hungry, ravenously hungry—this body of hers was hungry. She wanted to make the oatmeal she liked to eat in her real life, but there was none in the cupboard. She stood for five minutes in the middle of the kitchen, wondering what in hell this woman ate for breakfast. In the end she settled on a banana.
She didn't know what to do after that, so she took another shower. When she reached the bedroom, Derek had woken up and gone downstairs. The clothes in the closet repelled her, but she put some on. Another skirt and blouse. In the mirror, she looked like a moderately attractive office worker.
Probably she ought to put on makeup, but she didn't.
Derek smiled at her in the kitchen but said nothing. They read the paper. She drank more coffee. She looked at the clock. She said, "I'd better get to work."
He appeared surprised. "Why so early?"
"There's some stuff from the conference I want to get in order."
Derek frowned. "I can go in early." He folded the paper.
"You'll drive me?"
"Of course," he said.
Now he comes out the door, locks it behind him. They get into the pickup. He pulls out and they head toward campus.
He says, "No makeup."
She can only beg, with a look, for his patience.
"You look pretty."
"Thank you."
She puts her hand on his thigh, just for a moment, then removes it.
When he stops the car, she opens the door, takes up her bag and binder. She kisses him—this seems expected.
"I'm sorry," she says. "About yesterday."
He nods, gently unsmiling.
"We'll have to... we'll..."
But his face tells her he's had enough. She shuts the door and he drives away.
She recognizes the building. It's gray cement with an angled roof and small square windows that don't open. She has come here from time to time, as part of her real job, to meet with researchers or deliver results for outsourced work. The entrance is on the other side, on the science quad, so she walks there. She passes a few graduate students, a man she thinks she recognizes, but nobody says hello. It's already hot and the wind blows her hair into her eyes. She wonders how she usually does her hair—probably not this way.
A directory inside the entrance tells her that administration is on the second floor. She takes the stairs. A moment later, here she is, in the hall outside the main office. Its double doors are propped open. Beyond them a receptionist or administrative assistant sits behind a desk, typing on a computer beside a nameplate reading BECCA SELGIN.
It's unclear what to do next. Is her office in there, part of a complex? Or is it out here, off the hallway, alongside what appear to be professors' offices and seminar rooms? She walks to the end of the hallway and back, then to the other end, looking for her name. She doesn't see it. There is nothing to do but go in.
The woman called Becca looks up. She is in her twenties, pale, overweight. A dish of candies sits beside the nameplate. She smiles at Elisa but not without some restraint, some reluctance. "Morning! You're in early."
Elisa holds up the binder. "Lots to read."
"Oh yeah, how was it?"
"Very informative." This is going well, she thinks. Then she says, "Any mail for me?"
The girl appears a bit flustered, as if this is not something she is often asked. "Uh... maybe? Check your box?" Her eyes dart to the left, as though that's where the mailboxes are.
"Thanks, Becca," and she walks left.
There is a small room off the main one, accessible from the hall, with wooden pigeonholes on either side, and after a moment Elisa finds her name. There is no mail. But she can see now that the administrative offices lie along two short hallways, one on this side of Becca's station, one on the other. She leaves the mailroom and walks down the hallway on this side, looking for her name on a door. The doors are all closed. She's glad she came in early.
"Are you looking for Judith?" It's Becca's voice.
Sweat is breaking out under her arms. Is she looking for Judith? She supposes that's what a person who would be walking this way should be doing. So, after an excruciating pause, she says, "Yes."
"She'll be late. She's got a doctor appointment. Oh geez, maybe people aren't supposed to know."
"It's all right."
This, then, means that her own office is not in this hallway. Correct? Because if you're going down the hall toward your own office, nobody asks you if you are looking for Judith. So she draws a silent breath, turns on her heel, and crosses in front of Becca with what she hopes comes off as a purposeful stride. She is wearing the pumps she wore to the conference. They're the only shoes in her closet that she is sure have been associated with work. Becca says, "You look different."
Elisa doesn't stop, it's a conversation she is not prepared to have. Over her shoulder: "Oh?"
"Oh God, I didn't mean bad. I'll just shut up."
She's down the hallway, peering at nameplates. "Don't worry!"
And here, finally, is her office. A note card, hand-printed with her name, is taped to the wall beside the door. The door is locked. She takes out her keys, finds one she's never seen before, shoves it home. The door opens. She pushes inside, closes it behind her.
16.
She stands with her back against the door, breathing shallowly. Her relief is profound.
The room is perhaps twelve feet square. Several plants, a coat rack, a desk. On the desk is a phone, a printer, a computer. There is a file cabinet, several chairs, bookcases covered with papers and binders. The shades are drawn, as if against afternoon light.
She crosses the room and opens them. Now she sees a photo, on the desk, of Derek and the boys, the same one that shocked her in the hallway last night. She opens a drawer and puts the photo in it.
Hours might pass before anyone knocks. This is what she hopes. She boots up the computer.
The computer desktop is uncluttered, the background image generic. It's like her machine at the lab, which in this life, she supposes, is someone else's lab. There are links to various web pages, which she double-clicks. They lead to university sites, administrative resources, that require a password.
She tries the password she used at the lab, a random series of numbers and letters. It doesn't work.
Though there is nothing in the world she wants less to do than open the door and go to Becca's desk, that's what she does.
"Hey!" says Becca. She's eating a granola bar.
"I am totally discombobulated today," Elisa tells her.
"Tell me about it!"
"And I am spacing on my password."
This gets her a pair of raised, excessively plucked, eyebrows.
"Seriously?"
"I know, right? Do you have a list somewhere?"
Becca shakes her head; her voice takes on a more businesslike tone, as if some line has been crossed. "You gotta go to the SRIT web page and enter your e-mail, and it'll ask you the secret question, you know, and then you can change the password. I'll send you the link." She turns to her computer and starts typing. "They tell you not to, but I write mine down. It's on a sticky under the desk."
Elisa goes back to her office. No sticky under the desk. On the computer, there's an icon for an e-mail program, and she opens it. And there's her day's work, laid out before her: thousands of e-mails, doubtless stretching back months, years, that will tell her what she said and whom she said it to, and presumably what on earth it is she is supposed to do here all day. The sight of this list, and the nested series of folders where the e-mail of the past has been archived, paralyzes her. She feels the way she did when, as a little girl, she pressed her nose to the glass of the TV screen to see what static really was: a mesmerizing and random and utterly boring thing that nevertheless compelled and frightened her. Then, as now, she felt fascinated and doomed. She opens the e-mail Becca has sent, and clicks the link. Enters into the browser window the e-mail address she has just learned. Hits "FORGET PASSWORD?"
The security question is "RULE 2."
This was all about the third rule, Derek said.
Shit, fuck, damn. What the hell are these rules?
Elisa is certain that, should all else fail, she could walk over to the IT office and act like a dumb bitch and make them hand over the password. It's a nice morning and already she is longing for a bit of fresh air. But she wants this finished now. She wants to crack this thing without getting up off her chair. She rubs her eyes with the heels of her hands and groans. Okay. Okay.
She picks up the phone and calls Derek's office, and he answers. He says, "Is everything all right?"
"Can you answer a question for me?"
"Sure," he says, after a moment.
"What's rule 2?"
The silence that follows is long.
Elisa says, "I just... I don't remember the order. And I forgot my password, and rule 2 is the security question."
"All right."
"I'm sorry."
Another silence. She can hear people talking in the background, perhaps in an adjacent office, and a truck rumbling by outside his window. He says, "Lisa, you understand that I am just... this is just completely baffling to me. I am just... going along with it now. Because I don't know what else to do."
"I'm sorry, Derek, I..."
"I know you're sorry, I can tell, but that's not the point. The point is..." There is the creak of his chair, an antique wooden office chair that his mother bought him when he accepted this job. "The point is you're something else too, not just sorry, and I don't know what it is."
She whispers, "Neither do I."
"That's not a comfort to me. Or an explanation."
Perhaps it's best to say nothing. She says nothing.
Derek says, his voice deepened by resignation, "Rule 2 is 'Blame yourself first.'"
She remembers Derek on the stairs, the way he closed his eyes, his anger giving way.
"Oh God, of course."
"Uh huh."
"Thank you, Derek."
Again, silence.
"Nothing yesterday," she says, "was your fault. But I can't explain now."
"I know it's not my fault."
"I'm not having an affair. There isn't anyone else. It's nothing that happened or anything you did, it's just me. Do you believe me?"
He laughs. "Yes. Sure. We'll just... sure."
"We'll just what?"
His breath catches; the chair creaks.
"We'll carry on... we'll just carry on."
17.
By nine thirty she has more or less worked out what she does for a living. Processes applications, deals with graduate student complaints, updates databases. Reminds professors how to do things: computer things. She has been reading the old e-mails, many hundreds of them, and in a spiral notebook she found in the desk has begun to take notes on each person she seems to have regular contact with. Her fellow office staffers; professors, students, the assistant dean. Every e-mail offers a few more small details, and each detail serves to confuse the overall picture of her job. She can see the parts but doesn't know how they fit together. The job is both wildly intricate and completely boring.
By ten she is wondering if she should take a leave of absence. But after that, then what? The sooner she learns the better.
People have been moving around in the hallway for half an hour now. Female voices. She has never much liked other women. Derek had wanted daughters both times, but she was glad to have boys. Even when things got bad with Silas, when Derek reiterated his wish that they'd had girls (and in a tone that suggested it might be her fault, that her contrary desire had somehow expressed itself through her womb), she remained glad. If Silas had been a girl, it might have been worse.
She has known for an hour now that only women work in this office. And that every last faculty member is male. There are two female graduate students, both with foreign names—she wonders if she knows them, if she likes them. Probably not. Probably she likes the professors. She likes scientists. She is one.
And it occurs to her to wonder if the other her, the real one, has continued to live her life, her real life. She feels a moment of panic. She is living my life! Or perhaps they've changed places, that Elisa and this one, and the poor soft housewife, the woman bound to her husband by rules, is now panicking in that bony body.
Ruining it—ruining her body with excess. And grief. Because that Elisa has just discovered that her son is dead.
Her jaw tenses and her heels drum the linoleum floor. Then there's a knock on the door and a woman's head pokes in and says, "How was it?"
Judith. This must be Judith. She is page one of the spiral notebook—the single most e-mailed person in the sent box. And Elisa recognizes her. Late thirties, bespectacled, curvy and loud, this woman has hovered around the edges of her real, her remembered, professional and personal lives for years. People at the lab know her. She's at the coffee shop or the supermarket, talking on her phone. Men Elisa knows like this woman, want to sleep with her. Larry knows her—she gets dumb art framed, pastel-colored prints of chickens and barns, old magazine covers. His gentle mockery of her that tells Elisa that he wants to sleep with her, too.
Of course Elisa dislikes her. Judith is one of those people you don't know but know you'd hate. Which for Elisa is most people. But now, here, they are friends—the best of friends, to judge from the e-mails. She has read at least thirty e-mails she herself has written to Judith, all of them in a tone—one of sly, wisecracking cheerfulness—that seems utterly alien to her own sensibility. Somehow this woman has awakened some undignified part of herself: gossipy. Sassy.
"Boring," Elisa says, and tries rolling her eyes.
Judith slips in, shuts the door behind her, flops down into the only other chair in the room. "Any hot guys?"
"Maybe a few."
"And didya fuck 'em?"
"Ah... not all at once."
Is this working? Elisa feels close to hyperventilating. She has the fingers of one hand looped through a drawer handle on her desk, and she is hanging on for dear life. Judith gives her a slow smile. "Does Derek miss it when you're gone?"
What does this mean? Sex? "Oh, God," she says, "I barely have time to put my bags down."
Judith laughs. She appears relaxed, as though this has been a normal exchange. Her hair is short and dark and frames a pretty but undistinctive face. What is it that men like? What is it that she is supposed to like?
But then Elisa gets an inkling of what this is—what it's like having a friend. This is something women, some women, need. This woman must know her secrets. This woman was her friend when whatever happened with Derek happened. Maybe she knows about the rules. Maybe she can tell her something about Silas.
Only an instant has passed, in which Elisa considers telling her everything. Listen to me, hear me out. Between friends. I'm not crazy. I'm someone else.
But when Judith says "What?" she changes her mind.
"What 'What'?"
"You got a look."
"Passing thought. I'm tired. I have a lot to do."
Sage nodding. "We must change our wasteful ways."
She is referring to the university-wide budget crisis. This is why she was sent to the conference, Elisa has discovered. She has been asked to cut corners. This means staff—consolidating jobs, firing people. It might have to be Becca—Becca would go, and someone from the back office would do her job at the front counter from now on. Callers would get a voicemail menu. In the long run, it won't make much difference. But nobody wants to give up her private office and sit out in front.
"Yep," she says. She ought to produce some witty banter, she knows, but the strain is too great. It's been less than twenty-four hours, and every moment has been an effort. To pretend. Her throat tightens and she gulps air.
Judith seems to get the message. She gets up, smacking the arms of the chair. "Back to the grindstone! Lunch later?"
"Sure."
Judith turns on the way out. Says, "Lisa."
"Mmm?"
"When you're ready. You can just go ahead and tell me whatever it is. Okay?"
She can only nod in response. When Judith is gone, she crosses her arms on the desk and lowers her head onto them.
18.
They eat at a campus café run by art students; Judith leads her there without asking where she wants to go. They must eat here every day. Judith talks about her latest conquest, a man who works in development and is "sort of married." Elisa knows about him already, from the e-mails. She has been worrying all morning about this lunch but it's easy—Judith will happily do all the talking, if she is allowed. Another good reason to like the woman.
Back at the office she selectively writes to certain people, complaining of an imaginary computer crash and asking for updates on various projects and situations. She clicks all the links on her desktop, figures out how to use things. It isn't difficult. Indeed, it's like her other job, except with less direct responsibility, thus easier to fake. By four o'clock she is feeling more confident that she'll be able to do exactly that. She has learned the names of the other women in the office—Linda, Tessa, Jane—and what they do. She has seen a few professors she recognizes, passing through on their way to the lab, and she says hello. It is good to have something dull and necessary to think of.
Because her real preoccupation is not, shouldn't be, this job. It's Silas. That he is in this world, alive. She can't shake the feeling that he has somehow engineered this: that he has brought her here, to show her something. To prove something. The internet has told her that he makes worlds. That's how he puts it. "I make worlds." At Infinite Games, he is known as a rebel. This is how he presents himself. She has found an interview with him, in an online trade publication, in which he flogs a new game he designed, called Mindcrime: Destiny's Mirror, and criticizes his rivals. The gaming industry, he says, is made up of emotionally stunted engineers with no imagination. Only he, Silas Brown, is doing anything of lasting value.
INTERVIEWER: But your projects don't sell. At least they don't sell compared to Berserker and the other big titles at Infinite.
SB: Sales aren't the point. Vision is the point. I'm trying to invent a new paradigm. Designers are stuck on the notion of story. As if it's the story that makes a game worth playing. But nobody gives a fuck about story. Nobody cares what happened in some guy's past, like if bandits raped his mom or kidnapped his sister or gunned down his buddies or whatever. That shit is stupid. It gets in the way. Games aren't stories, they're games. They have to invent themselves. Like life.
INTERVIEWER: But isn't life made up of stories?
SB: No. Stories exist to make sense of life. But they're a pointless exercise. Life is inherently nonsensical. Drawing strands of meaning together is for idiots. All there is, is right now, this moment. Noticing things and doing things. Making things happen. Building a tower of blocks, kicking them, making them scatter. Do it again and again, the pattern of blocks is different every time. You can't replicate it. That's what I want to evoke in a game. The first-person shooter, in its current conception, is moribund. Nobody gives a fuck about missions, about assuming some dumbass motivation some other guy thought up for you, like having to assassinate an arms trafficker or getting revenge on some guy or whatever. It's a fake moral justification for what the gamer really wants, which is to make shit happen. To manipulate the controls and watch things die and be born. To make worlds with your hands.
INTERVIEWER: But obviously people do want missions. Those games sell better than your games.
SB: People don't know what they want. I do.
If there was any doubt in her mind that this world was real, that this Silas was real, that interview has put it to rest. Silas is alive. That's him. She remembers a discussion she had with him one night, while he lay in his bed, a handheld video game on pause in his lap, an impatient expression on his face. He was thirteen. She was asking him, begging him really, to change his behavior at school. Because, when he got into trouble, it made trouble for his brother. Because the people who cared about him got upset. Because he had a future, and everything he did now had an effect on that future. Didn't he understand? He did not live in a vacuum. Everything he did had an effect.
By this time, Silas had begun to assume the imperious air that he would carry with him for the rest of his short life. He betrayed little emotion aside from stoic endurance. He looked at her and said, "That's not my problem."
Weakly, with profound exasperation: "How can you say that?"
And Silas said, "If I have an effect, then so do other people. So they can have their own effect to push against my effect. Can't they?" And he looked at her with real curiosity, as though truly interested in the answer.
"Some people can't."
"Then that makes it their problem."
"But Silas—it's not all about you. It's about other people, too. Who are close to you. Don't you want to help them with their problems?"
He frowned, turning back to his game. "When have they ever helped me with mine?"
As was often the case when she dealt with him, the rage came fast and hot, and she clenched both fists and pressed them into her thighs to suppress it. She said, "We try, and you don't accept it."
Bleeping, digital music, the sounds of explosions. "Well then that makes it your problem."
Video game design. Why didn't they think of that? They might have gotten him on that track early, won his respect by giving him the opportunity. Of course they never considered that such a thing existed. Games were distractions, unconnected to real life. They did not think of them as made things, as designed things. Another blind spot. They might have saved him.
But here, in this world. Did they save him? What was different? What had been different? Was there a split, a single place where the universes diverged? Did they—did she—make a different choice here, a choice that kept him from climbing into that van? What small thing, what word or deed, would have been enough to change this?
Or perhaps there was no single place where the worlds diverged. Maybe many things separated the two. Maybe it isn't a matter of cause and effect, but of random variation. Brother universes, forever at odds.
19.
She is staring at the ceiling, thinking, when she hears her name being called, a pounding on the outer door. She jumps up, hurries into the hallway. Everyone else has gone home. It's Derek.
"I've been waiting for twenty minutes."
"I'm sorry! You should have called."
"I shouldn't have to," he says. He is following her back down the hall, to her office. He stands, peering around, while she gathers her things. She notices him looking at the empty space on the desk where the family photo should be.
"You're right."
"We're going to be late," he says.
And without thinking, she replies, "For what?"
He stares at her. "Our session."
"Oh God," she says. "I'm sorry."
She follows him out and pulls the doors shut behind her. On the stairs, after a moment's thought, she says, "The conference—I thought today was Monday."
"It's all right," he says without turning around. It is obviously not all right.
Derek drives ten minutes in silence until they reach the city limits, and then says, "It has not escaped my attention that you are only forgetting the most important things."
She sits with her hands folded in her lap. She wishes she were back in her office.
"The things," he continues, "most germane to the survival of this marriage."
"I don't—" she says and then stops.
"It feels like sabotage. I am not saying that it is. I do not know your motivations. But it feels to me like you are trying to sabotage us. For some unseen purpose."
"I'm not."
"I don't think I believe you."
Eventually they arrive at a renovated farmhouse on a lonely stretch of road between villages. The driveway leads to a barn in back that has been fitted with sliding glass doors and a discreet wooden sign that reads AMOS FINLEY, MFT. Derek slides open the door and steps in, and Elisa follows.
They are in a bright carpeted room with wood paneling and an unmanned reception desk. From behind a green-painted door come plaintive voices. A clock reads 4:56. They are not late.
After a moment the door opens and a young man and woman walk out, the woman leading, red-faced, the man trailing behind with his hands in the pockets of his jeans. They pass by Elisa and Derek without a glance. A small bearded man emerges now; he is thin and long-faced, about fifty, dressed in tan pants and a big floppy cotton sweater. Elisa has never seen him before. He says, "Derek, Lisa. Welcome."
They follow him through the door. The room is capacious, but comfortable. Wide windows look out onto a meadow. Rag rugs lie on a polished wide-plank floor. The man sits on a small sofa, tucking one shoeless foot under the opposite knee. Derek sits in an upholstered chair half-covered by a blanket, and Elisa takes a seat beside him, in a similar chair.
"How are you this week?" the man asks them.
Derek looks at her, and the man follows suit.
"Fine," she says.
From Derek, a quiet exhalation.
The man gazes at him, then at Elisa. He says, "Lisa?"
"I forgot about our session today. I thought it was Monday."
"She was on a trip," Derek offers. "Over the weekend. So she missed work yesterday."
The man waits, expectant. Derek doesn't speak. Elisa is developing a headache.
"Is one of you forgetting the second rule?"
The man, the therapist, is almost smiling. He is filled with life—this conflict seems to delight him.
"Elisa," he says, "perhaps you'd like to remind us of the second rule."
At least, a question she knows the answer to. But she remains silent.
"Derek?"
"'Blame yourself first, circumstance second, your partner last.'"
The man turns back to Elisa. "Elisa, Derek seems to think there is a problem this week. Do you want to claim it?"
The headache comes into focus just over and behind her left ear. She tips her head back. A crack seems to run diagonally across the skylight, then disappears. A twig, perhaps, blown by the wind.
Elisa could panic, if she wanted to give herself over to it. She hoped to be heading home around now—at least there she has already had a few small successes. She has made coffee, she has found her favorite nightgown.
But what is happening now seems impossible to navigate; it makes no sense. Of course they have been in therapy before, separately and apart. But that was about the boys. And it wasn't with this man, this strange, almost jolly creature. She is inclined to think of him as sinister, the instrument of her impending downfall. But there is a part of her that likes him, liked him immediately upon seeing him. He feels to her like the closest thing to an ally she has in this room, maybe in this life, at least so far.
She'll take the path of least resistance. To the ceiling, she says, "I broke a rule. A different one."
"And which did you break?"
"I... refused intimacy."
"You can refuse intimacy."
Elisa tips her head forward and looks at the therapist.
"But," he goes on, "you have to offer something in its place. Did you forget to do so?"
"Yes."
"Perhaps Derek would like to suggest a substitute for intimacy?"
Derek shakes his head. He is clearly annoyed by this avenue of discussion. "She is leaving things out."
Amos Finley has continued to gaze at Elisa. "You have the floor now, Lisa, do you want to explain?"
She does. She wants to tell him everything. Instead she says, "Last night. I... panicked. I don't know why. I thought something was wrong with me. That I was sick. So we went to the hospital. But it was nothing."
Derek is agitated. It's clear that he would like to leap to his feet, pace in front of the judge's bench, take her story apart. But he resists.
The therapist stands up, smoothes his pants and sweater. He gazes over their heads, out the window. He sits down on the sofa again, this time with his legs crossed, Indian-style. He says, "Something has changed this week. Let's get to the bottom of it. Lisa, you went to a conference?"
"Yes."
"Did something happen there? Something that has caused your behavior to change?"
"No."
Derek breaks in. "There has to be. You're different."
This is true. She can't speak. She reaches down and rummages in her bag for her aspirin, but for some insane reason there isn't any there. When she looks up, the therapist is holding out two tablets for her. She takes them. "Water?" she asks, and he stares at her a moment before pointing to a second green door in the corner. It's a small restroom with a sink and toilet. In the mirror above the sink, she looks red and disheveled. She swallows the aspirin and comes back into the room.
"Thank you, Doctor," she says.
The man laughs.
Derek rubs his forehead. "Are you mocking me?"
She has not sat down. She stands in the middle of the room and feels sweat blooming under her arms.
"No!"
"She said she had a stroke," Derek says, gesticulating at the therapist. "She didn't have a stroke! We went to the hospital, and we came back, and she made me tell the whole story of how we met. Of our entire marriage, the trouble with Silas, everything. She is forgetting, or pretending to forget, everything. She forgot the rules. She forgot about this appointment."
The therapist is not looking at Derek, but at her, gazing at her with a strange intensity, as though for the first time, as though she's naked. She is still standing.
"Lisa," he says quietly. "Have you forgotten these things? Or are you trying to... make things difficult for Derek?"
"I'm not trying to be difficult."
"Why did you call me Doctor?"
"It was a reflex, I'm sorry."
Derek says, "She is doing everything differently. She sat at a different place at the table. She wore an old nightshirt." He seems embarrassed at the last, as if suddenly aware of some deep pettiness once hidden from himself.
"May I sit down?" Elisa says, then takes her place without waiting for permission.
The exchange has changed the mood of the session. The fight has gone out of Derek. The therapist steers the conversation around to other things, he is talking more generally now, about trust in the marriage, restoring and maintaining trust. It sounds like a canned speech, some shtick from a book he wrote. Elisa ought to be paying attention, there is likely plenty to learn here, but her attention drifts to the window, to a cluster of whitetail deer browsing at the edge of the woods, and to the motion of the wildflowers and grasses in the breeze. The sky has clouded over, everything appears warm and lush. She longs powerfully for her real life.
The session is ending. Derek and the therapist are standing up and so Elisa does too. The therapist asks Derek to go out to the car so that he can speak with her alone for a moment.
Derek's compliance is instantaneous and disconcerting. He turns and walks out the door and is gone. The room is very quiet now. The deer are gone from the meadow. There is a movement to her right and when she turns the therapist is standing there, holding his eyeglasses in his hand.
"Lisa," he says.
She remembers, just in time, not to say Doctor. But when she utters his name, quietly, it comes out wrong, blunted and slurred. Amos. They face each other, breathing in and out.
"There's nothing you need to tell me?"
"No."
His eyes are large and tired and faintly wet. They blink. "But there is something you want to tell."
She doesn't answer.
"I think we should meet one on one."
She is heavily conscious of her body in the room and, though she is dressed, she wishes she had something—a bathrobe, a blanket—to cover herself with. There is an intimacy between them—between the therapist and his idea of her—that makes her feel disturbed and excited and envious. Because it's her intimacy, the other Elisa's. She pictures that woman in her own real body, trapped in it, in her thin body and vacant marriage, driving out here, pulling over at the side of the road, gazing at the therapist's house and office with loneliness and longing.
And like that, she has an epiphany. In her real life, she is lonely.
"We've come a long way together," says Amos Finley. "You know you can tell me anything. Don't you?"
"Yes."
"When you're ready."
She is moved, suddenly. She sort of wants to hug him. He seems to notice and draws back, almost imperceptibly.
"All right," she says.
He smiles, slides his glasses back onto his face. His eyes recede and focus and he holds his palm out, inches from her shoulder, and says, "Go to your husband."
PART TWO
20.
She tells Derek she'll be going in to work early every day, in order to reorganize things. He agrees to drive her, though she wishes she could just drive herself—but never mind. This still buys her an extra hour and a half alone, at the office, studying... this job, ostensibly, but actually, this life.
She has found several very old e-mails to Sam in her sentbox, overly cheerful exhortations to take care of himself and not drink excessively and not be hurt by the things his brother says and does. The tone embarrasses her—it's artificial, desperate. These chipper notes stopped more than a year ago, and don't appear ever to have been answered. He is living in Santa Monica, apparently the same place Silas lives. Why?
She needs to call him.
It is Wednesday morning. She is sitting in her office. Her inbox is filled with responses to her queries: people have sent her project updates and status reports, as she requested. She reads them, and begins work on a slate of recommendations for the reorganization of the office. It isn't hard. They won't have to fire Becca. She will have Judith read it at the end of the week.
Elisa is disturbed, a little bit, at the ease with which she is occupying this position, one she had no inkling even existed just two days ago. Does this say something about her? Are her talents so generic as to be adaptable to any job? Or is this job so generic that anyone could do it?
More vexing still is the way the job foregrounds, in her mind, the reasons she left science in the first place. She was convinced, remains convinced, that if the world of biology had consisted solely of herself, a lab, and an unlimited budget, she would still be doing it. Indeed, she would be very, very good at it. As an undergraduate, she had been part of a genomics project right on the forefront of the field; she was the only undergraduate on the team, and the only woman. At first she thought she'd arrived—that to have crossed those lines was to have been accepted.
But of course it didn't work that way. To the other students, even to her professor, her inclusion gave them license to let their guard down, to indulge their pettiest selves. She was mocked the day she wore a skirt, then mocked the rest of the time for wearing jeans. She was told, many times, that her safety goggles flattered her, or didn't flatter her. A tittering tour group, passing through the building, was presented with "the girl scientist." Then she made the mistake of dating a grad student for a couple of months, a man of lesser intelligence to whom she was supposed to defer; and when they broke up, his friends on the team ignored her questions, sabotaged her work, talked about her in stage whispers when she was in the room. "Hey!" she called out one night in the crowded lab, "Will somebody please look at me and answer my fucking question?" It was her ex who answered, to the amusement of his friends, "I thought you didn't like people looking at you. We're just trying not to be sexist."
Her professor, who before had treated her with a certain amount of respect, now came on to her. "You look pretty today," he said, and when she replied "Please don't talk to me that way" a heavy curtain came down between them and he never asked her to do anything again.
The project was tainted. She tried to get on another, to no avail. People told her she was crazy to want to quit. Her adviser, the only woman in the department, told her this. "That project is going somewhere," she said. To Elisa's complaints of sexism, of bullying, of disrespect, the woman turned steely. "I'd like to send you back in time twenty years and see how you like it."
She hated the bureaucracy. She hated the corporate money, the suggestion that certain outcomes were acceptable to the administration and certain outcomes weren't. She hated the fact that all the people working in the department office were women and all but one of the professors were men.
Now, here, she has become one of the women in the all-woman office, in a department where all the professors are men. They are decent people, the professors who are here this summer, but she is not real to them: they don't acknowledge her as anything more than an office worker. Even at her lab job, in the real life, even when she is doing the same kind of organizational work she does here, she is regarded as a person who does science. But this job seems to have been perfectly okay with the Elisa Brown who accepted it.
Then again, why should anyone treat her differently? And maybe part of being a good scientist is knowing how to parry insults, how to navigate bureaucracies. How to seize authority and respect, rather than waiting for them to be conferred. Indeed, maybe these are the defining traits of a good scientist—maybe the things that made her think she was a good scientist thwarted, were actually clues that she was not a good scientist at all. And maybe this Elisa has, for better or worse, accepted that.
She considers, briefly, that perhaps the same could be said about being good at marriage. Or parenthood.
It makes sense, she supposes, that in this world where Silas is alive, she isn't in touch with him. But Sam? It's hard to imagine the offense that would drive him away—that would compel him to move to California, to live near his brother, and never answer her e-mails. Maybe they write letters—there could be letters somewhere, tucked into a drawer or cubbyhole somewhere at the house. She'll look later, when she gets home, but somehow she doubts it.
When things were at their worst—with Silas, and with Derek, too—she could always talk to Sam. Even when he was small, when he could barely speak, he was interesting to talk to—habitually quiet, he became voluble when they were alone, asking questions, remarking on the scenery, trying to figure things out. He would be the scientist, she thought. He wanted to know how things worked. He had ideas about how they might—compelling ones. By the time he was five or six his questions fell outside her ability to answer, what-ifs about the nature of time, existence, the limits of the universe. He did excel at science in school, for a while, but his grades suffered as Silas intruded more and more upon their lives, and they never really recovered, even after the accident. By then Sam had come to think of himself as an indifferent student, and so he was.
But he was still smart. She would like to tell him what had happened to her, to ask his opinion. He would certainly have one. He'd dismiss immediately the notion that she had lost her mind, go right for the science. "Where did Han Solo go?" she asked him once when he was six, having watched him, in play, tuck his action figure behind a sofa pillow. "Sucked into a hole in the space-time continuum!" was his reply.
Twenty minutes seem to have passed, here in her office. The document file open in front of her makes no sense. She closes the word processor, opens a web browser, and clicks in the search box.
She is thinking of Sam when she types the words "parallel worlds," then adds "physics" and "study" and "university." This is the kind of thing they talk about these days, in her real life, summer nights out on the porch when he has come over for dinner and Derek has gone to bed, Sam smoking with his chair tipped back and his feet up on the railing. Sam would take the idea seriously, at least for an hour. But what the internet gives her is a lot of science fiction. Television shows and novels. There are scientists who have theorized the existence of parallel universes, but nobody who is actually trying to figure out what is in them. This is the purview of cranks, dreamers, and Hollywood screenwriters.
She wants to look harder, to find somebody who is actually looking into this idea, who thinks of it as palpably real. Instead she finds herself googling Silas again, digging more deeply into the search results. There is a reference to "Silas Brown of Infinite Games, minefield on gamedev.org." She searches for minefield, gamedev, and ends up on an internet forum for hardcore gamers, video game testers, and game developers. And there is indeed a member named minefield.
Minefield appears to be belligerent, impulsive. He has been banned multiple times. In the hierarchy of the game development world, he appears to be among the most successful professionals who still post on internet forums. There are celebrities in this world, and he is not among them, but he has some small fame here, in this community of hobbyists, wannabes, and up-and-coming professionals.
He is clearly among the most knowledgeable members of the forum, and Elisa can tell, after half an hour of reading the threads he visits, that questions are often asked with the sole purpose of luring him into the open. At times he is very generous with what must be basic advice. He answers a question about coding convex rotating polygons. He advises somebody on creating realistic intelligent motion in virtual crowds. (None of this makes any sense to Elisa; it's like a map of a foreign country. She has never played video games, beyond Pac-Man once or twice, at the pizza shop, with a boyfriend, in 1984.)
But every now and then minefield will respond to a question with withering personal criticism. You're fucking pathetic. You come here expecting something for nothing. That is the stupidest fucking question I've read on here in six months. He tells people they'll never amount to anything. He mocks their ideas, their spelling mistakes, their screen names, their graphic avatars. His victims fight back, weakly, trying to justify themselves. The forum breaks into camps, attacking or defending minefield. Members say they've had enough, threaten to quit the board. Moderators are summoned and issue warnings. Minefield mocks the moderators. The thread is locked. Minefield is banned. Minefield comes back, feigning contrition, and it all begins again.
It has to be Silas. He is everything she remembers. He is as charming as he is vitriolic; you feel proud when he accepts you, and when he turns on you, you blame yourself.
Elisa remembers something now, something from the months leading up to the crash. Silas brought a girl home after school one day. She might not ever have known it had the girl not sneaked out of his room to use the bathroom: Elisa saw her at the end of the hallway, darting across from one door to the other. They must have come in while she was out, or entered quietly through the back. Whatever the case, they were here, together: Silas and a girl, in the house.
She heard the girl return to Silas's room. She waited a few minutes, then sneaked down the hall.
They were very quiet. Elisa could hear them kissing, hear them shifting on the bed. Then the girl said, "I really need to do this, Silas."
"Shh." Silas groaned quietly, then spoke to her in a tone she hadn't heard before: commanding, certainly, perhaps condescending, but gentle. He said, in a near-whisper, "All right, let's see it."
The bedsprings creaked; Elisa heard a zipper. Not her clothes, her backpack. Then a grunt of effort as something was hauled out of it. The pages of a book, being turned. "What don't you get?" was Silas's question, weary, skeptical.
"The whole thing," she said.
"Dude," he said, "this is easy."
"Dude," the girl said, not without sarcasm, "for you. Not for me."
Their conversation continued, with Silas dominating, talking about straight and curved lines, the slope of a line. F of X, F-prime of X.
Derivatives. He was explaining calculus to her—helping her with homework.
Elisa withdrew. She had never seen or heard him with a girl. He never brought one to the house. This one must have insisted. She withdrew to the kitchen, where she could still hear the doors open and close, and idled until the girl seemed to be leaving. Then she headed down the back stairs and busied herself in the storeroom, pretending to look for a particular can of paint, so that she could catch a glimpse as the girl headed out the back.
It worked. She was alone, a small thing with a giant backpack. She started when she saw Elisa there, gasped.
"Oops, sorry," Elisa said.
"Yeah, sorry, hi," the girl said, hiding her face behind her hair and scuttling away, through the door, across the patio, and into a gap in the hedge. Elisa was surprised again—she wasn't pretty. Plain, in fact, the face flat and broad, lank hair, thick glasses. She was done up like a punk, with pierced nose and lip and a black leather jacket. Her jeans were tight on her heavy hips; her exposed hairy ankles must have been cold, as it was a dry clear winter afternoon, near dark.
Were they having sex? Something she would wonder, later, after he was dead: had he ever slept with a girl, or with girls; this one or others? She wouldn't find condoms or any other evidence among his possessions, afterward; but then again she would never look very hard. Some of his things are still there, in boxes, in the other life. They might contain anything at all, or nothing.
A week later Elisa was at the supermarket and saw the girl there, buying a can of Coke and a chocolate bar. She was with a friend, a skinny blond-haired thing in a ratty knitted hat. "Oh, hey," the girl said, hiding her face behind a curtain of hair, and Elisa said, "How are you?"
"I'm good," Silas's girl said. She appeared eager to leave; Elisa was on her way in, these two on their way out.
And then Elisa said, quietly, conspiratorially, "I hope he's treating you well," and the girl's face froze, and her friend's mouth made an O of delighted shocked surprise.
"Uh huh" was the reply, and they were gone. Elisa imagined them gasping and cackling in the parking lot, and felt as embarrassed as she had ever felt in her life. She was worse than her mother, worse than Derek's even. She had visited humiliation upon this poor child, and brought it down on herself as well. But she had wanted to warn the girl. Silas was trouble, after all.
After he was dead, she came to believe it was only Sam they failed—they allowed his brother to dominate him, to wear him down. Maybe if Silas had been the older one, they would have pushed back harder. Maybe there was some part of them that believed Sam ought to be able to fend for himself, being bigger and (so they thought, at first, and maybe they were right) smarter. Maybe they knew all along he was gay, maybe on some level it made them uncomfortable. C'mon, don't be a pussy, fight back. Later, when Sam came out to them, the first Christmas of his college career, she wondered if Silas had ever abused him, had subjected him to incest, if Silas had somehow made him gay. Nothing in her memory would suggest that Sam had ever been anything but what he was, yet she thought it nevertheless. Derek too. Sam's sexuality a problem, Silas its cause.
But that's what Silas was like. Even in death, he dominated their thoughts, their way of seeing. The likelihood of Silas's influence simply seemed greater than the likelihood of random chance; Silas was easy to blame things on, when so much was his fault.
Wasn't it?
Elisa's palms are starting to itch. Minefield is mesmerizing, mercurial; to read his helpful posts is harrowing. Does this make sense to you? I'm not sure that's the best example. If that doesn't work, let me know, I've got another idea, too. It's too perfect, too reassuring, too accommodating; reading it, you wait, hushed, for the explosion, you sense everyone on the forum waiting with you. And then someone, some hapless newb, betrays his ignorance—wait uh I dont get it what?—and Silas pounces.
She wonders now what she did then—is it all theater? Is the charmer merely a stalking horse for the bully? Or is the charmer for real, a genuine part of Silas that he simply can't sustain? If it's real, what does it feel like for him, proffering this part of himself, then having to snatch it back? Is he ashamed? Does he feel remorse?
They should have asked themselves that more often, she thinks. When he was bad, they punished him, or tried to. But maybe it hurt him to be bad. Maybe he wasn't so hard, in the end—maybe it was all armor, a way of defending his most sentimental, most vulnerable self.
But she doesn't like to think about that. If that's true, then they did it wrong. If that's true, then Silas is more like her than she wants to admit.
21.
Elisa tells Becca she's having a business lunch with somebody from Killian Tech, and will be back at two. Becca doesn't seem surprised or concerned, and Elisa walks out into the July sun.
Killian Tech is her lab—the one that, in her memory, she was managing just a few days ago. It's a dozen blocks from the Levinson Center, closer to the lake, and she walks briskly, suffering in the heat.
The people, the houses all look familiar. She is alert for differences in this world that do not involve her husband and sons. But none are evident. That's not to say they don't exist, only that she has never been as observant as she might have liked to think. How could she be? Her thoughts have always been more interesting to her than the world itself. She remembers what she liked best about her lab work: data. Having data to pore over, on the filthy computer in the corner by the window, while the other students fussed at their experiments behind her. She loved that world, the world of abstract representation. She loved making numbers make sense. That's what she did here, at her old lab.
The Killian Tech building is a flat, one-story structure with windows on all sides. It used to be dental offices and is wedged incongruously into a residential neighborhood. Its owner, her employer, lives in fear of zoning complaints from neighbors. But so far there haven't been any. They're quiet and unobtrusive.
Elisa's office is on the northeast corner of the building, facing a bus stop and a NO PARKING sign. From the sidewalk, through the windows, it looks more or less the same: a desk and computer, some plants. She made few changes to it when she took the job, and whoever is working there now clearly took the same approach. Where Elisa had hung a black-and-white photograph of a cobblestone street, this person has hung a diploma. The filing cabinet is larger—Elisa had always meant to do that. There's a photo of a blond-haired woman on the desk, and a little wooden box containing some sand, some stones, and a tiny rake.
She pretends to wait for the bus but continues to watch the workings of the office. The techs aren't visible from here—most of the lab facilities are behind an interior wall, where the light can be controlled. But she doesn't recognize the people coming and going through the lab door. Of course she hired many of the techs herself and wasn't here to do so, in this world.
Eventually a thin man walks into the office, smiles at her through the window, and sits down. He begins to read his e-mail. She remembers doing the same thing: meeting the gaze of bus passengers, turning to her work.
Something occurs to her that surely she has thought of before: the world doesn't need her. The highest compliment you can be paid at work is that you are indispensable. We couldn't do it without you. But of course they can always do it without you. Killian Tech is doing it without her. Her sons are doing it without her. In her world, the real world, when Silas died, the world knitted itself back together behind him, and all that was left were a few scars, nearly all of them in their house. Maybe his girlfriends were sad for a little while. Then they moved on.
Perhaps the reason she is afraid of Derek here, in this world, is that she knows he doesn't need her. She's seen it. He can find other people to talk to, to have sex with. So can she. His attachment to her here is superfluous: there is a part of her that wants to just get it over with, to betray him or be betrayed, to leave him or be abandoned. Because it can happen, and the fact that it hasn't is a hanging thread, an unresolved note.
The man in her office looks up again and she realizes she has been staring at him. He is not her, looks nothing like her, but in this world, to those people, he is her. His brow is creased; he is gazing at her frankly, and with puzzlement. Elisa feels self-conscious now, and turns to go. At this moment a bus arrives, and she impulsively steps onto it.
There is a moment of frustration and embarrassment, as the bus pulls away, that she cannot find her bus pass. Indeed, it would appear that, in this life, she doesn't have one. She smoothes out a crumpled dollar and feeds it into the cash box, then takes her seat.
This is the number 20. It goes down to the park by the lake and then out to the western edge of downtown before circling back. The air-conditioning is on high and a soft-rock radio station is playing a song by Journey. She shivers. No, she didn't board the bus on impulse: she intended this all along.
She gets off at the main depot by the library and walks east. After three blocks, she turns south. The frame shop is the fourth building on the right, a small converted cottage, again on a residential street. From the outside it appears identical to the one she knows—cedar shakes stained a slightly artificial brown, louvered windows with gold text: AURORA FRAMING.
He's there, standing at the main counter, a broad work table arrayed with mat board samples. Behind him, the walls are covered with frame corners. He is typing on a small laptop computer.
He looks up at her with a completely generic smile and says, "May I help you?"
Larry. He's trim, early fifties, and an inch shorter than Elisa. Clean-shaven, with hair cropped close. He looks subtly different to her—perhaps it is his failure to recognize her. She hasn't seen this expression on his face, one of expectant politeness, directed at her since the day they met.
Of course, here, this is the day they met.
"It's silly, really," she says, and hates the sound of her voice. "Maybe not worth your time." She draws from her purse the family photo that used to stand on her desk. "I'd like this put into a decent frame."
He takes the picture, turns it over in his hands. "Do you intend to hang it? Or stand it on a surface?"
"Hang it," she says.
He turns it over again, as she knew he would. He is deliberate, thoughtful. This is the sort of movement that, in the old life, would quicken her breath, make her tongue find her lips to moisten them. But her body doesn't want to catch up with her memories, not yet.
"Are you after a contemporary or more traditional look?"
"Traditional. But not gaudy."
This makes him glance up. Or maybe he just wanted a second look at her. Another small smile, then he turns around and reaches for one, then another, frame sample. His shoulders are broad, for a small man's, and his jeans hug his hips. He pushes some papers out of the way, removes the photo from the frame and places it on the countertop, facing Elisa. He selects a sample piece of cream-colored mat board, lays it down over the corner of the photo. Arranges a frame sample at its edge.
"This will give it a warmer look than what you had it in," he says. "Now this..." He swaps out the mat board for a different piece, this one a pale gray. "This will complement the colors of the stones here, behind the figures."
"Ah," she says.
"Your family?"
"Yes."
He looks up at her. "Handsome boys."
"Thanks."
"Do we know each other?"
"I've seen you around town." There is a quaver in her voice. She extends her hand. "I'm Lisa."
"Larry."
Their palms touch, his dry, hers clammy. He holds on exactly as long as is appropriate. She says, "I like the cooler one. And the plainer frame."
"Good choice, I think. We can have that for you in a few days."
He opens a drawer, pulls out a form, takes pen in hand. She is reminded of the way, after sex, he picks up his glasses and puts them back on. She is trying to remember, with unexpected difficulty, what he is like in bed, what his body feels like close to hers. He writes "Lisa" on the form.
But Elisa has already backed up. She is nearly at the door. She says, "No need for that. I'll be back."
Larry looks up, apparently surprised. Doubtless there is a deposit policy. But in the end he says only, "All right then."
Back on the street, she admits to herself how much of this encounter she planned out ahead of time, how many hours some small ashamed part of her mind spent blocking out the movements, working on the script. Perhaps this is why she expected something more—a feeling of something gathering momentum. Instead it feels like the first steps up a steep hill.
Was it really that different, in the old life? Did she want him right away, or did it take time? Maybe she is misremembering the sudden, inalienable erotic pull, the feeling of inevitability.
Or maybe this is the real life, and that one was fantasy. Erotic fantasy.
No. It is just that lust is physical. Maybe it bypasses the mind, like the impulse that pulls the hand away from a hot stove. Lust is in the body, she thinks, boarding the number 20, shoving her dollar bill into the slot, and this is not my body.
22.
It's a few nights later. She's home alone, Derek is out buying a six-pack of beer from the supermarket, and the phone rings. The caller ID tells her it's Lorraine. She decides to let the machine get it.
But then she picks it up. She can't stop herself: she's curious. "Hello, Lorraine."
"Hello there, dear, how are you?"
"I'm good, I'm good..." Already it's different: Lorraine never calls her dear, doesn't ask her how she's doing.
"I expected to hear from you—how did the conference go?"
"Fine, it was fine." She is analyzing every word for hidden content: the "I expected" is typical Lorraine, but not in this context. Lorraine expecting to hear from her? The question does not sound sarcastic. The intent does not seem to be to mock and humiliate.
They make small talk. Or Lorraine does, and Elisa tries to. Elisa doesn't know how to do small talk. This is why they never have anyone over. Or perhaps in this world they do—maybe they have parties, lavish boring parties. Lorraine is telling her about her book group, a novel they read, the comments some of the other women made. She describes a television show she watched the night before, and offers some details about the child of some neighbors who has just graduated from Harvard.
Elisa feels as though she's supposed to offer something in return. Some little morsel. She mentions getting the photo reframed.
Silence. Then, "Maybe you should just take it down, dear."
"Maybe," Elisa says. Another silence follows, so she continues: "Do they... do you ever speak with them?"
"You know that I don't." This quietly, uncertainly.
It feels like an opportunity to Elisa, but she isn't certain what kind. She says, very quietly, "Lorraine. It's all right if you do."
She thinks she must have made a mistake—Lorraine says nothing, then draws a deep, ragged breath.
"Thank you," she says, and nothing more.
A moment later she hears Derek coming through the door. Does Lorraine want to talk with him? No, she doesn't. "I just wanted to catch up with you, dear," she says, with audible relief.
"Well, thank you."
Some pleasantries, which Derek listens to as he arranges beer bottles in the fridge. "My mother?" he says, when Elisa hangs up.
"Yes."
He shakes his head. "All she ever wanted was a daughter. Sometimes I think she likes you better than me."
23.
She starts bringing work home. The binder. Budget materials. It's all begun to make sense, but she wants it to be intuitive. Natural. Out of all the things that are alien to her, she thinks, if I can master this, then I can master it all. Evenings, Derek cooks, she clears the table and puts the dishes in the machine and wipes the table down. Soon she has her glass of wine and is working under the bright kitchen lights. Surely this is not quite right—there are things she's missing, things Derek expects her to do. But he says nothing. Sometimes he looks at her too long: this is when he's waiting for her to discharge some obligation. Nights he appears anxious, frightened even. He says good night, then waits. She might get up and go with him, and in bed he will reach out and touch her shoulder, then withdraw. She might remain seated in the kitchen. They haven't had sex yet.
Friday night, she stays up after he goes to bed. She has been in this world for five days. Laptop open in the kitchen, she searches the internet for references to Silas. Then she composes an e-mail to Sam. It reads, We want to see you. She doesn't know if this is even possible—if they or he have the money to travel. Surely they do—they did in the real life. The e-mail sits on the desktop, unsent.
She wonders what the other Elisa is doing. If she is adapting, or if she is floundering. The latter, Elisa tells herself—this Lisa, the one she is impersonating, is soft, pliable, defanged. Or perhaps she is merely flattering her real self. In any event, if that Lisa has adopted her old life, Silas is being mourned again.
The memory of Silas's death—his funeral, his burial, and her subsequent months, years of grief—no longer compels strong emotion in Elisa. The way a factory worker, forced to listen to the same clang of metal against metal for years and years, will lose that frequency for good: that part of her is worn away. But the thought of that woman, her doppelgänger, experiencing this for the first time, stops her breath. A little hiccup of misery convulses her body; the stool barks on the linoleum. The poor thing.
But then again, maybe the Lisa who once inhabited this body, who endured the challenges of a living Silas, has had to evolve a more agile mind. Maybe I'm the weak one, Elisa thinks—the woman who couldn't hold her marriage together, who cheated and lied, who couldn't get along with her mother-in-law. And now the smarter Lisa, the stronger Lisa, is making better use of her freedom than the real one ever could. Maybe the fake Lisa mastered Killian Tech even more quickly than the real one mastered the Levinson Center. She has seduced Derek and resisted the advances of the creepy frame shop guy, who keeps calling her, though they've never met. She is working harder, thinking more clearly, feeling more deeply. What does Silas's death mean to her? In this world, the wrong one, he might as well be dead. He isn't in her life anymore.
She still hasn't sent the e-mail. After a moment's thought, she signs it Love, Mom and hits SEND.
As if in reaction, she reaches out and snaps the computer shut. A wave of terror rises up in her and then subsides. On the chair beside her lies her purse. She has carried it with her everywhere all week and has barely looked into it, except to remove and replace her wallet. Now. This is a good time for it.
She pushes the computer out of the way and hauls the bag onto the table. She begins to remove its contents, item by item.
First, makeup. A compact, a lot of lipstick. Eyeliner and rouge. Lipstick she has worn before, but eyeliner? What on earth is this woman supposed to look like? There are Tic Tacs, orange and mint, and two half-empty packets of tissues. There is a pocket calendar. It gives her a moment of panic, until she realizes that it's from 2007 and is completely empty after the month of May. Lots of receipts, some itemized and listing items that are also in the purse. An eyeglass case containing her same old glasses, which she has kept ever since switching to contact lenses years before. Contact lens case and fluids. Wallet. Keys. Phone.
There are pockets on either side of the main compartment, fastened with a snap. One is filled with pens, as it was in the other life, where the second compartment was empty.
Here, though, the second compartment contains a piece of graph paper folded twice over on itself, rounded and furred at the corners. She removes it, pushes the bag out of the way, and unfolds it on the table in front of her.
It's in her own handwriting, and it reads:
1. Pay compliments and express gratitude.
2. Blame yourself first, circumstance second, your partner last of all.
3. If you must refuse intimacy, offer something in return.
4. Account for your time.
5. Do not use the children to attack your partner.
24.
Elisa lies awake beside Derek wondering what else might be different. She remembers the moment of the switch—has been trying to remember it more precisely—and is certain that there were more or fewer clouds, or that the clouds changed position. And if the weather was different, then a lot of other things must be different. Right?
But then wouldn't Silas himself, Silas's existence, be the thing that changed the weather, changed everything that is different now? No. There was something before the van crash that kept him from going along for the ride, and something before that, and something before that. And so the world was already altered, and Silas's death or survival trivial. The world must be indifferent to Silas, to her affairs and her appearance and her personal demeanor. She is a casualty of circumstance, not the center of the universe. And so this change, this transference, cannot be meaningful. It's something that happened by accident: a glitch.
For this universe not to be about her, not to be made with her in mind, then, there would have to be more than just the one where Silas survived. There would have to be one where Silas was maimed, and one where Silas was never born, and one where he was a delightful and well-adjusted child, and one where he was a houseplant.
How many would there have to be? All of them.
Of course it's easier just to say she's nuts. She's the glitch. At least that way, she can imagine how she might ever return, if that's what she wants to do. The mind is more malleable than the universe, as far as she knows. You don't need to be a genius or a video game character or from the future to manipulate the mind. The mind is made to be fucked with.
This, then, is the most likely, the most sensible, explanation—the world is the same, and Elisa Brown is the thing that is different. There's a phenomenon she has read about, not déjà vu, but déjà vécu, false memory. You think you remember something, but it's your unconscious mind that has created it, for purposes the conscious mind can't fathom. The thing you remember seems real. But in fact only the thing before you is real. This room. This bed. This man.
She wonders what time it is. The clock is on Derek's side of the bed, turned away because its glow is too bright. Has she slept yet tonight? She feels that she hasn't, but there's light outside, almost. Or maybe that is the moon. At this hour, whatever hour it might be, anything and everything seems possible, anything at all could be true. What has happened seems real. The possibility of return seems real.
If she isn't insane, and if she could return, what would that mean? That is, if her consciousness leaped from one body to the next, then who is to say the body it left is still alive? Maybe the other Lisa isn't in it, mourning now for Silas; maybe that body is dead. Maybe it ran off the road and through the guardrail and there is a funeral in that world, for her, going on today. Maybe Derek is waking up alone today to bury her.
Or who is to say that this consciousness wasn't just copied, that the other Lisa is the same as ever, working at the lab and carrying on with her lover? Even if she could go back, maybe there is no place for her to go. Her body there is already occupied by the original.
And if this is so, then what happened to the woman who occupied this body? If Elisa leaves it now, perhaps it will drop dead, too.
Of course all of these thoughts are predicated on the idea that the consciousness in question is, in fact, a thing. A thing that has to be somewhere. And this is not a notion Elisa Brown has ever believed in. The soul. The God who tends it. All of it, nonsense. The soul is chemistry, it dies with the body. Consciousness is an illusion, a piece of software. When the machine shuts down, the program isn't merely resting. It doesn't exist.
What has happened to her does not fit into this worldview. Yet there is none she knows of that will accommodate it.
Derek stirs beside her. She doesn't want to wake him. She slides out of bed, pads to the toilet, and then down the stairs to the kitchen and her computer. She is hoping that when she powers it on she will find an e-mail from Sam. But what she finds instead is one from Amos Finley. It reads, Lisa. Have you thought about seeing me one on one as we discussed?
The e-mail has been cc'ed to Derek.
Her reply goes to the therapist only. No, I don't think so.
She is startled, first at this apparent breach of privacy, and then even more startled that the therapist is awake and sending e-mail at 5:00 a.m. on a Saturday.
Elisa sits and considers. There is some code here, some message that she does not understand. What is the value of betraying her to Derek in this small way? It is an effort to force her to reform. They are ganging up on her, the men.
As if the e-mails have summoned him, Derek appears, wrapped in his thin cotton robe.
"Who are you e-mailing?"
His back is to her. He has opened the coffeemaker and is dumping the grounds in the trash under the sink. He does this with the brusque, exaggerated movements that suggest displeasure at her not already having done it.
She says, "The therapist is suggesting a private session. With me. He cc'ed you."
"What therapist?" he asks, filling the coffee urn with water.
She says, "Our therapist."
"Why are you calling him 'the therapist'?"
"Sorry," she says, after a moment. "Amos."
He doesn't respond. Something in the curve of his back indicates that he believes he's being fucked with. Please, Derek, she wants to say, it's me! But she says nothing.
He asks her, "So when are you going?"
"I'm not."
"Why not?" Derek turns, straddles the stool across the table from her. Behind him the coffeemaker begins to pant and moan.
Instead of answering, she says, "It's strange that he's e-mailing at this hour."
"It's strange that you are."
Their eyes meet. His hand reaches for a mug of coffee that isn't there yet. Fluidly he transforms this motion into a gentle stroking of the table's surface.
And just like that she would like to have sex with him. She is startled by the thought, and he seems to notice, and to notice what it is that caught her attention. He moves, just barely—the head cocked a fraction of an inch, the single eyebrow twitching over one eye. She closes the laptop, slides down off the stool, and walks out of the kitchen without a glance behind her. She takes off her nightgown on the way up the stairs. By the time she's standing naked by the bed he has crossed the threshold of the bedroom, his robe billowing around him.
It's quick and intense. His body is better here. Stronger. He doesn't seem younger—indeed, he looks and feels his age. But this is better than the concealing softness of the other Derek. Her own, heavier body feels more erotic. She makes no sound other than a single deep involuntary groan.
For those moments there is nothing to apologize for and everything makes sense.
They lie beside each other, touching at the hip. Down in the kitchen, the coffeemaker beeps five times.
"You're different," he says, this time without hostility or accusation.
"Yes."
"Do you know why?"
"Not entirely."
For a moment she thinks he's going to let it go. Then, "Do you want to tell me what part you know?"
"Not yet," she says.
"Will you tell Amos?"
"No."
Immediately she feels this is the wrong answer. Does it violate a rule? But he slides his hand up onto her hip and across her belly. Perhaps this is something they haven't had together for some time: a small secret.
25.
Two hours later they have eaten breakfast and drunk their coffee in comfortable silence. It's almost possible to consider the situation normal. There's a newspaper to read, so she stares at the words for a while, turns the pages. She tries and fails to empty her mind.
In the afternoon Derek goes out on errands. She opens her laptop and finds an email from Sam:
Mom
Nice hearing from you
No time/money to travel
Sorry
Sam
She reads the message several times. It feels heavy with meaning. After a few minutes' thought, she writes back: I want to come see you. She sends it before she can change her mind.
A few minutes' idle clicking later, she shuts the laptop, paces around the kitchen. They keep a phone list magneted to the fridge in the old life, but here there's nothing. She opens and closes several drawers, then finds an address book—neatly maintained, implausibly, in her own hand—in the drawer under the phone. She looks under Brown and then Sam, and finds an address, two phone numbers.
She picks up the phone, stares at the keypad. Puts it down, goes to the front window. No Derek. Goes back to the phone and calls the first number.
There's a click, and then she can hear a city street. For a moment, no one speaks.
Elisa says, "Hello?"
Sam says "Hi," elongating it into a question.
"Are you all right?"
Another pause, and then, "Sure. Are you?"
"Yes. Yes, I'm fine." Elisa lets out breath and experiences a moment of lightness, dizziness. Relief. She hasn't understood until now just how anxious she has been, how terrified of this moment. And for what? That he wouldn't be real. But here he is. She hears a car horn, voices speaking Spanish. "I'm sorry I've been... out of touch," she says.
"Uh huh..."
"This must seem... it must be strange to hear from me."
"Yeah... yeah..."
"Thanks for your e-mail. Did you get mine? I guess you didn't, I just sent it."
"Oh really?"
"I want to come to see you." She hesitates. "You and your brother."
He doesn't respond to that. He seems to be speaking to someone else—buying something at a store, perhaps. Somebody thanks him in a foreign accent. After a moment, she goes on.
"I hope that he hasn't... I hope Silas isn't... hurting you, Sam. Or making things... harder for you."
There's a long pause. Elisa feels the need to fill it.
"I realize that things haven't been good between us. For a while now. I... want to rectify that. Start to rectify that. If you'll let me. Some things have happened to me—" And here, she is suddenly and unexpectedly moved, and her throat closes. She has to pause. She takes a breath, says, "Some things have happened to, to change the way I see my life. And I need to try to make things right. With you and Silas."
He laughs, but it's fake—forced.
"Sam?"
"That guy." The voice has changed now.
Elisa says, "I don't—"
"Silas."
Here in the kitchen, she becomes aware of the ticking of the wall clock, and it seems very loud. Her voice is very loud, even though she is nearly whispering. "Is there a problem with—"
"That guy is such a dick. Seriously. He's so mean to me."
A chill goes through her. The receiver creaks in her hand: she is holding it tight enough to break it.
The voice that comes out of it says, "He's so mean! I hate him. I wish he would just leave me alone."
She says nothing.
"Boo hoo hoo," says the voice.
"Silas," Elisa says.
"I do hope you'll visit, Mom. Maybe you can tell him to stop bothering me? He never listens to me. Whoops," he says, "gotta run. It was great talking to you!" And the conversation is ended with a click.
She has just spoken to her dead son on the phone.
Forty-five minutes later, the sound of Derek's truck rouses her from sleep. She is on the sofa, her head tipped back on the cushions at a strange angle. Her neck hurts. Derek walks in and his body language communicates disapproval. He dislikes napping in general. In the other life, he overcame this prejudice, with effort. She sits up. "Hi."
"You're not getting enough sleep."
"No, I guess not. Where were you?"
"Home Depot," he says. He's wearing a very clean pair of jeans and an old tee shirt and standing on the living room carpet with a plastic sack dangling from his hand. It is about to start to seem strange, his standing there, when he lets out a small grunt and sits down. He drops the bag on the floor and takes her numb hands into his. "This morning," he says, "was nice."
"Yes."
"I want you to go see Amos. Please."
She makes herself sit up, cranes her neck, trying to get the kink out. Derek releases her hands and begins to rub her shoulders.
"Thank you."
"Will you go?"
She wants to answer him, but it's as though the whole charade has hit a wall. She exhales, feeling his hands on her. Her throat tightens and her breath catches and she tells herself that she isn't going to cry.
"I'm sorry," he says. "This is... everything has gone insane this week. I don't understand what's happening."
"We have to put this behind us," she says, finally. A shot in the dark. "Amos. Everything."
Her eyes are closed, she can't see his reaction. But there is despair in his voice as he says, "I don't understand how you can say that."
"We can do it. We have done it."
"I haven't. I haven't, not yet."
But she meant in the other life. For a moment she wants, very badly, to tell him. To come clean. She remembers how it felt last night, the secret between them, the information she planned to conceal from Amos. A path to a new life, a third one, that begins right now.
Isn't this something she has wanted, every now and then? Hasn't everyone wanted this? To just throw it all overboard, the bad decisions of the past, and start over? Her life, at this moment, is a nightmare: she is tired of pretending, she is tired of trying to figure things out. Let it go. Derek would do it with her, he would let it all go. Wouldn't he?
A new life. Starting now.
The phone rings once. Neither of them moves. It doesn't ring a second time. She feels it, that ring, as if it's a nail the universe has created, that has pierced her and fixed her to the sofa. She doesn't move or speak, other than to squeeze Derek's hand, a gesture that could mean anything. Sorry. Please. Love. Help. Get closer. Go away. In the end, he goes away. He kisses her sweating forehead and disappears with his plastic bag. A few minutes later she hears, from a distant part of the house, the sound of an electric drill.
That night an e-mail arrives from Sam: I don't get it. What are you doing? When Derek goes to sleep, she books a plane ticket for California.
26.
By the end of the following week she has mastered her job. It hasn't been hard. To other people, she has just appeared forgetful. She's made up for it with excessive cheerfulness. Half her day she spends online, following Silas on messageboards. She thinks she has found iterations of him in other places—other gaming forums, a forum about motorcycles, another one about serial killers. The latter, she discovers, is famous for rumors that some members of it are actually murderers—twice in the past, apparently, new members who joined discussions of killings turned out to be the killers. Silas is very interested in this phenomenon—he is always accusing people of being killers, and then, as on his game developers' forum, ends up temporarily banned. Finding him has been easy—though he uses different names on every forum, he always has the same signature line on his posts, an unattributed quotation: He saw himself in a strange city with his friend, except that the face of his friend was different.
The first part of the week went this way: after work on Tuesday, she appeared at the usual time to be picked up by Derek. But instead of getting into the car, she leaned in the window and told him she wasn't going to therapy. If he was going home, she would ride with him. But if he was going to see Amos, she would walk.
He appeared stricken. He gazed through the windshield and gripped the steering wheel with both hands and said, "I'm going to go."
"I'll walk home."
"In those shoes?"
"I brought sneakers."
They looked at each other. She should have kissed him. Instead she took a step back. He said, "Lisa. Are you going to leave me?"
For too long, she said nothing. Because that's another thing she could do with this third life. She could start over on her own. But she did eventually say, "No."
"Do you love me?"
"I love you."
She sounded certain of it, to her own ears, but she didn't know if she meant it or if her conviction came from her need to say it. Derek accepted it, accepted her love. He drove away. And she changed into her sneakers there on the curb and walked home. And in fact she has continued to walk home every day, even in the rain.
Now it's Friday, not quite noon. She could leave if she wanted to, but this office has slowly turned into a sanctuary. She likes it here: it's the place where she knows what's going on. She has a set of papers in her hand—printouts of some budget material for an interdepartmental committee, on which handwritten notes have been made. It needs to go to physics. She reaches for a campus mail envelope, then stops herself. It's a nice day.
Physics is on the other side of the science quad. She walks in a perfectly straight line, ignoring the cement paths, cutting through the grass, diverging only to avoid hitting trees. A young Asian man holds the heavy wooden door open for her; she nods her thanks.
The office is hot. They don't have air-conditioning, for some reason—just fans. There is a sense here of confusion and dishevelment—she likes it. She finds the department's equivalent of herself and hands her the papers. But of course that's not really why she came, is it. She doesn't admit it to herself until she is standing at the front desk, where the assistant, a woman around her own age, is chatting with a second woman, this one younger. They are talking about a movie they have both seen.
"Sorry to bother you," Elisa says to the assistant. "But I have kind of a strange question."
Both women look at her.
"Is anyone in this department—any of the scientists—studying the concept of... other... that is, parallel, or multiple, universes?"
The assistant says, "Like science fiction?"
"Well—not really. Or I guess maybe, but I mean in reality." She feels like a fool.
"Oh, geez, I really don't know," the assistant says.
Elisa claps her hands together, steps back from the desk. "Thanks. Sorry to bother you." Only once she's nearly to the stairs does she let out breath—she doesn't know what made her think that was a good idea.
But a voice stops her—"Hey! Uh, hello?" It's the young woman. She is half-jogging down the hall, in the manner of someone who rarely moves faster than walking speed.
"Hi."
"Hey!" The girl is out of breath. She arrives where Elisa is standing, gets a panicked look in her eye, takes half a step back. "I'm a postdoc here?"
"Ah." She is small and cute—cat-eye glasses, blue hair. A physics woman.
"I'm Betsy. I'm... yeah. So the multiverse?"
"Oh! I'm Elisa. Is that your field? Parallel... whatever?" She's embarrassed now—embarrassed to utter these words in this building, embarrassed to say "whatever" to a woman half her age.
"No, no. I'm doing experimental stuff—radiation patterns from quarks."
"I see."
Betsy takes a wallet out of the pocket of her jeans—a man's wallet—and slides a business card out from behind her driver's license. Her hands are trembling slightly. Elisa is impressed, for some reason, that she has business cards. It reads BETSY OROSCO, PHYSICS. And a web address. "That's my site," she says. "It's kind of a... physics blog?"
"Cool," Elisa says, and feels herself blush.
But the girl is encouraged. "Yeah, and, ah... I'm kind of interested in this stuff. The multiverse stuff. I mean, I post about it sometimes. It's a hobby."
"Physicists have hobbies?"
"Yeah, more physics!"
Betsy invites her into her office, and Elisa follows her to the end of the hallway, up a flight of stairs, and around a corner. They have to duck as they pass beneath an inexplicably low section of ceiling, and then they're in a little cul-de-sac with three doors. Betsy opens the one on the left and leads Elisa inside.
The office is intimate, cramped even, but higher than it is broad, with bookshelves on three walls covered with textbooks, papers, and unusual objects: toys, oddly shaped bits of wood, machine parts, circuit boards. The fourth wall is empty save for a tall narrow window. In front of the window there is a green aluminum desk, and, on a filthy Oriental rug, two stained aluminum-frame upholstered chairs. Betsy climbs behind the desk. It bears two laptop computers, a cell phone, and many stacks of books and papers. Elisa sits in one of the chairs, which is familiarly uncomfortable. She says, "How long have you been working here?"
"A year."
"You've really made yourself at home."
Betsy's face reddens. "I like to make a nest." And then, surprisingly, she says, "I'm thirty-two, you know."
"Oh?"
"Did you think I was younger?"
"I admit I did."
"My mother says I'll look like I'm twenty until I'm forty, and then I'll suddenly look sixty."
Elisa says, "Thanks, Mom."
Betsy settles in her chair until its back rests against the windowsill. The motion upsets a small cactus there, and its pot clanks faintly against another one sitting beside it. The younger woman doesn't seem to notice. "I know, right?" she says, laughing. "But she thinks I'm a lesbian and that science is a phase I'm going through. So yeah but..." She leans forward. "Do you know you could make a universe? Like, in a lab?"
Elisa doesn't say anything.
"Or your house even."
"Really."
"In theory! I mean, I can't. You can't. Well, maybe you can, what do I know."
Betsy is excited. She seems to enjoy having Elisa here. Out the window behind her a corner of the building is visible, this very building, and beyond it the science quad. They are in some kind of wing or extension. Yet this seems miraculous somehow, an optical anomaly. A building that is visible from inside itself.
"How?" Elisa asks her.
"I was just reading about this. You'd need a seed. A little tiny thing. Ten pounds of matter, packed into a really tiny space. And if it's all packed in there enough—so that it's basically a black hole—then the repulsive component of gravity, which yes there is such a thing, should be enough, once you trigger it, to expand that matter into a whole other universe."
Elisa says, "A little tiny one."
But Betsy is shaking her head before Elisa has even finished speaking. "No, regular size. Like with stars and galaxies and everything. Like this one."
At the words like this one, Elisa experiences a chill. For a moment, she can't believe it—she is actually talking about this.
She says, "Wouldn't it... blow everything up?"
"Nope. It would occupy its own space. Another space."
"And you could make this. A person could make this."
"Yeah. Well—in theory. You would need to smash the right particles together. To make the seed. And then, to trigger the expansion—it's tricky. I mean, nobody has done it. As far as we know."
Elisa leans forward. A cloud has covered the sun, the quad is in shadow, but light is still striking the corner of the building that is visible from the office window. "But maybe somebody has."
"Maybe this is it! The universe somebody made."
After a moment, Elisa says, quietly, "A person could go there?"
"In theory."
"But in reality?" Elisa asks. "Is that possible? Can you go there? To the other universe?"
"Well..." Betsy says, and there must have been something in Elisa's tone, some excess of hope, that is causing her to pull back from her initial enthusiasm. "You'd have to go through the black hole somehow. Which of course there's all kinds of complications there. Like it would compress you into a stream of atoms, which is to say you're dead. And then, you know, it's a black hole. So."
"So?"
She shrugs. "Even if by some miracle you survived the trip. You could never come back."
27.
"But listen to me," Betsy is saying, "blathering like an idiot."
Elisa shakes her head. "No, this is exactly what I wanted."
"I am kind of giddy, having a nonstudent visit my office. So, wait, you're... what's your deal then? Do you work on campus?"
"I'm an administrator," she says carefully, "in the biology department." After a moment, she says, "I used to be a scientist, too."
This seems to please Betsy. "So okay, wow. Physics? Not physics."
"Plant biology. Genetics."
"So how did you get interested in this? This stuff?"
Up until now, the meeting has seemed like a lovely bit of serendipity—the realization of a fantasy she didn't realize she'd harbored. Betsy Orosco is perfect: a probing intelligence wrapped in a sheath of innocence, good humor, and charming clutter. Indeed, Elisa could not have invented a better person to explain these things to her. It's almost as if she has created this strange little room, up in this obscure dusty corner of campus.
Elisa's palms are sweating. She grips the greasy burlap armrests of the chair. If this universe, if any universe, could have been created by someone, then who? Could she have created it herself? By accident? Is there a universe where she stuck with science? Moved from biology to physics? Worked on a particle accelerator? Smashed the right things together? Created new iterations of herself, her husband, her sons? Could this be only one of many? Could this be only one of an infinity? Did she mean to do it, or was it a mistake? Does she even know she did it? Maybe she didn't even notice that it happened. Maybe she thinks the experiment was a failure.
She is vaguely aware that an awkward silence has sprung into being. She looks up to find Betsy looking at her, biting her lip.
Elisa says, "I'm... trying to understand something. That happened to me. That is happening."
Betsy's response is quiet and tentative: "What happened to you?"
"I'm not sure I want to say."
The two of them gaze frankly at one another for a moment, and then Betsy turns away, leans back in her chair. The cactus pots clank.
Here it is again, the moment to tell or not tell. She thinks of the billions of women throughout history who have silently endured this same moment of indecision, the little fermata before confession. A last breath before the uncomfortable intimacy is forced onto the friend, or the lover, or the mother, or the sister. I was raped. I'm married. I'm in love with you. I'm gay. She's lying to Betsy: she is quite sure that she does want to say. What she isn't sure about is whether she wants to be heard. Because there are only a few possible good outcomes, and an infinitude of bad ones. Sorry, I have to go. No, that's crazy. Why are you telling me this? You need help. It's narcissistic, isn't it, this need to tell—to hear oneself give voice to one's feelings, to watch them register on another person, to watch the person shoulder the burden. There's no rational reason for it, just the relief from solitude. Betsy says, "Then maybe you shouldn't."
Her face is alert, the eyes wide, the lips pulled back revealing straight white clenched teeth. She appears alarmed—whether at the possibility of further, perhaps unwanted, intimacy, or at the sound of her own words, Elisa can't tell. Both, probably. The words are not unfriendly; Elisa senses, strongly, that Betsy likes her, likes that they are both women, both scientists. There is a great deal, it seems, that they might understand in one another, that they wouldn't have to explain, should they become closer. Elisa would like coming here, to the physics building, to meet her friend for coffee. She would like to hear more about Betsy's work, about her strange ideas, her speculations.
Too close, too soon: that would ruin it. Elisa is disappointed and relieved. She nods, grips the armrests of the chair, readies herself to get up and leave.
"No, wait," says Betsy, "I'm sorry."
"I should be sorry," Elisa says. "I'm taking up too much of your time."
Betsy's half out of her chair, her hand extended over the desk, the fingers splayed. "No, no, no. Please."
They are frozen like that for a moment. Okay, then, Elisa thinks, we're going for it. She relaxes back into her seat, and Betsy returns to her chair.
"You were going to say something that's important to you. I shouldn't have interrupted."
"You don't know me," Elisa says.
"Maybe that's good." They both sit in silence for a minute. Somewhere a door slams. "So."
"It'll sound crazy."
She's hoping for a No, no, it won't, but instead Betsy shrugs. Her face is expectant, but what is she expecting? To feel interested, compassionate? Or for a good story to tell her boyfriend, about the crazy lady who came to her office?
Well. No matter. Elisa sits up straight and looks out the window where the corner of the building is visible and says, "I was on a road trip. A few weeks ago. And everything changed. Everything around me. And me. My job, my body. My car. Things in my family are different."
She ventures a glance at Betsy, who is scowling in concentration.
"I mean, everything changed at once. It was all different. Instantly. The whole world. Or that's how it seemed to me."
They are both quiet as Elisa gathers her thoughts. The room is very hot. Now somebody walks past in the hallway outside. By the sound of it, this person is dragging a large cardboard box along the floor. Eventually there is silence.
"It still seems that way. It's still happening. It's like amnesia. Things have happened that I don't remember. Except I remember a whole other life in its place. My real life." She gazes directly at Betsy. "It feels like I switched from one life to another. And I'm not crazy, I don't think. Do I seem crazy to you?"
"No," Betsy says, but she is still scowling, still thinking, thinking, and it's not clear if she means it.
"I started doing research. Into different explanations. I'm... I thought maybe this... this theory could explain it."
Betsy's expression hasn't changed. She's sitting very upright in her chair, with one leg tucked underneath her, like a child. One hand hangs out of sight by her side and the other rests flat on the desktop. If she lifts it up, there will be a print there, outlined in condensation. Elisa closes her eyes and waits, and eventually Betsy says, "You mean you think you're in a parallel world?"
"I didn't say that."
"You think... you think that could be a possible explanation for your situation."
"I'm saying that's what it feels like. I am trying not to draw any conclusions."
Elisa opens her eyes again: it's time to look. The younger woman wears a curious expression, head tipped back, eyes unfocused, her lips slightly parted. Both hands lie flat on the desk now, and she is biting her lip. She says, "It's hard to see how it's possible."
"I'm just saying how it feels."
"No, no, I know, I get it." Betsy appears to lose herself in thought for a long minute. When she turns back to Elisa, she says, "You know this whole multiverse idea, you know where it came from? Who thought it up?"
Elisa shakes her head.
"William James. The psychologist. He wasn't talking about physics—he was talking about morals. Like, he was rebelling against the idea of predestination, of a universe that was a finished creation. A finished world. Where you have a role to play. And a God that cares about that role." Betsy is sitting up straighter now and gazing directly at Elisa. "For him, in a moral multiverse, your choices matter. You have free will. And what you do means something. It makes something happen."
The physics Lisa, she thinks, smashing the right things together. "Are you saying I made this happen?"
"I'm saying... that something made something happen."
"You think it's all in my head. Not out in the world."
Betsy is shaking her head, but there doesn't seem to be a lot of conviction in it. "I don't know what it is or where it is. Maybe your head and the world are the same thing."
Elisa is exhausted. Her hands are shaking and she is slumped in the chair. She is given to think of Silas, the interview she found. Games, he said, have to invent themselves. Rain, briefly, spatters the window, then abruptly stops. Did she do that? Did she make the rain? She is suddenly very confused; it's as if she is drunk, or high. "I don't think I know what you mean."
"Me neither," Betsy admits. She seems resigned to something, it is not clear what. "But let me have your e-mail. I know a guy. He would probably find you... interesting to talk to."
28.
Since Monday she and Derek have kept their distance from one another. When they have spoken, it has been politely. There has been no more sex. They're too nervous. He appeared thoughtful when he returned from the therapy session she didn't attend, but he has said nothing about it. So far she has resisted asking, but now, Friday night, over dinner, she looks up at him with the intention of doing so.
He's looking at her. They both turn away, then turn back. She smiles. He doesn't.
She used to talk a great deal, she remembers. Before Silas. When she drank, she would talk even more. She loved it—it felt... low class. They would go out with friends, or with another couple, and Elisa would find somebody to talk to, to talk at, and she would just go for it. Sometimes she would be off-putting to this person, and the person would notice, and would shift her attention to the group, or to someone else. Sometimes her interlocutor would be patient, would endure her. Sometimes something would click and this person would respond with equal enthusiasm. If the person was a man, the encounter would sometimes feel sexual. Derek both liked and didn't like this. He liked that she relieved him of the need to make conversation, he liked her energy. He didn't like it when she became too intimate, too quickly, with strangers. His heavy hand would grip her leg under the table, midway between the knee and the waist. This was a warning but it, too, was sexual. His fingers would land close to her crotch and they would stay there for a while. Sometimes she dialed it back a bit; sometimes she kept going, just to bother him. At these times his grip would tighten. They would argue on the way home, then go to bed.
She is wondering where this person went. This talkative, combative Elisa. She wonders if this Elisa has come back, in this world—if Amos Finley has brought her back, and now here the "real" Elisa is inhabiting the poor woman's body, dragging her back into reticence, into the realm of mystery. She wants to think of something to say to Derek that will evoke the old days, her wilder self, but she's at a loss. He lowers his gaze, sets down his fork, draws breath.
She says, "Have you ever played one of Silas's games?"
He's surprised. "Silas's games?"
"Have you?"
"You would know if I had."
She says, "Where can you get them? Do they have them at the mall? Are they for regular computers, or do you need a thing for your TV, or what?"
Derek shrugs, eyes wide.
"I want to try one."
He doesn't say anything, though it appears that he is trying to.
"Come to the mall with me," she goes on. "There's a game store."
He stares at her. Then, as though after long calculation, he nods.
He drives them to the mall in his truck. (In this world, they don't seem to like her car. She's glad; she doesn't like it either. The truck feels good—there is only enough room for the two of them in the cab. For their marriage. It's their marriage truck!) Elisa is surprised how many people are at the mall on a summer Friday night. She would expect they'd be out having a good time instead. Or perhaps that's what this is. She and Derek make their way past clothing and gift shops. Elisa, suddenly ebullient, takes Derek's hand.
He gives her a strange look but doesn't let go.
The video game store is adjacent to the food court. They walk in and are instantly confused. The walls are lined with little boxes depicting heavily armed and graphically stylized men and women. The games are made for different systems, but they all look the same. Everyone else in the store is under the age of twenty. To Elisa's surprise, there's a pretty girl behind the counter. She's wearing a nose ring and asks if she can help them.
"We're looking for something by the company Infinite Games."
The girl nods. She wants to know which game.
"Uh... Mindcrime's Mirror or something, is that one of them?"
"Mindcrime: Destiny's Mirror. Yup. It's pretty okay." Then she recommends a different game and gives them an appraising look. "The other one's kind of confusing. If you're newbs. Are you?"
"Definitely," Derek says, and Elisa is mildly surprised he even knows the term.
"I think we really want that Mindcrime, though," she says. She can feel Derek's eyes on her.
"Okay..."
"Does it run on a regular computer?"
"It's a console game," the girl says, more kindly now that she understands how clueless they are. "Do you have an Xbox?"
"No," Derek says quietly.
"Do you sell them here?" Elisa asks.
"Oh yeah, sure." The girl shows them a display of boxes. The consoles are expensive but not as expensive as Elisa assumed they would be. She says she'll take one, and a copy of the game. The girl suggests an extra controller and Elisa says fine. Derek is staring at her.
"You're serious?"
"It's one of Silas's," she says. "The game."
He nods as if any of this makes sense. He appears so confused here, among these strange young people, and all the light and color. He belongs in a library, surrounded by brown things. She touches his shoulder, kisses him.
The girl looks on in apparent amusement. "Don't get discouraged," she says, as she packs the boxes into a large shopping bag. "This is like the worst first game you could ever play. Do all the training. You need to figure out how to work the controls. After that, it's about a twenty-hour game."
"I'm sorry," Derek says. "What does that mean?"
"That's how long it takes. To finish."
He appears flabbergasted. "You're kidding."
"For you, though," she says with a smile, "longer."
Now they are driving home. It's hot and the sun is in their eyes. Derek drives with the stiff, silent precision that indicates there are questions in his mind. Elisa takes the opportunity to gaze at his face. She has not looked at him directly for more than a few seconds at a time since whatever is happening to her happened.
He is harder here, to be sure—cleaner, more controlled. This was always a part of his personality, of his physical self. This advanced containment. She met him, or rather saw him for the first time, at a party a boy had taken her to. The boy was a law student, an undergraduate. The party was mostly grad students. Her date was proud to be invited—he went around introducing her to people he barely knew and tried to burrow into conversations that were over his head. She didn't find this appealing. One of the conversations was with Derek and two other men, and while the other two men bantered with and gently mocked Elisa's date, Derek merely stood still, sipping his drink, his face hard. Not angrily so. Impassively. He struck her as a passionate man who had mastered his passions. She couldn't keep her eyes off him and didn't learn his name.
The boy took her home and she went to bed with him but never returned his phone calls after that. She started studying at the law library, a place she had never previously so much as entered. At first it was in the hope of seeing the man from the party. But eventually she came to like the anonymity of the place, the inscrutability of the information it housed. All of the facts were there, but none could be seen, not immediately. Not without searching for them, without knowing where to look. This was not like science. Scientists had to generate the data with experiments. The law, its precedents and interpretations, were written down. The law was here—all of it, right here, all around her.
Scientists, of course, didn't hang out in this library. She was the only one. One afternoon she was sitting at a table in the third-floor reading room and looked up to see Derek coming toward her from the stacks. Deliberately, almost defiantly. When he arrived he crouched beside her, crossed his arms on the tabletop. The hairs of his forearm were touching her notebook. He said, "You were at a party last semester." Elisa nodded. "Come get a drink."
(Later, months later, he would tell her, "I was terrified." This was the first time he ever disappointed her, not because he had been terrified, but because he hadn't been, but found it necessary, or perhaps just advantageous, to lie that he was. He wanted to appear more susceptible to strong emotion, and more experienced at managing it.)
This Derek, the one now driving home from the mall, is more like that Derek than any Derek she has seen in years. In the other life, raising the boys broke down his defenses, made him transparent, but threw his flaws into sharper relief. His willingness to blame others, Elisa in particular, for shared problems. His incapacity to accept a problem as chronic and unsolvable, and to readjust his expectations accordingly. By the time of Silas's death, he was worn out. The world had disappointed him. If you asked he would have said he was happy, and he wouldn't have been wrong. But it was the kind of qualified happiness that he never expected he would have to accept.
Here, though, in this world, the fortress of Derek has been partially rebuilt. It must have cost him real effort. And he doesn't want it to crumble again—surely, if it did, he would lack the will, the energy, for another recovery.
She loves and pities him. He is ill equipped for this life, for the other life, for any life. Though she supposes you could say that about anybody.
29.
It takes Derek about ten minutes to hook the video game console up to their television set. It's black, and the controllers are black—to Elisa they look like amoebas, with the various buttons and sticks as organelles. The game comes in a DVD case, with a picture on the cover of a man, a young man in tee shirt and jeans, seen from behind, peering into a distant yellow light. The box is hard to open, so Derek goes to the kitchen for a knife.
"This is stupid," he says, but he's laughing at himself.
"This is what people do on Friday nights now," she tells him, though what does she know about what people do on Friday nights?
Inside the box is the game disc, with the title printed on a black background. In a slot on the inside cover is a glossy black paper that reads "INSTRUCTIONS: FIND YOURSELF."
Derek turns the paper over, looks again at the box. "That's it? That's the manual?"
"I guess the game tells you how to play it."
"Ay caramba."
They sit cross-legged on the carpet in front of the TV, power up the console, and slide in the disc. It whirs. They expect some kind of credits, some title sequence, but instead they hear a click and a whine, and a yellow dot appears in the center of the screen. It expands, rapidly, like the picture on an old television, into the image from the cover. This version of the image is subtly in motion—the man is panting, his shoulders heaving, the muscles of his back trembling slightly. He is scratched and bleeding; his clothes are dirty. A bass chord sounds, and then a male voice, echoing as though in a cave: "Who am I?" Then the chord evolves into a slow orchestral dirge, and options appear on the screen. GAME. EXPLORE. CONTROLS. OPTIONS. EXTRAS.
"Explore?" Elisa asks.
"Nah," says Derek, and expertly manipulates the joystick until PLAY is underlined. He presses a button and the menu screen disappears.
"How did you know how to do that?" she asks him.
Derek shrugs. He seems mildly embarrassed and pleased. It's the sort of thing he can do—pick up a tool, or a gadget, or a computer program, and just use it. It is, she must admit, a bit of a turn-on, as is his pleasure at his own facility. A few notches more self-satisfaction, though, and it would be insufferable. This is a key to his appeal: confidence bordering on, but not crossing over to, arrogance.
Their Man is standing in a forest, alert and subtly breathing. A message appears on the screen: PREPARE TO FIND YOUR DESTINY. This is the training the girl told them about—they have to teach themselves to run, climb, throw, hit. On-screen icons indicate which controls to use. The screen shows their character, the Man, in various environments—a forest, an empty house, a road, a ship—where some obvious task, like scaling a wall or opening a safe, has to be accomplished. He is always shown from behind, his muscles working, his thin hair flopping on his head. Elisa is amazed at the photographic quality of the graphics—it looks like a movie. The training is fascinating and vexing, and the game proper doesn't seem to have really started yet. Derek says, "Jesus. Are they all this complicated?"
"The girl said no."
"Jesus."
When they decide that they've mastered the controls, or come close enough, Derek gets up and comes back with a bottle of bourbon and two glasses. They haven't drunk the stuff together in years, at least not in the world she knows. The ceremony in his bearing suggests they haven't in this one, either. He pours the drinks over ice and they clink them together.
"To an unusual evening," Derek says.
"To Silas."
The warmth drains from his face and he turns away from her, to the screen. "So are we playing this thing or not?" He drinks.
"Let's."
As it happens, they can't both play at once—it's single-player only. They take turns manipulating the controls and exploring the world of the game.
It is, of course, a mystery. The player is a nameless Man who wakes up bruised, beaten, and starving somewhere in a forest. He has nothing in his possession save for a compass—which doesn't appear to work here in the woods—and a creased and torn photograph. The photograph is familiar to them both: it is clearly based on the family picture from the lake, the one that Elisa remembers she has left at the frame shop with Larry. Though she hears Derek draw in breath, neither of them remarks on the resemblance. The faces are different, but the poses are the same.
During the first fifteen minutes, they die repeatedly by falling into holes or out of trees. Only when morning breaks in the game are they able to make their way out of the forest; they do so by following the sound of passing traffic. When at last the Man climbs onto a road, Derek says, "So, how long are we going to be doing this?"
They are both on the second bourbon. Elisa is shocked. It's only ten. She has no intention of stopping. "Oh, I don't know," she says. "Another hour?"
Derek shrugs. She can see that the game has caught his interest, but he doesn't want to stick with it. It's bothering him. She gets up and refills her glass. She is beginning to get drunk.
They flag down a pickup truck driven by a grizzled old man. Where to? he wants to know, and they hear the Man say wherever. Eventually the driver buys him breakfast at a diner. Then a waitress comes on shift, tying her apron around her waist, sees the Man, and screams.
Derek actually cries out and drops his controller. "It can't be!" the woman is saying. She backs into the kitchen, weeping.
"Rosie, you got a problem with this guy?" says a voice, and the Man turns to find a burly biker type standing behind him, cracking his knuckles. There is something sickening about this sound: it is wet and deep, like the popping of greenwood in a fire.
"Fight him! Fight him!" Elisa shouts, and when Derek makes no move to pick up his controller, she punches him on the arm. He flinches. "Derek. Derek! We have to fight this guy!"
"Let's pause," he says. "Can you pause?" They both try various buttons until they hit upon the right one, and a menu pops up: EXIT. OPTIONS. Derek exhales, seeming to shrink to half his size. He says, "I think I've had enough for tonight."
"You're kidding!" Yes, she is a little drunk.
He pats her shoulder. "I'm beat, Lisa. I wrote lectures all day."
"Oh."
"You kind of hurt my arm."
"Sorry," she says. She glances at the screen, then back at him. "Mind if I stay up?"
"Go ahead." He stretches. "Don't forget to save the game. In case you die."
"Thank you," she says. And then, "Derek. I'm going out to see them."
"See who?"
"I bought a plane ticket."
He's silent. He is staring not at her but at the biker on the screen.
She says, "In three weeks. I'm taking the Friday off."
His response is very quiet. "Why?"
"Do you want to come? You aren't teaching. You can come."
"I don't know."
"When was the last time you flew out to see them?"
He gets that look—the fear that he doesn't know what's going on. "You know."
"When?" She feels reckless. "Just say it!"
He doesn't reply, just stares at the screen. After a moment he rubs his face. Says, "He could be funny sometimes. Silas."
Derek has half gotten up to go to bed and then sunk down again, so that his legs are folded under him, as though he's getting ready to pray.
"Just cuttingly, shockingly funny," he goes on. "The way he would impersonate people. Walking by. Do you remember that?"
She nods.
"Even when he was, Jesus, eight or nine. He would do those conversations between people. At restaurants? Do you remember the old man with the cowboy hat and the girl with him? In, what the hell was it, the place that used to be where Subway is now?"
"The Arbor? No, the Terrace."
"Yeah, with the plastic ivy. He did the Texas accent, 'Weaaauuul now, darlin', y'all lemme know if ya find a tooth in them french fries.' Remember that?"
"Of course."
"And then the girl, 'Oh, Pappy! Oh, Pappy!'"
It's true, sometimes they laughed at Silas until they cried. He knew exactly what to mock. God forgive them, they even laughed when he made fun of Sam. At the dinner table, faced with a food he didn't like, Sam's chin would drop and tremble, his cheeks collapse, his eyes narrow and moisten. Silas had it down. Sometimes he would even beat his brother to it—as soon as the plate of beets hit the placemat, before his brother had a chance to react, Silas would pull The Face, and Elisa and Derek would convulse with laughter.
Oh, God, it was wrong. It was so wrong to encourage, but it was so funny. She thinks of minefield, Silas's online alter ego, and she wonders where that part of her son went. By the time he died, it was gone. His mockery wasn't funny anymore. And here, this game. There's no humor in it. Yet it compels her all the same.
Derek has stood up to leave. He says, "Maybe you could turn the sound down a little."
"Sure."
He drops to his knee, kisses her on the head. "Good night," he says, and he grips her shoulder as though to fix the moment, as though she might run away. "I still don't understand what you're trying to do and I'm afraid you're going to ruin everything. But I love you." And then he goes to bed.
30.
She only means to play for another hour, then join him. But she ends up playing all night.
She doesn't fight the biker, in the end—instead she backs off, leaves the diner, and while she's in the parking lot the waitress comes out to find her. "Is it really you?" she says. She's young and pretty, and Elisa can't help but see her through Silas's eyes. Is this a girl Silas made? A girl he wishes were real? There is a tenderness here in the way the girl's features are rendered: her red-brown hair, tied back loosely, a nose slightly too large for her face. The way the clean apron nevertheless bears faded stains that can never be washed out, and the way it creases when she gestures, the fibers frayed and weak with age. How is it even possible to evoke these details in a video game? It is impossible, immersing herself in this world, not to feel that she has missed something about Silas.
Of course every aesthetic decision here cannot have been Silas's. There is probably a team of programmers, graphic designers, and the like. And yet she feels, powerfully, that this character, this girl, is the product of her son's mind. This world, the world of the game, is Silas's, as well—it is as though she has been allowed to enter into his consciousness, to see the way he sees. It is something he made to satisfy himself. And it feels realer, fuller, than any version of him she was ever permitted to see in life.
But then again, how hard had she looked? It's true that, up until his adolescence, his emotions appeared to live on the surface—his problem, as they saw it, was impulse control, an incapacity to keep himself in check. His actions, as they saw them, were the unfiltered product of his subconscious. They lived in the world he made, by necessity, and he refused to enter the world they wanted him to live in.
But he did change, around the time he turned twelve. He fell quiet and began to brood. And though this made their days more orderly—fewer messes, less violence, fewer shouting fights—it also threw Elisa and Derek into a state of paranoia. What was he thinking? What was he going to do next? His quiet hostility became more disturbing than the acting out once had been, and Sam, who had endured their shared childhood with his wits largely intact, began to show the signs of deep anxiety and, eventually, depression: sallow complexion, sunken eyes, bitten fingernails. Sam hid in his bedroom much of the time, while Silas haunted the halls with a kind of regal insolence. He frightened them.
But perhaps there was nothing to fear. Maybe it wasn't merely his demeanor that had changed, but his desires, his emotional aims. Maybe he was waiting for Derek and Elisa to discover them.
Instead they nurtured their paranoia, let it grow and spread. Each accused the other of trying to sabotage the family, of giving up on Silas, of giving up on Sam. Each accused the other of infidelity before either was actually guilty of it, and each used the accusation as justification for the act. There was a strong sense, in that household, of impending dissolution. Both of them were tired. They indulged the part of themselves that just wanted to get it over with.
But what if, behind Silas's seemingly impenetrable affect, lay a nascent empathy? What if he was trying to find a path out of the wilderness of his childhood, through the games he played, the books he read? What if the Silas who made this game was, in fact, that Silas—the one they chose never to get to know?
She has stopped drinking, but the night has the quality of a bender, with periods of sudden strong emotion, and of blankness, and of pointless hilarity. The girl, it turns out, knew the Man from a period, some months before, when she was under the control of some thugs who lived in her trailer park. Then, the Man had gone by the name Jack, and though he was only passing through, he managed to chase the men out of town and help the girl, Rose, pay her mother's medical bills. Jack, she explains now, wouldn't say where the money came from, and indicated that Jack was not his real name. He said goodbye and good luck, and disappeared.
But now he's back. What happened to him? The Man shakes his head—he doesn't know. Rose gives him the only thing he left behind—a torn sheet of notebook paper with a six-digit number written on it in an unfamiliar hand.
"I have to get back to work," she says. "Don't mind Rocky—he's just trying to help me and Mama." If the Man is looking for information about himself, Rose tells him, he should try the motel in the center of town—that's where he stayed. And when he finally figures it out, "come back for me" she says and runs away, into the diner.
The motel gives way to a bus bound for the city, which, as the hours of night pass by, leads Elisa down a spiral of increasing ridiculousness: the criminal underground, domestic spying, terrorists, government conspiracies. The number on the paper is a combination—there's a safe, a post office box key, an encrypted document.
It is all, of course, adolescent in conception, but beautiful in execution: lavishly detailed, astonishingly full. Maybe all games are like this now, what does she know. But every time she tries to enter a building, there is something there for her to see. Every time she approaches a character, that character has something to say. Silas, whose motivations, whose desires, were always so inscrutable, has created a world and left it open to whoever might wish to enter.
When the sky framed by the living room window begins to lighten, she glances at the clock and sees that it is nearly five in the morning. She climbs the stairs to bed, sleeps until noon, eats a sandwich and drinks a glass of orange juice, then returns to the game.
She sees Derek a few times as the hours pile up; sometimes she can hear him in the kitchen, preparing food; at one point he walks past with a hammer, and several minutes later she hears pounding somewhere in the house. He pauses behind her a few times to observe her progress through the game—a visit to a mental hospital, then a run-in with the FBI, digging in the woods for a buried time capsule and riding the bus to a distant prison for a visit with an inmate—and she tries to fill him in, glancing occasionally over her shoulder to make sure he's paying attention. But mostly he seems to be waiting, waiting for her to finish.
It is late on Sunday morning when she finally comes to the end. A slow trudge up the driveway of a white-clapboard house in an affluent subdivision. A knock on the door. There they are, the family in the photo, the people who rejected him, who sent him away because of his choice to become a government spy, a killer. But there was no choice—he'd been framed and it was the only way out. It doesn't matter, he is told by the family patriarch, who refuses to let him over the threshold, behind which his mother and sister cower in fear. They want nothing to do with him. Go away, they tell him. We don't want you here.
Elisa can retaliate, if she wishes. She has learned how to do things. She can beat these people to death with her fists, burn their house to the ground. She understands enough now about the way the game thinks to know that this is a possible ending, an acceptable ending. Instead, she turns and walks away. Returns to the diner to collect Rose and take her and her mother away from their terrible little town. At the end, they are standing out in the road, hand in hand, facing the sunset, but not yet moving toward it. The Man, exhausted by his efforts, is panting, just as he did at the beginning; the women's hair is lifted by a breeze.
And then, a closing sequence. The camera, though of course there is no camera, lifts up as though on a crane, and a distant landscape is gradually revealed: not mountains, not the sea, but a trailer park, lavish in its dilapidation, pickup trucks and dirt bikes moving through its ragged streets; a commercial strip, populated by crumbling big-box retail spaces and empty cracked parking lots; a landfill, dump trucks and bulldozers swarming over it like ants, and tiny gulls swooping overhead; a cemetery, weedy and overgrown.
That's what lies ahead for the Man and Rose and her mother: suburban decay and ennui, pollution and filth, death and obscurity. Game over.
Elisa wonders what sequence she would have been shown had she chosen to set fire to the family home.
She turns off the television and the game console and lies on her back on the living room carpet. She ought to be tired, but she isn't. Instead, her mind is clear. She closes her eyes and thinks. When she hears Derek enter the room, she says, "You're right."
"About what?"
"I'm going to go talk to Amos."
There's no response. She opens her eyes, tips her head back. There he is, upside down. He's staring at her.
When she calls the therapist, he tells her to come right over.
31.
She gets lost. All she remembers is the name of the road—Orton Road—and that it's on the west side of the lake. Eventually she finds it, goes the wrong way on it, drives for miles in confusion. Then she backtracks and everything begins to look familiar.
Clouds have moved in and the temperature has dropped. There will be a thunderstorm. The meadow behind the therapist's office is empty of deer and the grasses are bent by the wind. When she knocks on the office door, nobody answers. She hears her name being called: he's behind her, on the back stoop of the main house. He turns and goes inside and she follows him.
The house is low and close and smells like frying meat. It has the air of being lived in alone—somehow clean and squalid at once. Books are everywhere and the old windows distort the outdoors. He leads her into a living room, dark and comfortable, with a sofa and coffee table and desk and easy chair. This is where he spends all his time, clearly. He stands in the middle of the room and gestures at the sofa.
Elisa sits down and waits for the therapist to do the same. Instead, he paces for a moment, as though measuring his thoughts. He seems smaller today, more intense and professorial. The kind of professor who doesn't get tenure. His demeanor, his house suggest a man who has withdrawn from active life, declared himself an observer.
At last he lowers himself into the easy chair. He knits his hands together. There is a wobble in his voice, the slightest sign of nervousness, as he says, "You were to have been open with me."
"I have been."
"Something is different."
She has dressed in business clothes, a skirt and blouse. She catches herself tugging the skirt down over her knees.
He says, "Do you want to tell me what it is?"
"I'm not sure I know."
He is agitated. His hands separate and rejoin. She is reminded of cell mitosis, the first time she saw it: a black-and-white film, in high school, that somehow seemed more real for its flickering jerkiness. The jittering little lives, straining to separate. The nucleus, exploding into two, pushing at the edges of its tiny world, stretching the cell walls until they broke. And then each half, identical now save for experience, drifting apart. As if they were never one.
Amos Finley twitches. Flinches, maybe. "The way you are—it reminds me of the way you were when you and Derek first came to me. You're being cagey—secretive. But nothing is supposed to be secret here. Didn't you promise that?"
"Did you make that promise, too?"
He tries on an expression of hardness. But what she sees in his eyes, his slump, is a kind of panic.
She says, "I can't remember. The promise, I mean."
In response he closes his eyes for several seconds. Sits up. His back creaks, or maybe it's the chair. When he opens his eyes again it occurs to her that he might be in love with her. The thought makes her sad, though not with pity. Rather with a sense of the impossibility of everything. The number of emotions in the world that can't find purchase anywhere and are wasted. The therapist, she can see, is at a loss. There is some authority he had over her that is clearly no longer accessible to him.
"Listen, Amos," she says. In her tone she is aiming for tenderness. "I'm not trying to be difficult. I honestly don't remember."
He is gripping his knees now, working at the fabric of his slacks. "What... precisely... have you forgotten?"
"Everything," she says. "Everything up until a few weeks ago."
He is gazing at her, frowning.
She goes on: "I don't remember coming to you, or any of the promises we made. When we were here a couple of weeks ago, it was like the first time I'd ever seen you. I don't recognize my life. My job. Derek—he's—he's not the same. Do you understand?"
He sits back, sighs. "You come back from a trip you took alone. To a business conference. And you begin to act nervous and confused, as though you're trying to conceal something. Then you cease therapy entirely. And now you want to tell me you have... amnesia?"
"It's more complicated than that."
She sits back, stares up at a corner of the room. She has a sudden desire to clean this house—to haul these old chairs and lamps out to the curb, rip up the carpet, scrub every inch of the place. She would like to reform this man, reform his life. She says, "Amos, were you ever married? Do you have children?"
The question seems to surprise him.
"Humor me," Elisa says. "Pretend I don't know anything about you."
"Twice divorced," he replies, quietly. "An adult daughter. From whom I am estranged."
"Have we slept together? You and me?"
The question startles him. "No."
"Okay. Good." There's an energy in the room now that she likes—she feels as though progress could be made. She feels like the woman who just finished that video game—a person who can find a path and walk down it. "Good. Maybe you can help me."
"That's what I am trying to do," he says.
"We need to make a new agreement. A new promise. Because the woman who promised things to you and Derek—that was somebody else. That wasn't me. We're going to make a new deal, just between us."
He shifts in his chair. It is clear that he doesn't like it. But he's curious enough to play along. He says, "Go on."
"When I'm here with Derek, I will try to be that woman, but you need to understand, I don't remember. You need to help me out."
"I am here to help you."
"Here's the thing, Amos. It's not that I forgot. It's like I never knew. I remember everything, but I have different memories. I remember my son, Silas, dying, nearly a decade ago. And Derek and me drifting apart. And I'm having an affair with a man named Larry who now, here, doesn't seem to even know me. And I have a different job, and I wear different clothes, and my house is messier."
He is scowling, concentrating.
She says, "A couple of weeks ago I was driving in my car, and it all changed, the car and everything, my clothes, my body, and I became this person. Who is in therapy with you. And is not having an affair, and has a marriage with these rules in it that I don't know where they came from.
"This is not my life," she says to him, leaning forward. She is excited, astonished, that she is managing to say exactly what she means to say. "This belongs to somebody else. Some version of me that you know and I don't. And I am pretending to be that woman."
She slumps back in her chair, lets out breath. "That's why I'm different," she says. "Because I'm somebody else."
There.
For a while, Amos Finley simply stares at her, and Elisa stares back. He closes his eyes, rubs his beard. He drums his fingers on the armrest of his chair, opens his eyes again, gives her a long appraising look. Then he gets up and leaves the room.
She hears him in the kitchen. Bottles rattle as the refrigerator door is opened. She hears ice cracking and clinking. A few minutes later he comes back into the room holding two ice waters in pint glasses and hands one to Elisa. She thanks him. He sits down.
He says, "You need to tell me what you want."
The question takes her by surprise. "I came here because you told me to."
But he dismisses this with a shake of his head. "You came so that you could say this to me." He is back in charge now. His voice is clear and directed. He says, "Do you want to be cured of this apparent delusion, this dream of a previous, or alternate, life? That is, do you want to live fully in the life you are now occupying? And that I have helped you create? Or do you want to hold on to what you believe is real, and alter this life to suit?"
Until now, this question might have been easy to answer, had anyone asked. She wanted to hold on to what she believed was real. But now she isn't sure. She admits this to him—she says, "I don't know."
"All right," he says, "then that's something we'll have to figure out. I want to make sure I understand: you believe that Silas is dead, and has been for some time."
"In the world I know, he is dead."
"In the world you know, yes. In your memory, you haven't seen him for years. You know nothing of him, this young man who is now an adult."
She says, "I've been following him online. I know what his job is. I know Derek and I rarely speak to him."
"Do you want to see him?"
"I'm going to," she says. "I've been in touch with Sam. I bought a plane ticket."
He appears surprised. He says, "You understand, much of the work I have done with you and your husband has involved helping you to detach yourselves from your children. Silas in particular. You have chosen your marriage over your children."
"No" is all she can say.
He nods. "That's what you've done. That's what this reality is all about. From my perspective—and I am just telling you what I believe based upon what I'm seeing—from my perspective, you are having a psychotic break. Some part of you is rebelling against the choices you've made. Your guilt has gotten the better of you and you are denying the reality of your children's estrangement from you."
She is beginning to feel uneasy now. "I did not choose my marriage over my children."
He is nodding, nodding, nodding. "The woman I know did exactly that. Sam allied himself with his brother, moved away with him. You tried to separate them, to persuade Sam to come home. You became depressed. You were hospitalized, more than once. This is not your first break. Your marriage nearly fell apart. Finally you gave up on your sons."
"I love my sons. I love Sam."
"That is not in question, Lisa, but you and Derek put them behind you."
It is like the moment on the road just after it happened, the semi blowing past, her head on the wheel, the undented soda can crushing itself in her mind. Panic is blooming in her. She opens her mouth to deny for the third time that she chose her marriage over her sons.
But the truth is that she believes it. That it is possible. The psychosis, the hospitalization. Making the choice. She is, was, capable of this. More so, certainly, than the universe is, of moving her from one reality to another?
This is not your first break.
She has rarely bothered to remember—allowed herself to remember—the year leading up to Silas's death. The van crash was like an exclamation point at the end of a cruel running joke—when they buried him, she took that whole year, rolled it up, and dumped it into the grave with him. They were of a piece.
But in truth, the crash was a fluke—an interruption of that time, not a completion of it. And she can remember feeling close to giving up. Lying in bed awake, making deals with herself. What would she sacrifice for it all to go away? To return to what she used to have with Derek, the kind of love that once defined her life, that was the point of her life, that moved her to give up the life she thought she wanted?
Elisa supposes that, in her world, in the world she knows, she suffered a first break after all. The year of blankness, of losing weight and embroidering until her fingers bled. But what was her first break here? What pushed her to it? What did it take her away from?
She looks up so suddenly from her thoughts that Amos gives a start. She says, "What was it? What broke me the first time?"
The question makes him uncomfortable. He fidgets. "You were addicted to the internet. Groups, forums. Many of them about children with mental illness. And politics, you became obsessed with politics, and you were angry all the time. You stayed up all night on your computer, posting on various forums under various names, and, according to Derek, you would only talk about those things, your children and politics. Sarah Palin, you became obsessed with Sarah Palin. You lost interest in sex. Derek nearly moved in with another woman."
Elisa says, "Sarah Palin? Seriously?" She barks out a little laugh.
Amos doesn't, or won't, smile. His hands are folded in his lap and he is watching her intently. "You became dangerously thin, and chain-smoked."
"I hate politics! I don't smoke."
"You chain-smoked. Your sons moved out west, and you engaged in loud and sometimes inebriated conversations with them on the phone, at strange hours. And you ended up in the hospital, after Derek found you knocked out cold on the kitchen floor. You hit your head on the corner of the counter and the coffee urn was shattered on the floor and you had been burned by hot coffee. You were treated for malnutrition as well as for your injuries, and when you got out and stabilized, that's when Derek threatened to leave. Instead, the two of you agreed to come see me. And we worked out the terms. That you would leave the boys alone, and quit the internet and smoking, and eat regular meals with him. And he would give up contact with the woman."
"Debra."
"Debra." He untangled his hands, wiped them on his knees. "Who, in any event, now lives elsewhere and is married, if I remember correctly."
"Forever?" she says.
"Sorry?"
"The boys, forever? We are never to have contact with them again?"
He groans, and his tone, when he speaks, is exasperated. "You were not to have contact with them. Derek could, if he liked, though he has not seemed inclined to do so. I don't know. You were to cut off contact, and it could be restored only through mutual agreement. Between you and Derek."
"I've broken the agreement."
He nods. "Yes, you have. Derek is afraid for you, Lisa. He isn't angry. Or rather, he has managed to control his anger. He's worried. He is afraid you're breaking down again."
There is a long pause as she tries to get it all straight in her mind. It makes a horrible kind of sense. All those aspects of her personality that she fears, over which she feels she has only the most tenuous control, those are the ones Amos has informed her led to her breakdown. It is how things might have gone.
"Lisa?" Amos says, sounding somewhat alarmed, and she is surprised to find herself standing up. "Where are you going?"
"I'm going home," she tells him over her shoulder, and then she's out the door and heading for the car. She expects him to be right behind her when she turns, but he's not: he's standing behind the screen door, looking out at her with his hands at his sides. She waves at him through the windshield, to reassure him, and he raises a hand.
He looks sad, as though he's failed.
32.
Instead of going home she drives up the lake to the state park. There is almost nobody there, because of the impending rain. A pickup truck is parked in the lot and a man is asleep on the rocky beach. A fat woman is listlessly fishing off the end of the jetty.
Elisa stands on the shore beside the stone benches and gazes at the power plant on the other side of the water. This is where the photo would have been taken. She tries to imagine what circumstances would have brought them here together—her and Derek and both boys. In her memory, it was hard getting the boys to do anything, let alone with the two of them together, by the time they were thirteen and fourteen.
Until now, Elisa used to think of those years, when she thought of them at all, as a time when Silas had to be endured. And indeed, he was unpleasant to be around—imperious, disdainful, he rarely opened his mouth except to mock or criticize.
But his mind was elsewhere—he was thinking about his life outside the house. By this time he had friends, sycophants, female followers; he had come into his bad-boy good looks, the James Dean cheekbones Lorraine was pleased to note had come from her; the thick black hair the wind always seemed to blow into an arrangement of perfect, studied nonchalance. He used the phone a lot, went out on his bike without asking, following just enough of the house rules to avoid open conflict. He showed every sign of not caring what they thought.
But Sam simply grew sullen, wouldn't come out of his room. Moved slowly. The collars of his shirts were always frayed and wet from nervous chewing. He licked his lips incessantly, leaving the skin around his mouth livid and peeling. There were bags under his eyes. The older of the boys, he nevertheless seemed like the baby of the family, fleshy and uncertain. Silas, of course, was uncommunicative: he was rarely there, or distant if he was; he answered their questions with the bare minimum of effort before ducking away, back into his private world. But Sam's unresponsiveness was more vexing. He didn't try to escape. He just sat there in the kitchen, or lay on the bed, eyes open and blinking, thin hair stuck to his moist forehead.
"Are you all right?"
"I guess."
"It's late, you should go to sleep."
A shrug.
"Do you want to talk?"
"No."
"Did you get enough to eat tonight?"
Another shrug.
The shrug had become a signature gesture. The shrug, the slump. Sometimes a limp. "Are you hurt?" "No." Elisa began to get the idea that Sam was feigning injury, not for their sake but for his own, for the small pleasure of privately comforting himself. She shared this theory with Derek. He seemed faintly repulsed by the idea.
Lorraine said, "It's genes. Nothing to be done. Luckily," she added, patting Derek's hand, "our family has never been moody."
Moody. Elisa's father used to call it "blue." "Your mother's a bit blue today." "Poor Lisa," he said during meals, when Elisa let her teenage hair fall into her face, and ground her teeth, and gripped the seat of her chair with both hands as though, if she concentrated hard enough, she might be able to fly away on it. "Poor Lisa is feeling blue."
Sam was not blue. He wasn't moody. He was depressed.
Elisa is exhausted from her weekend of gaming and the session with Amos. She finds a picnic table and sits down at it. It's astonishing that it hasn't rained yet; the clouds are heavy and black and the lake surface is lashed by the wind. She will sit here until the first drops fall. She imagines herself making a run for it, hands over her head. It seems important not to take cover now—she wants to be pushed to shelter.
Poor Lisa is feeling blue. Though when she reflects upon the way she actually feels today, it comes to her as no visible color, nothing as natural as sky. She feels like something blinding and artificial and impossible to look at directly. Ultrasomething. Infrasomething. She closes her eyes hard and hears the muscles tightening in her head. Betsy the physicist seems less real to her today, the possibility of madness more palpable.
When she thinks about that last year, before the accident, what she remembers is the desperate, guilty feeling of her love for Sam beginning to fray around the edges. The component of her love that was pity, curdling and turning into a kind of disdain. Resentment at his weakness. At the ways he was like her—or, rather, the ways he was like the parts of herself she disliked. His willingness to give himself over to other people's ideas of him, his willingness to give up.
She stayed up late with him while he lay sweating, and smelling of despair—she sat hunched over with her elbows on her knees, trying to find the right combination of words that would make him talk to her. What is it? Did Silas say something to you? Sam would talk—he wanted to talk—but he wouldn't talk about his brother. Instead he spoke in abstractions, in philosophical conundrums. And not very interesting ones. Why, he wanted to know, should he get up in the morning and go to school when nobody cared whether he showed up or not? (But your father and I care, Lisa told him, over and over. As if that would matter.) What was the point of it all? (She could not pretend to have an answer. The only one she knew was: the pursuit and expression of love. And she couldn't say this to her son, who loved no one and, to hear him tell it, was loved by no one but her and Derek.) Why did people like Silas when Silas was a dick who mocked and belittled them? Why did those same people turn around and mock and belittle him, Sam? And why didn't Silas tell them to stop?
She doesn't remember what she said. But the answer, of course, was that power attracted and weakness repelled. At times, that power could manifest itself as charm, as intelligence: as the positive attributes that people pretended to seek in one another. But it wasn't the manifestations that mattered, it was the power.
(Elisa remembers better what she wanted to say than what she actually did say: Goddammit, Sam, you're older and bigger than he is. Don't be a fucking pussy.)
Silas was powerful. And Elisa respected him for it. His evident indifference to, even disgust for, his own mother: she respected it. She respected it because she felt it herself, about her own mother, who liked to preempt criticism of her weakness by calling attention to it, with feigned pride. "Lisa's toes have been poking out of her shoes for three months and I didn't even notice!" "I forgot, completely forgot, to make Lisa lunch!" "I was so thoroughly drunk that night I had to send Lisa to the corner for cigarettes, if you can believe that."
It was true that at the time Elisa liked it. She liked the squirrelly little threesome she made with her parents—the shield of nondescript scruffiness and cultural superiority they suspended between themselves and the world. She felt proud to be her parents' daughter: she thought there was something real, some empirically verifiable quality, that justified their stance of amused condescension against other people. Now she knows it was fear.
It wasn't until she was in college, when she began to meet confident people, powerful people, that she understood. It was Derek's confidence, his ability to approach others to ask for what he wanted, to put the past behind him with finality, that crystallized her desire: he was the antithesis of her parents.
Of course Sam was not Elisa's mother. Elisa herself wasn't even like her, not really, and there was as much Derek in Sam as there was Elisa. But it was impossible not to see him as a manifestation of Gemma Macalaster, casually exerting her influence from afar. Or, rather: exuding, seeping. She was like a fog, like the mildly acrid cloud of cigarette smoke that had always surrounded her, that followed you out of the apartment in your clothes and hair and was with you wherever you went. She could still smell it sometimes, or believed she could, when she was drifting, finally, off to sleep: she could smell her mother's cigarettes in the pillowcase, in her nightgown, as if the old woman had visited just long enough to lie here and imprint her particular brand of passivity on the bedclothes.
She didn't want to think of her mother when Sam slouched off to bed, uncomforted, unconvinced. But she did, she did. And Silas, though he was, she understood, a deeply flawed young man, reminded her of nothing so much as the boys she loved, the bad boys her mother loathed, and the hard and obsessive parts of herself that she most valued.
She is pulled out of her thoughts by a change in the light. Behind her, from the west: sunshine. The wind is steady and slow now, and the clouds have moved on. The sleeping man on the beach is gone and families are arriving, laying blankets on the grass, unpacking baskets. It didn't rain after all. It isn't going to rain.
33.
Derek begins to seem slightly afraid of her. He has stopped asking questions and doesn't attempt to initiate sex. Elisa doesn't either. She feels as though her existence is a cup filled to the brim; she is trying to stand very still until her trip. For this reason she doesn't accompany him when he goes to see Amos on Monday.
When he gets home he looks at her with confusion and distaste.
"You're thinner."
"Yes."
"I thought you looked good. Before."
She shrugs. Did Amos tell him what they talked about? Does she care if he did? She knows she should tell him herself. But she is afraid to.
Derek says, "And you're not wearing makeup anymore."
"No," she says. "You're just noticing now?" He doesn't respond, and after a moment he goes upstairs to change his clothes.
There is a part of Elisa, an increasingly prominent part, that wants to follow him, to undress him, to make him love the woman she is turning into, which is to say the real her, the her of the other life. And there is a part of her that wants to push him away for good. The strange thing is, if she could have the Derek of the other life right now, the one from whom she is estranged, the hard one, the one whose love for her is rote at best, vestigial, she would love him, she would take him back and love him. But she does not want to let herself love this Derek, the one who has chosen her over their children.
She resists the temptation. She doesn't follow.
Instead she e-mails Sam to tell him when she'll be arriving—the trip is in a couple of weeks. He writes back within the hour: I don't think you should come.
Why not? she replies. But he doesn't answer.
Wednesday morning, still half asleep, she reaches out and wakes Derek by stroking his arm. It's five thirty. He gasps, leaps out of bed, then stands facing the window in a kind of ready crouch.
She sits up. "What is it?"
He turns and says, "I was dreaming." But he doesn't get back into bed, he stands there staring at her, blinking, still in a state of near-violent attention. In the gloom, backlit by dim gray sky, he looks like some kind of animal, or worse, something half human. For a moment she actually hates him.
That day she takes a long lunch and uses her new bus pass to go downtown. She walks the few blocks to the frame shop. Larry isn't behind the counter. But on the low shelf beneath the frame samples lies a small flat package, wrapped in brown paper, with a yellow sticky note attached, and she knows it's her picture. She asks the girl who's there, "Is Larry in today?"
"He's on lunch break."
"I'll come back," Elisa says, and walks back toward the bus station.
But it's lunchtime, after all. She goes into a Korean café. Was this place here in the other life? She doesn't remember it. The worlds are blurring—she is slowly merging with the woman she was before, and she can't remember all the differences. The thought fills her with desperation. Soon they will be so intertwined she'll never get them apart. She sits down by the window. A crack runs through the glass. She stares through it at the people passing, the buses collecting and releasing passengers.
A man is crossing the street toward her, carrying a plastic drugstore bag. It's Larry.
"Can I help you?" says a voice.
There's a woman standing beside her, holding a notepad. Elisa stammers out an order. Larry walks in the door of the café and sits down at a table across the room.
"Anything to drink?"
"Just water."
She has to crane her neck to see him. The café is almost empty—a silent couple in the corner, a man text-messaging over the remains of his meal. She shifts herself to the other chair at her table, in order to face Larry. He is reading the menu, though not for long. He speaks to the waitress for just a moment. A regular. Then he takes a magazine out of his drugstore bag and begins to read it.
He won't look up. She's sure of that. His ability to concentrate is tremendous. It's irritating to go for a walk with him, because he doesn't want to stop and look at anything, he just wants to walk. But when that concentration is trained on her, he sees nothing else. The intensity of his attention, in fact, can be overwhelming. Which is why he's good to have as a lover. But not necessarily to be married to.
The magazine is about music—a jazz magazine. Does she already know this—that this is an interest of his? Perhaps this is one of the things that's different.
She could go over there. If she waits until the food comes, it will be awkward. She could sit down and ask to join him. If he's the Larry she knows, he will be disoriented, will appear annoyed. Then he'll capitulate, adjust, accept. Enjoy.
Yet she must force herself to gather her satchel and glass of water and cross the room. She tries to convince herself that her reluctance emanates from anxiety. That she wants him so much, it is making her lose her nerve. Larry looks up.
"Mind if I join you?"
He is definitely surprised. "I—do we know each other?" But before he has even finished saying it, he recognizes her. His expression is one of puzzlement and slight relief.
"Elisa Brown. From the shop."
"Sure. You didn't give us your contact information."
"I just stopped at the shop, just now."
"Good. So you like it?"
"Ah—no, I didn't—I didn't pick it up. You weren't there—I didn't want to explain myself."
He takes a quick look at the door, then back at Elisa.
"I was already here," she explains. "Sitting by the window. And recognized you."
She thinks, Why are you making this so goddamn difficult? She wants, very badly, her real life right now. Where all this has already been accomplished, and sex is already being had. He says, "Of course. I'm sorry—please sit down."
She sits. He closes his magazine with some displeasure and tucks it back into his bag. Then he faces her squarely and folds his hands in front of him.
Even sitting still, he appears agile, efficient. His features are fine, the eyes alert. He isn't her type, isn't what she once thought was her type, which is men like Derek—larger, broader, more solid, as though they're made of the same thing all the way through. Their strength obvious in their posture, their movements. Larry is more like a machine, a collection of moving parts. You can see every muscle in his face. He could be an adventurer, a traveler. Not that he actually is. In fact he's a homebody. He lives in a tiny spartan house wedged between the lake and the train tracks, west of the park. Or at least he did, he's supposed to.
She isn't sure what she wants to say to him.
"I hope I'm not interrupting."
"Not at all. I can read anytime."
"You like jazz?"
He raises an eyebrow and it seems to pull up the opposite corner of his mouth. "I didn't always. About six months ago I started thinking I needed a new interest. So I began to study jazz. I bought a turntable and amplifier—for some reason I thought I should learn via records, rather than CDs."
The waitress comes out with Elisa's food, looks temporarily puzzled, finds her, brings the meal to their table. Elisa thanks her.
"You were saying."
"Yes—I've become preoccupied with jazz recordings. On records, I mean." This is characteristic—interrupted and asked to continue, he will do so without the slightest hesitation. As though to be offended is beneath him. "There's something about the physicality of it, I think. The needle dragging in the groove."
She had used to find this kind of conversation pretentious; Larry taught her to enjoy it. She's missed it these past few weeks. But now it is irritating her all over again. She sits on her hands and draws a breath.
"You should go ahead and eat," he says.
"What did you get?"
He smiles. "The same."
34.
They walk to the frame shop together and Elisa pays for her frame. She doesn't open it, and he doesn't ask her to. Unspoken between them is almost everything. He's single. He saw her wedding ring.
"We should do that again," he says, as she leaves.
She hesitates before saying, "We should." The hesitation is almost, not quite, enough to be embarrassed by. She is bewildered by the effort it is requiring to achieve the proper mindset for infidelity. It seems important to want him. We should. A necessary obligation.
She returns late to the office and nobody cares. At her desk, she unwraps the photo. It's a nice frame. Silas leers out of it—it's as if he's in a rock band, posing for an album cover. The cocky self-satisfaction. The pleasure of knowing that the family has shaped itself around him. (Stop that: He's just a teenager. Of course he's cocky. So were you.) She puts the picture back into the drawer it came from.
That last year, the year before he died, Elisa and Derek considered separation, seriously enough to make plans, to announce them even. They would separate the children, as well—in fact that was the entire point, or so they told themselves. Derek actually volunteered to take Silas. It was like him to do this. He was better at managing his aversion to Silas than he was his pity for Sam. Silas had begun to display the crowing satisfaction of the winner of a card game. He tidied his messy room, packed up some old things, had a friend drive him to Goodwill to drop it all off. (Maybe it was Ricky Samuelson, his killer. Maybe in the van he died in.) He was preparing for the next phase of his life, one he had created for himself, that he appeared enthusiastic about and eager to get under way. And Sam resigned himself to Elisa with depressing immediacy, looking up from the book he was reading, nodding once at her tear-stained face before returning to the book.
They had said too much, she and Derek. They had pointed out each other's shortcomings, using the children as illustrations. They indicated what qualities of each parent had been brought to bear upon the suffering of each son, which problems might have been avoided but for which habit of being, which blind spot. And each had accused the other of the very thing they feared the most about themselves: that they regarded their own child as frightening and repulsive.
Rule 5. Do not use the children to attack your partner.
They changed their minds, of course. Self-disgust was punishment enough. Silas made his disappointment known, and so, cruelly, did Sam, though it barely had a chance to register before Silas's death rendered it all meaningless. Did it happen in this life, too—the decision to separate, the announcement, the retraction? It is too exhausting even to speculate.
She peers at the clock in the corner of her computer screen. Two forty. She's glad it's still early. She doesn't want to go home. She stalks Silas online for a little while, does a web search for his sig line, trying to find another online iteration of him.
It would be a relief to be mad, wouldn't it? To accept Amos's diagnosis and embrace this notion, that the events she remembers with such intensity and conviction are the products of an imagination broken by guilt and grief. She could submit to more therapy, to medication. She would be given paid time off from her job. She could wave goodbye as Derek went to work in the morning, spend the day catching up on her reading, allow herself to be treated gently and a little fearfully, as any sick person would be. She could put that weight back on and give herself over to Derek's carnal needs. And her own, for that matter.
What does a crazy person look and sound like? Certainly not like this—showing up on time at the office every morning, staying until five, sending and receiving e-mail, taking meetings in clean and tidy clothes. No—she wants to be, she feels like, a person to whom something inexplicable has happened. If there is madness, it belongs to the universe, not Elisa Brown. The mind is not enough to explain it.
This reminds her of something. She does a search for "William James multiverse."
Betsy was right, James coined the term. There are many hits. One of them is a forum, MetaphysicsNet. She reads it for a while. Past lives, alien intelligence, magnetic energy, parallel worlds. The parallel worlds subforum is crowded and extremely active. It's a hot topic, thanks to recent movies and television shows. There are a lot of threads discussing its plausibility, based on scientific and psychological research.
She bookmarks the site, turns off the browser, does a bit of work. Judith comes in, closes the door behind her, whispers "I fucked him," in reference to whom Elisa can't remember, then describes the encounter in detail. Elisa has come to like Judith, she has to admit. Judith is full of life. Judith is abidingly real. In this world, Elisa clearly has come to appreciate things that are alive and present. This Elisa is more accepting. Talking with Judith, she thinks she ought to adopt this way of being, then bristles at the notion that, in yet another way, the worlds are bleeding together.
When Judith is gone, Elisa tips her head back, gazes at the sprinkler and water pipes overhead, falls asleep. She dreams that she is performing oral sex on Larry, that he fills her up like a water balloon, and she begins emitting heat and light like a sun. She wakes up gagging and gasping for breath. Her neck hurts.
The working day is over. Derek picks her up. They go home, eat, drink, end up making love, though without particular intensity, and for no apparent reason other than that it has been a while. They go to sleep. Drifting off she thinks, All of this is impossible, we're doing impossible things. People do impossible things, all day long.
35.
A few days later she is reading an e-mail she has received from a man named Hugo Bonaventure. It's full of exclamation points. He'll be on campus all day Tuesday and Wednesday! He would love to talk about the multiverse! There's a cell number.
Clearly this is Betsy's friend. When she gets to work, she calls him.
"Yes, yes!"
"Mr. Bonaventure?"
"Yes, yes!"
They agree to meet the next day. His office is in the Keller Center, about as far from the biology department as it's possible to get and still be on campus. When it's time to go, she wishes she'd brought shorts and a floppy hat, as the air is very hot under a cloudless sky. She is glad, however, that she has abandoned the pretense of bringing formal shoes to work, and now wears sneakers all day.
She remembers the last long walk she took in Wisconsin, before the switch—around the cemetery and the park down the road. She stood over Silas's grave and for the first time didn't cry. She felt sadness, but also acceptance and relief. The memories this act stirred up were mostly memories of other visits to this cemetery, when her feelings had been more profound. (This is what happens, she supposes, to dramatic events: they create feelings that create other feelings, memories that give way to memories of having them. The older you get, the more life seems like a tightening spiral of nostalgia and narcissism, and the actual palpable world recedes into insignificance, replaced by a copy of a copy of a copy of a copy. The sunshine today agrees: it has rendered the town in high relief, grainy and posterized, the colors too bright. So fake it's a new kind of real.)
But the main thing that day was the trip itself, the way it fit into what her life had become. A ritual for her to wrap her guilt and grief in, so that she could separate it from the rest of her days. And then, eventually, this package, this bundle, began to feel familiar. Comfortable to carry, easy to set aside.
This was the year she realized she had moved on. As much as that was possible. She realized that she had moved on, that her life had been restored to her. And then the thing that happened happened.
She wishes she'd brought a bottle of water, though the walk is less than a mile. By the time she's nearly there she's sweaty and her bra and sneakers are chafing. She needs new clothes. She needs some air-conditioning.
The Keller Center for Theory and Practice is a kind of science-meets-humanities think tank, housed in a nineteenth-century brick mansion. Professors inside and outside of the college apply for fellowships there; they are supposed to get new ideas about their work by talking to one another. They hold monthly lectures and receptions, which she has never attended. Or maybe this version of her has—she doubts it, though. She arrives ten minutes early and stumbles in through the heavy oaken front door, expecting to find a receptionist, some cubicles—an office. Instead the place has the look and feel of somebody's house—someone unusually tidy but blind to the ravages of time. The front room, a parlor really, contains too many sofas, all of them worn and lopsided and from the seventies. She flops down on one and spread-eagles herself, her burning limbs.
The building is silent, save for a slow pulsing drone that must be an air conditioner: it is very cool here. A broad low table before her is covered with academic journals and, oddly, back issues of a glossy men's magazine. Light is blasting through the leaded windows but the room still seems gloomy. She likes it—she wants to move in.
The next thing she knows somebody is poking her in the shoulder and she is reflexively wiping drool from her face.
"'Ello? 'Ello? Is Missus Brown?"
"Oh my God. I'm so sorry."
She gathers in her arms and legs, blinks, tries to look over her shoulder. But he isn't there.
He's in front of her now, on the sofa across from her, a tall gangly man in shorts and button-down shirt. She can barely make out his face, as he is backlit by the blazing afternoon sun. But his hair, his massive ball of curly reddish-blond hair, glows so brightly she can see the shape of his skull beneath it.
"Is very comfortable here, no?"
"I guess the heat did me in."
"So you are tired, yes, I see. We will talk right here then, okay?"
"Where is... are you the only one here?"
"Yes! They have all gone away in summer!" He flutters his hands.
"Of course."
"So! So! I am speaking to Betsy Orosco! You want to talk about the multiverse! This is good, people like it, it's in the TV shows and movies a lot, you know?"
"Yes."
He's so tall, and the sofa he sits on so old and sunken, that his knees come up to the middle of his chest. He keeps them spread far apart and his legs are very hairy and she can see right up the leg of his shorts to the outline of his balls underneath a pair of white briefs. Somehow this seems cosmic, profound. Why does a man need two? Why not just one, a big one, dangling beneath the penis like a veined and whiskered egg? No, it has to be twins. She thinks of the poor redundant sperm, born daily and reabsorbed, unused, by the body: a cycle, the two balls in tandem, equal partners, competitors perhaps, in this Sisyphean undertaking. She laughs, but it comes out as something more like a sob.
"So..." says Hugo Bonaventure. "Something unusual, correct, you are saying is happening to you, yes?"
"Something unusual, uh huh." She sits up straighter now, rubbing her face. She still can't quite make out Bonaventure's features.
"You are professor of what?"
"It's not... I'm not a professor. I work here. At the college."
"Okay, okay..."
"I am just a regular person," she says, and marvels that those words would ever come out of her mouth. "I just want to know."
"Okay, okay..."
He nods, nods, is waiting for her to speak. And it occurs to her that she has no idea what she wants to say. There is a long silence.
"How much did Betsy tell you? About my... situation?"
"Just, how do you say," he replies, making curves in the air with his hands, as though illustrating a voluptuous woman, "you give me the, not the silhouette?"
She doesn't understand, and then she does. "Outline."
"Yes, the outline, thank you, yes. I make the recording?"
He taps his shirt pocket, where there is a bulge the size of a pack of cigarettes. She understands that he has a tape recorder in there and is requesting her approval. Without thinking she tells him sure.
She collects herself and goes through it all for him. Her life before it happened, the trip to Wisconsin, the drive, the moment of change. The differences. She tells him about telling Amos Finley, "my psychiatrist," she calls him. Toward the end of this monologue Hugo Bonaventure appears to grow agitated, impatient. She has discovered that if she closes her eyes for a moment, then opens them while they are trained on his face, she can catch a glimpse of it before her pupils contract from the light. His nose and chin are long, his eyes deep. He appears handsome and a little bit frightening, an exemplar of that extreme kind of effete masculinity accessible only to men without the slightest awareness of its existence. "Okay, okay," he says. "Okay, okay."
"Yes?"
There is a pause, punctuated by nodding, as though he is charging himself up. He says, "Okay, you say this moment, there is a change, can you give the description again? Of what it is like?"
"All right." She tells it again, more slowly this time, trying to add detail. The positions of the clouds. The rivets on the guardrail. The shape of the crack in the windshield. She nearly chokes up describing the crack, realizing that she might never see it again. The chip, like a leaning triangle, where a rock struck it, and the strange way the crack rises from it, ruler-straight, for six inches before it veers off to the left, then right, and heads for the upper corner of the glass. Hugo Bonaventure nods, taps his bare knees with his long fingers.
"Yes, yes, I ask you questions now, okay?"
"Sure."
"When this happens, yes?, there is a sound?"
"What kind of sound?"
"I don't know this," he says, "only you know this, the kind of sound. Maybe there is a ring, a vibration, something—" He claps his hands together and the claps echo off the high ceiling. "—something like a pop, a bang?"
She had not considered this possibility. A pop? A bang? She says, "The window was open, and then it was closed. So the wind noise was gone—the wind shut off."
"Yes?"
"But it was smooth. The change. The old car, it was noisier, and the new one is quiet. More solid. Everything went quiet."
"So a lack of sound. But no pop, bang, ring."
"No," she says. "Nothing like that."
"Okay, tell me, okay, do you smell something different? Or in the air, yes, there is some kind of crackle? Current?" He raises his hands and wiggles his fingers. She can hear them whispering against one another. "You feel anything on your skin? Something electrical? Or maybe some flash, there is light, not outside light but inside, on your retinas, you see?, the pop of light, the impulse, pop!, you see?"
"Yes, I see. Let me think," she says. She is trying to remember. The smell. Yes, it changed. There was the smell of the road and of the dusty interior of the Honda and then the stale recirculated air and plasticky odor of the new car. And the temperature changed, and there was the movement of air, of the hairs on her arm. But there was no pop, no flash. There was no smell that didn't seem to come from what was around her.
She tells him this. She says it was smooth, the transition: sudden, but clean. "There were no... artifacts. From the change itself. It didn't feel like something was happening. It was just, there was one thing, and then there was another."
If he is disappointed in this, he doesn't let on. He says, "Okay, very good, now I tell you straight from the bat, yes, maybe I think you are delusional? But perhaps not?"
"Uh... all right."
"But this is a thing we, that is, science, this notion, this idea, it is something real? We think, there are many ways for an event to transpire, the laws of physics allow this, there are probabilities, and all these probabilities are perhaps real, they have the same chance of being real, you see? A thing happens, any thing at all, it creates the universe of happening and the universe of not-happening. Do you understand? Always there is the branching, and every branch is a universe, and they are all real."
Her skin is puckering as the sweat evaporates from it and she shivers. She stops trying to make out Hugo Bonaventure's face. He's just a blank surrounded by light. That's enough. She says, "So this is real? The worlds are real?"
"Well, okay, sure. This idea of real, maybe this is not so important for the physics, do the math, maybe it's a little fanciful, one can make it with the math, sure, but to test it, how do you do this, okay?, how do you make the experiment?"
Her shivering has intensified. She feels very strange—as though her body is making energy. As though she is not quite in control of it. She clenches herself, clamps down on herself from the inside, in an effort not to melt here, to lose herself. The sofa is very soft and she feels very far down in it.
Hugo Bonaventure says, "This group, my colleagues, they do an experiment, okay? They take the tiny metal, like a tongue, a tiny thing, you barely see it with your eye, yes?, and they make it very cold, it is called the ground state, the lowest possible energy, yes? And here, at this place, the metal, because it is small and cold, here you don't have the disruption to the quantum state, correct? So they can make the measurements. And they pluck this metal—" He reaches out and flicks a finger. "—they pluck it, and they see that it is vibrating, yes? But it is also not vibrating."
He pauses, as if to let this sink in. She is nodding now, nodding the way he is nodding: they are nodding at one another.
"This is, then, okay, the universe where the object is plucked and the universe where the object is not plucked. My friends, they test it, they see both states."
"Both universes?" she asks him. "At once?"
"That is correct, yes, yes."
"You're saying they have seen this? Another universe?"
"They do, they see this."
She says, "Could they see me? The other me?"
He gets up. He paces for a moment in front of the coffee table. She is still fixated on the spot he has left: the image of his head, the void his head has imprinted on her vision, is left behind.
"Not quite, no, this is not possible. You are far too large." He lets out a kind of cackle. "Ha ha!, no, I cannot say that to a woman, but yes, you are too large for the physics to see. The outside forces, they push against you, yes?, they disrupt the quantum state. But maybe they can test!"
"Test me?"
He suddenly lopes around to her side of the coffee table and sits down beside her. She can see him clearly now: here he is. In profile, he is a study in extremes: his nose and chin appear even longer now, and bent, his brow a shelflike protrusion. His body gives off a moist nervous heat, and she realizes that the air-conditioning has made her really, really cold. The long pants that tortured her on the way over are now woefully inadequate.
He points to her bag, which she has brought with her and which is sitting on the floor between her sneakered feet. He is snapping the fingers of both hands. "You can give me, what do you say, two possessions?"
"I... possessions?"
"Two things you own, okay? One thing you have in both universes, the other one you have only here. Ha!, we would like the thing only from the other universe, but you can't give this!"
Elisa picks up the bag, holds it on her lap like an animal. "You want to take my things and... test them?"
"Not me to test them, my friends in California! No lab for this here, ha ha, sorry New York State, it's not so good. But my friends, they are doing this research, they can test maybe, okay? You give me two things, I send them, maybe we find something out, you never know." And he puts out a palm and beckons with his long fingers.
She gets it. One thing from here, one thing from there. She opens up her bag, roots around. The first thing she sees is a tube of lipstick she bought for an academic dinner years ago, some gala thing involving a guest of Derek's department, to which she had been persuaded to go. She used it that one time but didn't like the color, and doesn't like lipstick, and she never used it again. But here it is. She removes it and hands it over to Hugo Bonaventure.
"This was in the other place," she says.
"Okay, okay, very nice," he replies, and wedges it into the breast pocket of his shirt, beside the tape recorder. She's a little disconcerted—shouldn't he place it in some kind of specimen bag, affix a little label to it, something? But he's the scientist here, not her, not anymore.
She peers into her bag again. The light is dim; it's hard to see. There's her driver's license, which is different from her old one, but she needs that. She opens one side pocket, then the other. Shoves her hand into each. Comes out with a piece of paper.
It's the list, the five rules. The paper is creased and furred, the words, in her own hand, blurry and fading. She reads,
4. Account for your time.
Her fingers fold the paper, pointlessly, in half, concealing the list, and she hands it to Hugo Bonaventure.
"And this is from only here?" he asks.
"Yes."
"It is, how to say it, it is not from recent, but it is before you see the change?"
She nods. "That's right. It's old. It was in my bag when I got here."
His acceptance of this strange frame of reference appears total. He tucks the paper, insouciantly, into the pocket with the lipstick.
And then, abruptly, he stands up and sticks out his hand. "Well! Okay! I send these things away!"
"I... well, all right then."
She takes the hand, thinking he means to shake, but in fact he pulls and she rises, involuntarily, to her feet. For a moment she thinks she might topple over, but she manages to right herself, with the help of his free hand on her shoulder.
"Ha ha!" he says. "It is like we do the dance."
Her limbs ache—the air-conditioning has frozen her muscles solid. She feels dizzy and tiny lights zoom across her field of vision. Hugo Bonaventure is saying something. Then he is withdrawing, crossing the room, climbing the stairs. He waves from the landing, and Elisa waves back.
She goes out into the insane light and begins the trek back to the sanctum of her office.
36.
Next Friday morning. It's August now. She rises early, showers, takes her already-packed bag out to the pickup. Derek would ordinarily be the first one in the driveway, standing next to the open driver's side door, his arm on the roof, or leaning against the hood and scrolling through his phone. But today he's trailing behind. There has been an air of desperation in the house, as though time is running out; his eyes have lingered on her too long, too many times. He's been waiting for her to do something, say something. She knows what it is, and hasn't. This is part of what makes him a good lawyer—his ability to draw out a rival with silence. But it isn't working with her, because she doesn't know what to say.
On the way to the airport, Derek says, "Look, Lisa..."
She waits. He waits.
He sighs and again falls silent.
For a moment, she considers telling him everything. What happened to her, what she remembers, Betsy, her session with Amos, even Hugo Bonaventure. Watching him drive, she remembers how much she used to enjoy being a passenger beside him. His calm at the wheel, his deliberateness. Economy of movement. Driving her places, he made her feel safe—or rather, made her feel comfortable indulging her desire to be protected.
But today these same qualities seem to her evidence of a deep conservatism, a fear of inadvertent revelation. She can imagine very clearly what his reaction would be if she tried to explain herself now—silence, a frown, careful consideration. I believe you, he would say. I believe that you think this happened. He would apply logic, lay out possible remedies, mostly involving the solicitation of outside professional help, the application of psychoactive drugs.
Elisa doesn't want to have that conversation. She doesn't want to embrace his frame of reference. Now, she just wants to live in this new world, delusional or not, and see it to its end.
They are almost there. His hands are wringing the wheel as he says, in a voice edged with desperation, "Will I ever know?"
She can't remember when she's heard him sound this way. Or known him to be in such a position. She is tempted to pity him. "Know what?" she says.
His eyes are filmed over, his voice strangled. He stares resolutely through the windshield, exits the highway. "What this is. Why you're doing this."
"I'm just going to see our sons. I have to see them."
He sighs. "That's not an answer," he says, nearly in a whisper. They arrive at the airport. She expects him to park in short-term and wait with her. But he pulls up in the white zone. He doesn't even put the car in park. He seems to have gathered himself; she can tell by the set of his face that he is shifting his anger and frustration away from her and onto himself. A dangerous state—it usually leads to a fight. Not now, though—she's leaving. He says, harder now, "You are coming back, right?"
She levels a serious look at him. "Of course I'm coming back." She kisses him. He accepts it with taut resignation. Then she gets out of the truck and he drives off without waving goodbye.
The flight is long. She hasn't brought anything to read. She didn't think she'd be able to concentrate. But she can't sleep, either. Everyone around her seems to have some kind of digital device they're using to entertain themselves. She wants one—it would be nice to watch television, something she hasn't done in earnest in twenty years. Though in all likelihood you couldn't get any stations on a plane. But then what are these people watching?
They land in Denver and she changes planes and then, hours later, they land again. The light in Los Angeles is very bright; the airport workers on the tarmac are orange blurs. It's as if her pupils can't contract enough. She's near the back of the plane. The man beside her stands up as soon as it's allowed, though it will be several minutes before he'll be able to move. He stands awkwardly at his seat, sighing impatiently at regular intervals.
Elisa expects to see bustle when she emerges into the terminal. People in sunglasses, on phones, men in African robes, strung-out rockers on tour. Instead, the concourse is nearly empty. Sam is supposed to have met her, but there's no sign of him. She stands in the middle of the broad worn carpet and squints in every direction. Then she sits down and rummages in her bag for her phone.
When he emerges from the men's room twenty yards down the concourse, she doesn't recognize him. Or rather she does—it must be him, she knows it as soon as she sees him—but tells herself she must be wrong. In truth, she has been preparing herself for this moment for weeks. She understood he would be different. She imagined him thinner than was healthy, unshaven, pale-skinned. Tired. This image frightened her at first, but she has gotten used to it. She is ready for it.
But Sam looks nothing like it. He's clean-shaven, tan, and overweight. He's wearing a pair of pleated khaki pants, running shoes, and a golf shirt. There's a cell phone holster on his belt and he walks with effort, as though sedated.
She stands. He acknowledges her with a nod. When he is standing before her, she cannot help herself—she throws her arms around his neck and pulls him close. "Sam!" she cries, and he's sweaty, he smells sour, she cannot believe this bloated creature is her son.
He stiffens and she feels his hands gingerly patting her back. His shoulders are big, they're rounded, they feel like someone else's. But no, this is Sam. He stumbles a bit, takes a step back; she releases him. His face is obscured by flesh, but the eyes are the same. They seem to search her face for some kind of purchase.
She stands very still and lets him look at her. He sounds tired as he says, "Hey, Mom."
"Sam."
"You seem different."
Her laugh is almost a sob. "I feel different."
Sam looks down at his feet. He says, "So... you know where you're staying?"
"It's a chain hotel. Near your house."
She gives him the information and he nods. "Wanna go, then?" he says, and they walk together down the concourse. He keeps his hands shoved into his pockets, and his effortful walk now reveals itself as a slight limp. One of his feet isn't quite straight, and he grunts, nearly inaudibly, with every step. She wants to ask him what happened, but is certain that she is either already supposed to know, or has deliberately been prevented from knowing. (She remembers his fake limps from adolescence, his indulgence in others' pity.) A smell comes off him, like an office cubicle out of which the custodian has not yet carried the remains of lunch. She gasps a bit, holding back tears, but Sam doesn't seem to notice.
They say nothing to each other as they leave the airport. He offers a questioning look at baggage claim, and when she shakes her head he nods and continues through the automatic doors. The air is brutally hot but dry, and she suddenly wishes she were in the sun. But in fact she's in the parking garage, getting into Sam's car. It's quite new, an SUV, and very large. He gets behind the wheel and puts on a pair of aviator sunglasses that match, almost perfectly, the curve of his face. He looks like a giant insect.
Sam had a toy spaceman helmet, once, that Lorraine had given him. It was yellow, like a hard hat, and covered his head entirely; a large hinged visor of reflective translucent plastic obscured the face. For a time, he wore this helmet almost all day long: he would sit on the sofa reading picture books through it, he would sneak goldfish crackers or raisins up under the visor. He wore it to dinner, wore it riding his scooter, when they went to the store. He usually took it off to sleep, but occasionally she would find him in the morning still wearing it, the visor fogged, snores echoing out from under it.
He must have been five. He didn't seem to have any kind of fantastic identity that he associated with the helmet, no invented narrative. He just wore it. The helmet was an accessory to his personality.
Its only apparent power, real or imagined, was that his brother ignored him when he was wearing it. Occasionally Elisa would catch Silas staring at Sam, gazing at the reflective surface of the visor, perhaps at his own reflection, or perhaps just lost in thought about the impression the helmet made. But he didn't bother Sam in any way—didn't push or hit, didn't steal any toys or books or food. If anything, Silas redoubled his disruptive efforts elsewhere in the house, but it was a relief to Elisa not to have to pull them apart, to settle disputes.
The era was short-lived, of course. The helmet was always a little too small, and then Sam hit a growth spurt. Washing his hair one night, Elisa noticed twin wounds on either side of his scalp: scabbed and bloody tracks, as though he'd been scratched by some animal. She picked up the helmet, which lay on the floor behind her: its interior was filthy, greasy, crusted over with black deposits on two reinforcing ridges that matched Sam's scrapes.
It had to hurt him even to put it on. The ridges worrying at the wounds all day long, the cuts struggling and failing to heal themselves. Sam winced as she applied disinfectant to the cuts, and then, later, silently watched her clean the inside of the helmet with rubbing alcohol, apply strips of felt to the ridges with double-stick tape. Elisa dreaded the day he had to give it up; she thought he would cry for weeks.
But it didn't happen. For a little while Sam carried the thing under his arm, like an astronaut walking across the launchpad after a successful mission, and then at last he relegated it to his closet. Silas took up, once again, his efforts to torment his brother. Elisa tried to interest Sam in other helmets, none quite the same, but he shunned them.
On the one hand, it was as if he were determined to put it all behind him—not just the toy itself, but his need for its protection. He seemed, in this one small way, very adult.
On the other hand, there was something perverse about the totality of his resignation. Come on, she wanted to say—if a plastic helmet made Silas stop, then anything could make him stop. The helmet isn't magical. This last, she did say to Sam, while she sat on his bed trying to talk him to sleep. It's not the helmet, it's you. But the boy shook his head—she could hear it in the dark, his hair scraping the pillow—and she thought, but didn't say, Come on, Sam, fight back. Fight back.
Now they're on the freeway and her son exudes that same grim acceptance of circumstance. She doesn't even know what the circumstance is, but she can feel herself bristling in the presence of his capitulation to it. It's midafternoon, not yet rush hour, and their progress is swift. Sam says, "I guess you're not going to talk, then."
It is the same conversation she had hours ago with his father.
"Fine then," he says, without giving her a chance to reply. "So you're here, and in a while you'll go home, and I'll never know what this was all about. And yet here I am driving you around."
She says, "It isn't anything in particular."
He has no response to this. Minutes pass.
She says, "Things are changing at home."
Nothing. She is fixated on his heavy thigh, sunk into the driver's seat. Tears are pouring down her cheeks, though there is no tightness in her throat; she is able to keep her voice steady.
"Forgive me, Sam. I don't really know why I'm here."
Possibly he makes a small grunt of acknowledgment. His body shifts against the vinyl seat. She goes on: "There's something... wrong with me. It might be best if you—if you pretend I've got amnesia. That I don't remember anything about the past few years." She turns to him. "I know that your father and I apparently decided to—to put you behind us. I don't understand how we came to that decision. I accept that we did, but... do I owe you some kind of apology? I think I do. May I apologize to you?"
It takes him a few seconds to come out with, "What do you mean something's wrong with you."
She lets out a sigh. "I don't know. I don't even know. But I don't remember anything that happened. Not for years now."
There's a minute during which his breathing seems to quicken and deepen. He is so strange to her, bovine and implausible.
"You can do whatever you want," he says quietly.
"Do what?"
"You wanted to apologize, go ahead."
"I'm sorry," she says. It doesn't sound like she means it, but she does, she does.
He nods. She takes a tissue out of her bag and wipes her face.
The car pulls up in front of a motel, and Sam parks. "I have to go back to work," he says. "Silas is having you over for dinner. Some people are coming. You should be there at eight—it's just a couple of blocks." He pushes his sunglasses up the bridge of his nose; his entire face is slick and sweating. "I dunno. It's fucked up. It's fucked up that you're here."
She can't speak.
"I don't mean to be mean. You know. But I just... I really don't get it."
"I don't either."
"Okay, well..." He puts the car back in gear. "I really have to get back to work. Silas needs me to do some shit."
She can't stop herself—she says, "You still work with Silas."
He lets out breath and tips his head back. "Jesus. Yes. I still work with Silas. I do the books at Infinite."
"I'm sorry."
"Me too," he says, but she doesn't know what he means by this and he doesn't seem to either.
She gets out of the car, takes hold of her bags, and shuts the door. The tires bark on the pavement as Sam drives away.
37.
The neighborhood they live in is called Silverlake. She learns this from the motel clerk, who is a young black woman with pierced nose and tongue and an ostentatious straightened hairdo. When Elisa asks if there's a place nearby to get coffee, the girl points and says, "Five blocks."
But first she checks in. It's four thirty in the afternoon. Her room is cold and dry. She takes a nap, then wakes up shivering half an hour later, her skin pulled tight, wondering where on earth she is.
The café is staffed by people who look like the girl at the motel. She sits at a wobbly table sipping her coffee and examines the maps she printed from the internet. The boys live on a narrow winding street several blocks from the main drag; she memorizes the route to their house, then tucks the papers into her bag and walks there.
A Spanish-style bungalow in apparent disrepair, the white stucco cracked and falling off, the shrubbery unwatered and half-dead. She stares at it from across the street for a while. No cars pass, though she can hear, faintly, the noise from the main street, blocks away. She contemplates returning to her motel but instead walks across the street and climbs the steps onto the wide shaded porch. A table and chairs are set up, and on the table lies a half-full ashtray and a bottle opener. A wind chime hanging in the archway is silent and still.
The driveway is empty of cars and the broad front window lacks any curtain or shade. She walks up to it and peers inside.
Elisa is surprised—the living room is quite tidy, perhaps to a fault. The Swedish-style furniture is new; the jute rug is clean and lies on even bleached floorboards. There's a glass coffee table, an insectile aluminum floor lamp, and a sofa and two chairs draped in matching white slipcovers. Beyond this arrangement is an open door frame that leads to a bright kitchen, with black-and-white tile floor.
There appears at first to be a blanket wadded up on the sofa, partially covering a reddish-brown pillow. Elisa blinks, and the blanket and pillow rearrange themselves into the image of a naked red-haired girl.
The girl is asleep. She looks nineteen or twenty and is dangerously thin. Her flesh is pale and she has no breasts to speak of. One arm hangs down to the floor and the other is pinned under her body. Her legs are spread, her genitals in plain view. Elisa draws in breath and the girl opens her eyes.
At first, the girl appears not to have seen Elisa standing in the window. She makes no move to cover herself, and anyway there is nothing, no robe or towel nearby, for her to use. But the longer she lies there, eyes open, the clearer it becomes that she has seen Elisa and is staring at her. The eyes are bleary and the face freckled; the girl's lips are parted and between them lie small yellowish teeth. She is like a rare, damaged specimen of some endangered species, some strange rodent or bird.
Is the girl drugged? There is no expression on the face; the eyes blink with exaggerated slowness. Elisa doesn't move—she wants to leave, but this would somehow be worse, a greater invasion of privacy than remaining still. Her ankle itches and her bladder feels heavy.
Finally the girl closes her eyes and rolls over and returns, evidently, to sleep. Her legs are sinewy, her feet large. Elisa watches her draw and exhale breath.
What precisely does she think she is doing here on this porch, in this city, interfering in lives she has evidently forsaken? Muscling this world onto some path where perhaps it doesn't belong. She's doesn't have a plan, not really. She is still groping, still doesn't belong.
But she felt this way, often, in her real life, too, didn't she? It was the signature emotion of parenthood. There was no way to know what actions had which results, whether all of it was her fault, or none of it.
She has been assuming that her memories of life before the crash are valid here, but maybe they aren't. Is any memory valid, really? Has she ever remembered anything the way it really happened? Has anyone, ever?
Maybe none of this world belongs to her. None of it.
She climbs down from the porch and wanders the streets, sweating. The light glinting off car windshields is again too bright. A greasy haze of unreality coats everything. She suddenly feels sick, as if the coffee she drank has just changed its mind, and she sits down on a curb over a street grate, expecting to vomit. But the feeling passes and is replaced by exhaustion. She finds her way back to the motel and asks the clerk to call and wake her at seven fifteen, but the clerk shakes her head and says it's all computerized, just push the wake-up button on the phone.
Elisa's hand is trembling as she does this, sitting on the edge of the bed, on the slick synthetic-fiber comforter that will not allow her body to find any purchase. When the operation is finished, she throws back the bedclothes and climbs in, shivering and fully dressed. Falling asleep makes a sound in her mind like a steel marble circling the drain.
38.
Two hours later she stands on the sidewalk again looking up at the bungalow. The driveway is full now; Sam's SUV is here, a small sports car, an old Volvo, and a motorcycle. Long shadows of palm trees and phone poles rake the street. Her breathing is quick and shallow and she wants to run away. But she climbs the steps for the second time that day and knocks on the door.
Sam answers. His body fills the doorway. There's a large glass of something in his hand, whiskey she supposes, with a couple of half-melted ice cubes floating in it. He manages a small smile that quickly fades. "You don't have to do this," he says, and she can detect in him, for the first time, some small reservoir of compassion. She thinks, This is really happening. This is how things are.
She says, "I came all this way."
After a moment, he steps aside and lets her in.
People, young people, fill the living room. Each holds a drink in one hand—straight-sided glass tumblers of something—and a cigarette in the other. They embrace no recognizable style. One boy looks like the counterpart of the girl at the motel—black jeans and tee shirt, piercings and wild hair. Another wears suit pants and an untucked oxford shirt and tiny round glasses with lenses not much larger than his eyes. There are a couple of guys who look like members of a biker gang—she recognizes one of them from photos of the Infinite Games staff. A fat girl with big breasts stands alone by the wall. And the girl from earlier is here, wearing a simple, baggy linen dress that is too big for her. Its hem drags on the floor.
It isn't that any of them is particularly unusual, set against the strangeness of humanity; taken alone, any one would qualify, at first blush, as mildly eccentric. But there is something abstracted about them as a group; the party seems conceptual, like a movie set. It is as if none of them knows any of the others, as if they have all just met.
Sam comes up behind her and says, loudly, "Everyone, this is Lisa."
A few people turn and say hello. The fat girl actually appears frightened.
She is trying to decide, in the half-quiet that follows Sam's introduction, whether or not to reply when a figure appears in the kitchen doorway, holding its own drink and cigarette. It is unmistakably Silas. His eyes travel first to the red-haired girl, and then follow her gaze to Elisa.
In spite of everything, she wants to cross the room and embrace him. Oh, Silas—I know you so well. Every year, in that motel room in Wisconsin, she has lain on the bed, eyes closed, and imagined what he would look like if he were alive. She has invented a hundred scenarios for him—lives he could lead, experiences that might transform him. He has been rich, he has been imprisoned. He has been married, itinerant, famous, missing.
She never imagined this one, of course.
Silas is advancing toward her across the room. The party has lost interest in her, conversation has resumed, but she can't shake the feeling that everyone is watching, that they all know what she has done, what has happened to her, what is going through her head. That they are witnesses to an experiment of which she is the subject.
He is standing before her now. It's him. "Lisa," he says.
His body is strong—that's what strikes her. Always, as a child, he was smaller than his brother, slighter. He is wiry now, muscled. He looks like he could climb up a wall, jump from rooftop to rooftop.
And his face, it is so familiar: the expression of impassivity, the features flattened by indifference, as though pressed against bulletproof glass. His eyes are half-lidded, his small mouth opened slightly, the skin tanned and lightly coated with sweat. He is twenty-four years old. She flinches when he takes one of her hands in both of his, lifts it to his thin lips, touches it to them. They are warm and dry.
"Silas..."
His eyes open wider and meet hers. They seem to know that she isn't his real mother. He releases her hand and it falls to her side. "Would you like anything to drink?" he asks. She can't read his tone.
"All right," she says.
His eyes bore into her, then blink. He doesn't ask her what drink she wants. He just turns and walks back toward the kitchen, with a spring in his step. She remembers him as a toddler, the sight of his small hard back retreating through a doorway or around the corner of a room, the feeling that something just out of sight was about to happen, something would be... disrupted.
There is a presence beside her and she remembers that Sam is here. That Sam exists. He's breathing loudly again, loudly enough to hear against the background noise of the party. She turns to him and he is gazing at her with mild curiosity and apparent exhaustion. His glass is empty except for the ice cubes, which are little more than slivers now. His face is red and puffy and she realizes that he is an alcoholic. Silas is doing this to you. I did this to you.
Sam turns away and follows his brother, heaving himself across the room. Elisa is left alone at the door. She can see the boys moving in the kitchen. She doesn't want to be standing here when they come out.
Everyone in the room is half her age, but they seem older somehow. She stumbled into parties like this in college—disdainful or insecure people, trying to act cool. She liked the law students for their politeness, their confidence. They were invested in the system, they felt comfortable there. Elisa never thought of herself as a misfit—she didn't like misfits.
Having children changed that. The pitying way other parents looked at her when Silas pushed their kids or stole their toys. She understood the outsider mentality. It came in handy in her old life, her real life, where she painted paintings and had extramarital affairs. She feels compelled to let these people know—I've been loathed and pitied, too. I paint paintings.
Instead, she goes over to the red-haired girl in the long dress and says, "I owe you an apology."
The girl blinks.
"I came to the house today and looked in the window. I'm afraid I may have embarrassed you."
The girl looks frankly at her, taking in her clothes and shoes and face. She says, "You're their mother." Her voice is surprisingly low.
"Yes."
The girl shifts her insignificant weight from one foot to the other and her dress moves against her body. It's obvious that the dress is all she's wearing. It's so large and hangs down so far that the upper edge of an aureole is visible at the neckline. Her nipples show through the thin fabric and Elisa feels cold looking at her, though the room is warm.
Elisa hears more people entering behind her—a small crowd in fact. She smells cigarette smoke and hears bottles clanking together inside a paper bag. And then she realizes that she has seen this girl before. The perfect round arcs of the eyebrows are the same, and the oval chin. The oversized, not big, nose.
"You're the girl from the game," she says.
"What?"
"From Mindcrime. Silas's game. You're the waitress at the diner."
The girl appears confused for a moment; her eyes film over. Then she seems to remember, and just as suddenly to lose the thread again. Elisa feels bad for bringing it up. The waitress in the game was voluptuous, full-hipped and large-breasted, radiant with health. Whereas this girl is sick. Did she ever look like the waitress? She blinks, again meets Elisa's gaze.
"What do you do?" the girl asks.
"I work at a lab," she replies, then corrects: "A college, I mean." But the girl isn't listening.
"No. What do you do?"
"I don't understand."
The girl licks her chapped lips. The lip-licking continues. She's smiling, a strange distracted smile that could be left over from some other conversation. The teeth are small and sharp with tiny spaces in between them, and a hard look has come over her face, a desire to inflict hurt.
"When you're not at your lab. At a college."
After a moment Elisa says what she fantasized saying moments before, she says, "I paint."
The girl says, "Waterfalls? Horses?"
"Abstracts." But she doesn't, of course. This Elisa doesn't. Her studio is an office now—Derek's office.
"Abstracts," the girl says.
"That's right."
"Do they match your shoes?"
There's a tug on her arm. It's Sam, rescuing her from the conversation. The girl looks at the floor and Elisa mutters goodbye. She is introduced to some new people, none of whom she notices. Suddenly she wants that drink. She detaches herself from her son and goes to the kitchen. Silas isn't there. It's small and dirtier than it appeared through the window, the stove burners lined with aluminum foil in which grease is pooling around bits of burned food. At least they're cooking, she thinks, that's a good sign. A bottle of bourbon is standing on the counter, and she helps herself to some, using a cracked tumbler from the drying rack. There's a jar beside it, a mason jar filled with some brown liquid that seems to be slowly swirling and glittering, and she stands there a moment, sipping the bourbon, trying to figure out what the jar contains.
Silas walks in from another room, running his hand through his hair. "Oh," he says, "that's right. Your drink." He brushes his hands together, as though he's just finished sawing some boards, and leans against the stove, arms crossed.
Elisa's heart is galloping. She takes a deep draught from her glass and says, "Silas, what is wrong with that girl?"
"What girl."
"The red-haired girl. The waitress from the game."
His eyebrows rise infinitesimally, but his expression remains otherwise unchanged. "Rachel? She has problems."
"She needs help."
"Going to unleash those crack mothering skills?" he says. "Worm your way into her heart? With all of your kindness and charm? All your experience helping people?"
She can't speak. He goes on.
"What with your life devoted to thinking about things besides yourself and your petty desires."
"You should talk," she says.
"I don't pretend it's any of my business. That girl is fucked up and she is dealing with it. It isn't my fault. Not everything is my fault. This will come as a surprise to you, Lisa. I'm not to blame for everything. Some stuff is even your fault."
39.
She feels it, the old helplessness. The feeling that she has run, suddenly, unexpectedly, out of options. This was the signature emotion of her years as a parent of small children, the feeling that at any moment the mother in her might simply expire, leaving her alone, in some private world of failure, with the parts of herself she had abandoned. There must have been some signal she gave off, some stale odor or subtle corporeal slump, that telegraphed this emotion, because Silas always seemed poised to exploit it, to see what he could accomplish inside the space it created: the time he learned the word bitch, and all day he said bitch, because he had heard her refer to another woman this way, and knew she disliked this woman, and knew the word would do something to her, would make her react. And so she didn't react, not at first. She turned the other cheek, like the pediatrician told them, like the child psychologist told them, like the piles of parenting and self-help books told them, she turned and walked out of the room.
But walking away was a reaction, ignoring was a reaction. It meant a real reaction was forthcoming. He followed her around all day, saying the word, until finally, the day nearly endured, her head aching and throat tight, she went into Sam's room to say goodnight, and Sam looked at her, his face quivering with uncertainty, and said, "...bitch?"
She slapped him, hard, across the cheek. And before he could react went into Silas's room where Silas lay on his bed beneath a galaxy of glow-in-the-dark stars laughing, and she slapped him too, harder, twice.
Silas screamed—not cried—as though she were sawing off his leg. She could feel through her feet the boy's body writhing in the bed as in an epileptic fit, and his head thumping against the wall beside it, something he had begun to do to drive them madder still (but there she goes, ascribing motivation to this strange act, when who could know, really, why Silas, or anyone for that matter, did anything?)—there were marks there, visible in the daytime, stains and depressions left by his sweaty head. And by now Sam was crying too, and Derek's feet thundered on the stairs and she felt herself being pulled out of the dark August-hot bedroom, away from the stink of boyhood and the wailing and the thumping of Silas's head against the wall.
Derek led her down the stairs. He laid her on the sofa, pushed her down, held her arms down at her sides. "Stop it, stop it," she was telling him but he wouldn't let go, and soon she was struggling, like Silas, thrashing her own head, trying to knee him in the back.
Eventually she gave in. He wasn't going to let go unless she gave in. So she lay still listening to the children cry and she said, "I need to apologize to Sam."
"What happened?" Derek said, and she could not stand the fucking sound of his fucking patronizing holier-than-thou voice, as though she was the only irrational person in the house. Fuck you, Derek, she thought, fuck you for eternity.
"Sam called me a bitch and I slapped him." Not true, a voice told her, he had said the word, he hadn't called her anything. He had said the word and she had assigned the intent. But she did not correct her story.
"Sam did?"
"Let go of me, Derek."
"And that's what he wanted," Derek said, almost to himself. "To make you slap Sam."
"Let go of my arms."
"Are you sure?"
Die, you fucking fuck. "I have to apologize to Sam."
"You won't go into Silas's room."
"Derek," she said between clenched teeth. "Let fucking go of me."
He did as she asked, slowly, as if she might spring up and attack him, and she climbed the stairs and apologized to her son. This would be the first time they almost decided to separate, she and Derek, the first of three. The second was right before Silas calmed down, right before he started doing well in school; and then there was the one right before his death.
They should have broken up sooner. Their love for each other was not important, but this wasn't clear to them at the time. They just wanted to be with the only other person who understood, whatever the consequences.
"He made me," Sam whispered to her that night, tears on his cheeks. "I know," she whispered back, "I'm sorry," and rocked him to sleep.
Two weeks later he turned seven.
40.
She tries to keep a steady voice, though her throat is tight and the words come out strained. "I see what you do, Silas. Online."
"Online?" She has caught him off guard. His face is long, bony, a man's face, not a boy's, and the word massive occurs to her, though he is not large. He is simply her son, a man.
"The forums you're on. You treat people badly. You invent new versions of yourself just so you can treat people badly."
Silas's eyes widen and he barks out a laugh. "You're webstalking me?"
She hears a noise behind her and in a moment feels a hand on her arm. It's Sam. "Mom."
"That is really something," Silas is saying. "I'm not nice enough to people. On the internet! Did you catch that? I'm a dick on the internet, and Lisa has flown out to LA to let me know."
"We ought to go," Sam is whispering.
"Listen to your good son, Lisa, he's right. You ought to go."
But I don't want to go, she's thinking. This is what I came for. Although it occurs to her that, ultimately, she doesn't really know what she came for. Except to see. Her eyes fall from her son's face to the counter, where the liquid in the jar is still swirling, though no one has touched it. The liquid is pearlescent. It appears to be illuminated from within. And she wonders, ludicrously, and for no clear reason at all, is this part of an experiment Silas is doing, part of his effort to create other universes? Perhaps it is in this kitchen's twin, in another universe, that he created this one. And now, from inside this universe, he is trying to create another. That's it, in the jar, that's the matter that will expand into it. Ten pounds of matter, Betsy told her, packed into a really tiny space. She wants to reach over and pick up the jar, see how heavy it is. She looks down at her glass and sees that it is nearly empty.
Silas is staring at her. If he has noticed her noticing the jar, he gives no sign. She allows Sam to lead her out of the room, through the crowd of people, out of the house. She'll come back. She needs to pick that thing up, to heft it and peer into its depths. The red-haired girl is nowhere to be seen. Elisa leaves her glass on the table on the porch, beside the ashtray and lighter.
Down the front steps and onto the sidewalk. Sam is marching her along the street, his fingers tight around her arm. She says, "Sam. Let go."
He releases her, tosses the arm back to her. It flops against her side. "I told you not to come." They're walking fast, toward her motel.
"That girl," Elisa says. "What's wrong with her?"
"She's Silas's." Dismissively, as though this makes speculation pointless.
When they reach the motel he follows her into her room. The curtains are open and the evening sun is blazing through the window, but the room still seems dark, and the air is clammy. Sam turns the air conditioner down. He has brought something with him, in his pocket, a bottle of whiskey the size of a beer. Elisa sits down on the bed and watches him unwrap the two tumblers from the tray on the nightstand and pour a few fingers of liquor into each. He still has great facility with his body, when he isn't around his brother. She remembers the towers of blocks he liked to make, such irresistible invitations to Silas. Even long after it became clear these projects would not survive his brother's attention, he continued to build and defend them. He would spend hours beside a tower of blocks, his body taut with attention as Silas walked by.
Sam takes the only chair in the room, facing her, and hands her one of the glasses. He says, "You have to tell me what you want."
She drinks. "I just wanted to see you."
"What did he think?"
His enunciation of the word he—an emphasis half mockery and half vestigial respect—tells her that he is referring to Derek. "I don't know."
"And the therapist guy. Who you hired to do all this to you."
"You know about all that?"
Sam doesn't respond.
"I didn't tell him I was coming, Sam. The therapist, I mean. I just came." Though she remembers, moments later, that she is lying. When he doesn't respond, she says, "I needed to see you."
"Is this amnesia thing a real thing, or is it just bullshit?"
She leans over to place the glass on the nightstand. She doesn't want it. She says, "It's real."
"You honestly have no memory of, what exactly?"
"I told you. The last... nine years. I do have memories. Some of them are... inaccurate."
The sun is sinking behind the line of buildings across the street. A car horn sounds, and somebody shouts. It's getting harder and harder to make out Sam's face. He doesn't speak; she thinks he's going to, for a moment, but then he doesn't. She remembers other hotel rooms at other times. A camping vacation that went to pieces: they ended up in a cheap roadway motel, sprung for an adjacent room for the kids so that she and Derek could quietly fuck with the TV on. Actually, that was a nice trip. Very nice. All four of them got along: united in failure. There was a trip to Derek's parents' place, the guest room under renovation (the guest room was always under renovation), the Best Western near the highway, with the indoor pool so overheated she fainted. They never seemed to plan to stay in hotels, they were only a substitute for real plans.
She says, "I remember Silas... dying. When he was fifteen. And then later, you're happy. You're not like this." She has to squint against the sun. She says, "None of this seems real to me, Sam. There was no shrink. You live near us. We share a car."
She says, "I want my Honda back, Sam."
He is staring at her.
Elisa can't stay upright anymore—it's as if Sam's gaze has released her. She lets herself lean toward the pillows, and then she's lying down and her eyes are closed.
"Tell me things," she says. "About yourself. Things I don't know."
He's quiet for a while. She thinks she's going to fall asleep, but her body is buzzing. Every now and then her fingers twitch, as if she's grabbing something in a dream.
When he speaks, it's with effort, as though he is forcing himself to play along. He says, "I don't know. I assume you don't remember the meeting."
She shakes her head no.
"The family meeting? The one where you told us?"
Now she doesn't want to hear it. But he has decided to tell her. He shifts his body on the chair, with effort, and when he speaks he is panting slightly.
"You scheduled it. Like it was a work meeting. It was at 5:00 p.m. on a weekday. Silas was off in his room coding or something. I was... I don't know what I was doing." He clears his throat, inhales through his nose like an old man. "You got us all around the dining room table and said basically you were kicking us out and were going to cut off contact for a while. And I said A while?, and Dad said Indefinitely. And Silas said, Who told you to do this, the shrink?, and you said, It's something we worked on with Amos."
"Oh, Sam..." she says.
"And Dad started rattling off the details, like we had two weeks to do this and a month to do that before you changed the locks, and Silas was just laughing and laughing. Because he had just heard about the job, like, that same day."
"This one? The one..."
"At Infinite. And he was like, You know what, how about we leave today? Like, right now? And he got up from the table and started carrying his shit out to the car. You guys just sat there like, what the fuck? And I could see your expressions just start to harden, like you must have promised yourselves you were going to do, and I sat there asking you if it was a joke, or what."
He has managed to drain his glass and pours himself another from the bottle on the floor.
"At some point Silas looked at me and said, Are you coming? And I looked at you and you just hung your head. Silas said, Can't you see, dude, they're done with you? Get up off your ass and pack. So I was like, fuck all this fucking shit, and I went with him."
"Where did you go?" Elisa asks him.
"Some motel. Silas called some girls. We got drunk. After a couple of days we flew out here. They paid for everything—they wanted him really bad. They gave me a job. We bought the house."
The room is quiet, but the air itself seems to be making a noise—a pink noise, a hiss with a low note in it somewhere. No, she realizes, it's just the air conditioner. And this is a disappointment to her, she wanted it to be the air, the sound the air made.
"Why did we do it?" she whispers, mostly to herself. But he answers.
"We were acting like assholes." And then, after a moment, "You were assholes. No, all of us are assholes, that's the problem." Sam is starting to sound drunk now, his teenage self is coming back, the self-pitying Sam, the whining Sam. She thinks, I'm an asshole for thinking that.
"Actually, that time, in the motel," he says, too loudly, "I cheated on Angie with some girl, Silas's girl's girlfriend or something. We made the girls go down on each other then I fucked the one, the other one. I called Angie and told her and she dumped me. Then we moved."
He wants to hurt her with this tableau of debauched sexuality. And she is hurt, but she is mostly confused. She says, "I don't know who Angie is."
There's a silence. Finally, "We were together for years. We were supposed to get married?"
"I'm sorry, Sam. Why—"
"Whatever."
She waits. He is panting, slumping in his chair. "Why did you call her? Tell her? Why were you... unfaithful?"
Sam doesn't answer. He closes his eyes and for a time he seems to be asleep. But he keeps his half-full glass upright and when he opens his eyes again he appears alert.
"You're not happy here. With Silas."
He doesn't respond.
"Surely it was about him, not you," she says, willing it to be so, begging him to agree. "What we were doing, it was about Silas, about breaking his... his hold on you."
The light in the room is lower now and illuminates the wall behind her. It reflects a green cast onto his face. He says, "I don't know. No."
"You could have been free of him."
He's quiet for a while. He says, "No. I came here. He... he got me a job. He took me in. Sort of like one of his girls."
"His girls?" Elisa asks.
"Yeah, he—well, you met one. He dates messed-up girls. He tries to, I dunno, fix them or something. Help them. It never works out, it ends in tears, you know."
"Maybe he's just taking advantage of them, Sam. Did you ever think of that?"
His eyes narrow. He appears puzzled. "No—I don't know what he's doing, to be honest. But taking advantage. No. He's not like that. With girls."
Elisa thinks, you don't know him like I do.
She consumes the contents of her glass in one gulp. There's something she wants to ask him. She can't. Then she does. "You... shared. Those girls. In the motel."
He's looking at the wall over her head.
"That girl," she says. "The red-haired girl. Do you share her?"
He shifts in his seat, drinks. "No."
"Sam, you don't like girls."
He's very still now. Quietly he says, "What do you mean?"
"You don't like girls. You're gay."
"I'm not gay."
"Yes," Elisa says, "you are. You came out. A few years after he died. You have a boyfriend. I think. You don't tell me everything."
The silence is much longer this time. She is in mourning now, mourning for the Sam she knows. Her friend, her only son.
His shoulders are hitching and his feet scrape against the floor. As if he's bound to the chair and is trying to get free. His empty glass falls from his hand onto the carpet.
"Sam?"
There are tears in his voice as he says, "I'm cooking the books!"
She waits.
"Silas is—we tried to start our own company. His games don't sell. You know—they're supposed to be arty. People hate him here... it's a fuckin'... it's a miracle we're not fired."
He draws a deep breath and his throat sounds a low and wandering note, a wheeze. When he speaks again, he has reined in his emotions.
"Silas is... he doesn't like working for somebody else. I mean, I can't blame him. He wanted to go independent. He found a partner, we were going to go in together on this space in La Puente... so we started... skimming off the top. I mean, Silas felt like—we both did—like we deserved it. They really are assholes, these people. But the guy, the partner... he fuckin' disappeared. We got robbed. It was so fuckin' stupid..."
They sit there in the darkening room, facing each other, Elisa on the bed and Sam on the chair. She reaches out and takes his thick pale hands in hers.
"Just come home," she says. "Leave it all behind."
He snorts. "Oh, for fuck's sake."
"I mean it."
"I can't just leave. I'm the accountant. It was all his idea but I'm the one. I could go to jail. I gotta try to make it right. We're living off of beans and rice."
"How much have you taken, Sam?"
He shakes his hung head.
"Your father and I... we could help you out. If you just quit now. Quit now and come home."
The shaking of his head, his shaggy head, slows and then stops. She has gazed at it, the whorls of fine hair, the hot slick skin, always a little oily, so many times in her life, every time she held him, comforted him. Whenever he was sick, whenever Silas hurt him. She wants to lean forward and kiss it, but holds back.
He says, "I still don't understand why you're here."
"You're my son. I love you."
"The things you said..."
"I don't remember!"
He withdraws his hands from hers, sits up straight, glares at her with his wet hard eyes. "That you couldn't help me. I was beyond help. I was weak to follow him. That I was ruining your life."
"No."
"That's what you said. You said I needed to break away. You'd be there if I decided to break away."
"That's what I'm saying now!" she cries.
"But you knew I wouldn't."
"Sam. I don't understand."
He's silent.
Her hands are tangled in her hair; her voice breaks: "It isn't supposed to be this way!"
But Sam is shaking his head again. "There's no way anything is supposed to be," he says. "Things can be as good or as bad as they want."
He gets up and heads for the door. But then he turns back, leans over her, and kisses her on the cheek. "This is so bizarre and fucked up," he whispers to her, "all of it." He picks up the whiskey bottle, holds it up to the light. He slides it into his pocket and walks out.
41.
In the morning, after dreamless unrestful sleep, she gets up, dresses without showering, and leaves her room. It's just after eight on a Saturday. Silas won't be out; she can rouse him from bed. She'll settle it with him. Come to some kind of agreement.
The air is cool, without a trace of moisture. Nothing is open but there's traffic anyway, as if people are driving for its own sake. She's walking briskly to combat the fatigue, her tennis shoes making no sound at all against the sidewalk, and it feels like the old life, tired and on the edge. She feels almost normal.
In the early light the house looks dumpier and more vulnerable. The gutters sag and the shingles are coming loose. She climbs the steps, tries the door, and when it proves to be locked she reaches for the bell. But she thinks better of it. Goes around to the back, lets herself into the yard through a low chain-link fence gate. A small screened back porch leads to another door, with a curtained window at eye level. She can make out a washer and dryer and part of the kitchen.
She tries this door, too, and it's also locked. But inside a rusted coffee can, on a bookcase slumped against the back wall of the house, she finds a key. It feels too heavy in her hand, and too cool, as if it has borrowed these properties from some other object, in some other place.
Silas used to lock his bedroom door, would lock her out, as early as age two. They found a way to jimmy it open—a tiny screwdriver or tweezers in the hole in the knob—but eventually Silas discovered this and stole everything in the house that could conceivably be used for unlocking. Derek began to keep a collapsible screwdriver on his key ring, but around the time he turned seven or eight, Silas managed to get his hands on a safety bolt, which he installed crudely on his side of the door, bypassing the knob entirely. They could have removed this, too, of course, when he was at school. If she remembers right, they may even have done so, once or twice. But the real answer was to give up.
This turned out to be the answer to a lot of things, with Silas.
Well, here are the results. They are cynical and unhappy here, on their own, and she and Derek are delusional and unhappy at home. She has created a family of miserable loners who seem incapable of helping one another.
Was there anything she could have done that would have resulted in a satisfactory outcome?
She needs to believe that the answer is no.
She slides the key into the lock, and a moment later she is standing in the messy kitchen, her heart pounding. Breaking and entering. But this isn't her real life, is it. This is where things are different. She is seeing if she can make things change. Hands shaking, she sets the key on the counter.
The house is quiet. She moves into the living room, which is as disorderly as the kitchen. The fingerprinted residue of cocaine smears the coffee table's filthy glass surface. Cut up plastic drinking straws and a razor blade. There's a wad of Kleenex stained by blood and an ashtray filled with cigarette butts and the twisted paper ends of joints.
A hallway leads away from here and she walks down it. The bathroom door is open, and three others are closed. No light is visible beneath one—a closet. She takes a breath and opens the next.
It is hard to imagine a sadder sight. Sam is lying asleep and fully clothed on an unmade bed, the sheets twisted together and trailing across the carpet, where movie and gaming magazines are spread within easy reach. The dresser drawers are all open and unfolded laundry spills out, mingling with the dirty shirts and pants that litter the floor. There's a smell of sweat and shit and come, and nothing hangs on the walls. The window shade is up. It looks out on an overgrown half-dead shrub and beyond it the back of another house.
She withdraws, pulls the door shut, then stands in the hallway drawing and releasing steady breaths. She remembers the game, Silas's game, a scene in which she walked through an empty apartment, with the intent of finding something, a handwritten letter, in a drawer. In her memory the experience seems real, no different from her memories of things she has actually done. If she stayed here, standing in the hallway, doing nothing, then nothing would happen, forever. This is how she feels. The world suddenly is very limited. So little seems possible—almost nothing, when you get down to it. She is struck suddenly by a powerful conviction, that she will exhaust this world's potentiality soon—that she will reach a city wall, or a high cliff, or a door that won't open, there will be nothing beyond this obstacle, and that will be the end. She can feel the membrane that separates this stunted universe from the next; for a moment it seems as though she can actually see it, the faint curvature of the circular world as it swallows its tail, closes in, sealing itself off from the horror of infinity. Her hand reaches out and finds the wall; the coarse unsanded surface feels very real. She walks to the end of the hall and opens the last door.
It is, by contrast, incredibly tidy. There are several computer screens set up on an angled desk, a case of reference books, an Oriental rug half covering a hardwood floor. The bed is made with bright red sheets, and the girl is curled underneath them, her red hair and cheek anemic-looking against the pillowcase. Beside her Silas is sitting up, his shirtless back against the headboard, with a paperback book in his hand. On his face is an expression of profound surprise.
Elisa feels a surge of excitement, almost joy. She stares at him. Then she turns and goes back to the kitchen and sits down at the table.
Silas appears a minute later, wearing boxer shorts and a clean white tee shirt. He has recovered from his astonishment. He says, "What the fuck."
"Sit down," she says, and the conviction in her voice surprises her; it does not match her emotions. "We have something to discuss."
He appears vulnerable here, without a drink in his hand. He's thin, she can see that he's unhealthy. But his eyes are alive with calculation.
After a moment he pulls out a chair and sits. On the counter behind him stands the object that preoccupied her yesterday, the mason jar filled with brown liquid. She intends to make demands, arrangements, to save his brother. Instead she says, "What is that?"
He frowns, turns, looks around. When he turns back to her, he's angry: she has surprised him again. "What's what?"
"That." She points. "In the jar."
Clearly he doesn't want to turn again, doesn't want to be tricked or thrown off balance. But he can't help it, he looks.
After a moment, he says, "I have no fucking idea."
"Really?"
"Some rotten shit from the fridge, I don't know. This is why you're here? You break into my house to ask me this?"
She says, "You have to let your brother go."
Silas squints at her. He drums his fingers on the tabletop; his other hand twitches on his bare knee. He looks like he should be smoking a cigarette and indeed glances around as if one might be there, freshly lit, waiting for his fingers.
"I don't get it," he says.
"Whatever you've done to him. Made him do. The money you stole. Let him go. Let him come home. Solve your own problems yourself."
Silas leans back, shaking his head. He holds out his palms. "What did he tell you—I made him do it? Steal?"
She waits.
"For what? My company?" He shakes his head. "No. I got problems, but I have a fucking life. He's the one with the real problems. Whatever he's stealing, it went up his nose or in his arm, it didn't go to me. And I'm risking my career to cover his ass. I would love for him to walk out that door," Silas says, pointing, leaning toward her across the table. "I'm sick of his shit. I share my house with him, my fucking work, everything, and he just fucks everything up. It's a good thing the startup fell apart, because he would have just fucked that up, too, right when it started getting good."
Elisa hesitates. Doubt is creeping in. To look at them, to look at both boys, the state of their rooms, she could believe him. But she doesn't.
"Your girl," she says. "Do you share her, too?"
Silas is shaking his head before she has even finished speaking. He throws his arm over the back of the chair. "Wow," he says. "Wow, that's awesome. Is that what he's telling you?
"Look," he goes on, "None of this is any of your business, Lisa, okay?" He's angry, his eyes are hard, but there is something strange in his voice, something she doesn't recognize, that is almost like sympathy. Was it always there? Even in the world she remembers? No, she tells herself, no, this world is different. It couldn't have been there. "You threw us over," he is saying. "That was your prerogative. But now it's mine to tell you to fuck off. If you want to take Sam with you when you do, more power to you. I would love it. Maybe then I wouldn't lose my job over him. But either do it, and leave me alone, or don't do it, and leave us both alone. My job is none of your business, my girlfriend is none of your business, the shit on my counter is none of your business. I could have predicted this," he says, looking tired and old. "That you would change your mind. What about Derek? Him too? Is he on board with this?"
She can't speak.
"That's what I thought. No, he makes too much fucking sense for that."
He stands up, pushes his chair in. She is surprised at how tall he is: he hadn't finished growing when he died. She wouldn't have predicted it. Again he gestures with his hand as though there's a cigarette in it. "You ought to go back to that shrink and have your head examined." He looks at the ceiling, then back at her. "I can't believe you did this. Came out here. Incredible."
"You should move your back door key," she says, and she is trying to sound defiant, but she just sounds like his mother telling him to put his things away. He is showing her his back, he is walking out of the room. "It took me ten seconds to find it."
Sitting there, illicitly, in her sons' kitchen, she is feeling her mind begin to rebel against it all: it would like to shut down now, reject everything, begin work on its own reality, some happy fantasy where it could exist in peace. But she has to resist: she has to consider whether or not Silas might be telling the truth. If perhaps she is wrong, and has always been wrong. About Silas and Sam and everything, in every possible world.
The answer can be found—Derek knows, Amos knows, Sam knows. Here, in this life, all the men have the information, and her role is to extract it, to wheedle it out of them, to beg them for it.
Silas is halfway down the hall now. She says, "Silas, wait!" And to her surprise he does, he stands there with his hand on the wall, close to where she touched it minutes ago. His back is to her. He's waiting.
"Where did you go? That time you ran away, when you were fifteen."
His hand drops from the wall and he half-turns toward her. He appears exhausted.
She says, "You remember—your lost weekend. You left school and didn't come back for days. Where were you? What did you do? We searched and searched." In her voice is real conviction, as if she actually remembers it, as if it actually happened.
But Silas is shaking his head. "I just wanted to be alone," he said. "That's all. I went off to be alone."
For a second he looks as though he's finished; he shifts his weight, he shows her his back. But then he turns back to face her. "And I didn't run away," he says. "I walked. It was easy. I just walked away and nobody followed me."
And now, as if in illustration, he does exactly that: walks down the hall, away from her, opens the bedroom door and passes through. The door closes, gently, and then everything is quiet again.
42.
She has one more day here. She has an idea about what to do, but isn't sure how to go about it. So she lies on her hotel bed, dozes for a short while, tries calling Sam several times. He doesn't answer.
Around noon she heads for the coffee shop she went to the previous day. There's a free internet connection—the first fifteen minutes are free, anyway. And her laptop battery is almost dead and she forgot to bring the power supply. But she works fast. Caltech, that's where Hugo Bonaventure teaches, and it's nearby. She tries to recall what he told her, the experiment his colleagues performed, and over her coffee finds the name of the man who heads the research team, and his office location. Buses go there: she plots a route. She is aware that it's Sunday, but scientists are in their labs on Sundays, she certainly remembers that much.
Of course this occurred to her before she left, though until now it hasn't coalesced into a plan of action. She has heard nothing from Hugo Bonaventure, and her offers to take Betsy out for coffee have gone unanswered. The plausible explanation she has been craving, the one that lies outside herself, has never seemed farther away. And now, in the wake of her conversations with Silas and Sam, she needs it.
This world is aberrant and wrong, and somebody needs to tell her this. Even if she can't go back. She will live in hell if she has to, if only somebody will please tell her that, yes, it's hell, and she does not belong here.
The buses are infrequent. She walks a long way to find the right stop, then is too late and she must wait in the heat for the next bus. She makes some transfers. She doesn't have the right change. She gets off, gets cash, buys some mints and asks for singles. It's an hour and a half before she reaches the campus, and she is exhausted and hot and feels like finding a tree to lie under and go to sleep. She wishes she'd taken the laptop out of her satchel, left it at the hotel. It's heavy.
Caltech is small, all of its buildings clustered within a single large city block. The buildings are dull and angular, sandlike, etched with strange patterns. She walks among them as if dazed, finds the one she wants, consults the directory board in the lobby. The air here is cool and almost nobody is around; she stands at the directory for a long time, listening to the sounds of her own breaths. Finally she climbs the stairs and finds the right office.
No one's there, of course; the door is locked. There is a schedule taped to it that refers to the spring semester already passed. She stands, thinking, for several minutes. Then she hears footsteps and looks up.
It's a tall young man, heavy and bearded, sweating, in a hurry. He moves past her and keys open a door a few offices down. When she reaches it, he is rummaging through a file cabinet, muttering under his breath.
"Excuse me?"
The boy looks up, startled—perhaps he passed her without even noticing she was there.
She says, "Where could I find Professor Simmons today?"
He blinks, stands up straight. "Uh. Lab?"
"Which building?"
"Or he might be at home. His wife's like nine months pregnant."
She waits.
"Downs? Like, right over there." He points. "I mean, it's the same building. Just, down the hall and through the double doors." She's about to thank him, but he says, "Wait, who are you?"
"He's working on something I'm involved in."
The boy blinks. He looks like the bass player from a seventies rock band. He says, "Hold on, I'll take you." He continues his search through the file drawer, seems to find what he wants in the form of a sheaf of heavily annotated, equation-covered papers. It is quaint, the idea that somebody would need to go somewhere to gather up some papers, in this electronic age. She likes it. The boy ushers her out into the hallway.
He walks a little too fast for her. She has to sort of run. They pass through the double doors, then another set, and then he stops and asks her to wait before turning to yet another door, this one protected by a proximity reader, in front of which the boy waves a lanyarded ID before entering.
She leans against a cinder-block wall with her eyes closed. The wall behind her is alive with some deep and thrumming energy. She doesn't fall asleep, or probably doesn't. When she opens her eyes a man is standing there. He is tall—are they all tall here?—and clean-shaven, with unkempt sandy hair just beginning to go gray. His cheeks are a bit sunken, his chin too long. He's dressed in jeans and a tee shirt that says, in a flowing script, Choose Rudeness. He says, not rudely, "You were looking for me?"
"Professor Simmons?"
"Yes."
Elisa licks her lips. "I'm Elisa Brown," she says. "A friend of yours sent you some things of mine. Some objects."
Puzzlement. "Who's your friend?"
"Hugo Bonaventure? He's at SUNY Reevesport for the semester. That's where I work."
The man's face is friendly, open, but she understands she doesn't have much time. The name Hugo Bonaventure does not seem to register with him.
"It's your experiment, correct, with the vibrating metal flange? That is also not vibrating?"
He appears surprised. "Yes, that's right."
"This man Bonaventure, he sent you some things of mine. To test. To see if... if..."
Simmons is waiting.
"He said you could determine... where they were from."
It takes a moment or two for his mind to churn through the possibilities, to decide that she is mistaken or not worth his time. She wishes there were something, anything, she could say to halt the process, go back in time by a minute or two, devise the perfect appeal to make him listen. But no, he's done with her, he is shaking his head, he is about to say goodbye.
Then he says, "Oh wait—that guy?"
She waits.
"Curly haired guy? From Belgium or something?"
"That's him."
Simmons sighs. There's a bit of sweat on his face, a light sheen, and he wipes it away with his hand. With the other he digs a phone from his pocket and glances at it, at the time.
He says, "Okay, follow me."
They trace her previous path through the building in reverse, and at equally high speed. At some point the phone in Simmons's hand rings and he begins talking into it about certain grocery items that need to be bought. "I got somebody here," he says, and pockets the phone, and then they're at his office door, which he opens with a key attached to a belt loop by a length of string.
It's a mess. Elisa is strangely gratified to see an academic office that adheres so completely to type. Books and papers are piled on every surface but two: a chair behind Simmons's desk, and a chair in front of it. He neither sits in the former nor offers her the latter. He is pawing through a giant pile of mail.
"He sent me some stuff. The guy."
She nods, though he isn't looking at her and can't see it. "He says you're colleagues," she says.
"No."
"But you know him?"
"He teaches here. He's not a scientist. He's more of a..." His elbow makes contact with a pile of mail, a different one, and it topples, spilling papers and packages onto the floor. "Fuck. He's more of a gadfly. He's in HPS."
"HPS?"
Simmons has found what he's been looking for: a brown padded envelope, torn open at one end. He sticks his hand in. "History and philosophy of science. Google him." He pulls the hand out, and there they are: her lipstick and list. The envelope is tossed back onto the desk. He offers her the objects.
"Guy thought I could test them or something. Or asked me to. Probably I'm just part of some little experiment."
She takes her things back. The envelope is lying there, still bulging slightly from Simmons's hand; it is addressed with a marker, in a bold near-scribble. There's a note inside, no doubt; she wants it. But she can't bring herself to ask for it. "Experiment?" she says.
"He pokes scientists, tries to make them react. You know."
She says, "So you can't test them."
And now Simmons seems to notice her for the first time. He looks directly into her eyes, scowling slightly, as if in order to figure out what precisely it is he's got here. She feels like a fool.
"You really think these things are from another world?" he asks.
Slowly, she slides the lipstick and list into her satchel. "I don't know what I think. Something happened to me. I'm just trying to figure it out."
He's nodding, nodding. Elisa is beginning to feel the full force of his concentration. She is attracted to him, to this intensity. It is akin to her own, she feels—or akin to what she once was. "I'm a scientist too," she blurts. "Or used to be."
But Simmons just shakes his head. "That guy isn't going to help you," he says, and shows her the door.
It's true, what Simmons told her—Hugo Bonaventure is a sociologist. She finds this out ten minutes later, in the computer lab on the first floor, after Googling him, like she was told to do. He is an eccentric, much beloved among undergraduates. He is interested in metaphysics, mass delusion, and the notion of science as religion.
His résumé is available on the HPS website. It's many pages long, listing dozens of papers and several co-authored books. Under the heading "Work In Progress" is a study titled "Science Faction: Why We Believe in Alien Abductions, Parallel Worlds, Superpowers, and More." The names of several collaborators are listed, and one of them happens to be printed on a business card Elisa already has in her satchel, along with a web address, the address of a physics blog she hasn't yet bothered to visit. She visits it now, and there he is, a collaborator himself. Hugo Bonaventure, collaborator on an ongoing study, "Science Faction," with Betsy Orosco.
43.
On the plane home she begins shaking. She feels no particular emotion; she's just shaking, as if she's very cold. But she isn't cold. The woman sitting beside her leans ostentatiously away and eventually presses the button to summon a flight attendant.
"Ma'am? Are you all right?"
Elisa's voice wavers as she says, "I'm fine."
"Do you need medical attention, ma'am?"
"No."
"Can I get you anything?"
"No, thank you."
A few minutes later her teeth are chattering and the hiss of the ventilation system is making her feel sick. The flight attendant comes back with a cup full of ice and a small bottle of gin. Elisa accepts it, pours it. Drinks it.
Only as she is falling asleep do the tremors subside. She wakes at landing with a desperate need to pee, and hobbles out of the airplane in actual pain. The feeling of release, when she reaches the women's room, is profound, and she presses her palm to her forehead and moans.
Derek isn't picking up his cell. It's Sunday, she doesn't know what he could be doing. She gets a cab. Out the window it's cloudy and cool and her body aches from sleeplessness and her bout of shaking.
At home, something's different, she isn't sure what. The truck is missing, but she calls out to Derek anyway. In the bedroom, she dumps her bags on the floor, then lies down on the bed. She falls asleep. When she wakes up it's dark and she notices that the closet door is open. It looks half empty.
She walks down the stairs. "Derek?" The only light is from the kitchen, where she finds a piece of paper folded into thirds with her name written on it in his handwriting. She picks it up, turns it over, unfolds it, and reads.
Lisa,
I had to leave for now. I need some time to think things over, and I think you do too. I'm staying with a colleague and will be in touch.
Maybe you're right and we should revisit the past. But I don't know why you would go to California instead of telling me what's going on. I assume you had a bad trip. I hope I'm wrong.
Sorry
Derek
When she's finished reading it a second time she folds it back up and drops it on the table. Through the open kitchen window, from somewhere far off, she hears the sound of a girl screaming, then the scream trailing off into hysterical laughter. She goes to the refrigerator, pulls out a block of cheese, and eats the entire thing standing there with the door open. Then she goes to the living room and plays the game again.
PART THREE
44.
Time begins to accelerate.
Elisa spends the next two months attempting to restore her life to the one she lost. She throws out most of the clothes in her closet, then buys new clothes in her old size, as an incentive. She is almost there anyway, having eaten little over the past two weeks, and continues walking back and forth to work.
She gets an apartment. Derek is shocked; perhaps this is her intention. They have lunch together every week or two, and each time seems, to her, more pointless than the last. Derek's shock gives way to hurt and eventually to acceptance. She doesn't tell him anything, and doesn't ask him anything, about the past. (She does not go to see Amos, either, despite the voicemail messages he's been leaving.) Soon, she is certain, Derek will realize that there is no longer any real reason to meet, and they'll stop, and their separation will harden into established fact. Her new place is downtown, four blocks from the frame shop. It's a one-bedroom apartment, and she has decided to use the living room as an art studio. Through one of the windows it is possible to see a little wedge of lake, tucked up against the diagonal of a church steeple: a real lake view, without having to climb onto the roof.
Sam doesn't reply to her e-mails. She tries calling him at Infinite Games, but is told he isn't in. She doesn't leave a message.
The bearded man who has her old job at Killian Tech is still there in the corner office, plugging away. She looks at the company website and finds out that his name is Wayne Pratt. He has a personal website where he posts close-up photographs he has taken of various plants. His CV is available for download, as well. She downloads it: it now sits on the desktop of her computer.
She stops by the frame shop twice. Both times, he isn't there. In retrospect, she thinks she probably knew he wouldn't be, at those times. Desire seems very far away right now. She doesn't miss him; something about being in this body, being this Lisa, has undone her desperation. But she misses desire itself, she misses need.
After reading the classifieds every morning for a month, she sees an ad for a blue 1991 Honda Accord with 153,000 miles on it. She buys it, then sells the Intrepid. She spends some of the difference on beaded seat covers like her old ones. The first time she drives the car with the seat covers on, she cries. She considers, then abandons, the notion of trying to crack the windshield in the same pattern as the old one.
Her separation from Derek deepens her friendship with Judith. She resists this for a couple of weeks, as it wasn't part of the old life, and the woman is annoying. But it is nice to have a friend. After a time she begins to look forward to their lunches with nervous excitement, joy even, and she doesn't understand why. It is almost a cleansing ritual. They usually eat in the food court at the supermarket, where Judith can talk as loudly as she likes and Elisa can scream with laughter. Elisa never talks about herself, just listens to her friend natter on.
After one of these lunches, picking up a few groceries before returning to work, Elisa passes Betsy in the tea and coffee aisle. Their eyes meet and the younger woman looks away. Did she recognize her? She's thinner now, so perhaps not.
She begins to spend much of her free time online, on the MetaphysicsNet parallel worlds forum. She actually registers and chooses a screen name: CrackedLisa. She regrets the name a couple of weeks later, but by that time she's already begun to develop her identity.
The forum is carefully moderated. No apparent crazies. They divide into two main camps, people who philosophize and theorize about the concept, and people who think they have evidence of its real-world existence. It is not common for people to believe they are in a parallel world; at least no one says so. But she senses they are there, lurking. She reads back through the archives, three years' worth, digesting it all. People recommend books and she borrows them from the library. It becomes her hobby. She starts painting diptychs: nearly identical panels, save for slight differences. She doesn't tell anybody what she means by them. Of course there's nobody to tell except Judith, who wouldn't understand.
If there is pain from her separation, she is not conscious of it. She doesn't long for Derek. It's as though he's a food that spoiled and that made her sick, and now she never wants to eat it again. She is certain this will change, but so far nothing.
It's as though she's suspended between the two worlds. Or living in a world that is one subtracted from the other. Nothing at all seems real now.
45.
October. It's unseasonably warm, even under a dark gray sky. The orange and yellow boughs of maples whip in a hot wind but the leaves don't fall, not yet. The streets seem empty. The details don't match. It's fall break at school so she gets a couple of days off, and decides it's time. She goes to the frame shop. He's there. He has grown his winter beard already. She asks him to come to lunch.
They eat at the usual place, the Asian café. Elisa watches him carefully as he eats. He is so familiar: the careful way he has of keeping his beard free of food: he opens his mouth a little too wide, takes smaller bites. (Or is this, in fact, familiar? Is this a thing he used to do? It's suddenly unclear what is memory of their past together, what she has generated as part of her fantasy of him.) He has asked for an extra napkin and holds it under his chin every time he lifts the chopsticks to his lips. His movements aren't stiff, they're controlled. Fluid and hard. He talks about jazz. She doesn't want him to talk about jazz, he's not supposed to be interested in it. But she nods, listening.
She mentions her separation and his brow furrows as it does every time he must process new information. He makes a sound, a kind of clicking with his mouth, like a hard drive being accessed. The clicking isn't right—he didn't used to do that. She'd like to point it out to him, to make him self-conscious about it, but it's too soon for that kind of intimacy.
Elisa buys. She asks him if he really needs to go back right away. His mouth clicks.
They go to her apartment for sex and she gives him exactly what he wants when he wants it. He is taken by surprise and doesn't last very long. He's embarrassed, in fact he apologizes. "Don't apologize," she says. Her Larry wouldn't apologize. Or would he? The truth is, she doesn't know anymore. She didn't have these experiences with her Larry. Maybe he would have apologized. Maybe he would have come even sooner. While he dresses, he tells her she should come over and get a look at this new turntable he bought. He corrects himself: "New old turntable." It's got a belt drive, and he just installed a new belt. She doesn't understand what this is, or why she's being told, but okay. "Call me, then, we'll make a date." When he's gone she savors the taste of him before making coffee. That much is right, anyway. She's nervous, her hands tremble, but her heart is steady. It's not what she expected, none of it is.
She sees Derek at the supermarket. His shirt's tucked in and he has combed his hair. She can tell he's trying to make himself attractive to women. To a certain kind of woman—not the kind she has become. The kind Amos Finley and he—and she—tried to make her into. He turns and sees her and she expects him to pretend he didn't, to turn away, the way Betsy did. But instead he muscles his cart around and comes right up to her, and appears disappointed when he arrives.
Their pleasantries feel ridiculous. They have never engaged in them with each other before.
She says, "You were going to say something."
He is gripping the handle of his grocery cart and his knuckles stand out in sharp relief. "I don't know."
This is not something he often says. He won't look at her. He looks softer, as if he's been eating more without her there to stop him. This should make him look more like the Derek of the old life, but somehow he doesn't.
"It's kind of ridiculous," he says, finally. "That we're apart. And apart from the boys." Now he looks up. "I'm not asking you to come back, I'm just saying. The point of cutting off the boys was so we could stay together."
"And?" she says.
It's a mistake. It makes him angry. "Jesus, Lisa."
"I'm sorry."
But he's already turning away, hauling the cart behind him. "Nothing was ever easy with you," he tells her over his shoulder, and as she watches his broad back recede through the crowd she thinks, Easy? Are things supposed to be easy?
46.
One day she is walking past Killian Tech and sees that the little Zen sandbox and photo of the blond woman are missing from the corner office desk. So is the diploma that used to hang on the wall. Impulsively, she walks in. She asks for the head tech.
She is a bit surprised at the man who appears: gray-haired, in his sixties, he is lanky, stooped, confident in his demeanor. He's got a bandage on his elbow; something tells her he fell off a bicycle. She has never seen him before. The head tech she remembers was stocky, in his forties, a man named Ronnie. She introduces herself and asks if they need a lab manager.
"Yes, we do," he says, surprised. "How did you know that?"
She explains: waiting at the bus stop, she used to notice the man in the corner, hard at work. And then today his things were gone. "I'm detail oriented," she says.
But he appears puzzled. "How did you know Wayne was the lab manager?"
"I have some experience with this kind of work." Which is not really an answer.
When he doesn't reply, she says, "Listen—do you have a few minutes? Interview me for the job."
"Why don't you just drop off a CV?"
"I will, I will. But let's talk."
He opens his mouth to say no. Then he hesitates, says, "I know you. You work in the biology department, don't you."
This gives her pause—she didn't anticipate being recognized. But she says, "Right."
At last he shrugs, invites her to follow.
Elisa experiences a rising excitement. She was going to wait on this, she was going to bide her time. She's unprepared. It's late afternoon on a Thursday, her office closed early because a construction project snapped an underground power line. She is dressed like her old self, in jeans and a cardigan sweater. Non-Ronnie leads her down a hallway and it isn't quite right, it isn't what she has in her head. The floor plan was different. Is different. But it's close enough, with its drop ceiling and muted patterned wallpaper and faint buzz from overhead fluorescents. This is where she used to work—it has to be.
Ronnie's office wasn't a separate room; he just occupied the corner of the storeroom that lay at the end of the hallway. He didn't spend a lot of time there, he was usually out on the floor working. The room was lined with steel shelves packed with boxes, papers, glass items, chemicals in bottles. When occasionally he would summon her for a meeting, he would enter in front of her and squeeze into the small space behind his desk, and gesture toward a plastic patio chair, inviting her to sit.
But this hallway doesn't lead to the storeroom. Or if it does, they don't go there. Instead this manager, the new manager, opens a hollow-core door into a cramped office cheaply lined with wood paneling. There's a file cabinet in the corner. The lights here are buzzing even louder than in the hall. He points at a plywood waiting-room chair, upholstered in worn gray fabric. Then he folds himself into one himself, behind the desk.
"So what experience do you have?" he says, when she has descended, nervously, into her seat.
"I managed a lab for eight years. Very much like this one."
"You know what it is we do?"
She nods. "Outsourced research, genetic testing, forensic contract work, that kind of thing."
The man shrugs. "How many clients did your old lab have?"
Same as this one, she thinks. "It varied, depending on the size of the projects. Anywhere from three or four big jobs at a time to a couple dozen small ones... all told, there were thirty or forty clients we had regular contact with, maybe five or six we had work from regularly, that made up the bulk of the business."
He is looking steadily at her, his chin supported on his pointer fingers. "This was where?"
"Madison, Wisconsin," she says without thinking.
"You really should bring by your CV."
But she can't resist. She is looking around this office, thinking about the time Ronnie confessed an affair to her—a woman he'd met at a seminar in Rochester, they'd been seeing each other for a few months, and he was racked with guilt but couldn't stop. As he talked, her gaze was fixed on a poster over his right shoulder, some kind of parody of the New York subway map. The stops had labels like WEIRDOS and PIEROGIES. No posters are hung in this room. It appears that, like Ronnie, this man doesn't spend much time in his office. Suddenly she wishes she smoked, that she was smoking. She wants something to hold. She says, "A typical day here would probably go like this. Somebody, let's say you, or me, if you hired me, unlocks in the morning. You take a clipboard off the rack and do a walkthrough, turning on the machines and computers, flipping on the lights, checking on the petri dishes and so on. Then the techs would start rolling in around a quarter to eight to fire up the mice, check the cages, what have you."
He is scowling at her now, concentrating deeply, and she feels she is making a mistake, but can't stop talking. This isn't like her, she thinks—but it is, it is, it's like the old her, the missing one, who liked to stay up late in the lab at night, the one who loved men too much, the one who gave herself to Derek, to motherhood, and never looked back. And she remembers why she never looked back. It is embarrassing to be this person. She is exuberant and imprecise and makes a fool of herself. She breaks things, ruins things. Elisa tells herself Stop, don't blow it—but she keeps on. "For the rest of the day," she says, "they'll be logging results and crunching numbers. You'll have work to do for the city, I'd imagine, environmental stuff, a few nonconfidential police jobs, and by ten in the morning I'll have fielded calls from a couple of clients, handled some inquiries, and so on. The office manager, I mean. Afternoons we prepare portfolios for people, and I would compile those and send them off electronically, or if they want them hand-delivered with an explanation I will go do that. We also take care of small jobs in the afternoons—well water testing, drug testing, that kind of thing, usually this falls to whoever the intern is, a college kid generally. And then you usually leave by five thirty, stop in and see your mother or pick up something to eat—" And here of course she is thinking of Ronnie, Ronnie and his stern and handsome wife Gwen, and Ronnie's mother at the nursing home and the sandwiches he used to tell her he was going to order before he left, and did Elisa want one?, and she realizes that this will not do, this man isn't Ronnie and the lab is different. "I don't know what you do," she goes on, "but I double-check that everything that's supposed to be powered off is powered off, and everything that's supposed to be running is running, and lock the mouse lab and storeroom, and then I lock the front door and I go home."
She is panting and feels faintly nauseous. They're silent together for a moment. The head tech's face is taut, his eyes bright.
"Well, we don't have mice here," he says.
She can only muster an "Oh."
"Also, I don't..." He screws up his face, tilts his head, gazes at her with one eye half closed. "I'm not sure... my mother doesn't live around here."
"That was just a... an example."
"Is this some kind of joke?" the man asks. "Did Dean put you up to this? I don't understand the bit about my mother."
She keeps very still. Her sweater itches her and she clutches her bag to her lap.
"Something like that."
"You seem to have worked at a lab. And we do some of those things. Environmental testing, work for the city. Not the police. You've done this kind of thing, you say."
"Yes."
"But... this is a practical joke?"
"No, no," she says. "I just meant... I didn't mean..."
He is leaning forward now, palms flat on the desk. "I find this whole encounter very odd," he says.
"I'm sorry."
"You are the woman from the biology department, aren't you? Laura?"
"I was only—I'm only looking for another job."
He stares at her for a long time, and as he does, his face grows longer and harder, like something that has melted and then cooled.
He says, "I'm thinking it probably isn't going to be this one."
He says, "Do you want to explain yourself? Should I call the biology department?"
There's nothing she can say that will explain anything. She is afraid that he will get up and block the door. She doesn't think he would, but what does she know? About this man, or about anyone? People are who you think they are until they do the thing that proves you wrong. Her head has begun to pound.
She says, slowly, "It's not a joke, it's just... I thought it might impress you."
Silence. This is inadequate. But she doesn't have anything else.
"How about," she continues, "if I just get up and leave now, and never come back. This was a mistake. I'm sorry."
His expression does not change. She gets up. She leaves and doesn't go back.
47.
It's winter before she hears from Derek again. Or not quite, really: the week after Thanksgiving, snow falling and blowing in wild circles in the street, several inches already on the ground. It's a Saturday and she is watching this spectacle out the window and thinking what everybody else in Reevesport is thinking, which is that their hope for a prolonged autumn without scarves and gloves is now shattered. Of course there's a part of her that likes this weather very much, likes the feeling of forced indoorsness, the excuse to drink more hot coffee. She is glad to be alone. Thanksgiving she spent, for the first time in years, in Chicago, with her parents, and though she expected to be depressed by their advancing age and eccentricity, she found them almost charming. They didn't comment on her separation from Derek. They seemed genuinely glad to see her. They appeared very firmly in love with each other and in the idea of isolation from the rest of the world.
The first thing Derek says when he calls her is, "Crazy weather, huh?" and before she can stop herself it makes her laugh.
"I don't think we'll ever get good at that," she says. "Small talk."
"I suppose not."
There is a moment of awkward silence. It's strange to experience: they have shared so many hours of companionable silence in a quarter century—more—that the awkwardness seems to belong to someone else.
Derek says, "What are you doing right now? Can we meet?"
"Drinking coffee. Come on over."
She says this without thinking—he's never been here of course. To this apartment. His silence is answer enough; she corrects herself. "How about the Edge?" This is a café not far from here, though of course he'll have to drive. Though this only matters to her—he likes driving.
Fine, he says, he'll see her in an hour.
Like everyone on the street, she hasn't gotten the winter clothes out of storage yet, so she puts on a hooded sweatshirt and a canvas jacket on top of that, then walks to the café with her head down and her bare hands curled deep into her pockets. There are the sounds of wind and traffic, but no voices; people passing say nothing to each other, nor to their phones. Some crows somewhere are freaking out. It feels like the end of the world.
The café is warm and moist, the windows fogged and dripping, and the staff are playing loud music, as though to compete with the wind. She imagines that Derek will be annoyed by the music and she's right; though he says nothing, he can't resist training a sour expression in the direction of the counter. They both order black coffees—her inquiring look at Derek, a lifetime milk-and-sugar man, in both worlds as far as she can tell, goes unacknowledged—and take a table far from the window. On the bulletin board behind Derek is a pristine pull-tabbed ad for bass guitar lessons and a lost cat notice. She thinks, We are still married.
"I'll get right to the point," he says. "I've stopped going to Amos."
"Why?"
He levels an annoyed gaze at her. "We went to him to stay together. We're not together."
Her instinct tells her to apologize now, but her instincts are bad, so she says nothing. After a pause, during which his body jerks the chair, loudly, into a new position, he goes on.
"I don't know if you think this is permanent."
Is it a question? She says, "I have no idea what this is."
"Well, I think this is a trial separation."
"Okay."
He scowls, sighs. "Don't do that. Capitulate."
"I'm not capitulating. I'm just encouraging you to get to the point."
"Okay. Sorry."
She savors the sorry as he gathers himself to speak.
"Now that we've been apart for a while, it all seems so..."
She waits.
"It's not that I regret this. But it's hard to remember why I was so upset that you went to see the boys. It made me... it felt like the ultimate transgression. Given our arrangement. But now it seems more sensible." So far he has been staring into his untouched coffee mug, but now he looks at her face. "Maybe the arrangement wasn't sustainable. Maybe it was time to change. I still have no idea what happened to you at that conference, and if we get back together—"
They're both surprised to hear him say this and he appears, for a moment, to be choking back tears. He sips his coffee with a wince before he resumes speaking, now with his head down.
"All I'm saying is, it doesn't matter what happened. What the situation is now is all that matters. And I'm thinking we should apologize to the boys. And try to start over."
"With each other?" she asks him.
"With the boys."
Neither of them speaks for a minute. The girls behind the counter are laughing at something. They have turned the music down—Elisa realizes now that she and Derek are the only customers.
"It just isn't right," he says, and there is no danger now that he will choke up. His face is hard; his head looks heavy, like a boulder. He's showing his age: the cheeks a bit sunken, the lines deeper. He carries it well. He has always looked best under the weight of some burden.
He says, "It isn't right that we're all scattered like this. I don't know how it happened."
Elisa pats his hand where it is loosely clenched on the tabletop, then crosses her arms over her chest.
48.
They decide to write a letter, a paper letter, and mail it. To Derek, this makes things more official. Neither of them suggests doing it together, in person; instead each of them writes a draft and they compare them via e-mail.
Elisa's reads:
Dear Boys,
Things have been changing in our lives and we wanted to talk to you about these changes. We have separated, but are in close touch with each other, and have come to realize that it was wrong to cut off contact with you years ago. We realize that it would be difficult for you to forgive, and don't expect you to. But we want to open the lines of communication. Will you talk with us about this?
We are so sorry. We hope that we can all be some kind of family again.
With love,
Elisa and Derek
It takes her ten minutes to write the letter and two hours to decide whether or not to sign it "Mom and Dad." When it's settled, she opens her e-mail to send the draft to Derek, and finds that he already has sent his to her:
Silas and Sam:
It is probably a shock to find a letter from your parents in your mailbox, and I hope you have opened it and are reading it now. If you haven't—if you instead threw the letter away unopened—then we can hardly blame you, given our recent history. We are writing to tell you that we now believe our decision three years ago to cut off communication with you was wrong: it was extreme, insulting, and unnecessary, and the worst part is, it didn't even work. It may surprise you to learn that we are now separated and living apart, and we are separated from you as well. And in a sense perhaps you are, and maybe always have been, separated from each other. This last is also our fault, certainly as much as it has ever been yours. We are finally beginning to accept that we were not good parents; we did not deal with your troubles well, nor our own, either.
This realization is particularly difficult for me, as I grew up, at first, without a good father, and later with no father at all. My father was a bad man—he was domineering, belittling, violent, and sadistic, and he beat my mother and nearly drove her to madness when I was a boy. It was a relief when he finally left, and over the years of my late childhood I watched my mother transform herself from a tired, beaten-down victim to a self-sufficient, strong, loving parent. I admire her deeply, and cherish the relationship I have with her today. I am glad she has been a part of your lives, and I know that I hurt her terribly with the decision your mother and I made together. Maybe she has been in touch with you—I have not asked her.
I never talked much about my father to you, because I didn't want him to have any effect on my family, but now I fear that he has had all too powerful an effect, and I have allowed his influence to ruin our lives together. I am sorry. I have lived a life of fear and passivity, and look at where it has brought us.
We would like to ask you to please consider restoring communication with us.
This letter is not signed, and neither is the e-mail he sent it to her in. Derek has never said these things to her. He never said anything about being afraid, or feeling passive, never told her that his father beat his mother.
The loneliness she feels, sitting in her apartment in front of her laptop, is so profound that she wants to go to Derek's house, to go home, take him to bed. Beg him if necessary. Instead she sends him her version, and a few minutes later he agrees that it is the better choice.
They send the letter; the boys do not reply. Eventually Derek sends his version. They are still waiting.
49.
Now she returns to her apartment. Now she gives up trying to remake this life into the old. She drinks, heavily, every Friday night with Judith. This was Elisa's idea, and Judith seems delighted by it, though it isn't as if she doesn't have lots of other friends, with more in common than she has with Elisa. Elisa should be more grateful, she thinks. Indeed, this new ritual is a kind of penance, for the days, some months ago, when she thought she might become friends with Betsy, the physicist. But clearly the woman has decided that she is some kind of freak. In retrospect this attempt at friendship seems silly, and an insult to Judith, her actual confidante and reliable, if ill-matched, pal. One of the things they talk about is Larry, whom Elisa sees again, several times more, with a growing sense of futility and effort and unease. It just isn't him, he isn't the man she loved, and she isn't the woman who loved him. It isn't even close, really, and soon she stops returning his calls. She avoids walking past the frame shop now and doesn't eat at the Asian café. She artificially maintains the sense, in her own mind, that theirs is a relationship coming to an end, so that she can have something to talk about with Judith. But in truth it never really got off the ground.
She thinks about Derek all the time. She would like to make amends but isn't sure what she wants to do with them. So she does nothing. They, too, have stopped getting together for coffee.
Elisa no longer wants to go back. Indeed, she is increasingly frightened, throughout the month of January, by the possibility that she might now be sent back against her will, in an instant, the same way she got here. She begins to think in terms of cause and effect: What did I do to cause this? What should I do to prevent it from happening again? She once feared the apparent randomness of her situation. Now she fears that some intelligence might be behind it, after all. She lies awake at night in her apartment with her jaw clenched, imagining having to mourn Silas a second time.
And as for Silas, he has disappeared. The forums say that he has left Infinite Games, though no one knows why. Minefield hasn't posted for weeks. She doesn't know where Sam is, either.
The one thing she does with any regularity that gives her some satisfaction, or at least some relief from her boredom and anxiety: she spends several hours a night on MetaphysicsNet. This allows her to transition from eating to sleeping without drinking too heavily, though she does drink. There is an almost frenetic level of activity on the parallel worlds forum. She begins to wonder if what happened to her happened to many people, at the same time, all of them conspiring in anonymous silence, afraid to speak out. At times she feels as though the claim is on the verge of being made, by almost everyone. And then, at other times, she feels completely alone.
Every day somebody seems to have discovered a new book, or study, or TV program, or blog on the subject. Every day the full membership gathers in a thread devoted to the latest thing and discusses it frantically. Elisa begins to think of the other forum members as actual friends. Joereilly lives in Palo Alto and in his avatar is posed, fat and bearded, in front of a sports car. Misstake is a lesbian with bangle earrings. Rare Fern is from Vancouver, British Columbia, and is supposedly a twenty-five-year-old woman whom all the men on the forum constantly flirt with. Of course she might as well be a man, any of them might be anyone. She has exchanged several private messages, and more recently e-mails, with a woman who calls herself DippedInSunshine, but whose real name is Patricia. Patricia is a divorced mother of three adult children, the youngest now in college. She is unfailingly cheerful, both on the forum and in private correspondence, but not, Elisa senses, frivolously so. Her cheer is genuine and stems from an actual, if groundless, belief that things will turn out all right for Elisa.
Elisa has told her about the letters they sent to the boys. She has told her about her guilt, Silas's disappearance. She doesn't tell her that Silas is dead, in a parallel world. Patricia's responses have been perhaps the only kindnesses she has been done in many months that actually have had any effect. They are written in an evident rush, in a kind of rolling, opportunistic grammar, punctuated only with ellipses. I know this is hard to accept... but you will love again... your wayward boy and romance as well... to be grieving... is good for the soul... you need to heal... it will take time... but believe me your life has just begun....
In this, anyway, Patricia is mistaken: Elisa turns forty-six in February and feels very much as though her life is mostly over. Derek actually takes her out to dinner. He gives her a handbag as a present. She is fairly certain a woman helped him pick it out—it's all wrong. She ought to be moved by this, by his sad effort, but she can't muster the proper emotion. Derek looks older; the planes of his face have shifted. All through dinner she thinks, We're going to die soon. Nothing is said about either divorcing or reuniting.
For some months Elisa has known that there is an annual conference of the MetaphysicsNet community. It is actually a combined event between MetaphysicsNet and a larger internet forum devoted to science fiction movies and television. There are presentations from cable TV networks, panels devoted to popular shows, and lectures from the more game or nerdy scientists and researchers in various cutting-edge topics—antigravity, rocket propulsion, theoretical physics, and the like. The scientists who actually populate the parallel worlds forum seem to regard the conference as a kind of vacation from their real lives—an opportunity to talk about the things their colleagues find strange or uninteresting.
Patricia tells her to come. You will love it... it could be the beginning of a new life for you... restore your faith in others... the people are wonderful... you will meet lifelong friends and companions... The conference is in July, in North Carolina. She buys a ticket and reserves a hotel room.
In March, Judith invites her to go out of town for the weekend. They drive together to Toronto and they do there the same thing they do in Reevesport—drink too much and talk about men. Or rather Judith talks about men while Elisa laughs. She finds it easier and easier to laugh with Judith, and this should make their time together restorative. But the laughter leaves her with a hangover—it feels fake, it hurts her throat and face, and in the morning, after Judith, she is prone to crying jags. She succumbs to one of these in their hotel room their second morning in the city and Judith climbs into her bed and holds Elisa in her arms. This is a fine gesture; nobody has touched her in months. But it just feels awkward. She stops crying, not because she's finished, but because she wants Judith to get out of her bed. Later they go to a museum, they go to a show. Then, on Monday, they listen to right-wing radio as they drive back to Reevesport.
In April, Elisa gets a lump at the back of her jaw. She thinks, This is the beginning of the end. The doctor says it's a swollen lymph gland and tells her it will go away. In May, it goes away.
In June she fucks a librarian who, when it's over, says to her, "You could have at least pretended to like it." Also she gets a call from Derek but the ringing stops before she can answer. Maybe he changed his mind. Maybe he dialed her by mistake.
Someone named highdigger appears on the parallel worlds forum. He seems to be a young man, with a young man's presumptions and confidence. He asks a lot of questions, and his sig line comes from Wilhelm Reich. He says he thinks maybe he'll go to the conference. When, in the last week of June, he calls somebody an idiot for assuming that a particular movie plot is scientifically plausible—Your naïveté disgusts me—the chill she feels reaches all the way to her toes. She is sure this is Silas. A wave of recrimination appears to drive him into hiding. Elisa sends a private message—Who is this?—that goes unanswered.
In July she takes a week off and drives to North Carolina for the conference.
50.
She's driving. A Thursday morning in July, hot outside, so the windows of her Honda are down and the highway air is rushing in. It's the third hour of a daylong trip from the town where she is living, barely, a life without apparent purpose, to the town where she will meet, for the first time, her imaginary friends.
Her name is Elisa Macalaster Brown. It has been a long time since she's driven alone on a highway for more than a few minutes, and she is surprised to find that she is frightened. Everyone is driving aggressively, coming up close behind her, flashing their brights, swerving into the passing lane and blowing by. Their cars are sleek, the windows closed, the engines making almost no sound. The big rigs, on the other hand, are extravagantly loud. Their trailers bounce and rumble and sway over her, and she hugs the wheel as they pass, terrified of being sucked into the slipstream. She regards this fear as good. It means she doesn't want to die.
She breaks for lunch at a truck stop outside Harrisburg. She orders a cheeseburger and a strawberry milkshake, but when they arrive she only wants the milkshake. Fat men in plaid shirts turn from the counter every few minutes and look at her. On the way out, she buys a bottle of Visine and a pair of aviator sunglasses. In the rearview mirror she looks like a character in a movie about the apocalypse.
Her satchel is open beside her on the passenger seat. Inside are magazines, books, her computer, and a folder of conference materials, including a schedule, some coupons, and an ID tag attached to a lanyard. The lanyard is printed over and over with the phrase TIME COP Thursdays This Fall on SciFiTV. Of this conference she has no expectations, no hopes. She is simply trusting Patricia. Judith is a bit jealous of this mysterious other friend from the internet—she seems to have been hoping she would be invited along. Not that she would have come if Elisa had asked.
Elisa has a picture in her mind of Patricia: a tiny woman, elfin, with big ears and an innocent, sprightly manner. In her imagination, Patricia wears red Keds, like a child, and speaks very slowly, in an even, breathy monotone.
Somewhere in Maryland, she has to stop and pee, but the rest stop she has chosen is entirely out of order. There are no signs, no caution tape, it's just abandoned. She does what many before her have apparently done: she follows a rough trail into the woods behind the restrooms and squats among the trees. The sound of the highway has nearly been swallowed up by the vegetation, even though she can still see the cars and trucks passing. Halfway through she thinks she hears a twig break and again is filled with mortal terror. But nobody is there. She again decides to regard her fear as good.
The conference is at the Holiday Inn in Chapel Hill. It is supposed to be a pretty town. But the hotel is just off the highway, and she doesn't know if there will be time to do anything else. She checks in at the desk and is given a key card, which she uses to let herself into a small room containing a large bed, television, end table, and upholstered chair. It is like every other hotel room in America: too lush. There are too many pillows, too many layers of curtains, patterns everywhere. Immediately she would like to strip everything away so that it is all simple. She does remove the comforter from the bed and stuffs it into the shallow closet, and this allows her to feel slightly calm.
It's time to venture down to the ballroom, where there will be an opening-night presentation.
Elisa puts on her lanyard (the tag reads CrackedLisa) and picks up her binder and rides downstairs in the elevator. No music: the elevator is a silent box. She listens to herself breathing. The doors open onto the lobby, where a sign marked METAPHYSICSNET/SCIFITV, with an arrow, stands on an easel. She follows it to the ballroom.
The room is enormous. In the center stand hundreds of folding chairs arranged into neat rows. Around the edge, buffet tables are covered with food. The front of the room is dominated by a low stage. A lectern stands in the center, with a giant screen behind it. There is a hum of loud conversation.
It's mostly white men, and most of the white men have beards. They are all holding plates of food and cans of soda. Nobody else seems to have brought down the binder. Elisa chooses a chair halfway back, along the inside aisle, and sets her binder down on the seat. Then she goes to the buffet and helps herself to a sandwich and a can of soda.
She wanders around the edges of the hall. More people keep coming in—there have to be 150 here now. Her arms are trembling a little bit: they are tired from the drive. Why did she come exactly? She wants to go back to her room and hook up her laptop and talk to these people on the internet. These aren't the people she knows—these people have faces and bodies, their personalities are manifest on their faces. A frizzy-haired woman, whip thin, cackles at a bearded man's joke. A chubby boy stands alone, wincing: he looks like a graduate student in some impractical subject. A pale man in a plaid shirt is swaying as if in a gentle breeze. Elisa keeps her smile carefully calibrated to deflect unwanted attention. And how is she supposed to eat her sandwich with this soda can in her hand?
She returns to her seat, balances the binder on her lap, and uses it as a table. She faces forward and waits. In spite of herself, she scans the room, in vain, for Silas.
Eventually the lights dim and grow bright again. People sit down. Somebody, a round-faced man, settles in beside her, wiggling his behind on the chair. She suppresses a wave of panic. The lights go dark and stage lights come on and people applaud. When a man walks onto the stage, they applaud again, louder this time.
He's lanky, easy, charismatic in a nerdy way. He wears khaki pants and a white shirt that looks like a tablecloth. He bought that shirt for himself, Elisa thinks.
"Good evening, and welcome to the seventh annual MetaphysicsNet-SciFiTV conference!" Applause. "I'm Peter Turner, founder of MetaphysicsNet, and I'm happy to say that this year's conference is our biggest and best yet!" More applause. Peter Turner describes what is in store, which is to say what is listed in the binder on Elisa's lap. We like things to be redundant, she thinks. It's a comfort to us to be told what we already know. Because we don't trust ourselves—we need to be reassured.
Indeed, Elisa feels reassured. She is grateful for the repetition. There is something mesmerizing about this experience: sitting in this large dark room with all these strangers, the carpeted floor and walls swallowing sound, so that there is no echo. The PA system on the verge of feedback but never reaching it. She can hear the hum of the air-conditioning and feel a faint vibration underfoot, as though powerful generators are operating directly below her. The man beside her is breathing evenly through his mouth, and every now and then the breaths give way to a chuckle, after which the breaths speed, then slow, then settle. The speaker begins, then ends, a sentence; when he's through he begins another.
All around her, the spectacle of humanity in control of its emotions and actions. All around her, calm anticipation. She tucks her unfinished meal underneath her seat and folds her hands together on her binder. She closes her eyes.
Peter Turner introduces the opening speaker, who receives a loud ovation. She hasn't heard of him—he works in Hollywood. He's the consultant for a famous TV series about UFOs. People laugh as he speaks but Elisa isn't hearing the words. She is thinking about the other Lisa, in the other world. She is convinced that this other iteration of her is also at this conference, that world's version of this conference, and that she is sitting in this same folding chair—that the two of them are still similar enough to have chosen the same seat. She feels that Lisa's hands on her own binder, feels them intertwined with her own. The other Lisa is thinking about her, too. Their hearts stutter against one another, then synchronize. Their breaths ease into phase. They have two sons and both are alive. They are married and they are separated. They work at a college and they work at a lab. They drive matching Hondas and are forty-six years old.
She is dimly aware that something has changed. There's noise. Somebody is touching her arm.
"Miss? Miss?"
It's the man beside her, the round-faced man. He's tapping her. She opens her eyes. The lights are on, and people are standing up. The man is younger than she is, but he is still calling her "miss." He says, "You're spilling your soda."
She looks down. Her soda can is leaning at a sharp angle in her hand, and a pool of liquid is flowing toward the edge of her conference binder. She stares at it in incomprehension. I don't drink soda. Maybe it was the other Lisa who chose it? Maybe she has switched—she's that Lisa now. She's back in the other world! Panic is rising in her chest; she gasps for breath.
"Uh... here," the man says, and he drops a paper napkin onto the spill. Then he takes the binder and can from her hands, brushing her thigh with his fingers in the process. "Sorry, sorry," he says. "You fell asleep?"
He sets the binder and soda on the carpet. She blinks at him. It's making sense now. She calms down. I'm myself, not her. His ID says RueTheDay.
"I don't think so," she says.
He's smiling at her now. He says, "Yes, you did. You're CrackedLisa!"
"Oh," she says. "I'm—yes, sorry." She holds out her hand. "I know you."
"What a pleasure!"
"Yes!"
"That talk was so awesome. Do you watch Depths on SciFiTV?"
"I—ah, no. I don't."
The man talks for a while. She remembers his avatar: it's a version of himself, rendered as a character from the cartoon South Park. The resemblance really is strong, uncanny even. He's very animated, around thirty. He wears a wedding ring and there are sweat stains under his armpits.
They stand up. He calls over a friend, an energetic woman it is clear he has a crush on, a crush that embarrasses him. She is curvy and pouty and also around thirty; her name is nottennis. Elisa knows her, too—she's the kitten wearing a jetpack.
Elisa listens to them talk. She answers a few questions. They seem excited to have an older person interested in the same things they are, although she hasn't recognized a single reference from either of them yet. She follows them out of the room and into the hotel bar, where she meets more people, shakes a lot of hands, and allows a tall, professorial type to flirt with her. His beard is prematurely white and there is a kind of flair to his personal awkwardness that she likes. She considers, then decides against, going to bed with him.
It occurs to her that she's wearing a wedding ring. She can't decide whether or not to take it off. If she leaves it on, maybe men will be less guarded with her, with less apparently at stake. But then again they might not even try. And is that what she wants, to hook up? Maybe a part of her does. She hasn't had much sex lately—why doesn't she want it more?
She leaves the ring on. She imagines that, in a parallel world, perhaps not the one she knows, she has taken it off and it has changed everything.
Several times throughout the evening a woman glances at her from across the room. She is around fifty, quite heavy, moon-faced. She wears round eyeglasses and a pink blouse with ruffled collar and sleeves and a capacious, coarse yellow skirt that reminds Elisa, in its thick folds, of the valance over the window in her hotel room. The woman isn't wearing a lanyard, and she seems to have a glow, like the moon itself. Her movements are slow and deliberate, as though they have been choreographed.
Elisa doesn't look for Patricia, because somehow she knows that this woman is her, though the woman is nothing like she imagined. They do not approach each other or introduce themselves: she isn't sure why. She feels disengaged in general from the conference, in fact—out of place and insufficiently interested. The bar is getting more crowded now and people keep jostling her from behind, reaching around her for their drinks. Bits of conversation intended for others are inadvertently shouted in her ear. really sucked after season three. and boobs out to here. which isn't in the remake. lifetime of gastrointestinal whatever. She looks around the room for Silas and could swear she sees Betsy Orosco exiting.
Betsy! Suddenly Elisa feels revitalized. She wants to talk to her, to get to the bottom of that whole thing. They really made a connection that day, last summer, didn't they? Surely Betsy doesn't think she's just some nut. Elisa doesn't mind, not really, what happened—she just wishes they'd been honest with her, that's all.
She gets up, mutters excuse me, pushes through the crowd. People keep staggering into her path carrying multiple drinks. Everyone's voice is loud, far louder than one might expect of nerds. Finally she's through and into the lobby, where the ambient temperature drops by five degrees, and where Muzak is drifting down from the ceiling. She looks around: there, down that hall. It must be her, the blue hair, the broad hips and purposeful stride. Elisa runs to catch up, sneakers squeaking on the fake marble floor.
It's not a hallway, actually, it's the foyer the elevators open onto, four sets of doors, four illuminated panels displaying the numbers of floors. One set is closing. Elisa hurries to it, peers inside as the strip of light narrows. There she is, the same rounded shoulders and cat-eye glasses. "Betsy!"
Betsy Orosco glances over Elisa's shoulder, looks right past as if she isn't there. Then the elevator doors close and she's gone.
51.
Sometime in the night she wakes up and tries to slide herself out of the big bed. She is bound up in the sheets, she feels them tugging out from under the mattress, and by the time one foot has hit the carpet the other has become stuck, and she flings her arms out for balance and finds the wall. She is standing there in the dark, in a frozen pirouette, her heart racing. She feels fully awake but knows she is not. The sheets release her foot. She collapses against the wall, pressing her face and both hands to it.
Elisa has no idea where she is. She doesn't know which direction to move in. She knows that she isn't at home: there is no sound from anywhere and no air is moving. She says Derek's name and then remembers she and Derek are no longer together, and then doubts that memory.
She thinks of the boys and experiences a moment of panic. In her mind they are five and six years old and in danger. This isn't right, she can't put her finger on how. She moves a step, then another, along this wall and suddenly fears moving further; she does not want to get closer to the boys. Whatever is the matter, she will make it worse. She says Derek's name again, and now it feels truly wrong: she's coming to. She's in a hotel. She went on a trip. Is she in Wisconsin? No—North Carolina. It's a Holiday Inn. The bathroom is just around the corner. She can move, now, in the dark.
Back on the bed she is sweating profusely. As if in response, the air-conditioning kicks on with a grunt. The clock reads 3:14. Then it reads 4:40. Then it's light and she is lying shivering with the sheets tugged off the bed and bundled in a heap beside it. She feels as though she hasn't slept at all.
She wears her lanyard to breakfast and sits with some people from the parallel worlds forum, including RueTheDay. They are mostly younger than she is, except for one very old man. His ID reads CharlesSmith. Elisa doesn't recognize the name. The group is animated and enthusiastic, and they are talking about the same things they talk about online, except that, in the absence of official moderation, they mention more television programs.
It's not quite what she was expecting. But she isn't certain what's missing. She finds herself peering across the banquet room, trying to identify other forum members, but she can't read their tags from here. She doesn't see Betsy anywhere. The woman she thinks is Patricia fills a bowl with scrambled eggs, then scans the room as though looking for a seat. She makes eye contact with Elisa, puts on a small demure smile, and walks in the opposite direction, to where there is an empty table. A few moments later a man walking on crutches sits down with her and the two sit facing each other in apparent silence.
The first major event of the day is a parallel worlds panel—it is one of the main reasons she is here. Her breakfast companions ask her if she is excited about it. Their attention takes her by surprise—it is strange that these unfamiliar people know something about her, about her preoccupations.
"I suppose I am," she tells them, and they all laugh.
A tired-looking man called part_human says, "You're just like you are on the board."
"What am I like on the board?"
"Reserved," says nottennis. She is clearly enjoying the attention of the men around her.
"Restrained," says PresumedInsane. He is her age, shockingly thin, Adam's apple, black beard spattered with gray.
RueTheDay says, "You're our resident grown-up."
"I'm not that much older than you."
"Not your age," says nottennis, "The way you are."
"Oh."
To her left, CharlesSmith silently works his jaw. He is alert but looks no one in the eye. After a time, he struggles to his feet and leaves.
Nottennis says, "Um, has anybody ever even heard of that guy?"
The parallel worlds panel is in a small conference room down the hall. There's a dais with four microphones set up on it, facing about a hundred folding chairs. Elisa considers waiting for her breakfast companions, but doesn't want to sit near nottennis unless she absolutely has to. So she hangs back and sits on the aisle in an otherwise unoccupied row.
The panel consists of a TV producer, a science fiction writer, a blogger whom everyone but Elisa seems to have read, and Betsy Orosco. Betsy has come in late; she is in fact eating a piece of toast. The other panelists, all men, steal glances at her that Elisa interprets as appraising. Betsy seems confident, in her element. She finishes the toast and sits with her hands folded, waiting. When ten o'clock arrives, they begin.
There is no moderator; the four speakers introduce themselves and each offers some opening comments on the subject. None of the four seems particularly comfortable around the other three. To Elisa's dismay, the TV producer dominates—he shares stories about working with particular famous actors. The science fiction writer clearly dislikes him—he denigrates the narrative logic of the producer's most popular show. The blogger tells jokes that fall flat, and Betsy, at first, appears as though she regrets coming at all. She tries, gamely enough, to talk about the actual physics of the multiverse, in much the way she presented it to Elisa in her office the year before. But here, it isn't going over so well. She explains in detail, too much detail, the complex quantum requirements for a universe to be created, and the audience shifts in their seats. And when she tells them that travel among universes is largely impossible, several people actually groan.
"But you never know, right?" says the blogger.
"That's right," the TV producer says brightly, to mild laughter, "you never know!" By the time the audience begins raising questions, everyone seems exhausted, as if it's midafternoon and they have been conferencing all day.
At some point Elisa feels a presence beside her and turns to find that the presumptive Patricia has taken a seat two down from her. She is wearing a floral print dress, clean new running shoes, and a crucifix around her neck. She is staring straight ahead. Her hands are folded in her lap and she remains perfectly still. Elisa smells perfume.
Someone in the front of the room, she thinks it's RueTheDay, is asking Betsy a question. "You say we can't travel back and forth between universes," he says. "But what about our consciousness? You know, our awareness?"
Betsy's answer, littered with finger quotes, is given with a wrinkled brow. Elisa is trying to concentrate on it. "I'm not sure if that's a question for physics. I mean, we'd need to define what 'consciousness' means, in terms of physics. If you want to get philosophical... in theory... I guess 'you' are already there, the iteration of you that is native to that universe."
"But is there... can you think of a mechanism... by which..."
"He's not taking no for an answer!" quips the blogger.
"... by which the consciousness could travel... could be transferred..."
"Into the Matrix!" the blogger says.
"... or I guess shared with that of the other you, or yous?"
Betsy leans forward. "Believe me, I want to say yes..."
"So say yes!" says the blogger.
"... but physics is concerned with the kind of questions that we can support with mathematics or experimentation. 'Consciousness' is a psychological notion, a philosophical notion. It is interesting, but it isn't something we can apply freely to our work. I can't say that something like consciousness can be transferred because I don't know what it consists of. And neither does anyone else."
Yes. Yes, Elisa is thinking, nobody knows, nobody understands. And this ought to reassure her, because it means that whatever she wants to be true about this experience, whatever she would like to believe has happened to her, is possible. It doesn't matter what Betsy Orosco or Hugo Bonaventure thinks, it doesn't matter what the guy at Caltech thinks, it doesn't matter what Amos Finley thinks. They can't tell her otherwise, can they?—because they don't know. They can't know. The only person who can decide what it is that has happened to her is herself: the experience is hers to define, and hers to explain or not. Her life, her consciousness.
But instead of feeling reassured, she begins to feel panicked. Because it occurs to her that what she wants—what she has wanted all along—is not simply to know. It is to be believed. She has placed her greatest need in the hands of other people—strangers in an alien world.
They are all strangers here, even herself.
Elisa senses a movement to her right. It is Patricia. She is placing a twice-folded rectangle of paper on the seat between them.
52.
Elisa leaves the paper there. She knows it is for her but can't bring herself to pick it up. The room is decorated in various shades of beige and gray and the paper is the whitest thing in it. It lies slightly open, the four corners lined up sharply, pointed at her. She can see into its maw, where a few lines of text have been printed.
The science fiction writer is speaking now. Somebody has asked him how parallel worlds should be depicted in stories, if the concept is bound by rules. His response is impatient; he speaks as though it is beneath him to be asked such a thing.
"Every compelling concept is bound by rules. But I can't sit here and tell them to you. They're determined by the story."
But Elisa is still staring at the paper. She detects movement and looks up to find that Patricia has turned her head and is gazing at her with moist and beatific eyes, smiling faintly, pitying her. Patricia blinks—no, she bats her eyelashes. The smell of her perfume is stronger now.
And now Patricia stands up and walks, floats almost, out of the room. To Elisa this seems disruptive, drastic: isn't there a kind of hush in the room just now, a suspension of movement and sound? But nobody seems to notice it happening. All that is left is the paper on the chair and the voice of the science fiction novelist.
"... for instance, in my last book, Familiar, which maybe some of you have read..."
"Very fine piece of work," says the TV producer.
"Why, thank you, Roland, have your people talk to my people. But in that book, the protagonist, a young man in search of his twin, enters parallel worlds through the pages of a book, a sort of enchanted book also called Familiar..."
The blogger says, "Everybody loves the po-mo," to scattered laughs.
Elisa is only half-listening. She reaches out and picks up the paper. Her dry hands make a sharp sound, sliding against it, unfolding it. She looks up to see if anyone has noticed. But nobody is paying attention.
The message has been printed on a computer. It reads:
I know what you are going through... I can help you... I will come to you... we will talk... a better life awaits... don't worry. There is an answer to all your questions... a solution to your problems... don't worry... soon. Patricia.
Somehow the message reads like a code—it seems to say more than is printed here. She reads it again and again, straining against the possibility of hope. Could Patricia be the one? Why not? Someone just said something about the rules being determined by the story. This is her story, isn't it? She, Elisa, can make the rules.
The voices of the panelists fall silent. She looks up and catches Betsy frowning at her from the dais, as though trying to figure something out.
Elisa's fingers begin to twitch. The paper in her hand crackles and she rises to her feet. She's angry.
"Betsy!"
The room turns to her. There's a wildness in her voice, a raggedness that is almost sexual. Her breaths catch in her throat and suddenly her heart is pounding so frantically against her blouse that she thinks she can see, on the periphery of her vision, the fabric moving. She tries to calm herself, to tamp down the desperation in her voice, but it's hopeless. She says, "I wonder if you might talk about my experience of this phenomenon. What we talked about last year. And where you stand on that."
Someone coughs. The silence deepens. Betsy opens her mouth to speak, then closes it again and glances at the papers in front of her. She looks up and says, "I know that... I'm not sure..."
"You sent me to your friend Hugo. He was going to do tests... well, he was going to have his friends do them. Tests. On my things."
She doesn't sound like herself at all.
"A tube of lipstick. And a list. One from both worlds, one from this one alone!"
Betsy is frowning again. She says, "I don't think..." and then trails off. She's gripping a pen in her fist and is clicking the nib in and out with her thumb.
"I went there—to the lab. They gave me my things back. They didn't know you, or Hugo."
Into the eerie quiet of the room, Betsy says, "I'm sorry, I don't know anything about that."
Elisa says, too quietly to be heard, "Did you ever believe me?"
And by now someone else has raised a hand, asked the TV producer a question. Elisa is still standing, still staring at Betsy, who is still clicking her pen. Eventually Betsy looks away, and Elisa sits down. She is not entirely sure what specifically she just said. The folded note is still in her hand. Her heart is still racing.
And then the room is empty, or nearly so, the neat rows of folding chairs have been disrupted and young people wearing eyeglasses and ID tags are bustling about pushing them back into place. Some people are standing near her, it's nottennis and RueTheDay. There's an electric, frightened intensity about them: at first she assumes it's because they are attracted to one another, that RueTheDay is contemplating an affair. And maybe they are, maybe he is. But then she asks them what they're doing next, are they going to the movie premiere in the ballroom, and they are strangely evasive. Nottennis takes a step back, bumping into someone to whom she must apologize.
"Uh... yeah," says RueTheDay, "I think maybe we'll make our way there eventually."
"We have to do something else first," nottennis adds.
"Maybe we'll see you there?"
Elisa nods, folding the paper in her hand into a still-smaller rectangle. "Sure. Sure."
The two of them retreat with evident relief, while Elisa stands blinking, wondering what just happened. She looks around the room. People are clustered in little groups, stealing glances at her. Betsy Orosco seems to have left, and the three male panelists are laughing about something at the dais.
She fears, is in fact quite certain, that she has made a fool of herself.
53.
That night she attends a talk on alien abductions and a panel on the possible alternate forms intelligent life might take. She meets a couple from the forum named Seth and Janet. These are their screen names. They tell her they just found the idea funny, giving themselves "normal" names to use online; they say they've taken to calling one another Seth and Janet around the house. Elisa didn't realize they were married. They don't seem to have been at the panel discussion this morning; they didn't witness her performance.
She goes with them to the hotel bar and the three of them drink. A lot. Seth announces at some point that he's going to kiss Elisa; Janet tells him to go ahead, in fact she dares him. He does it, and the two of them kiss for a while. He's only a few years younger than she is, and is quite attractive, with broad shoulders and a narrow waist and a bit of hair poking out of his collar from his back and chest. Janet whoops and laughs and then takes over, kissing Elisa with evidently equal enthusiasm. It isn't as unpleasant as she might have imagined, though it is indeed unpleasant. They invite her up to their room and Elisa says no at least twenty or thirty times. At some point they leave the bar. "But we've had such fun!" Elisa shouts after them. They are laughing too hard to hear.
Then she's with RueTheDay and nottennis, and they're laughing at her too. At times they whisper things to each other and then look at Elisa and crack up. She finds herself asking anyone who will listen that it is imperative that they wake up CharlesSmith and bring him down here immediately, and if they don't do it, by God, she's going to go do it herself. Then she is in the elevator and her hand is flapping uselessly against the glowing numbers. Somehow she manages to hit her floor and staggers back into the corner.
She hasn't been this drunk since... college? She can't remember very far back. The elevator heaves and sways. It stops, and the doors open, but she doesn't get out, she just remains pinned to the back wall, staring out at the hallway: a vase full of fake flowers on a round wooden table, a seascape hanging above it. There are voices. The doors close, and then, a moment later, open again. Three people get in, a man and two women; they are talking and laughing, the man looking over his shoulder.
"... she was like, 'Okay, fine!' And I was like, 'Fine!'"
"Of course she's like that."
"Did you meet her mother?"
"Oh, God."
"And then that coat."
"She called it 'vintage.'"
"Well, we shouldn't make fun."
"Oh, yes we should!"
The doors open. The people get out. It's the lobby—the elevator has gone back down without Elisa noticing. She hits the button for her floor again. This time she'll do it—she'll get out of the elevator and go to her room. As the doors close and the elevator begins to rise, she studies an advertisement affixed to the wall above the buttons. It reads, "Good times, good friends. Your one-stop dinner solution on game day!" The phrase seems hilarious; she snorts and giggles. Then she sighs, loudly, and begins to feel as though something in the elevator is different.
It's a change in the light, a change in the space. She groans a little and it sounds wrong. For no reason that she can fathom, she says "Ow." She fixates on the spot where the horizontal crack between the doors and floor meets the vertical one between the doors. It's sort of sexual. The elevator stops. The doors open and the spot vanishes and she says "Whoa."
Three people get in, a man and two women; they are talking and laughing. The man is looking over his shoulder.
"... she was like, 'Okay, fine!' And I was like, 'Fine!'"
"Of course she's like that."
"Did you meet her mother?"
"Oh, God."
"And then that coat."
"She called it 'vintage.'"
"Well, we shouldn't make fun."
"Oh, yes we should!"
Elisa tries not to move or make a sound. She has backed into a corner of the elevator, in an effort not to be seen. She is terrified. None of the people look at her. The door opens onto the lobby and they get out and Patricia gets in.
"Patricia?" she says, and her voice sounds very small and far away.
Patricia smiles that same beatific smile. She nods and presses the button for Elisa's floor. How does she know? Maybe she doesn't, maybe it's her floor too.
As the elevator rises, Elisa's breaths become shallower, faster. "Patricia," she says, "if they're out there..."
The elevator stops. The doors open. "Is there? Anyone there?"
Patricia shakes her head no, still smiling. She holds out a hand to Elisa and Elisa takes it, and allows Patricia to lead her into the vestibule.
But they're there. All three of them. And one of the women is saying, "And she said, 'Maybe you shouldn't come to the party after all.'"
"You have got to be kidding me," says the other.
"I am not. So I said, 'Fine, then,' and she was like, 'Okay, fine!' And I was like, 'Fine!'"
"Of course she's like that."
"Did you meet her mother?"
And as they enter the elevator the man looks over his shoulder, back at Elisa, who stares at him in horror. He blinks, and then the doors close and the people are gone.
There's a hand on her elbow. There's a smell of perfume. She is being guided down a hallway. She has had the presence of mind to dig her key card out of her pocket and now she is fumbling to slip it into the lock. But Patricia's soft hand is there to guide her. She is led into the room, to a chair, she is pressed down into the chair and then a soft shape is in the near-blackness of the room pulling back the comforter and sheets on the bed.
"Thankyou," Elisa is saying, "thankyou," and then her shoes are being removed, and her socks, and she's lying in the bed on the cool rough sheets and it feels so incredibly wonderful that she wants to cry.
"Is this real?" she wants to know, but no one answers.
Then she's awake again, the room is spinning, and she is kneeling in front of the toilet vomiting, with a warm hand, a hot hand actually, pressed into the middle of her back. "Gedditoff," she says and tries to brush it away, and the hand disappears.
She manages to brush her teeth and drink some water. Beside her, someone is cleaning the toilet with a wadded-up bit of toilet paper. She has the impression that it is perhaps Elisa, the other Elisa, come to visit this world. (Wouldn't that be nice, she thinks—we could be friends.) A peculiar sensation overcomes her—as she is nearly awake enough now, nearly sober enough now, to be disturbed by the presence of a stranger in her hotel room—of not quite being disturbed, or of contemplating being disturbed; she is aware that she can make, if she wishes, a decision about how she will feel. Her thoughts, though, are close and cluttered and bloated, jostling against each other in her head, and she can't keep them still enough to follow any one of them to its conclusion. She really just wants to get back into bed. She has taken her pants off, or somebody has, so she is standing here in the nightlit bathroom in her underwear and a linen blouse stained with flecks of her own sick, and the figure at the toilet rises, and the toilet flushes, and then she is led back to the bed and is asleep again.
54.
When Elisa wakes it is not yet morning, or it is morning and the heavy curtains have been closed, and Patricia is sitting in the chair, which she has moved to the side of the bed and in which she seems to be praying silently. The only light comes from the night-light in the bathroom, and the glowing alarm clock, which is blinking 12:00, and the strip at the base of the door. Elisa feels hollowed out and queasy; her mouth is filled with paste. She sits up. A glass of water stands on the end table, so she picks it up and drinks from it. Patricia is staring at her.
"I'm here to help you," Patricia says.
Her voice is unexpectedly rough and deep, like a heavy smoker's. Perhaps, in a previous life, that's what she was. She is holding something in her hands—a rosary? Elisa can't quite make it out, only the pale thick fingers working at it.
She is so tired—more tired, if this is even possible, than she was when she collapsed into bed. She wants this strange woman to leave so that she can go back to sleep. It would be simple enough to tell her so. To ask her to leave. Elisa draws a deep breath. She says, "I don't think you can."
"Take my hands."
"Patricia. You are Patricia, right? I'm sorry, I just need to sleep. I need to sleep."
The desk chair is positioned two feet from the edge of the bed. Elisa yawns. It is possible that she falls asleep again, sitting up with her head against the bedboard, for a few minutes, or maybe an hour. When she opens her eyes, Patricia is still waiting. Her hands are empty now and lying, palms up, fingers spread, against the floral print of her dress. Her glasses are hanging around her neck on a chain. The gloom makes her seem both more and less real.
That's it: Elisa must submit, or the woman will never leave. She drags herself to the edge of the bed, keeping the sheets twisted around her waist, covering her thighs. She drops her legs over the side, so that her knees are even with Patricia's; just an inch separates their bodies. Elisa reaches out and places her two hands, faceup, onto the other woman's palms.
"Okay," she says, "help me."
Patricia nods, once, and her heavy features rearrange themselves into a small, slow frown, a scowl of concentration. Her fingers close on Elisa's hands.
Elisa is so disappointed now: both in the conference, and in herself, for the way she has behaved. Why did she kiss those people in the bar? Everyone saw her do it. Why did she ask Betsy that question? It seems to her that she has run out of chances, not just here, but in her life. That this is the way it will be now: things will keep being left behind and never returned to, and life will take on a depressing, inevitable forward momentum, like a glacier's, slow and inexorable. And then it will end.
(She is wrong, of course. Things will change, they always do. There will be times when she looks back a few months, a week, even a day, and wonders at how innocent she was then, when the fabric of life felt so different. How could she be so oblivious to things that, in the here and now, are so obvious? Of course she will go back to Derek, eventually. He will, in fact, beg her to do so, going so far as to propose marriage to her a second time, though they won't have divorced.
It is true that the boys will stay disappeared, that she won't see them again for a long, long time. But they'll be in touch, after a fashion. She and Silas will seek each other out online. He will use his sig lines as signals to her. He will go by various names, on various forums; he will track her interests, her obsessions, hovering at the boundaries of her attention, moving out of view when she gets too close. He won't respond to messages. This will be their private game. They will follow one another's iterations, they will die off and respawn in unexpected places, and she won't tell Derek. And when at last he appears to her in the flesh, when he brings his brother to her as well, to forgive, it will be as sudden and inexplicable as everything else he's done, yet it will feel perfectly natural to her, familiar and unfamiliar at the same time, like everything else in the strange, enormous, echo-filled room that is her life.
There was only ever one Lisa, she'll tell herself then. There was only ever one life. It was just larger and more peculiar than she expected.)
But for now Patricia's fingernails are digging themselves into her flesh, and Elisa feels herself tugged forward, her elbows popping and cracking with the sudden strain, and out of Patricia's mouth comes a stream of loud, incomprehensible speech. Elisa cries out, tries to pull away from the other woman's grip, but it's too powerful, and Elisa is too tired and sick. "That hurts," she says between clenched teeth, but Patricia isn't listening, she's talking, it's all gibberish and Elisa can feel spittle landing on her bare arms.
And then Patricia's voice rises; she's shouting now. Surely this is waking up the other guests? Surely one of them is calling the front desk? The woman's hands are crushing Elisa's, she seems to be trembling; Elisa can feel it through the floor and can make out a glint of light from the dangling eyeglasses, shuddering in the air. Why doesn't she pull away? Why did she let the woman into her room in the first place? It doesn't seem like something I would do. That just doesn't seem like me. Here she is, though: Elisa has given up on escaping her, this woman called DippedInSunshine.
And now she tries to concentrate, she tries to listen. Because that was her problem, wasn't it—she didn't listen. The roaring in her head was always louder. So she concentrates on her fear and on the pain in her hands and on the torrent of sounds spewing from Patricia's mouth: these twisted vocalizations, the vowels elongated and shrill, the consonants clacking and popping in her face. It's language. It's saying something, something just for her, something that will help. She's so tired, but she digs in: she grips back, hard, and the two women sit there, trying to break each other's hands, one trying to understand, the other trying to make herself understood. Elisa wants to stop, but she can't, she knows that the second it ends will be the second right before it would have started making sense. And so she bears down, telling herself it will work, that in the end it's actually possible to believe in your dream so completely that you can drag the rest of the world into it with you.
Acknowledgments
For various kinds of assistance with this novel, the author would like to thank Tom Bissell, Jennifer Brice, Rhian Ellis, Brian Hall, Kristine Heiney, Fiona McCrae, Ethan Nosowsky, Jim Rutman, Ed Skoog, and Steve Strogatz.
J. ROBERT LENNON is the author of seven novels, including Castle and Mailman, and a story collection, Pieces for the Left Hand. His fiction has appeared in the Paris Review, Granta, Harper's, Playboy, and the New Yorker. He lives in Ithaca, New York, where he teaches writing at Cornell University.
The text of Familiar is set in Adobe Garamond Pro, drawn by Robert Slimbach and based on type cut by Claude Garamond in the sixteenth century. This book was designed by Ann Sudmeier. Composition by BookMobile Design and Digital Publisher Services, Minneapolis, Minnesota. Manufactured by Versa Press on acid-free 30 percent postconsumer wastepaper.
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Director: Vidhu Vinod Chopra
Genre(s): Thriller
On the board between Mexico and the United States, something big is brewing. A war between the police force and Cartel drug-runners is ready to explode into something cataclysmic. When concert violinist Jacob (Anton Yelchin) returns to his family's home with his fiancé to see his brother, Buddy (Chris Marquette), spirits are high. Buddy is unable to attend the upcoming wedding, but has invited his brother down to see the new ranch he has built. Things don't add up, however, as there is no way that Buddy would ever have been able to afford to pay for all of the work that has been done. Slowly, the plot begins to unravel, and it turns out that Buddy isn't all that he seems.
'Broken Horses' is an upcoming film from director Vidhu Vinod Chopra ('3 Idiots'), with principle photography beginning on 29th October 2012 in the Los Angeles area. The film received tremendous critical praise from directors James Cameron and Alfonso Cuarón, both of whom are quoted in the trailer. 'Broken Horses' is set to be released theatrically in the US and UK on 10th April 2015.
Starring: Anton Yelchin, Chris Marquette, Maria Valverde, Vincent D'Onofrio, Thomas Jane, Sean Patrick Flanery, Wes Chatham, Sadie Alexandru, Greg Serano, Jeremy Luke, Juan Riedinger, Nicholas Neve, Jordi Caballero, Christian Elizondo, Alizabeth Hamer, Sergio Garcia, Stewart Skelton, David Namminga, Henry Shotwell, Chad Bishop, AJ Meijer, Matt O'Neill, Eric Sharp, Steve Luna, Ray Julian Torres, Peter Vinding, David Lautman | {
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} | 8,014 |
The European power market was caught in a storm in 2022. The ongoing recovery from the Covid-19 pandemic as well…
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Two years on: How Energy Monitor is documenting the race to net zero
As European Commission President Ursula von der Leyen unveiled fresh proposals to combat the energy crisis on 14 September, around…
Nuclear lifetime extensions: Is the juice worth the squeeze?
The Fukushima nuclear disaster in 2011 resulted in the displacement of 160,000 people. As with Chernobyl in 1986, the world…
Weekly data: Shift in Germany's perception of nuclear energy
In Germany, energy insecurity has sparked discussions about keeping the country's last three nuclear power plants running longer than planned.… | {
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} | 4,294 |
{"url":"http:\/\/mathhelpforum.com\/pre-calculus\/21017-find-area-coordinate-triangle-print.html","text":"# find the area coordinate triangle\n\n\u2022 October 21st 2007, 03:24 PM\nsubzero06\nfind the area coordinate triangle\nHello\n\nGiven 3 points:\n(3,2) (-3,-3) (5,-1) are the vertices of a triangle. Find the area of the triangle.\n\u2022 October 21st 2007, 03:58 PM\ngalactus\nDo you have to use a particular method?.\n\nTry using the distance formula to find the lengths of the sides, then use Heron's formula.\n\nYou could show off and find the equations of the lines that make up the triangles sides and then integrate.(Nerd)\n\n$\\int_{-3}^{3}\\left[(\\frac{5x}{6}-\\frac{1}{2})-(\\frac{x}{4}-\\frac{9}{4})\\right]dx+\\int_{3}^{5}\\left[(\\frac{-3x}{2}+\\frac{13}{2})-(\\frac{x}{4}-\\frac{9}{4})\\right]dx$\n\u2022 October 21st 2007, 04:16 PM\nsubzero06\ni tried using ABS formula\nLet (x1,y1)=(3,2) ; (x2,y2)=(-3,-3) ; (x3-y3) = (5,-1)\nABS(x2*y1-x1*y2+x3*y2-x2*y3+x1*y3-x3*y1)\/2\n=ABS[(-3)(2)-(3)(-3)+(5)(-3)-(-3)(-1)+(3)(-1)-(5)(2)]\/2\n=ABS(-6+9-15-3-3-10)\/2\n=ABS(-28)\/2\n=-14\n\nfor some reason i get -14...am i suppose to get a \"-\" for an area??\n\u2022 October 21st 2007, 04:20 PM\nJhevon\nQuote:\n\nOriginally Posted by subzero06\ni tried using ABS formula\nLet (x1,y1)=(3,2) ; (x2,y2)=(-3,-3) ; (x3-y3) = (5,-1)\nABS(x2*y1-x1*y2+x3*y2-x2*y3+x1*y3-x3*y1)\/2\n=ABS[(-3)(2)-(3)(-3)+(5)(-3)-(-3)(-1)+(3)(-1)-(5)(2)]\/2\n=ABS(-6+9-15-3-3-10)\/2\n=ABS(-28)\/2\n=-14\n\nfor some reason i get -14...am i suppose to get a \"-\" for an area??\n\ni didn't check your calculations, but if you take the absolute value of a negative number, the answer is positive. so ABS(-28)\/2 = 28\/2 = 14\n\nso provided your other calculations are correct, you would get +14 as the answer, which makes more sense than a negative answer\n\u2022 October 21st 2007, 04:31 PM\nsubzero06\noooh yea i get it now\nthank you very much!","date":"2016-05-28 21:43:16","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\": 1, \"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\": 0.9304429888725281, \"perplexity\": 2413.535130177828}, \"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-2016-22\/segments\/1464049278091.17\/warc\/CC-MAIN-20160524002118-00133-ip-10-185-217-139.ec2.internal.warc.gz\"}"} | null | null |
Assignment Kandahar: After bin Laden
"It's a great day, Bin Laden being dead and all," said an American civilian Monday morning
Assignment Kandahar: Weekend attacks in the city, with new updates
Taliban insurgents launched a string of coordinated suicide and IED attacks on Kandahar city Saturday, purportedly in retaliation to the U.S. commando raid almost seven days ago that killed Osama Bin Laden in Pakistan.
Canadians in secret unit hunt al-Qaeda terrorists
Sitting in their unlit house, behind six-metre-high mud brick walls in the heart of a village so hostile to outsiders even heavily armed international troops rarely venture down its narrow roads and lanes, the al-Qaeda militants must have felt safe.
National Post editorial board: Please don't shoot, we're Canadian
Prime Minister Jean Chretien's admission that Canadian ground troops will not be sent to Afghanistan if their deployment entails fighting casts serious doubt
National Post editorial board: A war of necessity
Once again, Canadian soldiers, sailors and pilots will be marching as to war, as Pierre Berton, the popular Canadian historian, puts it in the title of his most recent book.
Through the peaks with gun runners in pajamas
Shortly before crossing into Afghanistan, National Post foreign correspondent Patrick Graham filed this report from Mach Mountain, Pakistan, a route long favoured by arms smugglers.
An Afghan shell game with bin Laden as the pea
In Peshawar, Pakistan, the Post's Patrick Graham talks to a former ally of bin Laden's about the terrorist leader's countless caves.
Canada in Afghanistan: 2001
The Post takes a comprehensive year-by-year look at Canada's presence in Afghanistan since 2001. Up first: 2001 | {
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} | 8,955 |
Q: How to remove "RE:" or "Fwd:" or "[EXT]" from an Outlook Appointment Invitation subject line when accepting? I'm using Outlook 2019 on Win10.
My company adds the prefix "[EXT]:" to the subject line of all emails received from outside our network. This includes invitations. Because of this, "[EXT]:" is in most of my calendar. It makes it hard to look at a busy calendar from my phone when 90% of the subjects start with [EXT]:
I can figure out VBA code to look for RE: or Fwd: or [EXT]: in subject lines and replace/delete them.
How do I trigger the VBA code automatically when Accept, Accept with Response, Tentative, etc. buttons/pulldowns are clicked?
A: When any of these buttons are clicked the response is sent back to the organizer and you may try to handle the ItemSend event of the Application class in Outlook which is fired whenever an Microsoft Outlook item is sent, either by the user through an Inspector (before the inspector is closed, but after the user clicks the Send button) or when the Send method for an Outlook item, such as MailItem, is used in a program.
You may also find the the Items.ItemChange event helpful, it is fired when an item in the specified collection is changed.
Public WithEvents myOlItems As Outlook.Items
Public Sub Initialize_handler()
Set myOlItems = Application.GetNamespace("MAPI").GetDefaultFolder(olFolderCalendar).Items
End Sub
Private Sub myOlItems_ItemChange(ByVal Item As Object)
Dim prompt As String
If VBA.Format(Item.Start, "h") >= "17" And Item.Sensitivity <> olPrivate Then
prompt = "Appointment occurs after hours. Mark it private?"
If MsgBox(prompt, vbYesNo + vbQuestion) = vbYes Then
Item.Sensitivity = olPrivate
Item.Display
End If
End If
End Sub
Another approach is to handle all incoming emails and remove the subject prefix. The NewMailEx event fires when a new message arrives in the Inbox and before client rule processing occurs. You can use the Entry ID returned in the EntryIDCollection array to call the NameSpace.GetItemFromID method and process the item. This event fires once for every received item that is processed by Microsoft Outlook. The item can be one of several different item types, for example, MailItem, MeetingItem, or SharingItem.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,261 |
\section{Introduction} \label{sec:intro}
It is now widely accepted that quasars are powered by accretion of material onto supermassive black holes (SMBHs). The continuum emission and the broad emission lines (BELs) often show aperiodic variations (e.g., \citealt{FitchEtAl1967}; \citealt{AndrillatEtAl1968}). Theoretically, the BEL fluxes are supposed to vary in response to the variations of the ionizing continuum with a lag of about light travelling time. Hence, via the cross correlation analysis of the BELs and the continuum, we are able to constrain the geometry of the spatially unresolved broad emission-line region (BLR) in active galactic nucleus (AGN). Notably, with the light-travel time delay, the distance of the BLR to the ionizing source is directly determined (e.g., \citealt{Peterson1993}). By an attempt to model that the BLR is virialized, the central SMBH mass then can be estimated (e.g., \citealt{WandelEtAl1999}).
So far emission-line reverberation mapping (RM; e.g., \citealt{BlandfordEtAl1982}) experiments have succeeded in measuring emission line lags in $\geq$ 60 AGNs; (e.g., \citealt{PetersonEtAl1998, PetersonEtAl2002, PetersonEtAl2004}; \citealt{WandelEtAl1999};
\citealt{KaspiEtAl2000, KaspiEtAl2005}; \citealt{VestergaardEtAl2006}; \citealt{BentzEtAl2009, BentzEtAl2013}; \citealt{DenneyEtAl2010}; \citealt{BarthEtAl2011, BarthEtAl2011a}; \citealt{GrierEtAl2012}; \citealt{Hu2015}; \citealt{GoadEtAl2016}; \citealt{JiangEtAl2016}; \citealt{ShenEtAl2016}). It is revealed that, the BLR size as measured for a particular emission line such as $\hbox{H$\beta$} \ \lambda4861$, is closely related to the AGN luminosity in the approximate form $R \propto L^{1/2}$ (the $R$-$L$ relation; e.g., \citealt{KaspiEtAl2000}; \citealt{BentzEtAl2006}; \citealt{ShenEtAl2012}). This relation offers the possibility of taking advantage of single-epoch (SE) spectra to determine the SMBH masses (e.g., \citealt{Vestergaard2002}; \citealt{McLureEtAl2002}; \citealt{VestergaardEtAl2006}). Over the past decade, several editions of these estimations have been developed (see, e.g., \citealt{McGillEtAl2008}; \citealt{WangEtAl2009a}). Resulted from the economical efficiency and operability, the SE virial SMBH mass estimation is a sort of praticable method on the determination of AGN SMBH masses compared to RM technique (e.g., \citealt{WooEtAl2002}; \citealt{McLureEtAl2004}).
RM studies, as they are known, have been traditionally performed mostly on low-luminosity AGNs at low redshift ($z <$ 0.3) using the $\hbox{H$\beta$}$ emission-line to measure SMBH masses. For AGNs at redshifts beyond 1, rest-frame ultraviolet (UV) BELs are required, such as $\hbox{Mg\,{\sc ii}}$, a crucial emission line of RM interest that can be presented in quasar spectra having redshifts between 0.3 and 2. However, RM results of $\hbox{Mg\,{\sc ii}}$ line (i.e., reliable detection of $\hbox{Mg\,{\sc ii}}$ lag) are quite scarce. This is initially interpreted that the $\hbox{Mg\,{\sc ii}}$ emission-line varies more slowly in response to continuum changes than $\hbox{H$\beta$}$ emission line, suggesting that the $\hbox{Mg\,{\sc ii}}$-emitting region may have larger-scale structure than that of $\hbox{H$\beta$}$ (e.g., \citealt{CorbettEtAl2003}).
The relationship between the BEL flux ($F_{line}$) and the continuum flux ($F_{cont}$) within an individual source is often expressed by $F_{line}$ $\propto$ $F_{cont}^{\alpha}$, where $\alpha$ is traditionally measured from emission line and continuum flux light curves from AGN monitoring campaigns. This is related to the so-called intrinsic \textquotedblleft Baldwin Effect" (see, e.g., \citealt{KinneyEtAl1990}; \citealt{PoggeEtAl1992}; \citealt{GoadEtAl2004}; \citealt{KoristaEtAl2004}).
In fact, $\alpha$ is commonly referred to as the response of the BEL to variations in the ionizing continuum flux. Formally, we can parameterize the correlation between the variations in the $\hbox{Mg\,{\sc ii}}\ \lambda2798$ line ($\rm{dlog}$$F_{line}$) and in the 3000 \AA \ continuum ($\rm{dlog}$$F_{cont}$) with a simple linear function of the form $\rm{dlog}$$F_{line} \propto$ $\rm{dlog}$$F_{cont}$. Given that the ratio in magnitude changes ought to be equivalent to $\alpha$, the value of $\hbox{Mg\,{\sc ii}}$ responsivity can be calculated by analysing spectroscopic monitoring data.
When determining the emission line responsivity parameter $\alpha$ = $\rm{dlog}$$F_{line}$/$\rm{dlog}$$F_{cont}$, it is of great importance to ensure that the emission-line flux is referenced to the correct (in time) continuum value (e.g., \citealt{GoadEtAl2004}; \citealt{GoadEtAl2014}). Generally, this parameter is determined from temporally well-sampled continuum and emission line light curves, and the correct reference continuum is determined by shifting the emission line light curve backward in time by the emission line lag. In practice, the $\hbox{Mg\,{\sc ii}}$ lag is not well-constrained in prior RM campaigns (e.g., \citealt{ClavelEtAl1991a}; \citealt{CackettEtAl2015}, but see, \citealt{ReichertEtAl1994}; \citealt{MetzrothEtAl2006}), and few robust measurements of its responsivity have been measured. Instead of temporally well-sampled light curves of a single AGN, in this study, we use 1210 data pairs of spectroscopic observations of 68 AGNs to {\em statistically} estimate the responsivity of the broad $\hbox{Mg\,{\sc ii}}$ emission line. Here, we posit that our ignorance of the emission line lag corrections averages out statistically, and an ensemble responsivity of $\hbox{Mg\,{\sc ii}}$ may be determined from many pairs of measurements from an amount of AGNs.
Reliable flux calibration is of significance to accurately determine the observed emission-line and continuum flux. The Sloan Digital Sky Survey (SDSSS; \citealt{YorkEtAl2000}) spectroscopy is routinely calibrated using a series of standard stars, particularly main sequence F stars.
For a single observation, it is assumed that the uncertainty of SDSS-I/II spectroscopic data is $\sim$ 0.04 mag (\citealt{Adelman-McCarthyEtAl2008}). With the smaller fibers, SDSS-III BOSS (e.g., \citealt{MargalaEtAl2015}; \citealt{HarrisEtAl2016}) spectroscopy is usually not as accurate as that of SDSS-I/II. In this work, we assume that the fluxes of narrow emission line have no variations during the spectroscopic monitoring due to the much large narrow-line region (NLR). Therefore, we attempt to use narrow-line fluxes to recalibrate SDSS quasar spectra. All the flux variations are measured using ground-based optical monitoring data from the observed flux of emission-lines and continuum in any two epochs during SDSS-I/II/III surveys.
The structure of this paper is as follows. In Section \ref{sec:style} we describe our quasar sample selection. In Section \ref{sec:sm} we introduce the details of our spectral measurements. We derive the correlation between the variations of $\hbox{Mg\,{\sc ii}} \ \lambda$2798 and of the 3000 \AA \ continuum in the SDSS quasars in Section \ref{sec:results}. We discuss the related results in Section \ref{sec:dis}, and a summary of our conclusions in Section \ref{sec:con}. Throughout this paper, we adopt a flat cosmology with $\Omega_m$ = 0.3, $\Omega_{\Lambda}$ = 0.7, and $H_0$ = 70 km s$^{-1}$ Mpc$^{-1}$, and use magnitude (rather than flux or luminosity) differences to characterize variations. Unless otherwise specified, the reported wavelengths (taken from \citealt{BerkEtAl2001}) and timescales are in the quasar rest-frame.
\section{THE Sample} \label{sec:style}
In this work, we use the quasar data from the compilation of the SDSS Data Release 7 Quasar catalog (DR7Q; e.g., \citealt{SchneiderEtAl2010}; \citealt{ShenEtAl2011}) and Data Release 12 Quasar catalog (DR12Q; e.g., \citealt{ParisEtAl2014}; \citealt{ParisEtAl2017}).
All the spectra were taken by the Apache Point 2.5 m wide-field telescope (\citealt{GunnEtAl2006}) during SDSS-I/II/III surveys (2000-2014). Each spectrum is stored in vacuum wavelength with a resolution of $R \sim 1500-2500$.
Our parent sample was compiled from the following 2 sub-samples: the DR7Q consisted of 105,783 objects that are brighter than $M_i = -22.0$, and the DR12Q including 297,301 quasars. There are 7,063 quasars from DR7Q and 28,105 quasars from DR12Q, each with multiple ($\geq$ 2) spectroscopic epochs, respectively. After confirming the quasar as a point-source in the SDSS image and rejecting the epoch with low-quality spectrum, we selected a sample of 2,374 quasars with $\hbox{Mg\,{\sc ii}}$ broad-line by requiring $0.65 \leq z \leq 1.50$. This requirement ensures that broad $\hbox{Mg\,{\sc ii}}$, narrow-lines (e.g., $[\hbox{O\,{\sc iii}}] \ \lambda\lambda4960,5008$) and the 3000 \AA \ continuum region are presented in the SDSS spectra.
We notice that additional flux deficit is confirmed in the SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS; e.g., \citealt{EisensteinEtAl2011}; \citealt{BoltonEtAl2012}; \citealt{DawsonEtAl2013}; \citealt{SmeeEtAl2013a}) relative to SDSS-I/II due to the difference in flux calibration from SDSS-I/II to BOSS. To obtain an accurate and reliable measurement of the intrinsic variations in $\hbox{Mg\,{\sc ii}} \ \lambda2798$ and in the 3000 \AA \ continuum for each quasar, we develop an independent correction to the flux variations, which is called narrow-line flux-recalibration (see Section \ref{subsec:vc}). This requires that every quasar in our sample not only has repeated observations but also contains a minimum signal-to-noise ratio (S/N; defined as the ratio between the emission-line flux and its error) of 5 for narrow-line(s). In addition, we rejected objects with unusual emission line profiles and/or continuum shapes (i.e., BALQSOs) from our final sample. The reduced $\chi^2$ values ($\chi^2$/dof) of our best-fit model for these sources are often fairly large during emission-line fitting (see Section \ref{subsec:mu}). More details about the sample-selection criteria include the following.
\begin{enumerate}
\item Multiple ($\geq$ 2) spectroscopic epochs/observations are included for each quasar in the SDSS-I/II/III surveys.
\item Quasar is confirmed to be a point-source in the SDSS image (take example for DR7Q, {\tt\string sdss\_morpho = 0}).
\item A minimum S/N ratio of 10 for quasar spectrum covering $\hbox{Mg\,{\sc ii}}$ through the 3000 \AA \ continuum is preferred.
\item A redshift between 0.65 and 1.50 should be possessed for each object.
\item A minimum S/N ratio of 5 for narrow-line(s) are required in the SDSS quasar spectra.
\item Quasar with peculiar $\hbox{Mg\,{\sc ii}}$ emission-line and continuum property is rejected.
\end{enumerate}
Table \ref{tb1} summarizes part of the final 1210 data pairs consisting of 68 quasars that passed all the selection criteria and will be used for subsequent relation analysis of the spectroscopic variations in the $\hbox{Mg\,{\sc ii}}\ \lambda2798$ emission-line and in the 3000 \AA \ continuum.
\begin{deluxetable*}{ccccccccccccccc}
\tabletypesize{\scriptsize}
\tablecaption{\label{tb1} Sample Summary}
\tablewidth{0pt}
\tablehead{
\colhead{ID} & \colhead{Object Name} & \colhead{Plate} &
\colhead{Fiber} & \colhead{MJD} & \colhead{$EW_{_{\rm{mgii}}}$} & \colhead{$\rm{log}$$L_{_{\rm{mgii}}}$} & \colhead{$\rm{log}$$L_{_{\rm{nev2}}}$} & \colhead{$\rm{log}$$L_{_{\rm{oii}}}$} & \colhead{$\rm{log}$$L_{_{\rm{neiii1}}}$} & \colhead{$\rm{log}$$L_{_{\rm{oiii2}}}$} & \colhead{$z_{_{\rm{vi}}}$} & \colhead{$\rm{log}$$L_{_{\rm{3000}}}$} & \colhead{Catalog} & \colhead{$N_{\rm{r}}$}
\\
\colhead{(1)} & \colhead{(2)} & \colhead{(3)} & \colhead{(4)} & \colhead{(5)} & \colhead{(6)} & \colhead{(7)} & \colhead{(8)} & \colhead{(9)} & \colhead{(10)} & \colhead{(11)} & \colhead{(12)} & \colhead{(13)} & \colhead{(14)} & \colhead{(15)}
}
\startdata
\input{table.tex}
\enddata
\tablecomments{\scriptsize Column 1: identification number of our data pairs in this paper. Column 2: SDSS object name. Column 3-5: plate, fiber, and MJD (i.e., JD-2400000) of the optical SDSS spectrum for each object. Column 6: the broad $\hbox{Mg\,{\sc ii}}$ EW relative to the underlying continuum at 3000 \AA. Column 7-11: luminosity measurement of narrow $\hbox{Mg\,{\sc ii}}$ doublet, $[\hbox{Ne\,{\sc v}}] \ \lambda3426$, $[\hbox{O\,{\sc ii}}] \ \lambda3726,3729$ doublet, $[\hbox{Ne\,{\sc iii}}] \ \lambda3869$, and $[\hbox{O\,{\sc iii}}] \ \lambda5008$ in this paper. Column 12: improved quasar redshift from catalog DR7Q and DR12Q. Column 13: the 3000 \AA \ continuum luminosity measurement for each spectroscopic epoch in our sources. Column 14: quasar catalog in which each observation is included. Column 15: the number of narrow-lines used in our final flux-recalibration.}
\end{deluxetable*}
For each quasar, we estimate the bolometric luminosity using the continuum luminosity at 3000 \AA \ and $L_{\rm{bol}}=5L_{\rm{3000}}$ (e.g., \citealt{RichardsEtAl2006}). The range of the bolometric luminosities of our quasars is 44.49 erg s$^{-1} \leq \rm{log}$$L_{\rm{bol}} \leq 46.31$ erg s$^{-1}$, and the median is log$L_{\rm{bol}} \sim$ 45.48 erg s$^{-1}$.
The distribution of the log$L_{\rm{bol}}$ of our quasars and the timescales $\Delta t$ of our dataset is shown in Figure \ref{f01}. Note that a small fraction ($\sim$ 30\%) of the SDSS-I/II quasars observed with SDSS-III is also taken into account.
\begin{figure*}
\includegraphics[angle=0,scale=0.975]{f1.eps}
\caption{Left: distribution of the log$L_{\rm{bol}}$ for our sample of 68 broad-line quasars. The range of the bolometric luminosities of the quasars is 44.49 $\leq$ log$L_{\rm{bol}} <$ 46.31 and the median is log$L_{\rm{bol}} \sim$ 45.48. Right: distribution of the rest-frame $\Delta \rm{t}$ for our dataset of 1,210 points. A large fraction ($\sim$ 75\%) of the timescales are located in the region of $\Delta \rm{t} \leq100$ days. \label{f01}}
\end{figure*}
\section{Spectral measurements} \label{sec:sm}
In this section, we proceed to measure the strength of $\hbox{Mg\,{\sc ii}} \ \lambda2798$ emission-line, the 3000 \AA \ continuum, and narrow-lines including $[\hbox{Ne\,{\sc v}}], [\hbox{O\,{\sc ii}}], [\hbox{Ne\,{\sc iii}}]$ and $[\hbox{O\,{\sc iii}}]$.
To derive the accurate flux of $\hbox{Mg\,{\sc ii}} \ \lambda2798$ and that of the 3000 \AA \ continuum emission, we used a pseudo-continuum model to fit the quasar spectra. This model consists of $\hbox{Fe\,{\sc ii}}$ multiplets, the power-law (PL) continuum, Balmer continuum, and high order Balmer lines. Before the local fits, we corrected the Galactic extinction in the SDSS spectra using the Milky Way (MW) reddening law derived by \citet{CardelliEtAl1989} and the derived $E(B-V)$ based on \citet{SchlegelEtAl1998} dust map, and shifted the spectra to rest-frame using the cataloged redshift as the systemic redshift. Following \cite{ShenEtAl2012}, we masked out narrow absorption lines for each source to reduce the uncertainties of our continuum and emission-line fits.
Though a $\hbox{Fe\,{\sc iii}}$ template was derived from UV spectrum of I Zw 1 (\citealt{VestergaardEtAl2001}), we do not include it in our pseudo-continuum model because it is routinely difficult to constrain this component in spectral fits (see, e.g., \citealt{GreeneEtAl2010}). The $\hbox{Fe\,{\sc ii}}$ template used in this work is exactly the same as the template \cite{ShenEtAl2012} used. That is, the UV $\hbox{Fe\,{\sc ii}}$ template is a combination of templates of \citet{VestergaardEtAl2001} in 1000-2200 \AA, \citet{SalvianderEtAl2007} in 2200-3090 \AA \ and \citet{TsuzukiEtAl2006} in 3090-3500 \AA, and the optical $\hbox{Fe\,{\sc ii}}$ template is \citet{BorosonEtAl1992} template (3686-7484 \AA). We independently fitted these two $\hbox{Fe\,{\sc ii}}$ templates, each with three free parameters, i.e., the normalization, the velocity dispersion, and the wavelength shift of the template. For the PL continuum model, the normalization factor and the slope are used as two free parameters.
For the contribution of the Balmer continuum, we follow the formula from \citet{DietrichEtAl2003} and \citet{TsuzukiEtAl2006}, in which the Balmer continuum is expressed by
\begin{equation}
f_{BC}(\lambda)=f_{3646} B_{\lambda}(T_e)(1-e^{-\tau_{\lambda}}); \qquad \qquad \rm{if} \ \lambda \leqslant \lambda_{BE}
\end{equation}
where $f_{3646}$ is the normalization coefficient at Balmer edge 3646 \AA \ and $\tau_{\lambda}=\tau_{BE}(\lambda/\lambda_{BE})^3$ in which $\tau_{BE}$ is the optical depth at Balmer edge $\lambda_{BE}$ (3646 \AA), and $B_{\lambda}(T_e)$ is the Planck function at an electron temperature $T_e$, which is assumed to be 15,000 K (e.g., \citealt{DietrichEtAl2003}, \citealt{JiEtAl2012}).
To improve the local fit for each narrow-line, high order Balmer lines up to $n =$ 50 are also included in our pseudo-continuum model, as \citet{JiEtAl2012} did. Utilizing the Balmer line emissivities for Case B, $T_e = 15,000$ K and $n_e = 10^8$ cm$^{-3}$ (\citealt{StoreyEtAl1995}), we constrain the relative strengths of these lines. We fix the flux ratios of the high order Balmer lines to the Balmer continuum flux at the edge (3646 \ \AA) according to the results in \citet{WillsEtAl1985}. Below we describe the detailed fitting procedures for $\hbox{Mg\,{\sc ii}}$ broad line, the 3000 \AA \ continuum, and several narrow lines.
\subsection{$\hbox{Mg\,{\sc ii}}$} \label{subsec:mgii}
For the broad $\hbox{Mg\,{\sc ii}} \ \lambda2798$ line, we first fitted the pseudo-continuum model consisting of the PL continuum, Balmer continuum, and the UV $\hbox{Fe\,{\sc ii}}$ template. All these components were fitted simultaneously in the following windows: 2155-2675 \AA \ and 2925-3500 \AA, which are devoid of strong emission lines.
We then subtracted the pseudo-continuum from the original SDSS spectra, and fitted the $\hbox{Mg\,{\sc ii}}$ line over the [2690,2910] \AA \ wavelength range. The broad $\hbox{Mg\,{\sc ii}}$ component was modelled with multiple Gaussians with up to three Gaussians (each with FWHM $\geq$ 900 km/s). As to the narrow component of $\hbox{Mg\,{\sc ii}}$, we used two Gaussians\footnote{Given that $\hbox{Mg\,{\sc ii}} \ \lambda 2798$ is a fairly widely separated doublet at 2795.530 \AA \ and 2802.704 \AA, a separation of $\sim$ 750 km s$^{-1}$ in Doppler shift, and $\hbox{Mg\,{\sc ii}}$ is often the narrowest of the broad emission lines, we apply the following additional constraints to the narrow component: FWHM $<$ 900 km s$^{-1}$ and flux $<$ 10\% of the total $\hbox{Mg\,{\sc ii}}$ flux, see, e.g., \citet{WillsEtAl1993}; \citet{McLureEtAl2004}; \citet{WangEtAl2009}. For narrow lines such as $[\hbox{Ne\,{\sc iii}}]$, $[\hbox{O\,{\sc ii}}]$, $[\hbox{Ne\,{\sc iii}}]$, $[\hbox{O\,{\sc iii}}]$, the upper limit for each line width is 1200 km s$^{-1}$, see, e.g., \citet{HaoEtAl2005}; \citet{ShenEtAl2011, ShenEtAl2012}.} and checked both possibilities of doublet ratio, 2:1 for optically thin and 1:1 for optically thick, and then chose the one that has a smaller reduced $\chi^2$ value.
\subsection{$The \ 3000 \AA \ Continuum$} \label{subsec:cont}
We used the PL continuum model in the above pseudo-continuum fitting procedure to estimate the 3000 \AA \ continuum luminosity $L$ = $\lambda L_{\lambda}$ at 3000 \AA \ (\citealt{ShenEtAl2012}). We found that our local fits for pseudo-continuum and $\hbox{Mg\,{\sc ii}}$ line are perfect and the reduced $\chi^2$ values from the spectral fits are close to 1, with the median value of 1.15 and 0.98.
\subsection{$[\hbox{Ne\,{\sc v}}]$\label{subsec:ne5}}
For $[\hbox{Ne\,{\sc v}}] \ \lambda3346,3426$ doublet, the pseudo-continuum model fitting wavelength windows are [2480,2675] \AA, [2925,3020] \AA, [3225,3300] \AA \ and [3450,3550] \AA. These wavelength coverages are not contaminated with strong emission lines. After subtracting the pseudo-continuum from the spectrum, we fit the wavelength range [3329,3446] \AA \ for $[\hbox{Ne\,{\sc v}}]$. Considering the spectroscopic S/N of $[\hbox{Ne\,{\sc v}}]$, we used two Gaussians for the $[\hbox{Ne\,{\sc v}}] \ \lambda3346$ and $[\hbox{Ne\,{\sc v}}] \ \lambda3426$ lines, and we tied their flux ratio to be $f_{3426}/f_{3346} = 3$ during the fit.
Since $[\hbox{Ne\,{\sc v}}]$ doublet are intrinsically weak lines, we did not adopt such restrict condition as S/N $\geq$ 10 in our spectral fitting procedure. Instead, we discard the rescaling factor differing significantly from the ones obtained from other narrow-lines in the spectra. Nevertheless, the doublet could provide reference to the flux-calibration correction factors that are calculated from other narrow-lines. (see Section \ref{subsec:vc} for more details).
\subsection{$[\hbox{O\,{\sc ii}}]$ \ $\& \ [\hbox{Ne\,{\sc iii}}]$\label{subsec:o2ne3}}
For $[\hbox{O\,{\sc ii}}] \ \lambda3728$ and $[\hbox{Ne\,{\sc iii}}] \ \lambda3869,3968$ lines, as there are no strong nearby broad lines, we simultaneously fitted the PL continuum, $\hbox{Fe\,{\sc ii}}$ emission (the optical $\hbox{Fe\,{\sc ii}}$ template), high order Balmer lines, $\hbox{He\,{\sc i}} \ \lambda$3889, $\hbox{H$\zeta$} \ \lambda$3890, $\hbox{H$\epsilon$} \ \lambda$3971, $[\hbox{O\,{\sc ii}}]$ and $[\hbox{Ne\,{\sc iii}}]$ over the wavelength range 3670-4020 \AA. The narrow components of these emission lines were each fit with a single Gaussian. On the other hand, the broad components of $\hbox{H$\zeta$} \ \lambda3890$ and $\hbox{H$\epsilon$} \ \lambda$3971 were each fit with multiple Gaussians up to three Gaussians.
We tied the flux ratio of the $[\hbox{Ne\,{\sc iii}}] \ \lambda \lambda 3869,3968$ doublet to be $f_{3869}/f_{3968}$ = 3 during the fitting.
Note that the $[\hbox{O\,{\sc ii}}]$ narrow-line is comprised of the blended $[\hbox{O\,{\sc ii}}] \ \lambda \lambda$3726,3729 doublet, which is rarely resolved in the SDSS spectra due to the inadequate spectral quality (for instance, do not have adequate S/N or low-resolution spectroscopy to unambiguously locate the doublet). We tried to fit the wavelength range [3680,3780] \AA using up to two Gaussians for the subtracted spectra (i.e., leaving $[\hbox{O\,{\sc ii}}]$ and $[\hbox{Ne\,{\sc iii}}]$ emission-lines) when the spectroscopic S/N ($[\hbox{O\,{\sc ii}}]$) $\geq$ 10. It is shown that the final $\chi^2$/dof value from our fitting procedure for each spectrum is usually close to 1.
\subsection{$[\hbox{O\,{\sc iii}}]$\label{subsec:o3}}
For $[\hbox{O\,{\sc iii}}] \ \lambda4960,5008$ doublet, we first used the optical $\hbox{Fe\,{\sc ii}}$ template and the PL continuum fit for each object. The PL continuum$+$iron fitting windows are [4435,4700] \AA \ and [5080,5535] \AA. We then subtracted the pseudo-continuum from the spectrum (leaving the emission-line spectrum). Considering the strong contamination of $\hbox{H$\beta$} \ \lambda4861$ broad-line on $[\hbox{O\,{\sc iii}}]$ lines, we simultaneously fitted $\hbox{H$\beta$}$ line and the pseudo-continuum model. We fitted the wavelength range [4700,5100] \AA \ and used one Gaussian with FWHM $<$ 1200 km/s for the narrow $\hbox{H$\beta$}$ component and up to three Gaussians (each with FWHM $\geq$ 1200 km/s) for the broad $\hbox{H$\beta$}$ component.
Considering the modest quality of the SDSS spectra, we fitted each of the two $[\hbox{O\,{\sc iii}}]$ lines with a single Gaussian. We tied the flux ratio of the $[\hbox{O\,{\sc iii}}]$ doublet to be $f_{5008}/f_{4960}$ = 3 to reduce possible ambiguities. It is true that asymmetric blue wings (see, e.g., \citealt{HeckmanEtAl1981}; \citealt{KomossaEtAl2008}) and dramatic double-peaked profiles (e.g., \citealt{LiuEtAl2009}; \citealt{WangEtAl2009}) often appeared in the $[\hbox{O\,{\sc iii}}]$ lines. However, these features can only be well constrained in high S/N ($\geq$ 10) spectra. For these spectra, we used up to two Gaussians for each of the narrow $[\hbox{O\,{\sc iii}}]$ lines.
\subsection{Measurement Uncertainties\label{subsec:mu}}
It is of immense significance to determine the uncertainties in the continuum and emission-line flux measurements. We followed \citet{ShenEtAl2012} and adopted the following Monte-Carlo method:
\begin{enumerate}
\item Using the given flux density errors, we perturb the original spectra randomly to generate mock spectra.
\item The mock spectra are each fitted with the same fitting routine to get the corresponding line fluxes.
\item We repeat 1 and 2 for 50 times and obtain a distribution of our spectral measurements.
\item The semi-amplitude of the range enclosing the 16th and 84th percentiles of the distribution is adopted as an estimation of our measurement uncertainties.
\end{enumerate}
Figures 2 and 3 show two examples of our fits to the spectra in presence of different levels of UV/optical $\hbox{Fe\,{\sc ii}}$ emission.
\begin{figure*}[t]
\begin{tabular}{llll}
\begin{minipage}[t]{3.4in}
\includegraphics[width=3.4in]{f2.eps}
\end{minipage}
\begin{minipage}[t]{3.4in}
\includegraphics[width=3.4in]{f3.eps}
\end{minipage}
\\
\begin{minipage}[t]{3.4in}
\includegraphics[width=3.4in]{f4.eps}
\end{minipage}
\begin{minipage}[t]{3.4in}
\includegraphics[width=3.4in]{f5.eps}
\end{minipage}
\end{tabular}
\caption{Example of our model fits to different emission-lines in the spectra of a broad-line quasar (J023025.03-004944.2). The original spectrum (black), the power-law continuum (gray), the continuum+$\hbox{Fe\,{\sc ii}}$ template fit (blue dashed), and the combined pseudo-continuum model (orange) to be subtracted off are presented in the upper panel of each plot. The corresponding bottom panels show the emission-line fits to $\hbox{Mg\,{\sc ii}}$ through $[\hbox{O\,{\sc iii}}]$ doublet, where black lines are the residuals and red lines are the combined model line profiles. For $\hbox{Mg\,{\sc ii}}$, we also show the model narrow-line emission in green and violet and the model broad-line emission in purple, pink and skyblue. \label{f02}}
\end{figure*}
\begin{figure*}[t]
\begin{tabular}{llll}
\begin{minipage}[t]{3.4in}
\includegraphics[width=3.4in]{f6.eps}
\end{minipage}
\begin{minipage}[t]{3.4in}
\includegraphics[width=3.4in]{f7.eps}
\end{minipage}
\\
\begin{minipage}[t]{3.4in}
\includegraphics[width=3.4in]{f8.eps}
\end{minipage}
\begin{minipage}[t]{3.4in}
\includegraphics[width=3.4in]{f9.eps}
\end{minipage}
\end{tabular}
\caption{Another example of our model fits to different emission-lines in the spectra of a broad-line quasar (J224412.33+240110.0) that involves stronger UV/optical $\hbox{Fe\,{\sc ii}}$ emission. The original spectrum (black), the power-law continuum (gray), the continuum+$\hbox{Fe\,{\sc ii}}$ template fit (blue dashed), and the combined pseudo-continuum model (orange) to be subtracted off are presented in the upper panel of each plot. The corresponding bottom panels show the emission-line fits to $\hbox{Mg\,{\sc ii}}$ through $[\hbox{O\,{\sc iii}}]$ doublet, where black lines are the residuals and red lines are the combined model line profiles. For $\hbox{Mg\,{\sc ii}}$, we also show the model narrow-line emission in green and violet and the model broad-line emission in purple, pink and skyblue. \label{f002}}
\end{figure*}
We visually inspected our fits and rejected BALQSOs.
\section{Results} \label{sec:results}
\subsection{Variations Calculation}\label{subsec:vc}
Now we start to determinate the flux variations of broad $\hbox{Mg\,{\sc ii}}$ and of continuum. During the calculation, we adopt narrow-line flux-recalibration to the line flux variations. Following the conversion between fluxes and magnitudes and applying the narrow-line flux rescaling factor to the flux of broad $\hbox{Mg\,{\sc ii}}$ and continuum for each source, we define the basic variation as:
\begin{equation}
\Delta m = -2.5 \mathrm{log} (f_2/f_1)+\Delta m_{\overline{r}}
\end{equation}
where $f_{1}$,$f_{2}$ denote the observed fluxes of broad $\hbox{Mg\,{\sc ii}}$ or continuum at two epochs, $\Delta m_{\overline{r}}$ represents the variation of our final rescaling factor cross the epochs, and $r$ is the rescaling factor from each narrow-line during two observations.
We calculate $\Delta m$ for the flux pairs of broad $\hbox{Mg\,{\sc ii}}$ and the 3000 \AA \ continuum separated by $\Delta t$ for each source. Assuming that one quasar was observed with $n$ spectroscopic epochs, the largest number of our data points (i.e., $\Delta m$ pairs) for this object would be $C_n^2$, if we take no account of the spectral quality.
For each $\Delta m$ pair, we chose the narrow-line rescaling factors in the range of 0.6$-$1.4. These data pairs occupy the vast majority of our dataset.
Given that most data points have different rescaling factors for different narrow-line, making the selection of the rescaling factors becomes one of keys in our variation calculation. To obtain a fairly reliable rescaling factor for each data point, we calculate error-weighted average value of all narrow-line rescaling factors (i.e., each rescaling factor is weighted by its uncertainty). In addition, we also require that the final rescaling factors is still in the range of 0.6$-$1.4, and the corresponding error is limited to 0.10. As a consequence, for the data pair with at least two S/N $\geq$ 5 narrow-lines, the median of the standard deviation from the mean (i.e., final correction factor) for these data pairs is $\sim$ 0.05, reflecting that the different narrow-line rescaling factors for each data pair are pretty tightly bunched together; for the data pair with only one S/N $\geq$ 5 narrow-line, our limited rescaling factor error requires that the S/N of this narrow-line should be $\geq$ 14, which is good enough to do flux-recalibration. The total number of our data points for the following variations correlation analysis is 1210 (see Section \ref{subsec:vcc} and \ref{subsec:sbc}).
Figure \ref{f03} shows the distribution and error distribution of our final rescaling factor in our variation calculation for all the data pairs.
\begin{figure}
\includegraphics[angle=0,scale=1.0]{f10.eps}
\caption{Distribution (top panel) and error distribution (bottom panel) of the final narrow-line flux rescaling factor for the epochs in SDSS-I/II (blue), SDSS-III (green), and SDSS-I/II/III (red) in variation calculation for all the quasars in our sample. The median are 0.98 and 0.05, respectively. \label{f03}}
\end{figure}
We notice that \citet{ShenEtAl2016} obtained a precision ($\sim$ 5\%) of the spectroscopy achieved for SDSS-RM in reference to the median asbolute deviation (MAD) of narrow-line flux variations; while here we try not to make deductions from this parameter, as we did a totally different rescaling factor measurement. More important, our sample includes the quasars observed with both SDSS-I/II and SDSS-III, where studies have confirmed a deficit in flux in BOSS of roughly 20\% relative to SDSS-I/II at long wavelengths (see, e.g., \citealt{HarrisEtAl2016}).
\subsection{Variations Correlation Coefficient}\label{subsec:vcc}
First we compare the narrow-line flux-recalibrated variations in broad $\hbox{Mg\,{\sc ii}}$ with those in the 3000 \AA \ continuum for all the quasars in our sample in the left panel of Figure \ref{f04}.
\begin{figure*}
\includegraphics[angle=0,scale=0.985]{f11.eps}
\caption{Left: the narrow-line flux-recalibrated variations of $\hbox{Mg\,{\sc ii}}\ \lambda2798$ against those of the 3000 \AA \ continuum on timescales of $\Delta t \geqslant$ 1 day for our quasar sample. The typical errors are shown in the upper left corner. Right: same as the left panel, but the variations of $\hbox{Mg\,{\sc ii}}$ and those of the 3000 \AA \ continuum are not corrected using the constancy of the narrow-lines. We do not show the typical error in the right panel, as our sample includes a set of quasars with observations in both SDSS-I/II and SDSS-III surveys.\label{f04}}
\end{figure*}
For comparison, we also plot uncorrected line and continuum variations on the same timescales in the right panel. The correlation between the narrow-line flux-recalibrated variations in broad $\hbox{Mg\,{\sc ii}}$ and the 3000 \AA \ continuum is tested using the Spearman rank correlation test. The null hypothesis is that there is no correlation between the input datasets. The correlation coefficient of this test is $\rho$ = 0.593 (0.456 for the dataset without narrow-line flux-recalibration). If we focus on the data on timescales of $\Delta t \leqslant$ 100 days, the result is 0.644 (0.453 for the other dataset), which is consistent with the result in \citet{SunEtAl2015}.
For the $p$ value (i.e., the probability of being incorrect in rejecting the null hypothesis),
we adopt a bootstrap resampling method. That is, we randomly (with replacement) select $\Delta m$ pairs from the observed sample. We then perturb the data pairs of the random sample by their uncertainties.
We calculate spearman's $\rho$ for the random sample and repeat the same process for many times (e.g., 10,000) to get an estimation of the significance level. The distribution of the Spearman's $\rho$ obtained from our 10,000 random samples is shown in the left panel of Figure \ref{f06}.
\begin{figure*}
\includegraphics[angle=0,scale=0.97]{f12.eps}
\caption{Distribution of the Spearman's rank correlation coefficient $\rho$ of 10,000 random samples due to the correlated error of narrow-line rescaling factor. Left: the random samples are deduced from flux-recalibrated dataset. Right: the random samples are deduced from partly flux-recalibrated dataset according to one certain criterion (see the text in Section \ref{subsec:vcc}).\label{f06}}
\end{figure*}
We reason that the $p$ value is far less than 0.01$\%$, indicating a significant positive correlation between the variations of broad $\hbox{Mg\,{\sc ii}}$ and the 3000 \AA \ continuum.
In addition,
we also attempt to recalibrate only the variations that the narrow-line flux-recalibration errors are small enough in comparison with the rescaling factor (e.g., $|\ln r| > $2 $\Delta r$) to generate a partly recalibrated dataset. Adopting the bootstrap resampling method to the two datasets on timescales of $\Delta t \leq$ 3 days, we find that our flux-recalibration process hardly biases our variation correlation analysis by comparing both significance levels. To make further validation, we also plot the distribution of the Spearman's $\rho$ for the random samples deduced from partly flux-recalibration on the timescales of $\Delta t \geq$ 1 day in the right panel of Figure \ref{f06}. The comparison of both significance levels infers that our flux-recalibration for all variations does not introduce a much larger uncertainty for the points with relatively large rescaling factor errors in our correlation analysis.
\subsection{The Slope Between Variations of Two Components}\label{subsec:sbc}
Given the correlated measurement uncertainties of the broad $\hbox{Mg\,{\sc ii}}\ \lambda2798$ emission-line and the 3000 \AA \ continuum variations, we perform a modified weighted least squares regression method to calculate the slope between narrow-line flux-recalibrated variations of both components. The covariance of the $\hbox{Mg\,{\sc ii}}$ line and the continuum variations, generated by the bootstrap resampling method, is also taken into consideration in our modified method. Assuming a linear model, we can get the following relation.
\begin{equation}
\begin{aligned}
\Delta m_{\rm{line0}}+\Delta m_{\overline{r}} = \Delta m_{\rm{line}} = a+b*\Delta m_{\rm{cont}} \\
= a+b*(\Delta m_{\rm{cont0}}+\Delta m_{\overline{r}})
\end{aligned}
\end{equation}
where in this case $\Delta m_{\rm{line}}$, $\Delta m_{\rm{cont}}$ represent narrow-line flux-recalibrated variations in broad $\hbox{Mg\,{\sc ii}}$ and in the continuum respectively. $\Delta m_{\rm{line0}}$, $\Delta m_{\rm{cont0}}$ are the corresponding original variations without narrow-line flux-recalibration, respectively. Clearly, both the $\Delta m_{\rm{line}}$ and the $\Delta m_{\rm{cont}}$ are correlated with $\Delta m_{\overline{r}}$, which is induced by the flux calibration. The total measurement uncertainty of our linear weighted least squares fit model is as follows.
\begin{equation}
\begin{aligned}
\sigma_{tol}=\sqrt{\sigma_{\Delta m_{\rm{line}}}^2+b^2*\sigma_{\Delta m_{\rm{cont}}}^2-2*b*\sigma_{\Delta m_{\rm{line}} \Delta m_{\rm{cont}}}^2}
\end{aligned}
\end{equation}
where $\sigma_{\Delta m_{\rm{line}} \Delta m_{\rm{cont}}}^2$ is the covariance of the variations of emission line and the continuum. In the calculation of $\sigma_{\Delta m_{\rm{line}} \Delta m_{\rm{cont}}}^2$, we assign normal distributed errors to $\Delta m_{\rm{line0}}$, $\Delta m_{\rm{cont0}}$, and $\Delta m_{\overline{r}}$ for each data point, as these parameters are independent of each other. Then we get an estimation of the covariance after repeating the same process for many times (e.g., 10,000).
Note that the variations of emission line and of the continuum are dependent on quasar luminosities, and hence we will discuss the slopes for different luminosity bins. Our sample is divided into 4 sub-samples by log$L_{\rm{bol}}$: 44.49 erg s$^{-1} \leq$ log$L_{\rm{bol}} <$ 44.95 erg s$^{-1}$; 44.95 erg s$^{-1}$ $\leq$ log$L_{\rm{bol}} <$ 45.40 erg s$^{-1}$; 45.40 erg s$^{-1}$ $\leq$ log$L_{\rm{bol}} <$ 45.85 erg s$^{-1}$; 45.85 erg s$^{-1} \leq$ log$L_{\rm{bol}} <$ 46.31 erg s$^{-1}$. Adopting our modified weighted least squares method (see equation (4)), we calculate the slope between the variations of both components for the quasars in each luminosity bin. The intrinsic bias introduced by narrow-line flux-recalibration process itself is estimated by the following method. 1) we constrained the intrinsic variability of the 3000 \AA \ continuum; 2) we assumed that variations (X) follow a Guassian distribution, whose RMS equals to the intrinsic variability of the 3000 \AA \ continuum; 3) we generated mock samples from such a distribution; 4) we calculated mock line variations by adopting Y = a + b * X; 5) we fitted the slope between these two mock variations; 6) we repeated this process for 100,000 times and obtained the distribution of the fitted slope. The difference between the mean of the slope measurement and the value of the input parameter b is then defined as the intrinsic slope bias. In Figure \ref{f07},
\begin{figure}
\includegraphics[angle=0,scale=1.0]{f17.eps}
\caption{Example distribution of slope bias (top panel) and intercept bias (bottom panel) for 100,000 random samples due to narrow-line rescaling factor error in the slope (and intercept) estimation for our quasars spanning a bolometric luminosity range of 44.95 erg s$^{-1} <$ log$L_{\rm{bol}}$ $<$ 45.40 erg s$^{-1}$.\label{f07}}
\end{figure}
we show an example of the intrinsic slope bias ($\sim$ 0.072) due to our correction to the variations of both components for the sources covering a bolometric luminosity range of 44.95 erg s$^{-1}$ $\leq$ log$L_{\rm{bol}} <$ 45.40 erg s$^{-1}$.
After eliminating the intrinsic biases introduced by the rescaling process, we obtain the corrected slopes between the narrow-line flux-recalibrated variations of broad $\hbox{Mg\,{\sc ii}}$ and of the 3000 \AA \ continuum for our quasars in different luminosity bins. Table \ref{tb2} presents our parameter measurements of the correlation between variations of $\hbox{Mg\,{\sc ii}}$ line and continuum. It is revealed that the slope decreases as the quasar luminosity increases. In Figure \ref{f08},
\begin{figure*}[t]
\begin{tabular}{llll}
\begin{minipage}[t]{3.43in}
\includegraphics[width=3.43in]{f13.eps}
\end{minipage}
\begin{minipage}[t]{3.43in}
\includegraphics[width=3.43in]{f14.eps}
\end{minipage}
\\
\begin{minipage}[t]{3.43in}
\includegraphics[width=3.43in]{f15.eps}
\end{minipage}
\begin{minipage}[t]{3.43in}
\includegraphics[width=3.43in]{f16.eps}
\end{minipage}
\end{tabular}
\caption{Comparisons between the narrow-line flux-recalibrated variations of $\hbox{Mg\,{\sc ii}}$ and of the 3000 \AA \ continuum for the quasars in different bolometric luminosity bins. The solid red curve in each plot is a modified linear weighted least squares fit to our data. \label{f08}}
\end{figure*}
we compare the emission line variations with the continuum variations for our 4 sub-samples. We also estimate virial SMBH masses using the 3000 \AA \ continuum luminosity and archive $\hbox{Mg\,{\sc ii}}$ FWHM data from SDSS-DR7Q and BOSS-DR12Q (e.g., \citealt{ShenEtAl2011}; \citealt{SunEtAl2015}). The median SMBH mass for each luminosity bin is $\sim$ constant (approximately $\sim 10^8$ $M_{\odot}$), meaning that the slope and Eddington ratio are also anti-correlated.
Additionally, we obtained an distribution of $\Delta m_{_{\rm{Mg II}}}/ \Delta m_{\rm{3000 \AA}}$ (equivalent to the responsivity $\alpha$) for our 1210 data pairs of spectroscopic observations of 68 SDSS quasars. The median $\alpha$ = 0.428 is roughly consistent with our average corrected slope (0.464 $\pm$ 0.013).
\section{Discussion} \label{sec:dis}
There have been some studies discussing the correlation between emission-line and continuum variations (e.g., \citealt{Baldwin1977}; \citealt{KinneyEtAl1990}; \citealt{PoggeEtAl1992}; \citealt{GoadEtAl1993}; \citealt{OBrienEtAl1995}; \citealt{GilbertEtAl2003}; \citealt{GoadEtAl2004}; \citealt{KongEtAl2006}). However, most of these correlation studies either focused on other emission-lines (e.g., $\hbox{Ly$\alpha$} \ \lambda1216$, $\hbox{C\,{\sc iv}} \ \lambda1549$, $\hbox{H$\beta$} \ \lambda4861$) or did not correct the ensemble flux variations using narrow-line fluxes. Thus we attempt to statistically investigate the relationship between the flux variations of broad $\hbox{Mg\,{\sc ii}}\ \lambda2798$ emission-line and of the nearby 3000 \AA \ continuum using narrow-line calibrated SDSS spectra.
Our results generally agree with earlier observational studies that the variations in the $\hbox{Mg\,{\sc ii}} \ \lambda2798$ emission-line is well correlated with those in the 3000 \AA \ continuum (e.g.,\citealt{WilhiteEtAl2005}; \citealt{Woo2008}; \citealt{BenitezEtAl2009}; \citealt{CackettEtAl2015}; \citealt{SunEtAl2015}). We analysed the data from the SDSS-RM project in \citet{SunEtAl2015} using the same method (i.e., modified weighted least squares regression) and found that the slope between the variations of broad $\hbox{Mg\,{\sc ii}}$ and of the 3000 \AA \ continuum for this dataset is 0.620 $\pm$ 0.028. Though different quasar samples are used in the two studies, both
\begin{deluxetable*}{ccccccccc}
\tabletypesize{\footnotesize}
\tablecaption{\label{tb2} Relation Parameter Measurements}
\tablewidth{0pt}
\tablehead{
\colhead{ID} & \colhead{Luminosity Bin} & \colhead{Spearman's $\rho$} & \colhead{Fitted Slope} & \colhead{$\Delta t$ Range} & \colhead{Median $M_{\bullet}$} &
\colhead{$N_{\rm{obj}}$} & \colhead{Slope Bias} & \colhead{Corrected Slope}
\\
\colhead{(1)} & \colhead{(2)} & \colhead{(3)} & \colhead{(4)} & \colhead{(5)} &
\colhead{(6)} & \colhead{(7)} & \colhead{(8)} & \colhead{(9)}
}
\startdata
1 & 44.49 $\leq$ log$L_{\rm{bol}} <$ 44.95 & 0.485 & 0.556 $\pm$ 0.026 & $\leq$727 & 7.6x$10^7$ & 8 & 0.036 & 0.520 $\pm$ 0.026 \\
2 & 44.95 $\leq$ log$L_{\rm{bol}} <$ 45.40 & 0.658 & 0.639 $\pm$ 0.031 & $\leq$4035 & 1.3x$10^8$ & 21 & 0.072 & 0.567 $\pm$ 0.031 \\
3 & 45.40 $\leq$ log$L_{\rm{bol}} <$ 45.85 & 0.697 & 0.467 $\pm$ 0.025 & $\leq$4781 & 2.8x$10^8$ & 21 & 0.012 & 0.455 $\pm$ 0.025 \\
4 & 45.85 $\leq$ log$L_{\rm{bol}} <$ 46.31 & 0.526 & 0.373 $\pm$ 0.050 & $\leq$3880 & 3.7x$10^8$ & 18 & 0.033 & 0.340 $\pm$ 0.050 \\
\dag & 44.49 $\leq$ log$L_{\rm{bol}} <$ 46.31 & 0.593 & 0.513 $\pm$ 0.013 & $\leq$4781 & 2.3x$10^8$ & 68 & 0.049 & 0.464 $\pm$ 0.013 \\
\enddata
\tablecomments{Column 1: identification number assigned in this paper. Column 2: divided quasar luminosity bins. Colomn 3-9: the Spearman's coefficient, fitted slope, range of timescale (in units of days), median quasar virial SMBH mass (in units of $M_{\odot}$), total number of sources, the intrinsic fitted slope bias due to the narrow-line rescaling factor, and the corrected slope that is equivalent to the responsivity $\alpha$ of $\hbox{Mg\,{\sc ii}}$ for the quasars in each luminosity bin.}
\end{deluxetable*}
results indicate a very small value in the responsivity of $\hbox{Mg\,{\sc ii}}$.
This suggests that the line not be expected to respond strongly to changes in continuum flux, and that it is might not easy to detect a plausible lag between the 3000 \AA \ continuum and the $\hbox{Mg\,{\sc ii}}$ variations (see, e.g., \citealt{CackettEtAl2015}).
On the other hand, using a sample of 101 quasars and 88 Seyferts with multiple International Ultraviolet Explorer (IUE) observations, \citet{KinneyEtAl1990} confirmed the existence of a correlation between the continuum and the $\hbox{C\,{\sc iv}} \ \lambda1549$ equivalent width with a slope of $\beta \sim$ -0.17 $\pm$ 0.04. They further concluded a similar relation for $\hbox{Ly$\alpha$} \ \lambda1216$ emission-line with a slope of $\beta \sim$ -0.12 $\pm$ 0.05. Given that $\alpha$ = $\beta$ + 1, the responsivity $\alpha$ of $\hbox{C\,{\sc iv}}$ is $\sim$ 0.83 $\pm$ 0.04 and that of $\hbox{Ly$\alpha$}$ is $\sim$ 0.88 $\pm$ 0.05. This indicates that, unlike $\hbox{Mg\,{\sc ii}}$, $\hbox{C\,{\sc iv}}$ and $\hbox{Ly$\alpha$}$ vary greatly in response to the variability of the continuum emission. Subsequently, \citet{ GilbertEtAl2003} and \citet{GoadEtAl2004} studied the $\hbox{H$\beta$}$ variability for NGC 5548 using the IUE archived UV spectra and/or ground-based optical monitoring data of 13-year-observations. They found that the responsivity $\alpha$ of the broad $\hbox{H$\beta$} \ \lambda4861$ varies from 0.4 (bright states) to 1 (dim states), anti-correlated with the flux of the incident continuum. This is consistent with our result of a negative relationship or inverse relationship between the emission-line responsivity and Eddington ratio.
In addition, our results are roughly in good agreement with theoretical studies on the intrinsic variability of $\hbox{Mg\,{\sc ii}}$. \citet{GoadEtAl1993} computed the response function for
$\hbox{Mg\,{\sc ii}} \ \lambda2798$ covering various ionization states originated from the spherical BLR. Assuming a panchromatically changing flux in the incident continuum, they found that the responsivity of the $\hbox{Mg\,{\sc ii}}$ emission-line varies throughout the BLR, but is generally small ($\alpha <$ 0.5).
Given that the shorter wavelength UV continuum usually has larger amplitude variability relative to the continuum at longer wavelength, our estimation of $\alpha$ might be slightly higher compared to (but roughly consistent with) the one predicted by photoionization models (e.g., \citealt{GoadEtAl1993}; \citealt{KoristaEtAl2000, KoristaEtAl2004}). Moreover, \citet{OBrienEtAl1995} utilized the multicloud BLR models to confirm the non-linear response of the $\hbox{Mg\,{\sc ii}}$ line, resulting in the continuum-level dependent response function. This is well verified by our results that the emission-line responsivity is anti-correlated with Eddington ratio (i.e., high/low states).
Furthermore, our results have important consequences for future studies of $\hbox{Mg\,{\sc ii}}$ RM and SMBH mass estimation for a large set of 0.3 $< z < 2$ quasars. The statistically derived average value of $\rm{dlog}$$F_{line}$/$\rm{dlog}$$F_{cont} \approx$ 0.464, suggests a weak $\hbox{Mg\,{\sc ii}}$ BEL responsivity to continuum variations. This indicates that minimizing spectrophotometric errors is essential to revealing the intrinsic variability of $\hbox{Mg\,{\sc ii}}$. Only high signal-to-noise flux of $\hbox{Mg\,{\sc ii}}$ in monitoring campaigns of long duration can we expect to determine reliable lags between the $\hbox{Mg\,{\sc ii}}$ and 3000 \AA \ continuum.
\section{Conclusions} \label{sec:con}
In this paper, we have statistically determined the relation between the magnitude differences of broad $\hbox{Mg\,{\sc ii}}$ and those of the 3000 \AA \ continuum based on the broad-line quasars taken from SDSS-I/II/III surveys, using a sample of 68 intermediate-redshift quasars, each with multiple ($\geq$ 2) observations and at least one S/N $\geq$ 5 narrow-line. The main conclusions are the following.
\begin{enumerate}
\item We found that $\hbox{Mg\,{\sc ii}}$ and the continuum variations are significantly correlated (Spearman $\rho = 0.593$). This is consistent with the idea that $\hbox{Mg\,{\sc ii}}$ varies in response to the continuum emission changes (see Figures \ref{f04}; Section \ref{subsec:vcc}).
\item Using the modified weighted least squares regress method, we confirmed that the slope between the variations in broad $\hbox{Mg\,{\sc ii}}$ emission-line and in the continuum (i.e., the responsivity $\alpha$) is not constant. But instead, we found that the slope anti-correlates with the quasar luminosity and/or Eddington ratio (i.e., high/low states; see Table \ref{tb2}; Figure \ref{f08}; Section \ref{subsec:sbc}).
\item We demonstrated the responsivity of $\hbox{Mg\,{\sc ii}}$ with an average $\bar{\alpha} \approx$ 0.464 (median 0.428) in our quasars, suggesting that high signal-to-noise flux measurements are statistically required to robustly detect the intrinsic variability and the time lag of $\hbox{Mg\,{\sc ii}}$ line (see Sections \ref{subsec:sbc} and \ref{sec:dis}).
\end{enumerate}
Generally speaking, a small slope would require high precision of the line flux measurements (i.e., small spectrophotometric errors and high S/N spectra) in order to obtain the time delay of $\hbox{Mg\,{\sc ii}}$ with respect to the ionizing continuum. One can also use the slope to constrain the physical parameters of the BLR. For instance, the optical thin gas would imply a negative response (see, e.g., \citealt{GoadEtAl1993}; \citealt{OBrienEtAl1994}). From this point, our results provide a useful diagnostic of physical conditions in the BLR for the sources with different luminosities.
\acknowledgments
We would like to thank the anonymous referee for constructive comments that led to an improved presentation. We are grateful to Chenwei Yang for giving friendly advice on our quasar sample selection and to Yue Shen for providing the UV and optical $\hbox{Fe\,{\sc ii}}$ templates used in this work. We want to thank Chenwei Yang, Yue Shen, Qingfeng Zhu, Xiaobo Dong, Tuo Ji, Jianhui Lian and Junxian Wang for their useful help. This work was supported by the National Basic Research Program of China (Grant No. 2015CB857005), the National Natural Science Foundation of China (Grant Nos. 11233002 and 11421303), and the NSFC-CAS Joint Fund (Grant No. U1431229). M.Y.S. acknowledges support from the China Postdoctoral Science Foundation (Grant No. 2016M600485) and the National Natural Science Foundation of China (Grant No. 11603022).
Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is {\tt\string http://www.sdss.org/}. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is {\tt\string http://www.sdss3.org/}.
SDSS-III is managed by the Astrophysical Research
Consortium for the Participating Institutions of the
SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven
National Laboratory, Carnegie Mellon University, University of Florida, the French Participation Group,
the German Participation Group, Harvard University,
the Instituto de Astrofisica de Canarias, the Michigan
State/Notre Dame/JINA Participation Group, Johns
Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University.
\newpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,777 |
New Pew 22-Nation Global Attitudes Survey Out
Steve Clemons - June 17, 2010
12 Comments Print Email Share
I just received the press announcement from Bruce Stokes and Andrew Kohut at the Pew Research Center on the just released Pew Global Attitudes Survey.
Fascinating stuff — and disturbing.
49% of Nigerians have a "favorable" view of al Qaeda, and a majority of Pakistanis "favor" an Iran nuclear weapons program.
The report also shows that President Obama is more favorable abroad than in the United States.
From the release:
The new survey by the Pew Research Center's Global Attitudes Project, conducted among more than 24,000 people in 22 nations April 7 to May 8, provides an in-depth look at attitudes toward major powers and world leaders; the global economic situation from free trade to financial regulation; Islamic extremism; and international problems such as climate change. Key findings include:
Budrus Is a Must-See
Pray for Oakley
12 comments on "New Pew 22-Nation Global Attitudes Survey Out"
Chuck Stevens says: June 22, 2010 at 3:42 pm
June 22, 2010 (LPAC)– Serious consideration for the design and tailoring of a peaceful nuclear explosive to seal the BP well must now be a highest level priority. On the basis of information available in the public domain, such preparation is mandatory. Testimony from the leading U.S. expert on peaceful nuclear explosions as to the efficacy of using a nuclear device to seal the BP well has now been made public. Evaluations of the probable compromised condition of the well bore and seafloor come from reliable professional sources, which can be checked. BP's presentation of the situation must neither be believed nor tolerated.
The political problem is that we have a President who is not in the real world. The very existence of the United States is endangered by the President's determination not to offend the British Empire, Wall Street, or both. But we can't let that stop us from saving the United States from a horrible fate. We can't wait two elections to save the United States from an incompetent President.
The prospect of massive flow of oil into the Atlantic within as early as 18 days, according to a projection by the National Center for Atmospheric Research, will make this a global disaster. There might be debatable features of such estimates, but lying by BP and its apologists is so severe we cannot base policy on such vast and portentous cover-ups. At the point this massive oil leak enters the Atlantic, it is a point of no return for North and possibly South America, and will rapidly move on to become a European and a global crisis.
This has become a major national security question, the only one more dangerous being the President himself.
For more details see
http://www.21stcenturysciencetech.com/Articles_2010/BP_nuclear-option.pdf
http://www.21stcenturysciencetech.com/Articles_2010/Nordyke_Interview.pdf
Chuck Stevens
David says: June 20, 2010 at 12:00 pm
Fascinating numbers.
PissedOffAmerican says: June 18, 2010 at 12:22 pm
Obama's Muslim outreach faltering
By Scott Wilson
It's been a year since President Obama addressed the Islamic world in Cairo, his most visible gesture to a people, culture and religion he believes the Bush administration antagonized unnecessarily.
But, according to a new poll, the Muslim world has not exactly grasped his extended hand. In fact, Muslims have lost confidence in Obama over the past year, the poll found.
And in Egypt, the Arab bellwether nation where Obama chose to deliver his speech, the U.S. standing today is worse than during the final year of the Bush administration.
Among Muslim publics "the modest levels of confidence and approval observed in 2009 have slipped markedly," write the authors of the 22-nation Pew Global Attitudes Survey released Thursday morning.
The poll, conducted in April and early May, found that public opinion of the United States rose in three of the seven predominantly Muslim nations surveyed.
But in two of those —Turkey and Pakistan — the rise remained within the margin of error and edged up from the lowest levels found in any of the 22 countries surveyed.
The poll reports that 17 percent of Turks have a favorable view of the United States, up from 14 percent last year. In Pakistan, the rating rose from 16 to 17 percent. (Nigeria grew slightly more adoring of the United States with 81 percent expressing a positive view.)
U.S. favorability ratings fell in Lebanon, Jordan, and even Indonesia — a place where Obama lived as a child, but has now twice postponed a scheduled visit. In Egypt, only 17 percent of the public approve of the United States, a 10-percentage point drop from last year and five points below the 2008 mark.
Muslim feelings toward Obama personally also have dropped sharply. Those expressing confidence in Obama dropped 10-percentage points in Turkey, Egypt, and Lebanon. Single-digit drops were reported in Jordan, Pakistan, Indonesia, and Nigeria.
"With regard to Afghanistan, Iraq and Iran, the polling found as many countries approving as disapproving of his handling of these issues," the authors write. "However, the American president gets his worst ratings for dealing with another world problem for which the U.S. is often criticized: the Israeli-Palestinian conflict."
In only three of the 22 nations surveyed (including the United States) did a majority approve of how Obama has handled the issue — the nations of France, Nigeria and Kenya.
And that was before the Gaza flotilla crisis.
http://voices.washingtonpost.com/44/2010/06/obamas-muslim-outreach-falteri.html
So, damned near the entire planet opposes how these groveling pieces of shit in DC are handling the Isr/Pal "conflict". So this enhances our security how??? Alienating the global community serves our best interests how???
Tell me, someone. What dividends are paid to the American people by supporting and condoning Israeli actions and policies that the rest of the world rightfully recognizes as being criminal and inhumane?
And how is the effort to stamp out anti-semitism served by supporting and condoning Israel's actions and policies, that the rest of the world rightfully recognizes as being criminal and inhumane?
And in considering these two questions, how can we ignore the fact that Israel proclaims itself to be a "Jewish State"?
http://www.foreignpolicy.com/articles/2010/04/22/a_state_for_all_its_citizens
A State for All Its Citizens
The United States should not be fooled by Israel's claim that it can be both Jewish and democratic.
BY NADIM N. ROUHANA | APRIL 22, 2010
In the conflict studies courses I teach, I expose my students to theories that claim state-sanctioned inequality is a source of perpetual conflict. I know this to be true not only from my academic research, but from personal experience: I also run a small research institution in the northern Israeli city of Haifa that focuses on the status of the Palestinian citizens in Israel and their relationship with the state. This population, with the silent complicity of the United States, has long been the target of official state policies of discrimination.
In spite of America's professed commitment to equality, the U.S. government makes an exception when it comes to Israel's insistence on being recognized as a Jewish state, which in theory and practice means privileging Jewish citizens over all other citizens. U.S. President Barack Obama declared his support at the United Nations last September for "two states living side by side in peace and security — a Jewish state of Israel, with true security for all Israelis, and a viable, independent Palestinian state." Similarly, Vice President Joe Biden told an audience at Tel Aviv University in March that negotiations should lead to "a Jewish state with secure and recognized borders." It appears that affirmation of Israel's identity as a "Jewish state" is becoming a routine part of U.S. discourse on the Israeli-Palestinian peace process.
But it would be politically and morally wrong for the United States to support recognition of Israel as a Jewish state. Israel's Palestinian minority makes up between 16 to 20 percent of the population, depending on whether the Palestinians in East Jerusalem are counted — a larger percentage than the African-American population in the United States. The total percentage of non-Jews in Israel — Muslims, Christians, and others — reaches approximately 25 percent. To recognize Israel as a Jewish state excludes this sizable minority from full and equal participation in Israel's political and civic life. This is a recipe for enduring social strife and conflict.
There are few honest observers in Israel who dispute that a Jewish state, by definition, privileges one group of citizens over another. This inequality is expressed in various ways, including in Israel's Basic Laws and its laws of land control, immigration, and resource distribution. The modern Israeli state belongs only to its Jewish citizens — and even to non-citizen Jews in the diaspora — but not to its Palestinian citizens. As a result, a sizable minority of Israel's citizens have no state to call their own. Israel's Basic Laws stipulate that "a candidates list shall not participate in elections to the Knesset … if the goals or actions of the list … expressly or by implication" negates Israel as a Jewish state. Thus a party that explicitly requires Israel to become a state for all its citizens and not a Jewish state runs the risk of disqualification.
Is this really what Obama wants? Has he contemplated the built-in inequality that accompanies a "Jewish state"?
The U.S. government's ironclad commitment to Israel's security is the result of international politics, on which there can be differing views. However, supporting Israel's continued privileging of one group of citizens over another on the basis of national identity or religious affiliation is neither morally defensible nor harmonious with America's founding principles. The concept of a "Jewish state" is not equivalent to the still-objectionable term "Christian state" used by some groups in the United States. Rather, it is akin, in the eyes of Israel's non-Jewish citizens, to the concept of a "white state" — a notion that is completely unthinkable in the West.
The United States has previously overlooked Israel's settlement policy for reasons related to its national interests and domestic political considerations. Now Israel is confronting the grave consequences of these policies: Difficult political choices over West Bank settlements have precipitated increasingly sharp divisions within Israeli society. Similarly, the diplomatic support the United States lends to Israel's ambition to be recognized as a "Jewish state" does not serve either country's long-term interests. Israel's welfare is best ensured by a system that guarantees real equality for all its citizens and national groups, rather than state-sanctioned ethnic discrimination.
Ron Stripe says: June 18, 2010 at 12:18 pm
I just saw this moments ago.
Gulf oil full of methane, adding new concerns
http://news.yahoo.com/s/ap/us_gulf_oil_spill;_ylt=Ao9IH5BvU4goKm0ZQOsngTqs0NUE;_ylu=X3oDMTNoOTB0azQ3BGFzc2V0A2FwLzIwMTAwNjE4L3VzX2d1bGZfb2lsX3NwaWxsBGNjb2RlA21vc3Rwb3B1bGFyBGNwb3MDMgRwb3MDNwRwdANob21lX2Nva2UEc2VjA3luX3RvcF9zdG9yeQRzbGsDZ3VsZm9pbGZ1bGxv
Blue Mooner says: June 18, 2010 at 8:33 am
on Environment —
Apparently these 24,00 people surveyed April 7th – May 8th weren't aware of a global disaster unfolding in the Gulf. Too early to ask their opinion if president Obama should declare the Gulf a natural disaster and have a low-grade nuclear explosion deep in the water (one that wouldn't even measure radiation)as a solution to a grave environmental situation that could spread across the ocean to northern Europe. The president looks quite incompetent in formulating a Gulf solution.
The deep-water nukes have been used elsewhere in the world with apparently very good success. Maybe Pew should re-do their environment survey before the "Dead Sea" will be taking on a new meaning and location.
"Proposed Solution(s) to Gulf BP Oil Spill"
May 31, 2010 by Dr Stephen A Rinehart
Email stevecin AT bellsouth.net
Background: The following comments are based in part on my 45+ years experience in structural dynamics/mechanical engineering from Georgia Tech and extensive design/project management experience including offshore oil platforms, oil pipelines, conventional/nuclear DOD weapons effects and combat weapons designs, and environmental fate and transport of chemical/oil plumes.
The following is the proposed technical and governmental approached(s) to immediately addressing the Gulf Oil Spill since BP cannot stop the well with a
Mr.Murder says: June 17, 2010 at 11:40 pm
*nondiscretionary
http://latentrecordings.com/cowboyjunkies/renmin-promo-1/
Meanwhile, a free download of the Cowboy Junkies newest music includes some wonderful talk of their visit to China.
"In late 2008, my family and I were given an opportunity to spend three months in China. We were boarded at an elementary/middle school in the small town of Jingjiang, situated on the Yangtze River about two hours from Shanghai. My wife taught English at the school, my three young kids attended a few classes and I spent my days exploring. We also did as much travelling as my wife
Mr.Murder says: June 17, 2010 at 4:02 pm
The oil cleanup would fall under dnoniscretionary(emergency) spending and thus should not count to the nominal budget. That removes the GOPers who are suppsedly worried about a balanced budget as an excuse to try and further stall any recovery.
Had Obama not raised minimum wage the results would be even worse at this time. He needs to implement a minimum wage raise for affected Gulf states as a way to spur increased development in scale economic measures.
The guard depolyment for Gulf States can be attached to the unemployment extension the GOP is fighting tooth and nail. Suddenly hotels are full of emergency deployments and the economy carries at an expected pace for the region. To curb the affect on our economy in terms of commodities we can introduce some normalization measures with Cuba alongside other trade measures to try and sustain a portion of the Gulf states character and unique cuisines.
The solution for BP is to become the new Standard Oil, grow into seven babies(each of those sprouting tentacles in the case of major markets such as North America) and become Exxon Lite. This allows them to carve a niche in which to dump liabilities in the form BP's Gulf account.
Wait, BP had a tax loophole for offshore platforms outside the USA waters? Who could imagine we'd have a Law of the Sea article requiring a designation for countries to sue the nearest available sovereign.
The map is changing, Obama has a popular opinion outside his own country. He could almost be called another Jimmy Carter, except for the fact Carter had a great record on the environment.
Obama's version of Rove is so slimy, that becoming drenched in oil, from advising him to carry the Bushco. water on energy policy, would be considered an upgrade. Oil and water don't mix after all.
PissedOffAmerican says: June 17, 2010 at 2:08 pm
"China on the Rise: A growing portion of global publics sees China, rather than the U.S, as the world's leading economic power"
And if you disagree they'll poison your baby's food with Melamine. Keep it up, and they'll get REALLY nasty and rot your plumbing with toxic and corrosive sheet rock. And if they still haven't fucked ya up sufficiently, they'll make damned sure your cabinets are exuding carcinogens. Amazing what you can do with a workforce of kids makin' ten bucks a week.
But to give credit where credit's due, at least the Chinese have the good sense to stop their tracked machines when a protestor stands in front of it. Too bad the only democracy in the Middle East doesn't quite get the wisdom in that.
"Islamic Extremism: There is no predominantly Muslim nation polled in which a majority of Muslims endorse suicide bombing, al Qaeda or Osama bin Laden"
Interesting. Yet the bulk of Israelis seemingly endorse torture, illegal detainment, incarceration of children, collective punishment, murder, international assasinations, land theft, targeting American citizens, and bubbling the skin off of women and children with white phosphorous weaponry rained on non-combatants.
"The Arizona Effect: U.S. favorability in Mexico has tumbled in the wake of Arizona's new immigration law – from 62% in polling conducted before the law's enactment to 44% afterward"
Excellent. Perhaps now they'll stop coming here in droves through the back door, and attempt to enter the country legally instead. But I doubt it.
JohnH says: June 17, 2010 at 12:48 pm
Link to disintegration of green movement:
http://www.atimes.com/atimes/Middle_East/LF16Ak01.html
Interesting analysis of Iranian attitudes:
"The disintegration of the Green movement and decline in political fortunes of its leaders has been widely attributed to a government crackdown. While government suppression has occurred, it is not the main factor.
To begin with, the green movement ran a dishonest presidential campaign. Its candidate, Mousavi, ran for president but refused to submit to the will of the majority when it became clear that he had lost the election. This has led many observers to believe that his presidential campaign was more akin to a coup attempt – or, more accurately, coup-lite, versus traditional military coups – than a bona fide election campaign.
This explains why Mousavi declared victory even before the polls were closed. It also explains why he claimed that the election was stolen the moment he learned that he had lost at the ballot box." | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,348 |
Maltę na Letnich Igrzyskach Olimpijskich 1928 reprezentowało dziewięciu zawodników (sami mężczyźni) w jednej dyscyplinie. Był to pierwszy występ reprezentacji Malty na letnich igrzyskach.
Piłka wodna
W turnieju piłki wodnej reprezentacja Malty zajęła 5 miejsce.
Bibliografia
Państwa uczestniczące w Letnich Igrzyskach Olimpijskich 1928
1928
Malta w XX wieku | {
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} | 4,147 |
\section{\label{IntroductionSection}Introduction}
The problem of particle motion in backgrounds of nontrivial geometry is
usually addressed by using some form of the Mathisson-Papapetrou method
\cite{Mathisson1937, Papapetrou1951}. One starts with the covariant
conservation law of the stress-energy and spin tensors of matter fields, and
analyzes it under the assumption that matter is highly localized. In the
lowest, single-pole approximation, the moving matter is viewed as a point
particle. In the pole-dipole approximation, its non-zero size is taken into
account.
The results found in literature can be summarized as follows. Spinless
particles in the single-pole approximation obey the geodesic equation. In
the pole-dipole approximation, the rotational angular momentum of the
localized matter couples to spacetime curvature, and produces geodesic
deviations \cite{Mathisson1937, Papapetrou1951, Tulczyjew1959, Taub1964,
Dixon}. If the particles have spin, the curvature couples to
the total angular momentum, and the torsion to the spin alone
\cite{Trautman1972, Hehl1976a, Yasskin1980, Nomura1991, Nomura1992}.
What we are interested in is a consistent single-pole analysis of spinning
particles in spacetimes with curvature and torsion. This is motivated by the
observation that single-pole approximation eliminates the influence of
particle thickness, and allows the derivation of the pure spin-curvature
coupling. In fact, this is the only way to see the influence of curvature on
the spin part of the total angular momentum. The ambiguous algebraic
decomposition of the total angular momentum into spin and orbital
contributions are of no help. What we need is a truly zero-size object. As
it turns out, the existing literature on the subject does not have this
sort of prediction.
The results that we have obtained are summarized as follows. Trajectories of
spinning zero-size massive particles generally deviate from the geodesic
lines. The deviation is due to the spin-curvature and spin-torsion
couplings. These turn out to be different from what has been believed so
far. In particular, the spin of the Dirac point particle does not couple to
the curvature. If it is viewed as a wave packet solution of the Dirac
equation, it does not couple to the torsion either. In fact, the wave packet
spin and orbital angular momentum disappear simultaneously in the zero-size
limit. We can say that Dirac point particles behave as spinless objects.
The layout of the paper is as follows. In section \ref{SinglePoleSection},
we define the conservation law of the stress-energy and spin tensors, and
introduce the necessary geometric notions. The algebraic part of the
conservation equations is solved in terms of the independent variables---the
spin tensor and the symmetric part of the stress-energy tensor. After the
brief recapitulation of the covariant multipole formalism, we define the
single-pole approximation for the independent variables, only. Section
\ref{EOMSection} is devoted to the derivation of the particle world line
equations. The actual derivation is only sketched, as the method has already
been analyzed in detail in \cite{Vasilic2007}. The resulting equations of
motion are compared to the pole-dipole equations found in literature
\cite{Yasskin1980, Nomura1991}. As it turns out, they coincide up to a
constraint that fixes the form of the spin tensor. This constraint is a
consequence of our single-pole approximation, and has striking consequences
on the dynamics of the Dirac particle. In section \ref{DiracSection}, we
discuss the important case of totally antisymmetric spin tensor, and obtain
a surprising result that spin of the Dirac particle does not couple to the
background curvature. To check the consistency of our single-pole
approximation, the wave packet solution of the free Dirac equation is
analyzed. It is demonstrated that the wave packet spin and orbital angular
momentum disappear simultaneously in the zero-size limit. In section
\ref{ConclusionSection}, we give our final remarks.
Conventions in this paper are the following. Greek indices from the middle
of the alphabet, $\mu,\nu,\dots$, are the spacetime indices, and run over
$0,1,2,3$. The indices from the beginning of the Greek alphabet,
$\alpha,\beta,\dots$, take values $1,2,3$. The spacetime coordinates
are denoted by $x^{\mu}$, the generic metric is denoted by $g_{\mu\nu}(x)$,
and $\eta_{\mu\nu}$ stands for the Minkowski metric. The signature
convention is $(-+++)$.
\section{\label{SinglePoleSection}The single-pole approximation}
We begin with the covariant conservation of the fundamental matter currents
--- stress-energy tensor $\tau^{\mu}{}_{\nu}$, and spin tensor
$\sigma^{\lambda}{}_{\mu\nu}$:
\begin{subequations} \label{ZakoniOdrzanja}
\begin{equation} \label{ZakonOdrzanjaTau}
\left( D_{\nu} + {\cal T}^{\lambda}{}_{\nu\lambda} \right) \tau^{\nu}{}_{\mu} =
\tau^{\nu}{}_{\rho} {\cal T}^{\rho}{}_{\mu\nu} + \frac{1}{2}
\sigma^{\nu\rho\sigma}{\cal R}_{\rho\sigma\mu\nu},
\end{equation}
\begin{equation} \label{ZakonOdrzanjaSigma}
\left( D_{\nu} + {\cal T}^{\lambda}{}_{\nu\lambda}
\right)\sigma^{\nu}{}_{\rho\sigma} = \tau_{\rho\sigma} - \tau_{\sigma\rho}.
\end{equation}
\end{subequations}
Here, $D_{\nu}$ is the covariant derivative with the nonsymmetric connection
${\mathit{\Gamma}}^{\lambda}{}_{\mu\nu}$, which acts on a vector $v^{\mu}$ according
to the rule $D_{\nu} v^{\mu} \equiv \partial_{\nu}v^{\mu} +
{\mathit{\Gamma}}^{\mu}{}_{\lambda\nu}v^{\lambda}$. The torsion
${\cal T}^{\lambda}{}_{\mu\nu}$, and curvature ${\cal R}^{\mu}{}_{\nu\rho\sigma}$ are
defined in the standard way:
$$
{\cal T}^{\lambda}{}_{\mu\nu} \equiv {\mathit{\Gamma}}^{\lambda}{}_{\nu\mu} -
{\mathit{\Gamma}}^{\lambda}{}_{\mu\nu}, \qquad {\cal R}^{\mu}{}_{\nu\rho\sigma} \equiv
\partial_{\rho} {\mathit{\Gamma}}^{\mu}{}_{\nu\sigma} - \partial_{\sigma}
{\mathit{\Gamma}}^{\mu}{}_{\nu\rho} + {\mathit{\Gamma}}^{\mu}{}_{\lambda\rho}
{\mathit{\Gamma}}^{\lambda}{}_{\nu\sigma} - {\mathit{\Gamma}}^{\mu}{}_{\lambda\sigma}
{\mathit{\Gamma}}^{\lambda}{}_{\nu\rho}.
$$
The derivative $D_{\lambda}$ is assumed to satisfy the metricity condition,
$D_{\lambda}g_{\mu\nu}=0$. As a consequence, the connection
${\mathit{\Gamma}}^{\lambda}{}_{\mu\nu}$ is split into the Levi-Civita connection
$\cfl{\lambda}{\mu\nu}$, and the contorsion $K^{\lambda}{}_{\mu\nu}$:
$$
{\mathit{\Gamma}}^{\lambda}{}_{\mu\nu} = \cfl{\lambda}{\mu\nu} +
K^{\lambda}{}_{\mu\nu}, \qquad K^{\lambda}{}_{\mu\nu} \equiv -\frac{1}{2}
\left( {\cal T}^{\lambda}{}_{\mu\nu} - {\cal T}_{\nu}{}^{\lambda}{}_{\mu} +
{\cal T}_{\mu\nu}{}^{\lambda} \right).
$$
We shall also introduce the Riemannian covariant derivative $\nabla_{\mu}
\equiv D_{\mu}({\mathit{\Gamma}}\to \{ \} )$, and the Riemannian curvature tensor
$R^{\mu}{}_{\nu\rho\sigma} \equiv {\cal R}^{\mu}{}_{\nu\rho\sigma}({\mathit{\Gamma}} \to
\{ \} )$. The relation connecting the two curvature tensors reads:
$$
{\cal R}^{\mu}{}_{\nu\lambda\rho} = R^{\mu}{}_{\nu\lambda\rho} + 2
\nabla_{[\lambda} K^{\mu}{}_{\nu\rho]} + 2 K^{\mu}{}_{\sigma[\lambda}
K^{\sigma}{}_{\nu\rho]},
$$
where the indices in square brackets are antisymmetrized.
Given the system of conservation equations (\ref{ZakoniOdrzanja}), one finds that
the second one has no dynamical content. Indeed, the antisymmetric part of
stress-energy tensor is completely determined by the spin tensor. One can use
(\ref{ZakonOdrzanjaSigma}) to eliminate $\tau^{[\mu\nu]}$ from the equation
(\ref{ZakonOdrzanjaTau}), and thus obtain the conservation equation, in which
only $\tau^{(\mu\nu)}$ and $\sigma^{\lambda\mu\nu}$ components appear. The
resulting equation reads:
\begin{equation} \label{GlavniZakonOdrzanja}
\nabla_{\nu}\left( \tau^{(\mu\nu)} + \frac{1}{2} K_{\lambda\rho}{}^{\mu}
\sigma^{\nu\lambda\rho} - K^{[\mu}{}_{\lambda\rho} \sigma^{\rho\lambda\nu]}
- \nabla_{\rho} \sigma^{(\mu\nu)\rho} \right) = \frac{1}{2}
\sigma_{\nu\rho\lambda}\nabla^{\mu} K^{\rho\lambda\nu}.
\end{equation}
This will be the starting point of the derivation of the particle world line
equations.
Let us now introduce the multipole formalism, which is necessary for the
derivation. It has been shown in Refs. \cite{Vasilic2006, Vasilic2007} that an
exponentially decreasing function can be expanded into a series of
$\delta$-function derivatives. For example, a scalar $V(x)$, well localized
around the line ${\cal M}$, can be written in a manifestly covariant way as
\begin{equation} \label{DeltaSeries}
V(x)=\int_{{\cal M}}\! d\tau \!\left[ M(\tau)\frac{\delta^{(4)}(x-z)}{\sqrt{-g}} -
\nabla_{\rho}\left(M^{\rho}(\tau)\frac{\delta^{(4)}(x-z)}{\sqrt{-g}}\right) +
\cdots \right].
\end{equation}
Here, ${\cal M}$ is a timelike line $x^{\mu} = z^{\mu}(\tau)$ parametrized by the
proper distance, $d\tau^2=g_{\mu\nu}dz^{\mu}dz^{\nu}$, and the coefficients
$M(\tau)$, $M^{\rho}(\tau)$, ... are spacetime tensors called multipole
coefficients. It has been shown in Ref. \cite{Vasilic2007} that one may
truncate the series in a covariant way in order to approximate the
description of matter. Truncation after the leading term is called
\emph{single-pole} approximation, truncation after the second term is called
\emph{pole-dipole} approximation. The physical interpretation of these
approximations is the following. In the single-pole approximation, one
assumes that the particle has no thickness, which means that matter is
localized in a point. All higher approximations, including pole-dipole,
allow for the nonzero thickness, and thus, for the nontrivial internal motion.
Apart from being covariant with respect to diffeomorphisms, the series
(\ref{DeltaSeries}) possesses two extra gauge symmetries. The first is a
consequence of the fact that that there are redundant coefficients in this
decomposition. Indeed, only three out of four $\delta$-functions in each
term of the multipole expansion (\ref{DeltaSeries}) are effective in
modeling particle trajectory in $4$-dimensional spacetime. The extra
$\delta$-function and the extra integration are introduced only to
covariantize the expressions. The derivatives parallel to the world line are
integrated out, as they should, considering the fact that matter is not
localized in time. As a consequence, the parallel components of the
multipole coefficients $M^{\rho}$, $M^{\rho\lambda}$, ... effectively
disappear. It has been shown in Ref. \cite{Vasilic2007} that the
corresponding gauge symmetry, named \emph{extra symmetry}~1, in the
pole-dipole approximation reads:
$$
\delta_1 M = \nabla \epsilon \,, \qquad
\delta_1 M^{\rho} = u^{\rho}\epsilon \,.
$$
Here, $u^{\mu}\equiv dz^{\mu}/d\tau$ is the particle $4$-velocity, $\epsilon
(\tau)$ is a gauge parameter, and $\nabla$ stands for the Riemannian covariant
derivative along the particle trajectory (\,$\nabla v^{\mu}=dv^{\mu}/d\tau
+\cfl{\mu}{\lambda\rho}v^{\lambda}u^{\rho}$\,). We see that the parallel
component of $M^{\rho}$ transforms as $\delta_1
\left(M^{\rho}u_{\rho}\right)=-\epsilon\,$, and can be gauged away. In fact, one
can show that the parallel components of the higher multipoles are also pure
gauge. In the gauge fixed multipole expansion, the only derivatives that appear
are those orthogonal to the world line.
The second extra symmetry stems from the fact that the choice of the line
$x^{\mu}=z^{\mu}(\tau)$ in the expansion (\ref{DeltaSeries}) is arbitrary.
If we use another line, let us say $x^{\mu} = z'^{\mu}(\tau)$, the
coefficients $M$, $M^{\rho}$, ... will change to $M'$, $M'^{\rho}$, ...
while leaving the scalar function $V(x)$ invariant. The transformation law
of the $M$-coefficients, generated by the replacement $z^{\mu}\to z'^{\mu}$,
defines the gauge symmetry that we call \emph{extra symmetry}~2.
The extra symmetry 2 is an exact symmetry of the full expansion
(\ref{DeltaSeries}), but only approximate symmetry of the truncated series.
In the pole-dipole approximation, it has the form
$$
\delta_2 z^{\mu} = \epsilon^{\mu}\,,\qquad
\delta_2 M = M u_{\rho}\nabla\epsilon^{\rho}\,,\qquad
\delta_2 M^{\rho} = -M \epsilon^{\rho}\,,
$$
provided the $M$-coefficients are subject to the hierarchy
$$
M = {\cal O}_0\,, \quad
M^{\rho} = {\cal O}_1\,, \quad
M^{\rho\lambda} = {\cal O}_2 \,,\ \dots \,,
$$
and the free parameters $\epsilon^{\mu}(\tau)$ satisfy $\epsilon^{\mu} =
{\cal O}_1$. Here, ${\cal O}_n$ stands for the order of smallness, and the condition
$\epsilon^{\mu} = {\cal O}_1$ ensures that the order of truncation is not
violated by the action of the symmetry transformations \cite{Vasilic2007}.
In the pole-dipole and higher approximations, fixing the gauge of extra
symmetry 2 defines the particle centre of mass. In the single-pole
approximation, the extra symmetry 2 is trivial.
Now, we shall replace the general function $V(x)$ with the stress-energy and
spin tensors of the localized matter. In order to describe a strict point
particle, we choose $\tau^{(\mu\nu)}$ and $\sigma^{\lambda\mu\nu}$ in the
form
\begin{subequations} \label{SinglePoleAproksimacija}
\begin{equation} \label{DeltaRazvojZaTau}
\tau^{(\mu\nu)} = \int_{{\cal M}} d\tau\, b^{\mu\nu}(\tau)
\frac{\delta^{(D)}(x-z)}{\sqrt{-g}},
\end{equation}
\begin{equation} \label{DeltaRazvojZaSigma}
\sigma^{\lambda\mu\nu} = \int_{{\cal M}} d\tau\, c^{\lambda\mu\nu}(\tau)
\frac{\delta^{(D)}(x-z)}{\sqrt{-g}}\,,
\end{equation}
\end{subequations}
where $b^{\mu\nu}(\tau)$ and $c^{\lambda\mu\nu}(\tau)$ are the corresponding
multipole coefficients. We emphasize here that this is not how single-pole
approximation is defined in the existing literature \cite{Yasskin1980,
Nomura1991}. There, the antisymmetric part of stress-energy tensor
$\tau^{\mu\nu}$ has also been treated in the single-pole manner. As
$\tau^{[\mu\nu]}$ is not an independent variable, this imposed unnecessary
constraints on $\sigma^{\lambda\mu\nu}$. In particular, the spin of the
Dirac particle was ruled out. To overcome this problem, the authors of Ref.
\cite{Nomura1991} abandoned single-pole in favour of pole-dipole
approximation. Their subsequent limit of vanishing orbital angular momentum
should have brought them back to the single-pole regime. In what follows,
however, we shall demonstrate that it is not quite so, and that such a limit
is not equivalent to the single-pole approximation as defined in
(\ref{SinglePoleAproksimacija}).
\section{\label{EOMSection}Equations of motion}
The particle equations of motion are derived in the following way. We insert
(\ref{SinglePoleAproksimacija}) into (\ref{GlavniZakonOdrzanja}), and solve
for the unknown variables $z(\tau)$, $b^{\mu\nu}(\tau)$ and
$c^{\lambda\mu\nu}(\tau)$. The algorithm for solving this type of equation
is discussed in detail in \cite{Vasilic2006, Vasilic2007}, and here we only
sketch it. The first step is to multiply the equation
(\ref{GlavniZakonOdrzanja}) with an arbitrary spacetime function
$f_{\mu}(x)$, and integrate over the spacetime. The resulting equation
depends on the function $f_{\mu}$ and its first and second covariant
derivatives, evaluated on the line $x^{\mu}=z^{\mu}(\tau)$:
$$
\int d\tau \Big[c^{(\mu\nu)\rho}f_{\mu ;\nu\rho} +
\Big(b^{\mu\nu} - K^{[\mu}{}_{\lambda\rho} c^{\rho\lambda\nu]} +
\frac{1}{2}K_{\lambda\rho}{}^{\mu}c^{\nu\lambda\rho}\Big)f_{\mu ;\nu} +
\frac{1}{2}c_{\nu\rho\lambda}\Big(\nabla^{\mu}K^{\rho\lambda\nu}\Big)
f_{\mu}\Big]=0\,,
$$
where $f_{\mu;\nu}\equiv (\nabla_{\nu}f_{\mu})_{x=z}$, $f_{\mu;\nu\rho} \equiv
(\nabla_{\rho}\nabla_{\nu} f_{\mu})_{x=z}$. Owing to the arbitrariness of the
function $f_{\mu}(x)$, the terms proportional to its independent derivatives
separately vanish. To find the independent derivatives of the test function
$f_{\mu}$, we make use of the particle $4$-velocity $u^{\mu}\equiv
dz^{\mu}/d\tau$, and the Riemannian covariant derivative along the particle
trajectory $\nabla$. The $4$-velocity $u^{\mu}$ is normalized as
$u^{\mu}u_{\mu}=-1$, and the action of $\nabla$ on a vector field $v^{\mu}(\tau)$
is defined by $\nabla v^{\mu}\equiv dv^{\mu}/d\tau
+\cfl{\mu}{\lambda\rho}v^{\lambda}u^{\rho}$. Next, we decompose the derivatives
of the vector field $f_{\mu}(x)$ into components orthogonal and parallel to the
world line $x^{\mu}=z^{\mu}(\tau)$:
\begin{subequations} \label{jna21}
\begin{equation} \label{jna21a}
f_{\mu;\lambda} = f^{{\scriptscriptstyle\perp}}_{\mu\lambda} - u_{\lambda} \nabla f_{\mu} \, ,
\end{equation}
\begin{equation} \label{jna21b}
f_{\mu;(\lambda\rho)} = f^{{\scriptscriptstyle\perp}}_{\mu\lambda\rho} - 2
h^{{\scriptscriptstyle\perp}}_{\mu(\lambda} u_{\rho)} + h_{\mu } u_{\lambda} u_{\rho} \, ,
\end{equation}
\begin{equation} \label{jna21c}
f_{\mu;[\lambda\rho]} = \frac{1}{2} {R^{\sigma}}_{\mu\lambda\rho}
f_{\sigma}\, .
\end{equation}
\end{subequations}
Here, the orthogonal and parallel components are obtained by using the projectors
\begin{equation} \label{jna22}
{P_{\orto}}\vphantom{P} ^{\mu}_{\nu} = \delta^{\mu}_{\nu} + u^{\mu}u_{\nu}, \qquad
{P_{\para}}\vphantom{P} ^{\mu}_{\nu} = - u^{\mu} u_{\nu} .
\end{equation}
More precisely, $f^{{\scriptscriptstyle\perp}}_{\mu\lambda} = {P_{\orto}}\vphantom{P} ^{\sigma}_{\lambda}
f_{\mu;\sigma}$, $f^{{\scriptscriptstyle\perp}}_{\mu\lambda\rho} = {P_{\orto}}\vphantom{P} ^{\sigma}_{\lambda}
{P_{\orto}}\vphantom{P} ^{\nu}_{\rho} f_{\mu;(\sigma\nu)}$, $h^{{\scriptscriptstyle\perp}}_{\mu\lambda} =
{P_{\orto}}\vphantom{P} ^{\sigma}_{\lambda} u^{\nu} f_{\mu;(\sigma\nu)}$ and $h_{\mu} =
u^{\sigma} u^{\nu} f_{\mu;(\sigma\nu)}$. Direct calculation yields
\begin{equation} \label{jna23}
\begin{array}{lcl}
h_{\mu} & = & \nabla \nabla f_{\mu} - (\nabla u^{\nu}) f^{{\scriptscriptstyle\perp}}_{\mu\nu}
\, , \\ h^{{\scriptscriptstyle\perp}}_{\mu\rho} & = & \displaystyle {P_{\orto}}\vphantom{P} ^{\nu}_{\rho} \nabla
f^{{\scriptscriptstyle\perp}}_{\mu\nu} - (\nabla u_{\rho}) \nabla f_{\mu} + \frac{1}{2}
{P_{\orto}}\vphantom{P} ^{\lambda}_{\rho} u^{\nu} {R^{\sigma}}_{\mu\nu\lambda} f_{\sigma} \, , \\
\end{array}
\end{equation}
which tells us that the only independent derivatives on the line
$x^{\mu}=z^{\mu}(\tau)$ are $f_{\mu}$, $f^{{\scriptscriptstyle\perp}}_{\mu\nu}$ and
$f^{{\scriptscriptstyle\perp}}_{\mu\nu\rho}$. We can now use (\ref{jna21}) and (\ref{jna23}) to
group the coefficients in terms proportional to the independent
derivatives of $f_{\mu}$. The obtained equation has the following general
structure:
$$
\int_{{\cal M}}d\tau \Big[ X^{\mu\nu\rho}f^{\perp}_{\mu\nu\rho} +
X^{\mu\nu}f^{\perp}_{\mu\nu} + X^{\mu}f_{\mu} \Big] =0,
$$
where $X^{\mu\nu\rho}$, $X^{\mu\nu}$ and $X^{\mu}$ are composed of various
combinations of multipole coefficients $b^{\mu\nu}$ and $c^{\lambda\mu\nu}$,
external fields $K^{\lambda}{}_{\mu\nu}$ and $R^{\mu}{}_{\nu\rho\sigma}$, and
their derivatives. In all the expressions, the external fields are evaluated on
the world line $x^{\mu}=z^{\mu}(\tau)$. Owing to the fact that $f_{\mu}$,
$f^{{\scriptscriptstyle\perp}}_{\mu\nu}$ and $f^{{\scriptscriptstyle\perp}}_{\mu\nu\rho}$ are independent functions on
the world line, we deduce that the $X$-terms must separately vanish.
The equation $X^{\mu\nu\rho}=0$ has a simple algebraic form
$$
{P_{\orto}}\vphantom{P} ^{\mu}_{\lambda} {P_{\orto}}\vphantom{P} ^{\nu}_{\sigma} c^{(\lambda\sigma)\rho} = 0\,,
$$
which is easily solved for $c^{\lambda\mu\nu}$. This yields
$$
c^{\lambda\mu\nu} = 2 u^{\lambda}s^{\mu\nu} + s^{\lambda\mu\nu}\,,
$$
where $s^{\mu\nu} \equiv - s^{\nu\mu}$ and $s^{\lambda\mu\nu} \equiv -
s^{\lambda\nu\mu} \equiv s^{\nu\lambda\mu}$ are totally antisymmetric, but
othervise free parameters. The equations $X^{\mu\nu}=0$ and $X^{\mu}=0$ are much
more complicated. The procedure goes as follows. First, we use the above
decomposition of $c^{\lambda\mu\nu}$ to perform a similar split of the
$b^{\mu\nu}$ coefficients. A new free parameter $m(\tau)$ appears to characterize
the leading term of $b^{\mu\nu}$. Then, the equations $X^{\mu\nu}=0$ and
$X^{\mu}=0$ are rewritten in terms of the undetermined parameters $m$,
$s^{\mu\nu}$ and $s^{\lambda\mu\nu}$, and properly rearranged. Skipping the
details of the diagonalization procedure, which has thoroughly been demonstrated
in Ref. \cite{Vasilic2007}, we display the final result:
\begin{itemize}
\item the world line equation
\begin{subequations} \label{Rezultati}
\begin{equation} \label{PrvaJnaKretanja}
\nabla \Big[ mu^{\mu} + 2u_{\rho} \left( \nabla s^{\mu\rho} + D^{\mu\rho}
\right) \Big] - u^{\nu} s^{\rho\sigma} R^{\mu}{}_{\nu\rho\sigma} =
\frac{1}{2}c_{\nu\rho\lambda} \nabla^{\mu}K^{\rho\lambda\nu},
\end{equation}
\item the spin precession equation
\begin{equation} \label{DrugaJnaKretanja}
{P_{\orto}}\vphantom{P} ^{\mu}_{\rho} {P_{\orto}}\vphantom{P} ^{\nu}_{\sigma}\left( \nabla s^{\rho\sigma} +
D^{\rho\sigma} \right) =0,
\end{equation}
\item the stress-energy coefficients
\begin{equation} \label{KomponenteTEI}
b^{\mu\nu} = m u^{\mu}u^{\nu} + 2u_{\lambda}u^{(\mu}\nabla s^{\nu)\lambda} -
\frac{1}{2}K_{\lambda\rho}{}^{(\mu} c^{\nu)\lambda\rho},
\end{equation}
\item the spin tensor coefficients
\begin{equation} \label{SveKomponenteC}
c^{\lambda\mu\nu} = 2 u^{\lambda}s^{\mu\nu} + s^{\lambda\mu\nu}.
\end{equation}
\end{subequations}
\end{itemize}
In these equations, the scalar $m(\tau)$, and the totally antisymmetric
tensors $s^{\mu\nu}(\tau)$ and $s^{\lambda\mu\nu}(\tau)$ are free
parameters. They determine the stress-energy and spin tensors via
(\ref{KomponenteTEI}) and (\ref{SveKomponenteC}).
The shorthand notation
$$
D^{\mu\nu} \equiv K^{[\mu}{}_{\lambda\rho} c^{\rho\lambda\nu]} +
\frac{1}{2}K_{\lambda\rho}{}^{[\mu} c^{\nu]\rho\lambda}
$$
is introduced to simplify the cumbersome expressions.
The obtained single-pole equations differ from the known pole-dipole result
\cite{Trautman1972, Hehl1976a, Yasskin1980, Nomura1991, Nomura1992} by the
presence of the constraint (\ref{SveKomponenteC}). It is a consequence of
our assumption that the particle has no thickness, and therefore no orbital
degrees of freedom. In the existing literature, an analogous but more
restrictive constraint appears in this regime \cite{Hehl1976a, Nomura1991}.
This is because the antisymmetric part of the stress-energy tensor
$\tau^{[\mu\nu]}$ has been treated as an independent variable, in spite of
the restriction (\ref{ZakonOdrzanjaSigma}). In our approach, the only
independent variables are $\sigma^{\lambda\mu\nu}$ and $\tau^{(\mu\nu)}$,
and the resulting constraint (\ref{SveKomponenteC}) is not so strong. In
particular, it does not rule out the free Dirac field, or any other massive
elementary field. Indeed, the formula $c^{\lambda\mu\nu} =
u^{\lambda}s^{\mu\nu} + \frac{1}{s}u^{[\mu}s^{\nu]\lambda}$ for the spin
tensor of the elementary particle of spin $s$ (see Ref. \cite{Nomura1992})
is a special case of (\ref{SveKomponenteC}).
In what follows, we shall examine the special case of spin $1/2$ pointlike
matter. Surprisingly, we shall discover that spin $1/2$ does not couple to
the curvature, leading to geodesic trajectories in torsionless spacetimes.
\section{\label{DiracSection}The Dirac particle}
The basic property of Dirac matter is the total antisymmetry of its spin
tensor $\sigma^{\lambda\mu\nu}$. As a consequence, the coefficients
$c^{\lambda\mu\nu}$ are also totally antisymmetric, and the constraint
(\ref{SveKomponenteC}) implies
\begin{equation}
s^{\mu\nu}=0.
\end{equation}
The vanishing of the $s^{\mu\nu}$ component of the spin tensor has far
reaching consequences. First, we see that the spin-curvature and spin-orbit
couplings disappear from the world line equation (\ref{PrvaJnaKretanja}).
Second, the spin precession equation (\ref{DrugaJnaKretanja}) becomes a
constraint equation. If we define the spin vector $s^{\mu}$ by
$s^{\mu\nu\rho} \equiv e^{\mu\nu\rho\lambda}s_{\lambda}$, and the axial
component of the contorsion $K^{\mu}$ as $K^{\mu} \equiv
e^{\mu\nu\rho\lambda} K_{\nu\rho\lambda}$, where $e^{\mu\nu\rho\sigma}$ is
the covariant totally antisymmetric Levi-Civita tensor, the equations
(\ref{Rezultati}) become
\begin{subequations} \label{jna6}
\begin{equation} \label{jna6a}
\nabla\left( mu^{\mu} + K^{[\mu}s^{\nu]} u_{\nu} \right) + \frac{1}{2}
s^{\nu} \nabla^{\mu}K_{\nu} =0 ,
\end{equation}
\begin{equation} \label{jna6b}
K_{\perp}^{[\mu}s_{\perp}^{\nu]} = 0.
\end{equation}
\end{subequations}
As we can see, the spin couples only to the axial component of the
contorsion, which means that \emph{Dirac point particles follow geodesic
trajectories in torsionless spacetimes}. At the same time, the absence of
torsion trivializes the equation (\ref{jna6b}), and no information on the
behavior of the spin vector is available. If the background torsion has
nontrivial axial component, a geodesic deviation appears, but also a very
strong constraint on the spin vector. Indeed, the equation (\ref{jna6b})
implies that the orthogonal component of $s^{\mu}$ always orients itself
along the background direction $K_{\perp}^{\mu}$. This unusual behavior
suggests that the spin vector of the Dirac point particle might be zero,
after all. In fact, the world line equations (\ref{Rezultati}) are derived
under very general assumptions of the existence of pointlike solutions in an
arbitrary field theory. They do not care about peculiarities of specific
theories or specific types of localized solutions. In what follows, we shall
analyze the wave packet solutions of the flat space Dirac equation, with the
idea to check if they can be viewed as point particles.
Let us construct an example. We start with the free Dirac Lagrangian
$$
{\cal L} = \frac{i}{2} \left[ \bar{\psi}\gamma^{\mu}\partial_{\mu} \psi - \left(
\partial_{\mu} \bar{\psi} \gamma^{\mu} \psi \right) \right] - m\bar{\psi}\psi\,,
$$
where Dirac $\gamma$-matrices satisfy the usual anticommutation relations
$\{ \gamma^{\mu}, \gamma^{\nu} \} = -2\eta^{\mu\nu}$, and are used in their
conventional representation ($\gamma_5 \equiv i
\gamma^0\gamma^1\gamma^2\gamma^3$). Then, we construct a wave packet. The
wave packet is a solution which is well localized in space, but resembles a
plane wave inside. To be viewed as a particle, its size $\ell$ is considered
in the limit $\ell\to 0$. At the same time, the particle stability is
achieved in the limit $\lambda / \ell \to 0$, where $\lambda$ is its
wavelength. We construct it as follows. At the initial moment $x^0=0$, we
choose the configuration
$$
\psi(\vect{r},0) \equiv Ae^{-\frac{r^2}{\ell^2}}\,\psi_p(\vect{r},0)\,,
$$
where
$$
\psi_p(x) \equiv \sqrt{\frac{k^0+m}{2m}}
\left[
\begin{array}{c}
1 \\ 0 \\ \frac{k^3}{k^0+m} \\ 0 \\
\end{array}
\right]
e^{ik_{\mu}x^{\mu}}
$$
is the plane-wave solution of the Dirac equation $\left(
i\gamma^{\mu}\partial_{\mu} -m \right) \psi =0$. It propagates along the
$x^3$-axis ($k^1=k^2=0$, $k^0 \equiv \sqrt{m^2 + (k^3)^2}$), and is
polarized upwards, for convenience. The exponential function $\exp
(-r^2/\ell^2)$ cuts out a small piece of the plane wave, and defines its
size $\ell$, while $A$ is the overall amplitude of the packet. The
wavelength $\lambda$ is proportional to $1/|\vect{k}|$. Using the Dirac
equation, we can calculate time derivatives and thereby determine time
evolution of this packet. In fact, we only need first time derivatives, as
neither $\tau^{\mu\nu}$ nor $\sigma^{\lambda\mu\nu}$ depend on higher
derivatives:
\begin{equation} \label{jna7}
\tau_{\mu\nu} = i \left[ \bar{\psi} \gamma_{\mu} \partial_{\nu}\psi - \left(
\partial_{\nu} \bar{\psi} \right) \gamma_{\mu}\psi \right]
-2\eta_{\mu\nu}{\cal L}\,,\qquad \sigma^{\lambda\mu\nu} = \varepsilon^{\lambda\mu\nu\rho}
\bar{\psi} \gamma_5 \gamma_{\rho} \psi\,.
\end{equation}
The wave packet expressions of these currents at $x^0=0$ are obtained
straightforwardly:
\begin{subequations} \label{RnjeTalasnogPaketa}
\begin{equation}\label{TauNulaNulaItauTriTri}
\tau^{(00)} = -2 |A|^2 e^{-\frac{2r^2}{\ell^2}} \frac{(k^0)^2}{m} , \qquad
\tau^{(33)} = -2 |A|^2 e^{-\frac{2r^2}{\ell^2}} \frac{(k^3)^2}{m} ,
\end{equation}
\begin{equation}\label{TauNulaAlfa}
\tau^{(0\alpha)} = -2 |A|^2 e^{-\frac{2r^2}{\ell^2}} \frac{k^0}{m} \left(
k^3\eta^{3\alpha} - \frac{x_{\beta}}{\ell^2} \varepsilon^{\alpha\beta 3} \right) ,
\end{equation}\label{Sigma123}
\begin{equation}
\sigma^{123} = - |A|^2 e^{-\frac{2r^2}{\ell^2}} \frac{k^3}{m}, \qquad
\sigma^{012} = - |A|^2 e^{-\frac{2r^2}{\ell^2}} \frac{k^0}{m},
\end{equation}
\end{subequations}
where only non-vanishing components are displayed. Now, we want to rewrite
the currents (\ref{RnjeTalasnogPaketa}) as a series of $\delta$-function
derivatives. We first fix diffeomorphisms by imposing the condition
$g_{\mu\nu}=\eta_{\mu\nu}$, and extra symmetry 1 by keeping only spatial
components of the M coefficients ($M^0=M^{0\rho}=\cdots =0$). The
decomposition formula (\ref{DeltaSeries}) is thereby reduced to
$$
V(x)=\int_{{\cal M}}\! d\tau \!\left[ M \delta^{(4)}(x-z) -
\partial_{\alpha}\left(M^{\alpha}\delta^{(4)}(x-z)\right) +
\cdots \right].
$$
In general, the line $x^{\mu}=z^{\mu}(\tau)$ is arbitrary, but the simplest
expressions are obtained if it coincides with the wave packet trajectory.
Thus, we choose $z^1=z^2=0$ in accordance with the the fact that the packet
propagates along the $x^3$ axis. As for the $z^3$ component, we do not need
its full $\tau$ dependence because we are only interested in the packet
behaviour at $x^0=0$. There, the proper length $\tau$ is chosen in
accordance with $z^0(0)=z^3(0)=0$, which is sufficient for the proper
definition of the $\delta$-expansion at $x^0=0$.
The multipole coefficients are obtained by multiplying $V(x)$ by a number of
$(x^{\alpha}-z^{\alpha})$ factors, and integrating over the $3$-space. In our
example, the simple integration of (\ref{RnjeTalasnogPaketa}) yields the monopole
coefficients $b^{\mu\nu}$ and $c^{\lambda\mu\nu}$, while multiplication with
$(x^{\alpha}-z^{\alpha})$, and subsequent integration gives the dipole
coefficients $b^{\mu\nu\alpha}$. The resultant non-zero monopoles are
$$
b^{00} = a\, \ell^3 (k^0)^2 \,, \qquad
b^{33} = a\, \ell^3 (k^3)^2 \,, \qquad
b^{03} = a\, \ell^3 k^0 k^3 \,,
$$
$$
c^{123} = \frac{a}{2}\, \ell^3 k^3\,, \qquad
c^{012} = \frac{a}{2}\, \ell^3 k^0\,,
$$
while there are only two non-zero dipoles,
$$
b^{012} = - b^{021} = \frac{a}{4}\, \ell^3 k^0 \,.
$$
The higher multipoles are of the order $\ell^5(k^3)^2$ or higher. Here, $a(k)$ is
the overall factor whose explicit form is not needed in the subsequent
discussion.
Now, we shall consider the single-pole limit $\ell\to 0$, while respecting the
wave packet stability condition $\lambda\ll\ell$. First, we choose the overall
amplitude $a(k)$ in the form
$$
a(k) \sim \frac{\lambda^2}{\ell^3}\,,
$$
thereby normalizing the monopole coefficients to be of the order of unity.
Then, using the single-pole behavior $k^3 \sim k^0 \sim \lambda^{-1}$, we find
$$
b^{00}\sim b^{33} \sim b^{03} \sim 1\,, \qquad
c^{123}\sim c^{012} \sim \lambda \,, \qquad
b^{012} \sim \lambda \,.
$$
The higher multipoles are of the order $\ell^2$, and thus, neglected. This
is a realization of the pole-dipole approximation. In the single-pole
regime, however, only the lowest terms are retained in the limit $\ell\to
0$. This means that, respecting $\lambda\ll\ell$, the terms proportional to
$\lambda$ must also be dropped. As a result, the {\it spin monopole coefficients
$c^{\lambda\mu\nu}$ vanish simultaneously with the orbital dipole
coefficients $b^{\mu\nu\lambda}$}.
The reason for this unusual behaviour is found in the constraint
$c^{012}=2\,b^{012}$. It is obtained by the integration of the more general
relation
\begin{equation}\label{SpinOrbita}
x^{\alpha}\tau^{(0\beta)} - x^{\beta}\tau^{(0\alpha)} =
\sigma^{0\alpha\beta} + div
\end{equation}
that is found to constrain the wave packet currents
(\ref{RnjeTalasnogPaketa}). It relates the wave packet spin to its orbital
angular momentum, so that the expected disappearance of orbital degrees of
freedom in the limit $\ell\to 0$ is followed by the unexpected disappearance
of the spin itself.
To summarize, we see that the spin vector $s^{\mu}$ vanishes in the
single-pole approximation, and the particle trajectory becomes a geodesic
line even in the presence of torsion. The validity of this conclusion,
however, demands some sort of equivalence principle to hold. This is because
the considered wave packet is a solution of the free Dirac equation, and the
inclusion of curvature or torsion may destroy it. What we can do is to
consider weak gravity, so that terms quadratic in curvature and torsion are
neglected. In that case, the free wave packets are a good approximation to
the exact solution, which implies that {\it Dirac point particles behave as
spinless objects in an external gravitational field}. They can still probe
the spacetime curvature, but for the probe of the background torsion, one
needs a thick particle.
\section{\label{ConclusionSection}Concluding remarks}
In this paper, we have considered the motion of point particles with nonzero
spin in spacetimes with curvature and torsion. Using the covariant multipole
formalism developed in Ref. \cite{Vasilic2007}, the world line equations are
derived in the lowest, single-pole approximation. This way, the particle
thickness, and the corresponding internal motion, have been eliminated. Only
mass and spin remained to characterize the particle internal structure.
In our approach, the single-pole behaviour has been adopted for truly
independent variables only. In particular, the antisymmetric part of the
stress-energy tensor has been eliminated from the conservation equations,
prior to imposing the single-pole regime. As a consequence, our single-pole
analysis turned out to differ from the existing literature.
The obtained equations of motion are found to differ from the known
pole-dipole equations by the presence of a novel constraint on the particle
spin tensor. With this constraint, the spin of the Dirac point particle
turned out not to couple to the background curvature, leading to geodesic
trajectories in torsionless spacetimes. In the presence of torsion, however,
a geodesic deviation appears, but also a strong constraint, suggesting that
the spin of the Dirac point particle might be zero. Being a consequence of
our single-pole approximation, this unusual result has been checked by the
explicit construction of the zero-size Dirac particle. To this purpose, a
wave packet solution of the Dirac equation is considered in the limit of
small size and wavelength. The expected single-pole behaviour has been
verified, but also, the spin tensor has been found to disappear in this
limit. In an attempt to explain this unusual behaviour, a relation between
spin and orbital angular momentum has been discovered to hold in our wave
packet example.
Before we close our exposition, let us comment on the possibility that the
disappearance of spin in the zero-size limit might be a general property of
all point particles. First, we notice that there is one more conserved
current in the Dirac theory---the $U(1)$ current $j^{\mu} \equiv
\bar{\psi}\gamma^{\mu}\psi$. In our wave packet example, it is proportional
to the wave vector $k^{\mu}$, and is related to the stress-energy
$\tau^{\mu\nu}$. By a close inspection, we find that the following
manifestly covariant relation holds:
\begin{equation} \label{TauJot}
\tau^{\mu\nu}j_{\nu} \propto j^{\mu},
\end{equation}
Its physical meaning is best seen in the rest frame where it reduces to
$\tau^{\alpha 0}=j^{\alpha}=0$. It tells us that the two currents are
mutually proportional, i.e. that the energy and charge flow in the same
direction. In the limit $\lambda / \ell \to 0$, $\ell\to 0$ it implies the
constraint (\ref{SpinOrbita}), and thus, explains why Dirac point particles
have no spin.
The relations analogous to (\ref{TauJot}) might exist quite generally. There
is nothing special about the statement that all the particle charges flow in
the same direction. This is something one would expect to hold for any type
of point-like matter. However, we need the relation (\ref{TauJot}) to hold
for thick particles, as well. Only then, and only for Dirac matter, the
consequence (\ref{SpinOrbita}) has been derived. If (\ref{TauJot}) were a
general property of the localized matter, the disappearance of spin in the
zero-size limit might be a feature of all massive point particles. Indeed,
when applied to other spins, the relation (\ref{TauJot}) implies
$$
x^{\alpha}\tau^{(0\beta)} - x^{\beta}\tau^{(0\alpha)} =
\sigma^{[\beta0\alpha]} + x^{[\alpha} \partial_0 \sigma^{00\beta]} + div \,.
$$
After the integration, the divergence term vanishes, while l.h.s. and the
second term on the r.h.s. give dipole coefficients of the order ${\cal O}_1$. In
the single-pole regime, both disappear, and we end up with the vanishing
spin.
| {
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Mark Christopher is a conservative/libertarian talk radio host. His show titled The Mark Christopher Show initially broadcast on 99.7 WWTN and then, 1510 WLAC both in Nashville, Tennessee where he was named the Associated Press talk show of the year for 3 straight years, and the #1 talk show in Nashville in 2006.
Since that time The Mark Christopher Show has aired on KTRS St. Louis and Knews Radio in Palm Springs.
References
External links
Official website
Year of birth missing (living people)
Living people
American libertarians
American talk radio hosts | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,422 |
Radio Caravana es una estación radial ecuatoriano en el 750 MHz del dial AM en Guayaquil. Inició sus transmisiones el 21 de julio de 1985 en la ciudad de Guayaquil, como un medio de comunicación enfocado solamente en los deportes de la ciudad y del país siendo pionero en la radiodifusión ecuatorianana, aunque con el tiempo se fue ampliando a partes de sección de noticias. Curiosamente, el primer programa se dio con un partido del Clásico del Astillero. El objetivo de la empresa se cumplió totalmente, pero además poco a poco la radio fue incorporando otros contenidos.
Actualmente, es una de las radios deportivas más importantes de Ecuador con gran cobertura a nivel de ciudades costeñas y serranas. También transmite vía Internet en el resto del país y en todo el mundo.
Su programación consta de información y actualidad noticiosa mayormente música y deportes, transmitiendo una gran cantidad de eventos deportivos, como partidos de la Selección Ecuatoriana de Fútbol, el Campeonato Nacional de Fútbol, las clasificatorias sudamericanas para el Mundial de Fútbol, la actuación de equipos guayaquileños en torneos internacionales como la Copa Libertadores y la Copa Sudamericana, entre otras competiciones deportivas.
Historia
Mario Canessa fundó Radio Caravana. Actualmente Grupo Caravana engloba a Radio Diblu y Caravana Televisión. Antes de funcionar en el norte, Caravana se inició en El Fórum y luego en P. Ycaza y Córdova. Actualmente, los estudios de radios Caravana y Diblu, funcionan en un moderno edificio ubicado en la intersección de la avenida Juan Tanca Marengo y la avenida 26 NO, en pleno Barrio Urdenor 2 de Guayaquil.
Referencias
https://www.eluniverso.com/deportes/2020/07/18/nota/7910190/patricio-cornejo-proximo-entrenador-ecuador-debe-ahorrar-tiempo/== Referencia de Radio Caravana.
https://www.elcomercio.com/deportes/futbol/emelec-prepara-accion-para-tener-publico-en-su-estadio.html Noticia de Radio Caravana sobre un club ecuatoriano.
https://www.extra.ec/noticia/farandula/andres-pellacini-radio-caravana-despedido-56524.html Polémica en Radio Caravana.
Enlaces externos
Sitio web oficial
Instagram oficial
Twitter Oficial
Caravana
Emisoras de radio de noticias
Emisoras de radio de Ecuador
Emisoras de radio fundadas en 1985 | {
"redpajama_set_name": "RedPajamaWikipedia"
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administrators, parents who homeschool using Montessori methods and Montessori advocates.
advancement in Birmingham, in Alabama, or in the southeast.
For more information, please use contact information below. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,963 |
Ed Sheeran debuts two new songs at intimate UK gig
Dan Martensen
Ed Sheeran played an intimate show in England Wednesday night and debuted two brand-new songs from his upcoming album, = [Equals].
At the Coventry gig in celebration of British music retailer HMV's 100th anniversary, he delivered the first live performances of the tracks "First Times" and "Overpass Graffiti."
In video captured of the "First Times" performance, Ed contrasts big moments in his career with special little moments in his personal life.
"I thought I'd feel different playing Wembley/80,000 singing with me/It's what I've been chasing, 'cause this is the dream," he sings.
In "Overpass Grafitti," he talks about a long-lasting love.
"I will always love you for what it's worth/We'll never fade like graffiti on the overpass/I know time may change the way you think of us/But I remember the way we were/You were the first full-stop love that will never leave/ Baby you'll never be lost on me," Ed sings.
Also included in Ed's 19-song set were his latest single "Bad Habits" and the song "Visiting Hours," which he released last week, as well as previous hits "Shape of You," "Thinking Out Loud" and "Castle on the Hill."
= [Equals] comes out October 29.
In Brief: 'Succession' takes all categories in Directors Guild noms for drama; 'Downton Abbey' movie moves, and more
Amy Schneider's historic 'Jeopardy!' run ends after 40 wins | {
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Ultralight And Compact. The collapsible backpacking chair coming in at just 2 lbs, and packs down to 13.8"(L)×4.5"(W) ×3.9"(H) in the included stuff sack. The lightweight chair is easy to carry and store, you can take it in your backpack or just leave it in your RV/cars. Great gifts for him/her..
High Quality & Durable :Framed by 7075 high-strength aeronautical alloy and high-quality waterproof nylon and mesh material make this heavy duty camping chair sturdy. Non-slip rubber covers on the feet ensure a firm stand. It can hold up to 330 lb.
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"redpajama_set_name": "RedPajamaC4"
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{"url":"https:\/\/cs.stackexchange.com\/questions\/85089\/precomputation-for-partial-products-in-monoids","text":"# Precomputation for partial products in monoids\n\nSuppose I have a sequence of elements $a_1,\\ldots,a_n$ in a group and I want to compute, for varying $1\\leq i\\leq j\\leq n$, the product $b_{i,j}:=\\Pi_{i\\leq k\\leq j}a_k$. Then I can do a precomputation of all $b_{1,j}$ and later determine each needed $b_{i,j}$ as $(b_{1,i-1})^{-1}b_{1,j}$.\n\nThe cost is $O(n)$ group operations and $O(n)$ additional time for the precomputation, and $O(1)$ operations and $O(1)$ additional time for each access. Clearly optimal. So far so good.\n\nNow what if I don't have inverses (that is I have a monoid instead of a group)? The best I can come up with is a binary-tree like precomputed data structure. The cost of access would increase to $O(\\log n)$. Can that be improved?\n\n\u2022 Interesting question! 1. Do you want a generic algorithm that works for any monoid (using only the monoid operations, but without knowing anything about the structure of the monoid)? Or do you have a particular monoid you are interested in? In the latter case it might be possible to do better than a generic algorithm. 2. There are probably uninteresting algorithms that spend a lot more during the precomputation in exchange for speeding up the access time a bit. I'm guessing those aren't of much interest to you, but do tell us if you are interested in those directions. \u2013\u00a0D.W. Dec 8 '17 at 2:57\n\u2022 The question is mostly from curiosity, so in principle I am interested in improvements along all kinds of directions. Originally, it came from matrix multiplication. \u2013\u00a0kne Dec 8 '17 at 19:15\n\nAs you say, with $O(n)$ precomputation you can arrange that each access takes $O(\\log n)$ time. You build a binary tree over the elements, and annotate each internal node with the product of the leaves under it. This takes $O(n)$ precomputation and $O(n)$ storage.\n\nIf you are willing to do more precomputation, you can reduce the access time. In particular, with $O(n \\log n)$ precomputation and $O(n \\log n)$ storage, you can arrange that each access runs in $O(1)$ time. The idea is outlined below.\n\nNotationally, we will let $b_{i,j}$ denote the product $a_i a_{i+1} \\cdots a_j$, as you define in your question.\n\nFirst, precompute the products $b_{i,n\/2-1}$ for $i=1,2,\\dots,n\/2-1$ and the products $b_{n\/2,j}$ for $j=n\/2,\\dots,n-1,n$. This can be done with a $O(n)$ precomputation. This will let you compute any product $b_{i,j}$ in $O(1)$ time if we have $i < n\/2 \\le n\/2$, i.e., if the range $[i,j]$ spans the midpoint.\n\nWe still need a way to handle ranges that don't span the midpoint. We'll handle that recursively. Basically, take the sequence $a_1,\\dots,a_{n\/2-1}$ and recursively build a data structure for it (e.g., find its midpoint $a_{n\/4}$, etc.). Also, recursively build a data structure for $a_{n\/2},\\dots,a_n$. This will let us compute all products $b_{i,j}$ where $j<n\/2$ or $i\\ge n\/2$, i.e., where the range $[i,j]$ doesn't span the midpoint. This covers all the cases.\n\nHow much time does the precomputation take? If $T(n)$ denotes the time for the entire precomputation, it satisfies the recurrence\n\n$$T(n) = 2 T(n\/2) + O(n),$$\n\nwhich solves to $T(n) = O(n \\log n)$. Similarly, we can see that the amount of storage needed is also $O(n \\log n)$. Finally, this data structure will let you compute any product $b_{i,j}$ in $O(1)$ time.\n\n(I'll let you figure out how to compute $b_{i,j}$ from this data structure in $O(1)$ time. It can be done with some clever bit-shifting tricks. It may help to think about the data structure in terms of the big-endian binary representation of the indices. For each index $i$, we compute $b_{i,i'}$ where $i,i'$ share a common prefix and then $i'$ is all ones after the common prefix; and we compute $b_{j',j}$ where $j',j$ share a common prefix and then $j'$ is all zeros after the common prefix. Given $i,j$, we can find the longest common prefix of $i,j$, then express as the product $b_{i,j} = b_{i,i'} b_{j',j}$, look up the values of $b_{i,i'}$ and $b_{j',j}$, and compute the product. All of these can be done in $O(1)$ time by placing the precomputed values in an array in the correct order.)\n\n\u2022 Thank you for your answer. I must admit, though, that I do not see how to achieve the lookup in $O(1)$. As far as I can tell I would have to compute $\\lfloor\\log_2(i\\oplus j)\\rfloor$, where $\\oplus$ is XOR. But for the logarithm (number of significant digits) I need time $O(\\log\\log n)$. It would still be an improvement, of course. \u2013\u00a0kne Dec 11 '17 at 17:24\n\u2022 @kne, I think when I wrote this answer I was thinking we can take advantage of bit-twiddling tricks like x & (x-1) but I don't have time right now to reconstruct what I was thinking to check whether that's right or not. It's possible I might have been confused. Some architectures have en.wikipedia.org\/wiki\/Find_first_set which makes this easy. \u2013\u00a0D.W. Dec 11 '17 at 18:44\n\u2022 Ah, OK, I was not aware of any RAM model that has (an operation equivalent to) $\\lfloor\\log_2\\rfloor$. \u2013\u00a0kne Dec 11 '17 at 21:35","date":"2020-01-27 10:49:00","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\": 0.82216477394104, \"perplexity\": 231.57983500492816}, \"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-2020-05\/segments\/1579251696046.73\/warc\/CC-MAIN-20200127081933-20200127111933-00185.warc.gz\"}"} | null | null |
Q: Why is CDF of binomial random variable step function I just read that the cumulative distribution function for a binomial random variable is a "step function where the function is flat and then jumps at each nonnegative integer value".
Can someone explain to me why the CDF is a step function with horizontal lines between each integer intervals and not simply points with probabilities at each integer?
Thanks.
A: The CDF is defined the probability that your random variable takes a value less than or equal to a real number:
$$ F\colon\mathbb{R}\to\mathbb{R}, x\mapsto F(x):=P(X\leq x). $$
And of course if $X$ only has probability mass on the natural numbers, the probability that $X$ is (e.g.) less than or equal to $3$ is the same as the probability that it is less than or equal to $3.3$ or $3.8$. Which is precisely the same as saying that the CDF is flat between $3$ (inclusive) and $4$ (exclusive) and only jumps at the natural numbers.
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# OUT OF TUNE
All New Tales of Horror and Dark Fantasy
Edited By
Jonathan Maberry
JournalStone
San Francisco
Copyright © 2014 by Jonathan Maberry
All rights reserved. No part of this book may be used or reproduced by any means, graphic, electronic, or mechanical, including photocopying, recording, taping or by any information storage retrieval system without the written permission of the publisher except in the case of brief quotations embodied in critical articles and reviews.
This is a work of fiction. All of the characters, names, incidents, organizations, and dialogue in this novel are either the products of the author's imagination or are used fictitiously.
JournalStone books may be ordered through booksellers or by contacting:
JournalStone Publishing
www.journalstone.com
The views expressed in this work are solely those of the authors and do not necessarily reflect the views of the publisher, and the publisher hereby disclaims any responsibility for them.
ISBN: 978-1-940161-69-3 (sc)
ISBN: 978-1-940161-70-9 (ebook)
ISBN: 978-1-940161-71-6 (hc)
JournalStone rev. date: November 17, 2014
Library of Congress Control Number: 2014953318
Printed in the United States of America
Cover and Interior Artwork: Robert Papp
Cover Design: Cyrusfiction Productions
Edited by: Jonathan Maberry
As always, for Sara Jo.
# INTRODUCTION
Ask most folks today what a 'ballad' is, and they begin humming power chords from groups like Journey, Whitesnake, and Meat Loaf.
But let's take a step back from hair bands and glitter rock. Let's go old school. Once upon a time, a ballad meant something more. They spoke to a different part of our soul. They tied us to our culture or gave us glimpses into the past. Sometimes they opened doors to strange places. The ballads of long ago told stories. Sometimes fanciful, often strange, constantly intriguing.
Those old ballads told of doomed loves and damned places, of murder and romance, of love lost and lives imperiled. The balladeers enchanted our imaginations with faerie folk and noble knights, with lonely witches and deeply unfortunate romantic choices, with the seen and the unseen. Some of them even told the truth. Or, a version at least.
More often the ballads conjured in our minds a place where something deliciously dreadful happened long ago. Maybe it was the very spot on which you now sit, or a land glimpsed only through a parted veil at purple twilight.
Many of the ballads are so old that no one can really claim ownership, and all provenance is suspect or apocryphal. Often these are songs and stories told and retold, changed and reshaped, with new tunes and new lyrics imposed upon the seed of a story. Scholars have confounded themselves with trying to trace the roots of Appalachian songs all the way back to Scottish glens or Irish grottos or overgrown English gardens. Some ballads are so old they seem half buried in the myths of the ancients. Others are as fresh as the rise of Jazz and Blues.
One cannot say, with any real degree of certainty, that there is even a thread that ties all ballads together. There isn't. And yet, there is. It's less a connection of form or origin, and more a feeling. An awareness that these old songs and stories evoke in each of us. Often, even at first hearing, we feel we know these songs. We've heard them somewhere before, we think; even when we likewise know we haven't. Their ghosts haunt the generations of songs that have come after them. Their dust is there. Their shadows.
OUT OF TUNE is not a collection of old ballads. No, sir. This volume contains only new stories. Prose, not rhymes. Stories, not songs. Fourteen tales spun by some of today's most talented writers. It's a witch's brew, no doubt. The stories are dissimilar in almost every way. Some are as bare as old bones, others are ripe to bursting. But they all share one thing. A thread. A ghost of a theme.
They are all inspired by old ballads. From England and Ireland, Scotland and Wales. And from America, too. Old songs, new stories. Not direct interpretations. No, those old ballads were whispers in the ears of these writers. Each writer took a thread from those timeless songs and in their own way spun new magic.
So, sit comfortably, pour yourself something nice, and dig in. And maybe—just maybe—you'll hear a spectral tune floating on the breeze as you read.
-Jonathan Maberry
# TABLE OF CONTENTS
Wendy, Darling—Christopher Golden
Sweet William's Ghost—David Liss
Black is the Color of my True Love's Hair—Del Howison
John Wayne's Dream—Gary Braunbeck
Bedlam—Gregory Frost
Awake—Jack Ketchum
John Henry, The Steel Drivin' Man—Jeff Strand
Fish Out of Water—Keith R. A. Decandido
Making Music—Kelley Armstrong
Tam Lane—Lisa Morton
John Barleycorn Must Die—Marsheila Rockwell and Jeffrey J. Mariotte
In Arkham Town, Where I was Bound—Nancy Holder
Driving Jenny Home—Seanan McGuire
Hollow is the Heart—Simon R. Green
The Contributors
Folklore Commentary following each song—by Nancy Keim Comley
# WENDY, DARLING
By
Christopher Golden
On a Friday evening at the end of May in the year Nineteen Hundred and Fifteen, Wendy spent her final night in her father's house in a fitful sleep, worried about her wedding the following day and the secrets she had kept from her intended groom.
The room had once been a nursery, but those days were long forgotten. She had stopped dreaming the dreams of her girlhood years before, such that even the echoes of those dreams had slid into the shadows in the corners of the room. Now it was a proper bedroom with a lovely canopy over the bed and a silver mirror and an enormous wardrobe that still gave off a rich mahogany scent though it had stood against the wall for six years and more.
Some nights, though...some nights the tall French windows would remain open and the curtains would billow and float. On those evenings the moonlight would pour into the room with such earnest warmth it seemed intent upon reminding her of girlhood evenings when she would stay up whispering to her brothers in the dark until all of them drifted off to sleep and dreamt impossible things.
Wendy had lived in the nursery with Michael and John for too long. She ought to have had her own room much sooner, but at first their father had not wanted to give up his study to make another bedroom and later—when he'd changed his mind—the children were no longer interested in splitting up. By then Wendy had begun to see the Lost Boys, and to dream of them, and it seemed altogether safer to stick together.
That day—the day before her wedding—there had been a low, whispery sort of fog all through the afternoon and into the evening. Several times she stirred in her sleep, uneasy as she thought of Jasper, the barrister she was to wed the following afternoon. She quite relished the idea of becoming Mrs. Jasper Gilbert, yet during the night, she felt herself haunted by the prospect. Each time her eyes flickered open, she lay for several moments, staring out at the fog until she drifted off again.
Sometime later, she woke to see not fog but moonlight. The windows were open and the curtains performed a ghostly undulation, cast in yellow light.
A dream, she thought, for it must have been. She knew it because the fog had gone. Knew it because of the moonlight and the impossibly slow dance of those curtains, and of course because the Lost Boys were there.
She lay on her side, half her face buried in the feather pillow, and gazed at them. At first she saw only three, two by the settee and one almost hidden in the billow of the curtains. The fourth had a dark cast to his features that made him seem grimmer, less ethereal than the others, though he was the youngest. She had not seen them in years, not since her parents had gotten a doctor involved, insisting the Lost Boys were figments of her imagination. She had never forgiven John and Michael for reporting her frequent visits with the Lost Boys to their parents, a grudge she had come to regret in the aftermath of Michael's death in a millinery fire in 1910. How she had loved him.
By the time of the fire, it had been years since she had seen the Lost Boys. After the fire, she had often prayed that it would be Michael who visited her in the night.
"Wendy," one of the Boys whispered now in her moonlit dream.
"Hello, boys," she said, flush beneath her covers, heart racing. She wanted to cry or scream but did not know if it was fear she felt, or merely grief.
As if grief could ever be merely.
She recognized all four of them, of course, and knew their names. But she did not allow herself to speak those names, or even to think them. It would have felt as if she welcomed them back to her dreams, and they were not welcome at all.
"You forgot us, Wendy. You promised you never would."
She nestled her cheek deeper into her pillow, feathers poking her skin through the fabric.
"I never did," she whispered, her skin dampening. Too hot beneath the covers. "You were only in my mind, you see. I haven't forgotten, but my parents and Doctor Goss told me I must persuade my eyes not to see you if you should appear again."
"Have you missed us, then?"
Wendy swallowed. A shudder went through her. She had not.
"As I'm dreaming, I suppose it's all right that I'm seeing you now."
The Lost Boys glanced at one another with a shared, humorless sort of laugh. More a sniff than a laugh, really. A disapproving sniff.
The moonlight passed right through them.
The nearest of them—he of the grim eyes—slid closer to her.
"You were meant to be our mother," he said.
Wendy couldn't breathe. She pressed herself backward, away from them. It was their eyes that ignited a terror within her, those pleading eyes. She closed her eyes.
"Wake up, Wendy," she whispered to herself. "Please wake up."
"Don't you remember?" the grim-eyed one asked, and her lids fluttered open to find herself still dreaming.
"Please remember," said another, a lithe little boy with a pouting mouth and eyes on the verge of tears.
"No," she whispered.
The hook. Soft flesh against her own. The pain. Blood in the water.
Her body trembled as images rushed to her mind and were driven back, shuttered in dark closets, buried in shallow graves.
"Stay away," she whispered. "Please. My life is all ahead of me."
She did not know if she spoke to the Lost Boys or to those images.
"My fiancé is a good man. Perhaps when we are wed, we can take one or two of you in. He is kind, you see. Not like..."
A door slammed in her mind.
"Like who, Wendy?"
Hook, she thought. My James.
"No!" she screamed, hurling back her bedcovers and leaping from the bed, hot tears springing to her eyes. "Leave me, damn you! Leave me to my life!"
Fingers curved into claws, she leaped at the nearest of them. Passing through him, chill gooseflesh rippling across her skin, she fell to the rug and curled up into herself, a mess of sobs.
In the moonlight, she lay just out of reach of the fluttering curtains and cried herself into the sweet oblivious depths of slumber.
When she woke in the early dawn, aching and chilled to the bone, she crept back beneath her bedclothes for warmth and comfort and told herself that there would never be another night when she needed to fear bad dreams. For the rest of her life, she would wake in the morning with Jasper beside her and he would hold her and kiss her until the last of sleep's shadows retreated.
The sun rose to a clear blue morning.
No trace of fog.
The world only began to feel completely real to Wendy again when the carriage drew to a halt in front of the church. Flowers had been arranged over the door and on the steps and the beauty of the moment made her breath catch in her throat. A smile spread across her lips and bubbled into laughter and she turned toward her grumpy banker of a father and saw that he was smiling as well—beaming, in fact—and his eyes were damp with love for her, and with pride.
"Never thought you'd see the day, did you, Father?" Wendy teased.
George Darling cleared his throat to compose himself. "There were times," he allowed. "But here we are, my dear. Here. We. Are."
He took a deep breath and stepped out of the carriage, itself also festooned with arrangements donated by friends of Wendy's mother who were part of the committee behind the Chelsea Flower Show. A pair of ushers emerged from the church, but Wendy's father waved them back and offered his own hand to guide her down the carriage steps.
George stepped back. He'd never been sentimental, and now he seemed to fight against whatever emotions welled within him. Amongst those she expected, Wendy saw a flicker of uneasiness.
"You look beautiful," he told her.
Wendy knew it was true. She seldom indulged in outright vanity, but on her wedding day, and in this dress...well, she would forgive herself. Cream-white satin, trimmed in simple lace, it had been one of the very first she had laid eyes upon and she had loved it straight away. Cut low at the neck, with sleeves to the elbows, it had a simple elegance reflected in the simplicity of the veil and the short train. Her father helped gather her train, spread it out behind her, and took her hand as they faced the church.
"Miss Darling," said one of the ushers, whose name she'd suddenly forgotten. She felt horrible, but suddenly it seemed that her thoughts were a jumble.
"I'm about to be married," she said, just to hear the words aloud.
"You are, my dear," George agreed. "Everyone is waiting."
The forgotten usher handed her a wreath of orange blossoms and then the other one opened the church door. Moments later, Wendy found herself escorted down the aisle by her grumpy-turned-doting father. A trumpet played and then the organ, and all faces turned toward her, so that she saw all of them and none of them at the same time. She smelled the flowers and her heart thundered, and she began to feel dizzy and swayed a bit.
"Wendy," her father whispered to her, his grip tightening on her arm. "Are you all right?"
Ahead, at the end of the aisle, the bridesmaids and ushers had spread out to either side. The vicar stood on the altar, dignified and serious. Her mother sat in the front row, her brother John stood amongst the ushers. And there was Jasper, so dapper in his morning coat, his black hair gleaming, his blue eyes smiling.
She no longer felt dizzy. Only safe and sure.
Until the little boy darted out from behind a column—the little boy with grim eyes.
"Stop this!" he shouted. "You must stop!"
Wendy staggered, a terrible pain in her belly as if she were being torn apart inside. She gasped and then covered her mouth, glancing about through the mesh of her veil, certain her friends and relations would think her mad—again.
But their eyes were not on her. Those in attendance were staring at the little boy in his ragged clothes, and when the second boy ran in from the door to the sacristy and the vicar shouted at him, furious at the intrusion, Wendy at last understood.
The vicar could see the boys.
They could all see.
"Out of here, you little scoundrels!" the vicar shouted. "I won't allow you to ruin the day—"
The grim-eyed boy stood before Jasper, who could only stare in half-amused astonishment. That sweetness was simply Jasper's nature, that indulgence where any other bridegroom would have been furious.
The third boy stepped from the shadows at the back of the altar as if he had been there all along. And of course he must have been.
"No, no, no," Wendy said, backing away, tearing her arm from her father's grip. She forced her eyes closed because they couldn't be here. Couldn't be real.
"Wendy?" her father said, and she opened her eyes to see him looking at her.
He knew. Though he had always told her they were figments and dreams, hadn't he seemed unsettled whenever she talked of them? Spirits, he'd said, do not exist, except in the minds of the mad and the guilty.
Which am I? she'd asked him then. Which am I?
Jasper clapped his hands twice, drawing all attention toward him. The unreality of the moment collapsed into tangibility and truth. Wendy breathed. Smelled the flowers. Heard the scuffling and throat-clearing of the stunned members of the wedding.
"All right, lads, you've had your fun," Jasper said. "Off with you!"
"Wendy Darling," one of the boys said, staring at Jasper, tears welling in his eyes. "Only she's not 'darling' at all. You don't know her, sir. She'll be a cruel mother. She'll abandon her children—"
"Rubbish!" shouted Wendy's father. "How dare you speak of my daughter this way!"
Wendy could only stare, not breathing as Jasper strode toward the grim-eyed boy and gripped him by his ragged shirtfront. She saw the way the filthy fabric bunched in his hands and it felt as if the curtain between dream and reality had finally been torn away.
"No," she said, starting toward Jasper...and toward the boys. "Please, don't..."
Her fiancé glanced up, thinking she had been speaking to him, but the boys looked at her as well. They knew better.
"She's had a baby once before," a pale, thin boy said, coming to stand by Jasper, his eyes pleading. "Go on. Ask her."
"Ask her what became of that child," said the grim-eyed boy.
Shaking, Wendy jerked right and left, trapped by all of the eyes that gazed upon her. Jasper frowned, staring at her, and she saw the doubt blooming in him, saw his lips beginning to form a question. Her father still glared angrily at the boys, but even he had a flicker of hesitation. In the front row, Mary Darling stepped from the pew and extended a hand toward her daughter.
"Wendy?"
Shaking her head, Wendy began to back away from those who loved her, retreating down the aisle. She tripped over her silken train and when she fell amongst the soft purity of its folds, she screamed.
"Ask her!" one of the boys shouted. Or perhaps it had been all of them.
Thrusting herself from the ground, whipping her train behind her, she ran. Her whole body felt flushed but she caught a glimpse of her left hand as she ran and it was pale as marble. Pale as death. At the back of the aisle, a few crimson rose petals had fallen, petals meant to be scattered in the path of husband and wife after the ceremony. To her they were blood from a wound.
She burst from the church, an abyss of unspoken questions gaping behind her, and she fled down the steps in fear that if she did not run, that yawing silence would drag her back. Pain stabbed her belly and her heart slammed inside her chest. Her eyes burned and yet strangely there were no tears. She felt incapable of tears.
At the foot of the steps, she tore off the train of her dress. When she glanced up, horses whinnied and chuffed. Her wedding carriage stood waiting. The driver looked at her with kind eyes and his kindness filled her with loathing.
"Wendy!"
Jasper's voice. Behind her. She dared not turn to look at him.
Racing across the street, she darted down a narrow road between a dressmaker's and a baker's shop. At a corner, she nearly collided with two more of the Lost Boys—names, you know their names—and she turned right to avoid them, racing downhill now. Another appeared from an alley to her left, but this boy was different from the others. He'd been badly burnt, skin and clothing charred, and unlike the others, he had no substance, flesh so translucent that she could see the stone face of the building behind him.
She wailed, stumbling in anguish, and fell to the street. Her dress tore and her knee bled, so that when she staggered to her feet and ran screaming—grief carving out her insides—a vivid red stain soaked into the satin and spread, the petals of a crimson rose.
"Mother," the burnt boy said behind her.
She did not look back, but glanced once at the windows of a pub as she bolted past. In the glass she saw their reflections, not only the burnt boy but the others as well, one with his head canted too far, neck broken, another beaten so badly his features were ruined.
Moments before she emerged from between two buildings, she realized where she had been going all along. Had she chosen her path or had they driven her here? Did it matter?
Wendy stared at the bank of the Thames, at the deep water rushing by, and all the strength went out of her. Numb and hollow, she shuffled to the riverbank.
Somewhere nearby, a baby cried.
Glancing to her left, she saw the bundle perhaps a dozen feet away, just at the edge of the water. The baby's wailing grew louder and more urgent and she started toward it.
She knew the pattern on its blanket. His blanket.
Kneeling on the riverbank, her bloodstained dress soaking up the damp, she reached out to pull the blanket away from the infant's face. His blue face, bloated and cold, eyes bloodshot and bulging and lifeless.
The sob tore from her chest as she reached for the child, lifted it into her arms and cradled it to her chest. Still she could not weep, but she pressed her eyes tightly closed and prayed for tears.
The bundle in her arms felt too light. Gasping for breath, she opened her eyes.
"No, please," she whispered as she unraveled the empty blanket. The empty, sodden blanket.
"Mother," a voice said, so close, and a hand touched her shoulder.
Wendy froze, breath hitching in her chest. This was not the burnt boy or the grim-eyed child from the church. This was another boy entirely.
Still on her knees, she turned back to see his face. Nine years old, now, his skin still blue, eyes still bloodshot and lifeless. Her boy.
"Peter," she whispered.
He thrust his fingers into her hair and she screamed his name—a name she had never spoken aloud before today. Wendy beat at his arms and clawed at his face as he dragged her to the water and plunged her into the river. She stared up at him through the water and his visage blurred and changed, became the face of his father, James, the butcher's boy. He'd earned his nickname with the bloodstained hook he used in handling the sides of meat in the shop down the street from the Darlings' home.
Her chest burned for air, the urgency of her need forcing her to strike harder at the face above her, which now became her own face, only nine years younger. The hands that held her beneath the water were her own, but she was no longer herself—instead she was a tiny infant, so newly born he still bore streaks of blood from his mother's womb. An infant conceived by a mother and father who were only children themselves, carried and borne in secret—a secret safeguarded by her brothers in the privacy of the room they shared, a secret which destroyed her relationship with them forever. A secret made possible by a father's neglect and a mother's denial.
Peter, she thought.
Starved for air, thoughts and vision dulling, diminishing, slipping away, Wendy opened her mouth and inhaled the river.
Blackness crept in at the corners of her eyes, shadows in her brain, and she realized she had stopping fighting him. Her arms slipped into the water and her hair pooled around her face. Bloodstained white satin floated in a cloud that enveloped and embraced her.
The hands on her now were larger. A man's hands. They dragged her from the river and for a moment she saw only darkness, a black veil for a cruel mother.
"Wendy," said an urgent voice.
She saw him then. Not the little drowned boy, but Jasper, her intended. He knelt over her, desperate and pleading and calling her name.
Gathered around him on the riverbank were the Lost Boys, those cast-aside children, each murdered by his mother. Those dead boys she had met once before on the night she had drowned her Peter in the Thames. They had been visible to the people in the church, dark dreams come to life, but now they were unseen once more. Jasper wept over her, unaware of their presence...
Wendy could only watch him, standing a short distance away. Her dress felt dry now, but the bloodstain remained.
"No," she whispered, as the darkness retreated from her thoughts and she understood what she saw.
Jasper knelt there, mourning her, grieving for the life they might have had. Wendy saw her own lifeless body from outside, her spirit as invisible to him as the Lost Boys. Others began to run toward the riverbank—her parents and her brother John, the vicar's wife and Jasper's brother, an aunt and uncle. They seemed like ghosts to her, these living people, their grief distant and dull.
The Lost Boys circled around her, dead eyes now contented.
"Mother," Peter whispered, taking her right hand.
Another boy took her left hand. She glanced down and saw the grim eyes that had so unsettled her in her dreams.
"You promised to be a mother to us all, forever," the grim-eyed boy said.
Wendy blinked and turned toward the river. Somehow she could still see the swaddled infant floating on the water, sodden blankets dragging it down, just as it had on that night nine years ago.
"Forever," said Peter.
They guided her gently into the river, where the dark current swept them all away.
On...The Cruel Mother
Child Ballad 20 (Roud #9), The Cruel Mother, is a 'murder' ballad. The heart of the ballad concerns a woman going into the forest and giving birth to two illegitimate children. She then murders them, either stabbing with a penknife or strangling them with a ribbon. She then buries them. In some versions she ties them together and buries them alive.
Later she sees two pretty children and rhapsodizes on how she would treat them most tenderly if they were hers, dressing them in fine clothing and feeding them only the best food. They answer that when they were hers she treated them very differently - though they will be forgiven she will not. Hell is her fate. In some ballads she must serve seven-year penance in hell.
In one significant variant a herdsman finds a child in the hollow of a tree. The child asks him to take it to the house where its mother is to be married that day. There the child announces that the bride is its mother. It then tells the assembled company that she has had three children. One she drowned, one she buried, and one she hid in the hollow of a tree. In what Child calls a 'Wendish' version (i.e. Sorbian), the bride/mother has had nine children.(1)
Various musicians have recorded The Cruel Mother with singers such as Joan Baez.
1) Child, Francis James. The English and Scottish Popular Ballads Vol. 1. Mineola: Dover, 2003.
# SWEET WILLIAM'S GHOST
By
David Liss
Maggie and I had been together for months, which is about as much as any relationship needs to drag on. Some guys, I know, are needy, and they're all about looking for that right woman. I don't have anything against that sensitivity of whatever, but let's just say I'm a little skeptical. I don't think there's any man alive who is happy with just one woman. Maybe he can settle, or trick himself into thinking he doesn't need anything else. More likely, I'm guessing, he's happy as shit for a while, and by the time he snaps out of it, he's got a mortgage and kids and responsibilities and all that bullshit. I saw that happen to my father, who looked at my mother like she was a piece of rotting meat. He didn't think I saw, but I did, and I vowed not to let that happen to me.
Even though I was on my guard, things went on with Maggie longer than I would have thought. It didn't hurt that she was a nice-looking woman. Pretty face with pale skin, black hair and big-wattage blue eyes. I hardly ever notice a girl's eye color, not even if I've been with her for weeks, but with Maggie, it was almost the first thing I saw. Almost, because she's got a really nice body, and she works out in the gym where I'm a trainer. A lot of girls built like Maggie would strut around in micro shorts and sports bras, letting the world get an eye-full. She could pull it off, but she dressed like a fat chick when she exercised. That's fine. I understand she wasn't looking for attention, but I'm a trained professional. I know a woman in good shape when I see one.
I'd been watching her for a while, trying to figure out my move. Guys tried to strike up conversations with her all the time, and she'd be polite, but she didn't want any part of it. She listened to music while she worked out, which made it easier to cut the chatter short. She'd pull out one ear bud when someone tried to get a conversation going, but stare at it the whole time, like she was itching to get back to the music. It made guys anxious, and they struck out.
I wasn't going to let that happen to me. When I moved in, it was going to count. I just had to wait for the right moment. One day I saw her at the squat rack, and I knew that was my in.
Not a lot of women bother with barbell squatting. It's technical and it's uncomfortable, and most people think it's for serious bodybuilders "only". Even so, there she was, at the rack, a pair of ten-pound plates on the bar, giving it a try. And doing it badly.
Your average lightweight gym member can't squat for shit, and as a trainer, I can tell you it's not worth teaching them. If a guy insists on hitting the squat rack when I'm training him, he usually cranks out half or quarter squats, and I don't tell him to do otherwise. Why bother? He doesn't want to hear he's doing it wrong. He doesn't want to hear that to do it right he's got to drop some weight and practice form. He wants to feel like he's moving a shitload of iron around, and he pays me to make him feel like he's the Hulk or something. That's the service I provide. I'm in the people business.
This woman wasn't paying me, though. I knew she was a class act. I saw the clothes she wore on her way in or out—skirt suits and shit and silk blouses and like that. She was some kind of professional. A lawyer or accountant or something. She was smart. I could tell from her eyes, from her hair, the way she carried herself. Whatever it was women like that talked about in their spare time—cinema and politics and the latest novel by some Indian lady—I wasn't equipped to handle. But squatting? That I could talk about.
"Excuse me," I said when she finished a set. "I hope you don't mind my saying this, but you're form is wrong. You'll hurt your knees."
She looked at me, like she wanted to hear what I had to say. Her eyes darted over my body in my shorts and polo shirt. She took out both ear buds, rested them in her palm. I'd passed the test, and with good reason. I am tall and I am fit. Women like to look at me, and I don't see the point of pretending otherwise. Her gaze shot up to my face, which isn't so bad either, and she tried to keep it there. She laughed self-consciously and shook her head.
"I'm not surprised," she said. "I read an article about squatting in a magazine, and I figured I'd give it a try, but it's harder than it looked."
"Harder than it looks, but easier than you think," I said. "You have to go all the way down. Thighs parallel to the ground. Can I show you?"
She let me show her. Just like that. Hands on her shoulders, then my palm pressed flat against the small of her back. She might as well have gotten naked right there, because even if she didn't know it, or pretended not to know it, I had just closed the deal.
* * *
A few minutes of my time there, a few conversations around the gym the next few times she comes in, and suddenly she hires me on for sessions twice a week.
I'm not an idiot. A woman in that kind of shape doesn't need a trainer. Unless you're a movie star or an athlete, no one in good shapes needs a trainer. This woman sure as hell had no business paying someone to toss her a medicine ball while she stood on a balance board. She didn't need for me to count off while she did some bullshit on the TRX. She hired me because she wanted to sleep with me.
Maybe she never intended to go through with it. Most people don't jump in the sack with every person they've got the hots for. I get that. Still, she was flirting—flirting with me and flirting with the idea. Maybe that's all she wanted to do. Maybe at first she thought the thrill of having a guy who looked like me put his hands on her twice a week was enough to scratch her itch.
And make no mistake, Maggie loved it when I put my hands on her. I felt her shiver and tense, I heard her let out her breath when I would adjust her form or correct her posture. Even so, she put up plenty of barriers. She mentioned her fiancé at least twice a session. William this, and William that—how kind, how smart, how sweet. It was as if to say she was spoken for, and she would not have any flirting. For all that, she asked me about my weekends. She fished for information about my social life. She touched my forearm with her fingertips during casual conversation. I know the drill.
It turned out I was right the first time. Maggie was a lawyer, though not like the kind you see on TV, talking to juries and solving crimes and shit. She was a corporate lawyer and sat in her office reading contracts all day, which sounded dull to me, and she admitted it was not the most exciting thing in the world. I was also right when I guessed she was smart. She liked movies, the more foreign and colorless the better. She liked to read books. Her fiancé, William, was some kind of high-powered engineer, specializing in environmental cleanup. He was a big deal. She was proud of him. He mattered. She loved spending time with him. She was looking forward to a life with him and starting a family with him. His awesome importance made her absolutely untouchable. No matter how much she might be attracted to another guy, no sane woman would mess up the sort of future she had with a guy like her sweet William.
Maggie and I were sleeping together within three weeks of her first session. Smart guys with smoking-hot girlfriends take note. I don't care how compatible the two of you are. I don't care how much your souls intertwine or whatever. If she's better looking than you are, she's going to step out with someone like me. Know that going in.
* * *
That's how it went for two months. William worked long hours, and he traveled all the time, so Maggie and I didn't have to work too hard to find opportunities to slip off to my apartment, which wasn't far from the gym or from the house Maggie shared with her fiancé. I had to do a little cleaning up for Maggie, throwing away pizza boxes and shoving laundry in the hamper, but whatever. She was worth it. After a while, she canceled the training sessions. They were too much of a tease for both of us. All that distraction with heavy weights—that's a recipe for an accident. In the interest of safety, we limited our interactions to me pounding the shit out of her whenever we could both slip away.
It wasn't like we were in love or anything. We were attracted. We liked to have sex with each other. That was it. She was out of my league, and I was out of hers, which made it hot. She felt like crap about what she was doing to William, while I didn't care, and that was also hot. Once a week she would tell me that we should stop, that she really loved him. I'd tell her that I would hate to let her go, but I wanted her to be happy. When I said that, I'd see something darken in her lovely face—the knowledge that I could simply walk away and not look back. I think that's what kept her around. Maggie wanted me to be desperate for her, and until I was, she wasn't going anywhere.
Then, one day, lying in my bedroom on a Saturday afternoon, Maggie looked up at me. "William's going to be out of town for three weeks," she said. "He's got an extended assignment in Glasgow. Some old factory has apparently been leaking who knows what into the soil for decades."
"Thank God for William," I said.
Maggie raised an eyebrow at me like I was being a dick. I didn't like that look, but as she wasn't wearing anything, I let it slide.
"You'll be able to come over to my house," she said. "I can cook you dinner."
"I can take or leave dinner," I told her, and that was true. I didn't want to play house with Maggie. I wanted to hook up with her a couple of times a week. Maybe more, but I wasn't interested in bringing her flowers and the two of us selecting the right wine to go with her French whatever. Maggie knew where I stood. I was taking advantage of her and helping her to shit all over her happy future, so I figured the least I could do was to be up front about it.
"Come on," she said, forcing a smile. "It'll be fun. It'll give us a chance to see."
"To see what?" I asked. But I already knew the answer.
She slapped my butt playfully. "To see how we do together."
"I think I already know," I told her, but that wasn't true. I didn't know, and I didn't want to see. I wanted her to keep sleeping with me. I didn't care if she broke up with William. That was her business. I had no interest in taking William's place.
* * *
Maggie lived in a snooty part of town where none of the neighbors knew each other, so my coming and going wasn't a problem for her. No one there gave a crap about her business. That's what she said. I saw a few housewives eye me as they pushed their baby strollers down the street, but if Maggie didn't mind, then neither did I.
Before William had been in Scotland a week, I was coming over every day, sleeping there every night. Maggie would text me when she was on her way home, and I would meet her at the house. As it turned out, she never made me dinner—it was always take out or prepared stuff from the grocery store. Then we would have to figure out what to with long hours together. It was a weird transition, at first. We had only ever flirted or had sex. Suddenly, we were a real couple, washing plates and choosing side dishes. I hadn't thought we were up to it, but it turned out I liked it. We would watch some old movie, and Maggie would talk about what this scene or that meant when the film first came out, or what the director was trying to say, and it was actually kind of interesting, stuff I hadn't thought about before. Then I would make some comment or ask some question, and Maggie would look at me like I was really smart, or at least smarter than she had given me credit for. It was like she was taking me seriously, and the truth was, I enjoyed it. I kept acting like I didn't, but I did.
One day over some salmon filets she bought she said to me, "What would you do if William confronted you?"
I shrugged. "What would you want me to do?"
"Would you hit him?"
I chewed my fish thoughtfully. "Yeah, probably. I mean, if I had to. If he got all in my face."
She looked down at her plate. I could tell she liked that. Smart women like that sort of thing because the guys they're usually with don't know how to protect themselves.
I was in her bed when the call came in. It was maybe just after midnight. I watched as she answered, as she went quiet and pale, as she put a hand to her mouth. I watched as she considered dropping the phone, or opting out by collapsing or something, but that wasn't Maggie. Maggie dealt with things. She took a long breath, and then grabbed a pencil and began to write down information. Then she hung up the phone.
"There's been an accident," she said, her voice mechanical, slow, precise. "William is dead."
But I already knew that. I knew it as soon as she put her phone to her ear, maybe as soon as it rang. I knew it, and I wasn't sure how I felt about it. Okay, maybe I was sure. I was glad.
* * *
I stayed out of her life for a few weeks. William's family decided he should be buried in Scotland, which it turns out was where he was from. I hadn't know that, but I liked it. He was buried far away. There would be no regular trips to set flowers at his grave. Perfect.
Maggie spent more than a week there, and then, when she returned, her time was taken up with friends and relatives and well-wishers. I understood that she didn't want to deal with me. Her fiancé was dead now. He had become Saint William, and all the things that had bothered her about him before—his being safe and unassertive and in mediocre shape—now seemed irrelevant. She had loved him and planned to live the rest of her life with him, and I was a reminder of how she had spurned and humiliated him, even if he hadn't known it. William was forever, but not me. A year from now, she wouldn't remember my name.
Except that a month after the funeral, she called me up. "I need to talk to you," she said. "Can you come over?"
When she answered the door, she was wearing sweat pants and an oversized t-shirt. That's woman code for "I'm not going to fuck you," by the way. I looked at her and her sad smile, and I knew that this was where I did not want to be. I thought of a million excuses I could use to get out of there now—an appointment forgotten, a friend with an emergency—but I let them slide.
Maggie poured us some white wine with a foreign name and sat across from me in her cavernous living room.
"I didn't want us to end on bad terms," she said.
"Who says we have to end at all?" I asked.
She turned away, and I could see she was crying. They were silent tears, and I could pretend not to have noticed.
"I can't be with you anymore," she told me, still not looking in my direction. "After what we did to William."
"We didn't do anything to him," I told her. "He never knew. He never had any feelings about us because we kept us a secret."
She turned her head toward me, hard and sharp. "I was going to marry him," she said. "Don't think I wasn't. If he showed up here, right now, I would marry him."
"Well, that's nice," I said as kindly as I could manage, "but he's not going to show up." I was starting to feel like I didn't even want to bother to sleep with her that night. It would be all sad and awkward afterwards, and it seemed like it was way more trouble than it was worth. The smart play was to agree with her, tell her I understood, give her a brotherly hug and get the fuck out of there. It was like there was this little guy in my brain, shouting into cupped hands what I should do, but I ignored him.
"William is gone," I said. "I know you feel guilty, but you never did anything to hurt him, and now he's not around anymore. But I am."
I came over and sat by her. She stiffened and pulled away from me, but I knew this was a game at this point. I began to kiss her neck. At first she shoved me away, told me to stop, but pretty soon we were upstairs and those baggy clothes were lying on the floor. Big surprise, right?
* * *
That's how I basically moved into her house. It was just a temporary thing. It had to be for a guy like me because I loved my freedom too much, but even so, I liked having Maggie around. At least, I liked the old Maggie. The post-William Maggie turned out to be kind of a mope. She didn't talk much. She didn't want to have sex unless I forced the issue. She sat silently at meals.
Why didn't I walk the hell out of there, you're wondering, and it's a good question. I'm not sure I have the answer. I think I wanted the old Maggie back. I think I believed if I waited long enough, the old Maggie would return. This half-dead thing would be a memory, and she would be fun again.
Two weeks into this, my patience was starting to wear thin. Maggie acted almost like she didn't want me around. I'd say, "Do you want me to move out?" and she'd say, "Yes," and I could almost believe she meant it. She'd tell me she didn't want to have sex, and then we'd have sex anyway. I started to think that I didn't understand women all that well.
Maybe that's why things started with Amanda. You can't argue that was wrong on my part. I mean, it was how I met Maggie, so was she in a position to complain? Anyhow, it's the way of things. People meet, they get together, they move apart. That's just how it goes, right? Amanda wanted more and more of my time, and she was new, a country not yet conquered. Maggie was just Maggie. Nothing she did or said really surprised me anymore. I still liked the things she did and said, but so what? I like tacos, but I'm not going to spend the rest of my life with a tortilla stuffed with chicken.
So, I was waiting for the right moment to let her know that I was done, that I was moving out, that it wasn't personal. I figured I'd keep the whole Amanda thing to myself, since there was no upside in that. But that was about the time the business with the ghost started.
* * *
I was about to start the talk. You know the one. I've been thinking, Maggie. Where is this really going, Maggie? That kind of bullshit. Not fifteen minutes earlier, Amanda sent me a text asking if I could meet her for a drink. A drink would lead back to her place—make no mistake about that—and I liked the idea. Amanda was tall, trim, big in the shoulders and chest, and I was getting worked up just thinking about her.
Maggie and I were sitting on the couch, with the TV on, but not really watching anything. Maggie was staring into the distance. This is it, I thought. This has got to be the end of our little experiment. I'm sliding into bullshit relationship mediocrity, and if I don't get the hell out now, I'm screwed forever.
I took a breath and opened my mouth, but Maggie spoke first.
"I've been seeing someone," she said.
This announcement surprised me. I won't lie to you. Maggie sat next to me, her legs tucked under her body, her arms wrapped around her chest. She kept her eyes down, and her loose hair covered her face like a veil.
"What the fuck?" I asked.
I know what you're thinking, and you're right to think that. Why was I making a fuss about this when I was the one who was about to kick her to the curb? She was making my life a whole lot easier right now, and I ought to accept a gift when it's offered. Somewhere in the recesses of my mind, that little guy with the cupped hands was trying to tell me all this, but I wasn't listening to him again. I was angry. How could Maggie be stepping out on me?
"I guess I can't be surprised," I said. "You fucked around on William, and now you're fucking around on me."
She raised her eyes, and I saw something there that scared me. Not physically scared, like she was going to stab me or anything. I saw a kind of darkness and chaos, the look of a person who warped all the rules when she was around.
"You don't know what you're talking about," she said. "I've seen William."
"William," I repeated, feeling more and more like a deflating pool toy. "Dead William?"
Maggie kept that dark, chaotic glare on me. "Yes," she said. "I've seen him. His ghost."
When I'd gone through the roster of how Maggie might respond to my leaving, I'd considered the possibility that Maggie might go crazy, but not that she might be crazy. Here's as much supernatural shit as I believed: she could sense something was up with me. She could feel me pulling away. Maybe she'd caught a whiff of Amanda's perfume, or maybe she had just seen a look in my eye, but whatever it was, she knew. She might not even have known that she knew, but some nugget of female predatory instinct was driving her to get her claws in me deep and not let go. This crap about a ghost was just her way of trying to nail me down.
"I don't believe in ghosts," I told her.
"Yeah," she said, picking up a glass of wine and draining half of it. "There's nothing like seeing one to make you rethink that position."
Here's what she told me. It was four or five days ago, after nine, and I was still at the gym, since I had a late client—go ahead, see if you can guess her name—and Maggie was at home. Her father and brother were in town for a visit, which I'd been avoiding, having clients at convenient times. They had been over for dinner, so when she heard a noise downstairs, she thought it was one of them coming back for some forgotten bullshit. It had sounded like there was something moving around the house, but when she looked around, there was no one there.
Then, she said, came the knock at the door. When she opened it, there was William.
She put a hand to her mouth to scream, but she says she didn't. Maybe it's true. I don't know. Women are always going to tweak those kinds of details, right, to make themselves seem less stupid. But the thing that matters here is that it was William, standing there like he had never died doing his important crap that no one cared about. He wasn't beat up or bleeding or standing with some industrial pipe sticking through his head. It was William, like he was still alive.
This is how the brain works. Not just Maggie's brain, you understand, but pretty much everyone's. Was William dead? Yes. Had he been dead for weeks? Affirmative. Did Maggie have any doubts about that? Negatory. Yet, she stood there and thought that maybe there had been a big mistake. Maybe William had never been dead at all. Maybe he had spent all this time eating haggis and oak cakes and now he had decided it was the right moment to head back to the US of A and rescue his fiancée from her meathead boyfriend.
I didn't get all the dialogue, but she said he talked like William. He sounded like him, both in his voice and his wording. Maggie invited him in, and he said no. He could not enter the house. He was there, he said, because he sensed she needed help. You believe that crap? She needed help. Like, from what? Did she need help looking at her stupid contracts at work? I don't think so. She needed help with me. This dead asshole can't even stay buried, right? He has to come crawling out of the grave to mess with my shit. Like he would have had the balls to mess with me while he was still alive. Don't tell me this doesn't raise your blood pressure too.
Maggie swears she didn't suggest it, but out of the blue William starts talking about why they can't get married. He starts talking about how he lives in a coffin now, and there's no room. He's getting all worked up as he lists all the reasons why two people can't move into a coffin, but the last isn't really necessary, is it? I mean, if you say you live in a coffin, I'm not about to suggest shacking up. You pretty much barred the gate at coffin, right?
Who knows what Maggie would have said. Maybe she'd have tried to talk him into some other arrangement, but they were both startled by the headlights of my truck pulling into the driveway. I wasn't looking up. I was minding my own business, thinking that maybe I should have showered before coming over. I kind of brushed past Maggie and hit the bathroom. By the time I came out, she was in bed with the lights off, pretending to sleep, and I pretended to believe it.
Now she was crying, but I'm actually good with tears. I hugged her, even though it was like hugging a goddamn statue. I made her some tea. Crying women love the shit out of tea. Who knows why? Coffee and tears do not mix, but tea and weeping go hand in hand.
"He says he'll be back," Maggie said, sipping her tea.
"He's not coming back," I told her, "because he was never here. You just dreamed it."
"I didn't dream it. It was real."
My phone started buzzing from an incoming text. Amanda wondering where I was. I was busy. That's where the fuck I was. Didn't the fact that I was not where I said I'd meet her pretty much demonstrate that? I never understand why people don't get the obvious.
I tried to take Maggie's hand, but she pulled it away. That made my nice guy routine a little harder to maintain, but I was trying. I was really doing my best.
"I know you're upset," I told her.
"No," she said, her voice even, sharp as a scalpel. "No, don't talk to me that way. He was here." Then she smiled at me, and for the first time, I was scared. "You'll see when he comes back."
* * *
I didn't believe her. Not really. Even so, I thought something was going on. Maybe Maggie was crazy. Maybe someone was trying to scam her. Maybe—and this one felt the most likely to me—she sensed I was pulling away and was trying to come up with a plan to keep me around. As plans go, this one was pretty fucked up, but women do some crazy shit when they're desperate.
In the meantime, I'd sent Amanda a few texts, politely telling her to fuck off, that I had things to do. At first she got angry, but when I didn't answer her messages, she just left me alone. I figured that was how it would go.
The next night, Maggie and I sat in near silence. She opened a bottle of wine, and we both drank a little too quickly. Then she opened another bottle of wine.
"Maybe we should dim the lights," I suggested.
"It's not a séance," she said testily. "He'll either come or he won't."
I drank some more wine.
At a little after 9:30 there was a knock at the door. It wasn't a Christmas Carol banging or a timid and ghostly tapping. It was just a knock, just like anyone rapping his knuckles against the door. I thought to say that maybe ghosts don't use doorbells, but I kept my mouth shut. I'm good that way. Instead, I watched as Maggie, now looking like a ghost herself, rose from the couch and shuffled over to the door. Her hand rested on the knob for a moment, and then turned it.
I had never met William, but I'd seen pictures of him, and there he was—thin and pale with his unkempt hair, too smart to groom himself. Death hadn't hurt his looks much, but they hadn't improved them either. The fact was he looked like a person. There was nothing floating or transparent about him. He was there, like he'd never died.
"I wanted to see you again," he said. His voice sounded normal, not ghostly or groaning or filled with weird echoes.
For all that, I knew this was a ghost. I was looking at a ghost. This was the spirit of a dead man, a soul, that which remained of the human form after crossing the barrier of death. I took out my cell phone and began recording the conversation. If nothing else, I figured I could take a video that would go viral. I'd be the guy who recorded a conversation with a ghost. That could get me on TV.
"I wanted to see you too," Maggie said. "I miss you so much."
Like I wasn't fucking sitting there.
I looked at the video feed on my phone. There was Maggie talking to William, and it occurred to me that this video was going to look like a whole lot of who cares. Two people talking. That's it. This was not going to go viral on YouTube because there was nothing special about it.
I stood up to walk over to them, but I found I couldn't move my feet. It was like they were glued there and I was watching some freaky movie. I was standing behind Maggie plain as day, but Dead William didn't seem to notice me. Neither did Maggie. It was like they were the only two people in the world.
"I want to go with you," Maggie said.
"You can't," William said. "My home is a coffin now."
"I'll come to you," she said. "If I come to you, will you marry me?"
"If I marry you, you'll die," William said.
"Then we'll be together," she said.
He shook his head. "It's not what you think," he told her. "You don't want to be dead."
"I don't care what it is," she said.
Suddenly I found I could move. I lunged forward, moving toward the two of them. For a minute William turned his gaze in my direction, but he didn't see me. He looked through me as though I were the one who was dead. Worse, he looked at me as though I didn't matter, as though I were insignificant to him.
Fuck that dead asshole, I thought. I slammed the door. Maggie looked at me, shock on her face. She threw open the door, but William was gone.
"What the hell was that?" she demanded.
"There was no point," I told her. "I thought we were going to get some ghost video, but he just looked like a regular person. And then I needed to stop him."
"Stop him from what?" Maggie said.
"He was, I don't know, hypnotizing you or something. You were talking like going to live in his coffin was an okay idea in your book."
"He wasn't hypnotizing me," Maggie said. She turned away from me, so I was talking to her back. "I want to go with him. I want it."
"Because he was doing something to do you," I said. "Messing with your head."
"No, it's because he's the man I was going to marry. You think I want to spend the rest of my life with you? You seduced me into cheating on William, you practically forced me to let you live with me. If I knew of some way to make you leave, I would do it, but you're the one haunting me. Not William. Going to live in a hole in the ground in Scotland would be a relief."
I stood there, staring at her, unable to believe what I was hearing. I had stuck around for her. I made her tea, and now she was turning against me?
"What are you telling me?" I asked. "That you would rather be dead with him than alive with me?"
"Go be with your other girl," Maggie said and turned her back on me. "I wish you had moved in with her already, but you won't. You'll come back here, like you always do. When William comes back tomorrow, I'm going with him. At least I'll be away from you."
* * *
I had the last of my shit in my gym bag and had already loaded up my truck. "You've gotten rid of me," I told her as I stood by the door.
She sat on the couch, staring ahead, not looking at me.
"If you didn't want me around, why didn't you say anything?" I asked.
"I did tell you," she quietly. "You didn't listen."
I shook my head. "Just promise me you won't have him coming around anymore."
"You want me to call an exterminator?" she asked.
I didn't have much to say to that.
I grabbed my bag and left. I was done with her. The shit she had said to me was unforgivable, and I wasn't going to put up with it. No fucking way. Except after about a week, I gave in. I called, but she'd disconnected her phone.
I decided to drive by her house, and there was a real estate agent—a real fat one—setting up one of those lawn signs. You'd have thought she was deadlifting the way she was huffing and sweating.
I don't like people to see that I give a shit about anything, so I played it all casual. "What happened to the lady who lived here?"
The fat real estate agent looked up and wiped at her forehead. "It's very sad," she told me. "The woman passed away. Her family wanted the house sold as soon as possible."
"That's fucking bullshit," I told her. I might have said some other things, pushed her around a little, but she was asking for it, laying that on me like that, so suddenly. It wasn't kind at all.
I went home and looked it up on my computer. There it was, the obituary in the paper. Maggie had died the night after I left. There had been an undetected heart problem, and she dropped dead, standing by her door. The paper played it up real tragic, like her fiancé had died a few weeks before. They treated it like it was some amazing story of love, like one of those plays you have to read in school or something. They especially got off on how Maggie had just changed her will, saying that when she died she wanted to be buried in Scotland, next to William.
I was not about to let that asshole win. No fucking way. Without getting up from the computer, without worrying about how much it cost, I bought a plane ticket to Glasgow for the next day. I'd buy a shovel or something there. They must have shovels in Scotland. Once I found that graveyard, I would show them both that getting away from me is not as easy as they think.
On...Sweet William's Ghost
Sweet Willam's Ghost is Child Ballad 77 (Roud #50). The Aarne-Thompson Tale Type Index lists it rather wonderfully as #365—Supernatural Opponents, Specter Bridegrooms.
A ghost, usually named William, comes to the window of his lover, usually called Margaret. There he asks for his 'faith and troth' back (i.e. her promise to marry him). Not knowing that he is dead, she asks him to come and kiss her, wherein he sometimes answers that if he does her days will not be long. In a few variations, she asks that they go to the kirk (church) and marry. He then tells her that he is dead. She returns the troth and follows him to his grave, asking if there is room in there for her as well. She then either dies on his grave, does not have long to live, or goes away weeping. (1)
In some versions she refuses to return his troth unless he tells her what happens after death to unbaptized children, women who die in child birth, and those who commit suicide.
This belief of the dead betrothed coming to visit their intended was evident even in the nineteenth century. Sir Walter Scott wrote of a woman who, arriving too late to see her dying fiancé, asked to see his body. She touched his hand and thus, superstition tells us, escaped a visit from his ghost. (2)
Sweet William's Ghost was recorded on Kate Rusby's 2003 album, Underneath The Stars and is the name of a musical project out of Portland, Oregon.
1) Child, Francis James. The English and Scottish Popular Ballads Vol. 2. Mineola: Dover, 2003.
2) Child, Francis James. The English and Scottish Popular Ballads Vol. 2. Mineola: Dover, 2003.
# BLACK IS THE COLOR OF MY TRUE LOVES HAIR
By
Del Howison
"...unstable souls ... they are the children of curse"
\-- 2 Peter 2:14, 1599 Geneva Bible
In the far corner of a forgotten stone garden stands a most unusual grave statue. A chiseled sandstone child, solemn and expressionless, she watches the seasons pass without comment. Dismissed in the dark extremity of this hallowed place by the statues of crying angels draped over the gravesite or stone urns and granite wreaths that are placed closer to the sunshine and front gate, the lone child stands erect, arms at her side, staring straight ahead. Nothing so simple has ever been so unnerving. Where other monuments were entwined by nature in this forgotten garden, the girl stands clean, the ground about her dead and brown as if Mother Earth has offered up the final rejection, allowing no peace. The original carving, although graced with great artistic emotion, is somewhat crudely set as if it was toiled upon by unskilled hands. From the direction she faces, the winds and rains have wiped away many details of her surface and the only inscription offered to the curious meanderer is hard to read and incomplete.
Mara L rr
orn 1889 - D ed 19 0
To soon was taken om hi wo
Bl k is th c l r of y tru ov s h r
Her face remains mostly intact though softened with time and weather. Her mouth wiped clean. If one were to walk around the likeness, they would notice very distinct long locks of hair that are carved with exquisite care, flowing down the backside of the image. The hair was so attended to by the sculptor that it feels like the predominance of his time was spent away from her face. The flow and placement of each shock fills one with awe for the workmanship that does not appear elsewhere on the effigy. The love that is exhibited here is wrapped in great sadness, as all extraordinary loves are in order to exist at all.
The shadow of maternity had hung over the shack for more than seven months before the fateful night arrived. Elwin could hardly wait for the birth of his first born. He had spent the last few years of his life anticipating and hoping this moment would come. The only woman who had been willing to shackle herself to a poor dirt farmer was lying in the next room, screaming with pain. It had been Hell on Earth living with her, and after their initial sexual encounter on their wedding night, the opportunities to procreate had been few and far between. There had been no lovemaking or joy. Any sexual satisfaction he encountered had been in the bed of a neighbor. He had rejoiced that one of the quick, awkward thrusts his wife had allowed him had actually worked.
As she felt the pain from the poison course through her body, she realized the time was nigh and revenge was tasting sweet. Her agony rose and fell like waves cresting and each time catching her breath was harder and harder. But she savored it, knowing that the last minutes of her life had been in her hands, her call. It could be minutes or might be hours but it would be over.
"Elwin, go," she said. "Fetch the undertaker and be as quick as you can."
"Undertaker?"
She almost sat up in bed with the pain and her declaration.
"Yes, the undertaker."
"What? What are you talking about?"
"I'm talkin' about I'm gonna die and I'm taking your sweet, precious baby with me. You wanted me to have your baby like some kind of breeding stock. Well, I did that. You took an oath to be true to me and you couldn't keep your damn pecker in your pants! I'm takin' her by taking me."
Elwin began to cry.
"What did you do!"
"You cheating slob. There's only one thing you love more than yourself. You love your baby, just not enough left over to love me for carrying it. I believe your baby is already dead and I'm just pushing it out. I will be following shortly and with that I take your heart for all eternity. I take what you wouldn't give me!"
She winced with pain and then looked him straight in the eyes through her sweat and the heat.
"It makes me happy. It makes me real happy. I'm taking your heart, Elwin."
She shouted and grabbed her stomach.
"Oh God!"
Elwin stuttered a moment, looking at his wife, and then headed for the door.
"I'm going to get Miss Orpha," he said
"Miss Orpha is a Yarb Doctor. She cain't help you none. It's too late. I don't need no witchin' around my family. If it was illegal to be peddling her mumbo-jumbo, I would have had her throwed her in jail. You are too late. Too damn late! Why don't you just run to your whore instead?"
She doubled up with pain and then grabbed her stomach as if it might explode. Her breathing was short, forced. Her face strained, turning darker in color, and sweat as if from a soaked rag ran from her face. Her voice came out soft between the exhalations of her breath.
"You gave me the original pain. Now I'm giving it back. I'm ending my suffering and that of the child. I swear again on the ghost of my father, as I did our wedding night, that you will find no pleasure in me or your offspring."
She practically raised up out of the bed in anger and spit at him as she spoke.
"I curse you! You will have nothing! You are a poor excuse for a man."
Then the pain grabbed her and knocked her back into the damp mattress.
"Death, be quick about it!"
He ran to the barn where the wagon sat and left to fetch the midwife. His wife grabbed the sideboards of the bed and held on as tight as she could manage. She prayed the child would try to emerge before they returned. Then, if it were living, she would kill it with her own hands. She could feel it was not dead yet. But he would have nothing.
While Elwin stood to the side, helping with clean water and towels, Orpha worked on the child. She leaned up between his wife's legs, and her hands and trained fingers manipulated the baby inside. She cocked her head as if listening to something but the only noise was the wife's laboring.
"It's coming," she said.
"The child?" he asked.
"Four, three, two..."
She paused and then the thunder rattled the cabin. She smiled at him.
"Wipe me," Orpha said. "The storm. Have you never waited on the storm by counting from the flash?"
He moved forward with a damp cloth and blotted the sweat from her forehead and out of her eyes.
"I hadn't noticed," he said, confused.
"You are negro scum," his wife shouted and spit at Orpha.
She shook her head more from what she was feeling rather than the running perspiration. She was unhappy with her discoveries.
"Kill me!"
His wife was screaming throughout the ordeal but Orpha paid her no heed. They had tied her ankles to the frame at the foot of the bed, legs spread wide. She would offer them no help and Orpha didn't want to be kicked.
"The child is kneeling breech," she said to Elwin.
"Is that good?"
"It is a sign, like the storm. There's evil in this birth. She'd be better dead."
Orpha watched his face. Clearly he did not understand what she was speaking of. "If the child lives, it will be evil."
He walked in circles trying to get his thoughts in order. "She must live! You must save her. She is all I've got."
"I have never willingly brought evil into this world."
"I'm the father. I can train her and teach her in the way of goodness."
"You are a poor excuse for a man," his wife told him for the second time that night.
"The bond is here," Orpha said, and with two fingers she pulled out a taut section of the umbilical coming from the opening of his wife, showing it to Elwin. He turned his head away. "Look, the child is not tight inside. There is room for the cord to slip along the sides of the walls and come out first. I am trying to push it back up, past the child. The feet and the ass are coming out together. If they pinch the cord, the child will suffocate before birthing. I will have no say in that. That would be God trying stop the child. If I try to pull the babe down and out before the cord can slip out sideways, the force will either pull loose the baby's limbs from the sockets or split apart your wife. I can't save them both."
"Save neither, witch!" his wife cursed at Orpha and tried to wiggle about.
"I am here to do what I have been trained to do."
"You are betraying my wishes, you old crone!"
She pulled out her hands and carelessly wiped her forehead, leaving a splattering of blood and fluid behind. "I am betraying my own soul. Your crazy wishes do not concern me at this point," she said to his wife.
Orpha wiped her hands with the gingham rag she kept in her apron.
"It won't happen. It can't happen," Elwin rocked his body and called to the heavens. "You must save her."
Orpha looked at him with disdain. "The child should be buried while there is still life in her, standing up, so that God can take the child's soul before he loses it to Satan. You are making a big mistake here."
The thunder took another shot at gaining their attention.
"You are so simple. Even for being a man, you are so simple." Orpha shook her head. "You understand nothing of the ways of the Devil."
"I understand that all I've ever wanted is a child. I will not have another chance. I'm poor, growing older every day and no woman looks at me. I have to save that child."
Orpha shook her head, then reached into the pocket of the wifery apron she wore and pulled out a handful of objects. She pushed aside the personal items from the top of the scarred wooden dresser and laid her items out to see. To Elwin it looked like some Ozark dirt and a few bones and jewelry laid scattered on the wooden top. His wife screamed and he finally found his voice.
"What are you doing?"
"What I've been prepared to do," Orpha said as she placed certain items in their own small pile. "The Devil has given us a choice and you are stupid enough to take him up on it."
She looked back at his wife struggling against the ties.
"The child has the chance."
She took a pinch of the dirt. With two fingers she spread the soil in a line crossways on his wife's stomach just above the hairline.
"What is that?"
"Blessed dirt to help counter the jimsonweed your wife ingested. If we are unlucky enough, one soul will remain with body tonight. The bone yard shall make the choice."
"But my wife...You must save her too."
"I must do nothing of the kind," she said and with her finger and thumb spread the belly dirt out into a wider line. "Salvation is elusive."
He was scared. Elwin stepped up and pushed at her. "I don't want any of your hoodoo magic in my house."
He was puffed, asserting himself. She spun on him, freezing him with a look. After the pause he shrank back against the wall into the shadows.
"You stay there," she hissed at him. "You stay there, little man, and let me do my work! You fetched me. Now I am here. I do your bidding, not mine! So leave me alone before I change my mind."
When all was still again, except for the vulgar screaming from the bed, Orpha turned back to her objects.
"Yes, I am a midwife," she said. "But I am also a Granny woman. I'll use every tool at my disposal. But on a night like tonight, it may not be enough."
The thunder answered in response. She spoke low and slow.
"You stay out of my way, little man, whilst I try to save at least one Missouri life this evening."
She took a necklace from the pile and placed it over her head, about her neck. Her movements were deliberate, unhurried and ceremonial as if there were none in agony within the confines of the room. Her fingers moved over the drop that hung from the beads, fingering the amulet's familiar surface, focusing her ritual. Her lips moved but there was no sound to be heard except for the high-pitched swearing of his wife and the drumming of the rain.
"Can you help her?"
Elwin's attention continued to be split between Orpha and his wife. He was growing more frantic by the moment, feeling helpless in this nightmare.
"I can," Orpha said. "But we don't know yet whether or not it is a she. Do we?" Her toothy smile shrank him back yet again.
"All we know is that the bairn is bedeviled and your wife has caused herself and the child great suffering. It must be removed and the pain will cease. Then we will find how the poison has affected it."
She kissed the periapt and turned the necklace around so that the pendant hung down her back. She faced the bed, her countenance taking on such a dark tone and emotion that Elwin would not have thought it was the same woman who had walked through the door with him. The umbra that bisected her face seemed to be part of her skin tone and gave a great lurid depth to the lines that etched her appearance. From the other pocket of her apron, she pulled a sizable knife with a serrated blade. She paused and then looked at Elwin.
"You must be strong," she said, looking him up and down. "Although, I am aware that is counter to your soul."
He looked at the glinting metal and then to his wife.
"For the cord?" he asked Orpha.
"No," she replied. "For the womb."
In the years that followed the birthing, Elwin was visited often by Orpha to "watch the child's progress. " While he was grateful for the assistance, he felt wary of their secret talks and long, woodsy walks. But he couldn't deny that there were things, "girl things" as Orpha called them, that he just didn't understand. Still, darkness draped the house and neighbors seemed to shun his land. He had lost his position at a local feed-seller and needed to make do with odd jobs and farm pickup work. To help out their income, he carved in wood and chiseled in stone making things for people. The house was poor in the poorest of areas.
Despite Orpha's continued warning to destroy the girl, she grew, thrived, and seemed happy. She would bring her father rabbits and squirrels along with any other small animals she had caught to help with their meals. She wouldn't trap them but instead killed them with sticks and rocks in the most violent manner. It put him off but he never turned down the food, for their needs were greater than his morals. Whenever he would speak to Orpha about this, she would ask him if he was ungrateful for the food. He wasn't, of course. Beggars could not be choosers, she said. He didn't want to lose his daughter's love and Orpha would shake her head, pointing at him, warning the day of reckoning was due.
"The Devil is a strong demon," she would say. "And you are a weak man."
If Elwin attempted to offer excuses, he was put off by Orpha and charmed by his daughter. Mara was a charmer. Her jet black hair carried the ghost of her mother and the shine in her eyes was mischievous and lost at the same time. She had but to touch her father's arm to calm him or stroke his cheek to take him to a happier, unquestioning place. Elwin was too tired to fight her as long as she was happy. It was all curious and the whiffs of a spirit that lingered in the cabin since the death of his wife rested like a blanket of silence and foreboding. He could not get out from under the weight and it kept him silent.
But he worried about Mara every day. There was more in his household than his mind could hold. In his heart he knew the time was near. Were he a religious man he might have prayed for guidance. Instead, once or twice a month, he would hitch the wagon and go to the Joplin library. There he found books on folklore and superstition and witchcraft and children born under a bad sign. He would stutter his way through them, moving his lips as he silently read to himself. He found that there were two options, the fight to win her soul, which he knew he couldn't win, and an out he couldn't force himself to take. Either he would drive out all that the blood of her mother and the poison that the jimsonweed had instilled in her or he would have to bury her, alive, pleading for forgiveness, her head pointed towards heaven. When the showdown for the soul of his daughter came, he would not lose for lack of preparation.
As Mara pinned the clothes to the line, Elwin watched the wind move her shiny dark hair across her back in small waves. The glistening was diamond in its sheen and the volume was unworldly in its fullness. When she turned and looked at him, Elwin smiled back in a learned reaction. Keep her happy. Let her know you were her rock, her only parent. He needed her to answer to him only.
As she had gotten older, she understood her power more and more. She really could do anything she wanted, even if she was a little unsure of what steps to take to make that happen. She removed the final clothes pin from her mouth and slid it over the dangling piece of material, holding it securely on the line. Then she stepped back to admire her work.
"Another job is finished, Mara," Elwin said. "Well done."
"Yes," she said. "That's the last time I'm going to do that."
Elwin watched her face cloud as she walked over and sat in the wooden chair next to him. The clothes danced on the line.
"That's the last time I do anything I'm really not parcel to."
"Ah, but were life that easy, child," he said.
She turned to him, intense and meaningful. "But it is, Papa." She played with her hands a moment, thinking what to speak. "I can make people do things if I want."
A childish statement all the more chilling for its innocence.
"What do you mean, Mara?" he asked.
But he knew what she meant.
"It's a feeling inside. It builds and builds until I want to pop. So I have to let it go to make myself better."
She turned her head slightly away and smiled at him with her eyes. She would be able to charm them all. He could feel the smile travel up his spine and sit between his shoulder blades, pecking the base of his neck.
"The other morning when you left to go to Alexander's Store, I sat out on the step and watched the woods. A squirrel, a fat red-brown, stood on that lower limb of the oak there."
Mara pointed at a branch some forty feet from where they sat. Her head cocked slightly as she spoke as though she were seeing the entire incident over again
"He chattered at me. I believe he was mad that I was sitting and watching him or maybe he was upset that I had killed and eaten his brother. I yelled out at him, 'Be quiet!' But he continued on with his talk."
She stood up from the chair as she told her tale and took a step toward the tree in reenactment.
"I said, 'Be quiet,'" she said and continued her slow step to the tree. "Finally I became so furious that I held out my hands, open wide like claws but flat, and screamed at the top of my lungs to frighten him off. He froze still for a moment and then dropped to the ground, still. I did it without stick or stone, Papa."
Mara turned back to her father, once again looking like the little girl she was.
"I just wanted to quiet him. But I killed him."
Elwin stood up and hugged her. "No, no, Mara. You didn't kill him. It was just a strange thing and he happened to die at the same moment you shouted. That's all."
Elwin put his arm around her shoulder and they turned to walk back into the house.
"Do you really think so, Father?"
"I do."
"That is odd. I thought it was me."
"No, no. Just a strange thing. Life is like that sometimes," Elwin said and stepped up on to the stoop.
Mara stopped and looked up at him above her. "Yes, I suppose that's true," she said, nodding her head. "So now that makes four strange things altogether this week."
She walked up the stairs past him and into the cabin. The chill returned to Elwin like an old companion.
When the first child turned up dead, the news spread quickly through the hollow. He'd been found fresh as life, like he was sleeping there in the grass. The was no sign of violence, no injuries, not even any out-of-the-ordinary bug bites. It was just an odd thing that the doctor couldn't explain.
"Sometimes these strange things just happen," was all he could say. "God calls us when we least expect it."
Elwin heard the talk at Alexander's Store and knew the boy's family. A layer of dread covered his heart as he walked the ridge to the cabin. When he arrived, he cupboarded the meager supplies and stepped out to look for Mara. She sat on a stump out back looking into the trees.
"What are you doing?' he asked as he sat down beside her.
"Watching."
"For what?'"
"For dinner."
"You don't have to today, Mara. I made enough chore money to buy us a few groceries."
Elwin pulled a long strand of grass from beside the steps and stuck it in his mouth. He looked at her face as he spoke.
"I heard about Paul today at the store."
She squeezed one eye tight for a split second as if he was giving her a headache but continued searching the trees.
"You know about your school friend Paul?"
She turned and looked at him. "He was not my friend."
"He was ten and he is dead," Elwin said. "He is much too young to be dead."
She watched his eyes and then looked back into the trees. "He was older than our dinner," she said.
"That's true. But he was a person not a rabbit."
"The rabbits never laughed at me," she said and stood.
She walked into the cabin, leaving Elwin alone on the steps. He sat trying to figure what he knew and what he did not want to know. It is hard for the most rational of people to digest things that hang just outside of daily experience and Elwin decided he had more library reading to do. That was it. That was all. He just needed more book learning.
Some weeks later, when Elwin had been planting most of the morning, he heard about the dead horse. He was sitting on a stone next to the road, drinking some water he'd brought to the field with him when the county agent went by and stopped to talk. No reason he said, just seemed like the horse's heart up and stopped. Strangest thing. Doc can't figure it. There had been mention of a young child being glimpsed in the pen shortly before it happened but nobody knew for sure. Elwin knew but he didn't say so, not even to himself.
"Took you too many years to admit it," Orpha said. "But now judgment is here, ain't it? You conceived her during your spell of sinning and your wife tried to destroy her because she knew of your sin. It all says 'Devil' in great big letters."
"Can you help me? Another child cannot die because I was too weak to stop what needed to be stopped."
Orpha went back to her stove and whatever she was cooking. She poked it with a fork to let the grease drip down its sides. It hissed when it hit the bottom. She looked up at Elwin from the open stove.
"You never want to let things heat up enough to pop. You can ruin a lot by letting it get that far." She stood up and sat the big fork on the cuttingboard. "I did warn you," she said. "It's true that all prophets are liars but I knew what I was talking about. I've been through this before."
Elwin leaned his head against the door jamb of her cabin and began to cry.
"Orpha, you've got to help me. I can't do this alone."
She jumped down on the ground next to him and grabbed his face with both hands. Inches from him he could smell the rot on her breath.
"I tried to help you! You rebuked me. Now things is dying and you come crawling to me. Well, damn you! Help yourself!"
He jerked and pulled away when some of her spittle sprayed his face from her anger. He wiped his lips with the back of his sleeve.
"You've got to help me. I've got nowhere else to turn."
Her eyes, like burning coals, seared through him and then she quickly turned away and climbed back into her cabin. She walked to the corner of the room and grabbed a spade.
"You know what you have to do."
She tossed the shovel at him and he grabbed it. He looked at it in terror, knowing it was the only solution to his problem.
"God has already damned your soul, Elwin. Salvation is a fickle thing."
She slammed the cabin door shut. He stood looking at his daughter's murder weapon.
Secretly digging the hole in the back corner of the cemetery was easy. There was no keeper, nobody to notice his activity. He had contemplated burying her somewhere out in the woods but knew he needed consecrated land so God would take her soul. Getting the reverend to bless new ground would have come with its own problems and suspicions so he had chosen ground that was already blessed.
The night of the deed was cloudless and still. It was darker than the inside of the well and Mara lay peaceful on her blankets. Elwin approached her with the clothesline he had torn from the trees out back. His work rag from the barn stuck out of one pocket. He tried to move without creaking the boards by stepping as close to the walls as he could, where the boards had less give. He crept to her side of the room and looked down at the only thing he had ever loved. She was beautiful. Her black hair even seemed to shine in the darkness, catching light from some unknown place. He stood for a long time looking at her, breaking up inside of himself.
With a final, deep breath, he committed himself by jumping on her and pinning her to the ground before she could awaken in the confusion. Being sure to stay clear of her hands, he tied them behind her and then began binding her legs together. She screamed and thrashed but he was too much for her. There was no one to hear her cries in the woods. He then ran the rope from the top of her leg bindings to her hands and over her shoulders, bringing it back down on the other side. She was now prevented from bending and had to remain stiff. He would lower her into the hole feet first, like a fence post, and she would remain erect, head toward heaven, as he shoveled the dirt in on top of her. He would save her and she would wait for him in heaven.
Elwin retrieved the work rag from his pocket and stuffed it in her mouth. He secured it by wrapping a length of rope about her face. Then he picked her up, slung her over his shoulder, and took her out to the wagon in the barn. He laid her gently in the back and climbed up top. Giving the reigns a shake, the horse began its journey to the stone garden and the final resting place of Mara.
At the gravesite he lifted her out of the wagon and carried her over to the hole. She had managed to slip a couple of her fingers free and touched his arm. He turned and looked into her face and almost abandoned the entire plan. Her eyes pleaded with him and the tears ran down Mara's cheek. The child was begging for life. The rope he had wrapped about her head to keep the rag in place had chafed her fine white skin and the redness was almost raw to bleeding. He stopped in the grass and set her down as his body heaved in agony over what he was doing. When his sobbing had subsided, he looked down at her and realized how much he loved her. He must save her. Her soul was all that mattered.
Picking her back up, he carried her to the hole and laid her next to it. Elwin grabbed her under her arms, and then he twisted her about until her feet were dangling over the edge of the cavity and he tipped her slowly in. He slid her down until she came to a stop with her head about a foot below the surface of the ground. Standing up, he looked around for Orpha's shovel. Using the side of it, he scooped the dirt into the hole and over the head of his child. It fell into the pit, filling up around. She wiggled and tossed as she began to breathe in the dirt through her nose. Then, suddenly, she was completely consumed by the soil.
He patted the mound flat and smooth, then set the spade aside. He knelt beside the dirt tomb and prayed to God. He was now a religious man.
"Thank you for giving me the strength."
There he left it in the Almighty's hands.
Three months later, Elwin was carving on the slab of sandstone that he had set on top of her resting place, hiding the churned ground from prying eyes. Some townspeople felt it was a memorial placed in the back of the old cemetery to a missing child who would never be found. He was a sad spirit and the local folks mostly left him to his own devices. It was that afternoon when he heard the news. They had found another dead child.
On...Black is the Color of My True Love's Hair
Black is the Color of My True Love's Hair is not in the Child Ballads—instead it is listed in the Roud index as #3103. The Roud Folk Song Index is a database of nearly two hundred thousand references to nearly twenty-five thousand songs compiled by Steve Roud. It includes all the Child material and is now maintained online by the English Folk Dance and Song Society. (1)
There are many different versions of this song, starting with the title—from Black is the Color, Black is the Color of My True Love's Hair, But Black is the Color of My True Love's Hair, Black Black is the Color of My True Love's Hair, and even Black Black Black is the Color of My True Love's Hair. (2). There are versions from both the male and female perspective.
The basic plot of the song is the singers' desire and longing for their love. The '"True Love" has, as the title alludes to, black hair. Their eyes, face hands, and even the ground under their feet is adored. The singer longs for the time when they can be as one.
Recent singers include everyone from Christy Moore on her 2008 album to Nina Simone. There is even a downloadable ring tone.
1) Roud Folksong Index. 2013. 25 September, 2013. <http://www.efdss.org/library-and-archive>. English Folk Dance and Song Society.
2) Roud Folksong Index. 2013. 25 September, 2013. <http://www.efdss.org/library-and-archive>. English Folk Dance and Song Society.
# JOHN WAYNE'S DREAM
By
Gary A. Braunbeck
"Music is the art which is most nigh to tears and memory."
— Oscar Wilde
"It (music) expresses that which cannot be said and on which it is impossible to remain silent."
— Victor Hugo
Man, it's bad tonight—sick, miserable, choke-on-this ugly—the worst it's ever been, worse even than when you were drying out in the hospital, and you need tonight's meeting before you decide to break the goddamn seal on the bottle stashed in the trunk of your car with those Other Things, Things you can't help but think of in upper case, that's how bad it is; it's so bad you can't bring yourself to name them, even silently to yourself, that would mean you're actually, seriously, no-shit-Sherlock considering the repellant thoughts and ugly pictures that are kicking so hard across your mind, these things you know damned well ought to turn your stomach but don't, not even a little (they've caused you to actually smile when you're alone and once even chuckle into your cold coffee), and if you decide to do it, the Things will make it so easy, easy, easy-peezy, so you have to think of those Things in upper case because to name them, to admit you have them because of the thoughts and pictures in your mind, that's what scares you right down to the marrow of the bones in this sad-ass body you've been walking around in for—what?—fifty-three years and however many months and days, this sad-ass body that at least had enough willpower to drag itself down here to the church basement and the all-too familiar door halfway down the hall, the one you're standing in front of right now; so you take a deep breath, close your eyes, force the thoughts and pictures to the back of your mind, hoping they'll stay there, God grant me the serenity to make it all fuck off, and as you grasp the doorknob and begin turning it, you see the piece of paper that's been hastily taped to the frosted glass—
AA Meeting Canceled
Tonight Only:
Ghosts Sing Sad Cowboy Song for the Whole Broken World — and you feel your jaw actually drop like the anvil-mouth of some cartoon character as you read the words a second time, hand still gripping the doorknob, but before you can react, you hear it start from inside the room: that song... and on a guitar, of course—a steel-faced dobro-style resonator, from the sound of it, just slightly out of tune, causing the melody to sound all the more distant and empty and hopeless; gritting your teeth, you pull your hand away from the doorknob and step back as if moving two feet away will stop the sound from reaching you, but reach you it does, just as loud, just as cheerless, no goddamned different from any of the hundreds of times you've heard it played or played it yourself by request, by request, by request only because that's the only way anyone in their right mind would play the song, sure as hell not by choice, nosiree, like King Lear always said never, never, Never, Never, NEVER.
"Are you going in?" says a voice to your left.
She's younger than most of the people you've seen at the meetings; her face is round, its skin slightly pink beneath the surface, no tell-tale loopy blue lines of broken veins in her nose, no too-bright sheen in her eyes, no crow's feet, nothing to indicate that she shares your disease—hell, she looks like someone who should be playing the Julie Andrews part in The Sound of Music, not a recovering drunk.
"The, uh... the meeting's been canceled," you say, not so much pointing at as absent-mindedly flipping a finger toward the makeshift sign.
"I know," she replies. "I'm here for the ghosts' concert. Isn't that why you're here, ___________?"
She calls you by name. You have never seen this woman before. She's so fresh, so genuinely pretty, so clean—no, it's more than being clean, she seems... absolutely unspoiled. You might have a small graveyard of brain cells relegated to the backplate of your grey matter, but you're not so far gone that you'd forget a face like hers.
She smiles and brushes past you, reaching down and turning the doorknob. She's pressed against your side, her hand almost touching yours, but doesn't seem the least bit shy or embarrassed by it.
"I always loved this song," Unspoiled says. "It's been one of my favorites for... oh, I don't remember how long."
You just stare at her, not knowing a courteous way to respond. C, F, G, A-minor and D-minor; the whole goddamn song's based on various combinations and repetitions of those five chords—hell, the intro alone looks and feels like the most goddamn boring tune ever written—
/ C - - / G - - / Am - - / G - - /
/ C - - / F - - / C - - / G - - /
/ C - - / G - - / Am - - / G - - /
/ C - - / F - - / G - - / C - - / - - - /
—Snoozeville, right? Yet somehow those five basic, dreary, mind-numbing chords—chords you could teach a genetically-retarded monkey to play—manage to give body, soul, shame, and voice to the pain of the saddest songs ever... and wouldn't you know, it was Dad's favorite?
Of course you know that, you know damn near everything about good old Dad; you know too goddamned much about good old Dad—Christ knows he talked enough during those last few days, lying there in his bed with a bedpan under his ass and a catheter running up into his urethra, trickling bloody urine into a plastic bag. Sometimes he'd give you a half-hearted smile, but he wasn't the same man, the one who used to berate, humiliate, and mock you, the man who could always find fault in everything you did, who knew just the right thing to say to make your accomplishments seem inconsequential in everyone's eyes including your own, who could diminish you with less effort than it took to pick his nose; no, this wasn't the same man at all. This was just a sick old bastard making a last-ditch effort to get his bad-tempered ass into heaven.
"Have you heard this song before?" asks Unspoiled, staring at you with a curious intensity that, goddammit, reminds you of good old Dad toward the end.
You'd take him to the county cemetery where his parents and sister were buried, and as he sat there in his wheelchair staring at their headstones, you'd study his face and see him wondering if there was something that he'd missed, something he could've done to spend a little more time with them, to save them from feeling alone and frightened during their last few days of life, maybe even wishing that one or all of them could still be here to comfort him and, for a little while, get his mind off the disease that now counted down the clock that told of the time he had left. You don't know, maybe he came here to study their graves the same way he'd study his own face in a mirror, naked and defenseless. Why won't you look at me and Mom that way, you'd wonder. We have some time left—not much, but some—and maybe we could repair some things—okay, okay, okay, be realistic—if not repair, then at least spackle over some of the cracks so that when we drop the bag of meat that used to be you into the dirt, we might feel some kind of loss instead of relief. Is that too much to ask at this point?
"Did you know," says Unspoiled, "that this song has actually been around for centuries, in dozens—maybe even hundreds—of variations?"
You nod your head. "Actually, yeah... I did know that." Because good old Dad, he'd told you about that dozens—maybe even hundreds—of times... when he wasn't a zombie.
He'd sit for long hours in front of his bedroom window, staring out at the same neighborhood he'd known for most of his life, searching for something hidden, something unnoticed until now, something that would reveal whatever secrets there were to be revealed. Close by, Mom and you waited for him to complete this final voyage into himself, hoping the old emotional wounds might at last heal so that he would maybe, maybe, maybe, please God just once turn and smile over his shoulder, telling the two of you, without the burden of words, that he'd returned from this last nightmare, this final batch of self-recriminations that had made him a sadistic stranger to you for too long; and now, now that he was returned to you for however brief a time, you would go outside into the warm spring light, your mother and you each holding one of his hands, and you would thank the day for its blessings as it fell into twilight, and you would remain there, in shadows, as before, hands joined. "I don't even feel sick," was among some of the last things he said, along with, "What I wouldn't give for a hamburger and cold beer right about now." And of all Final Requests, he asked you to get out your old guitar and play that fucking song for him because, he said, it helped him to relax, to breathe easier, to fall asleep the way a man ought to fall asleep—no drugs, no siree, just a good, hard-working man falling asleep because the good, hard work took it out of him today. He would say that the Duke, the Duke never took to pills or liquor to fall asleep in his movies. The Duke was a Man, a Man's Man, and boy, wouldn't it be nice to just once have the kind of dream that the Duke must've had when his head hit the pillow at night, the kind of dream only a strong, solid, all-American Man's Man dreamed? "Maybe I'll have it tonight," he said to you that last night as you strummed those five horrible chords on that shabby guitar you'd bought second-hand from a pawn shop when you were twelve. "Maybe tonight I'll dream John Wayne's dream."
Part of you wanted to smash that guitar to pieces right there and then, go all Pete Townshend on the thing and scream "Don't you remember his last movie, Dad? The Duke, your Man's Man, was a goddamn walking corpse, being eaten away bit by bit from the bottom of his bowels by the same thing that's gobbling up your guts—fuck! He spent most of that movie fractured on Laudanum. That's how he went to sleep, but you don't remember that, do you? No—all you remember is that Jimmy Stewart told the Duke it was not the way he'd choose to go out, and that the Duke, your Man's Man, the great All-American Cowboy, he took Stewart's advice and went out in a blaze of glory, guns blasting away and bad guys dropping all around, blood and bodies littering the saloon floor, and you sure as hell aren't going out that way, are you, Dad?"
No, he sure as hell didn't. But that doesn't have to mean you'll go out the same way, his son who was such a private embarrassment to him; his son who never walked into the sunset with his best girl at his side; his son who never counted off twenty paces with a cheating gambler in the street before whirling around and putting the scoundrel down with a quick, single, justified shot; his son who never rode high in the saddle or talked real slow and deliberate or strolled with the swagger of a Real Man who made decisions and stuck with them out there where the tumbleweeds blew across silent, dusty streets where the womenfolk and children could walk in safety once again, knowing it was Real Man who'd done what needed doing.
"Hey, Dad, look at those bad guys fall. Look what I decided to do. Am I a Man in your eyes now? Am I worthy to sleep the sleep of the Just, of the Heroic? Am I worthy to dream John Wayne's dream?"
Unspoiled leans her head to the side a little and says your name again. "Are you all right?"
"I was hoping for, you know, the other meeting that was supposed to—"
She grabs your hand as she pushes the door open. "Oh, this will be much better than one of those dreary meetings. Those meetings are for everyone." She pulls you into the room. "This, tonight, is just for unfortunate rakes—just one, actually. This is just for you."
You stare at her, hoping her crazy won't explode and get all over you. "Imagine what that means to me." What the hell is that supposed to mean? You have no idea, but it's too late to take it back.
She laughs and pulls you into the room. It's mostly dark except for one bright circle of light shining down onto the middle of the floor; in the center of the circle is a folding chair. She leads you there, pushes you down, and kisses the top of your head.
"Now don't you move. We went through an awful lot of trouble to put this show together."
Before you say anything, she's gone—snap!—into the shadows. You stand up and walk back to the door, but it isn't there any longer. Okay, okay, okay—maybe you just lost your bearings. That has to be it. You're just a little confused because of Things in your trunk and the unbroken seal on the bottle and the thoughts and pictures in your head, that has to be it, you're just... Shit—you're probably just losing it, finally, just like everyone said you would someday. You stick out your arms and start feeling around the walls, doing your best Helen Keller to find the door because it has to be here someplace, but after a minute or two, you find nothing and you're even more confused than you were before you stood up, so you go back to the chair and you just sit because you don't know what else to do.
The sound of an old-fashioned projector clatters to life somewhere behind you, and your gaze follows the beam of light to the far wall where someone—Unspoiled, maybe?—has pulled down a screen, and you watch as a grainy black-and-white home movie comes into focus and shows you a scene that you know damn well took place dozens of times in your youth but could not possibly have been filmed because your family never had the money to afford a home-movie camera:
You see a variation of yourself—so much younger but only a little stronger than the man he eventually became; he's standing in the corner of a room, head down, studying his feet as if expecting some great revelation to come thundering up from the earth's core and show him a Great Truth that will set his spirit free.
Sitting a few feet away in his favorite chair, good old Dad is pointing at the young man who would have to settle for becoming you; he's got a beer in his hand and a cigarette dangling from the corner of his mouth.
A conversation repeated so many times and with so few variations it has surpassed the realm of mantra and become the refrain of the inharmonious tune that has been and is your life:
"A man makes a decision and he sticks with it, boy."
"Yessir."
"A man does what he says he'll do. No more, no less."
"Yessir."
"How old are you now?"
Christ, you really didn't remember, did you? You think now "Twenty-three, sir."
"Twenty-three, and what have you got to show for it?"
A decent enough music career—playing clubs, upscale restaurants, sometimes getting enough put aside for some studio time, a couple of self-produced albums selling okay for digital download on some minor music sites. Friends. A place of my own. "I think I've got a lot to show for it."
"You ain't famous, now, are you? I mean, what's the point of doing what you do if all you're gonna settle for is being small-time? A big fish in a little pond?"
"I like my life."
"You're weak, boy. A real man, when he makes a decision to do something, he does it. Your problem is you never decided to be anything more than second-rate because you ain't got the guts to be a real man and decide to be something more."
Why didn't you just say you were ashamed of me? I could've worked with that. "I thought you'd be pleased that I make a decent living at something I love to do."
"Don't talk to me about doing something you love. Doing something you love don't get you shit in the long run. It just makes you weak, makes you a fella who's happy to settle for something. Hell, the Duke never settled in any of his movies. I thought you'd learn something from you and me watching all his movies when you was a kid. The Duke's movies, they taught good lessons. Taught you how to be a man."
Still looking down at the floor, waiting for revelations to thunder. "Because when he made a decision he always stuck with it."
"Goddamn right he stuck with it. Duke always went out a winner, a hero, a man you could respect. Because he knew, the Duke did. A real man, he makes a decision and sticks with it."
"Yessir."
"Does what he says he'll do, no more, no less."
"Yessir."
The film finishes, the light goes out, and the projector shuts off. Deep down in your bowels, you feel the need churning. Leave. Open the bottle. Do it. Take the Things and just do it. Don't go out like good old dad. Don't go out like the Duke. Christ, it hurts! If someone had jammed a knife-blade entwined with barbed-wire into your stomach and twisted it, it wouldn't hurt as much as this need.
"Goddammit," you whisper to yourself. "I want a drink. I want a drink. Just one. One drink. That's all."
The unseen guitar begins playing again, the same five chords, and each chord snarls into your head and your balls like a diamond-tipped drill. You want to get up and get the hell out of the room—and if you can't find the door, you'll beat a goddamn hole into the wall with your bare fists ("Just like the Duke would do," comes the echo of a voice that sounds like your own), you'll kick and claw and chew your way out if you have to, you'll—
—Unspolied appears as the screen is raised up. She's sitting on a stage several feet off the floor. Her body is all wrapped up in a white sheet of some kind, its material thicker than something made from cotton; it looks almost like a tarp. Only her face and one hand are visible.
The music from the guitar fills the room, a sentient force, and as you look into Unspoiled's eyes, you can feel her grief, even from this distance. She smiles at you, a smile that bespeaks an errant wish—that a young woman might never grow old, never lose the radiance that kissed her face when a suitor came to call, never see her beauty dissolve little by little in the unflattering sunlight of each morning, and never know a day when the scent of fresh roses from an admirer did not fill her rooms; as she begins to sing, you stare into her eyes, eyes with sad dark places around them that tell you she has often hid behind a scrim of gaiety to conceal a lonely heart, and both she and her song become every night you've sat isolated and alone, wishing for the warm hand of a lover to hold in your own as autumn dimmed into winter and youth turned to look at you over its shoulder and smile farewell.
"When I was a young girl, I used to seek pleasure
When I was a young girl, I used to drink ale.
Right out of an alehouse down into the jailhouse,
Right out of the barroom down to my grave."
As she sings, other forms move forward from the shadows: a knight in the remains of ruined armor, his sword in one hand, his bent and twisted visor in the other; behind him comes a cowboy, classic and tall, spurs jangling with each step, holding his stained and tattered hat in wind-burnt hands; an older woman dressed in mourning black; a group of soldiers in uniforms crisp and funereal, carrying the shroud-covered form of a fallen comrade. The words each of them sing are different, but the melody—the morbid, heartsick, soul-beaten melody—remains the same; it doesn't matter if Unspoiled is singing of one morning in May or the soldiers are singing of Saint James' Hospital or if the knight sings of the maiden fair who passed on her physical ruin to him under the guise of love; it doesn't matter a damn if it's a cowboy dying in the street or a young woman perishing alone in the countryside; it doesn't matter if it's a soldier who got a dose from a passing lady of the night or a mother discovering her prodigal daughter by the side of the road; it doesn't matter how the words are changed or if the rhythm is ever-so-slightly altered or even if the words are in English, it doesn't matter if it's a bad girl's lament or a sorrowful young girl cut down in her prime or if she's riding on horseback or the man is a soldier loyal and true or the cowboy knows that he's done wrong and so must pay penance; it doesn't matter if the pipes and fifes play or if anyone bangs the drum slowly; all of them eventually arrive at the same place. Send for the preacher to come and pray for me/Send for the doctor to heal up my wounds/For my poor head is achin'/my sad heart is breakin/My body's salivated and I know I must die/Hell is my fate/I'm a-feared I must die/There goes an unfortunate lad to his home/I'm shot in the breast and I'm dyin' today/All gone to the round-up/The Cowboy was dead/For I know I must die/die/die/die/die....
The lights snap to black and the figures on the stage are gone, as is the music. The ghosts have sang their ballad, they have revealed the truth that a young man once, while staring at the floor, wished would thunder up from the core of the world and set his spirit free.
You feel a hand on your shoulder and look up into Unspoiled's dimming eyes.
"Do you understand now?" she asks.
"It doesn't matter," you say. "It doesn't matter if your intentions were good. It doesn't matter if your heart was true. It doesn't matter if you understood right from wrong."
"Yes..." Her voice is filled with bliss.
"It doesn't matter if you loved well or not, if you kept that love or lost it, it doesn't matter. You—all of you—all of us—the whole goddamned broken world—it doesn't matter, because we can't help but be what we are, and what we are, in one way or another, will end us before we're ready, before we can be forgiven, before we can feel worthy of the life that is inside and around us."
"Yes..." Her voice is now Bliss itself.
"None of us will ever measure up," you say, rising to your feet, feeling tall and proud and strong. You smile at her, running your rough hands through the curls of her hair.
"Am I your best gal?" she asks.
"Always by my side."
"It's hard to be a man."
"To be a man means you make a decision and stick with it." When had you left the building and gotten back into your car? When had you driven here, to this restaurant/bar full of people who have no idea that it all means nothing?
You raise the bottle to your lips and drink deeply of the whiskey, just like the Duke would, whether or not the streets of Laredo waited outside the saloon doors or not. The bad guys were everywhere and always would be.
Hey, Mom! Hey, Dad! Look at me standing proud.
You climb down off your horse and pull the Things from the saddlebag. You rack a round into the shotgun, and chamber rounds into the four pistols. No hesitation, no doubts.
For I am a lonely cowboy, and I know I've done wrong...
Jimmy Stewart told the Duke he wouldn't want to go out that way, and the Duke didn't, but good old Dad couldn't lay claim to a blaze of glory at the end, could he? No, he couldn't.
But you can.
You can be a Man, a Man's Man who doesn't make his mark with guitar strings and meaningless words in best-forgotten songs.
Time to be a Man.
You push open the doors to the saloon.
Hey, Dad—look at the bad guys fall!
As you walk and talk real slow, just home from the prairie green, tall and proud as the villains and scoundrels perish, tall and proud like a man ought, tall and proud, with guns blazing, the soft light of home beckoning welcome, welcome home, home at last, home in John Wayne's dream.
On...Streets of Laredo
The Roud Folk Song Index proves that #23650, usually known in America as 'The Streets of Laredo,' is a ballad that people have been loving and adapting for centuries. For example The English Folk Dance and Song Society lists no less than 195 versions. (1)
The well-known ethnomusicologist and folklorist John A. Lomax wrote that it may have gone back as far as 1790.(2)
In its first incarnation as 'The Unfortunate Rake/Lad' the song is about a soldier dying of a 'disorder' that should have been treated with 'pills and salts of white mercury'—aka syphilis. He asks that he be brought to his grave accompanied by young girls singing, drums and fifes playing, and muskets fired over his coffin. (3)
Especially after its arrival in America, the ballad evolved. In some the gender changed from male to female and instead of the sympathetic "Unfortunate Rake," the name became "The Bad Girl's Lament."(4)
Possibly the most famous version of this song in America is The Streets of Laredo, which is often referred to as a 'Dying Cowboy Lament.' Lomax writes that as it moved to the Southwest, it took on local references that obscure its British origin. Now it mentions cowboys, gamblers, a dram house, and a card house, among others, and features the protagonist instead of dying of syphilis, unsurprisingly, dying of a gunshot. The protagonist asks that cowboys and whores (in the bowdlerized version, pretty 'maidens') carry his coffin to the grave. Yet, however obscure this ballad's origins became, there's still a glimmer of the British military in the line "Oh, beat the drums slowly, and play the fife lowly." (5)
This ballad is popular with many musicians. Johnny Cash, Don Edwards, Joan Baez, Eddy Arnold, Arlo Guthrie, Prefab Sprout, Susanne Vega, and Bing Crosby have all recorded versions.
1) Roud Folksong Index. 2013. 2 October, 2013. <http://www.efdss.org/library-and-archive> English Folk Dance and Song Society.
2) Archive of Folk Culture. 2013. 3 October, 2013. <http://www.loc.gov/folklife/LP/CowboySongs_opt.pdf> Library of Congress.
3) Mainly Norfolk: English Folk and Other Good Music. 2013. 3 October, 2013.
<http://mainlynorfolk.info/lloyd/songs/theunfortunaterake.html> English Folk Dance and Song Society.
4) Smithsonian Folkways. 2013. 4 October, 2013. <http://www.folkways.si.edu/the-unfortunate-rake/american-folk-struggle-protest/music/album/smithsonian> Smithsonian Center for Folklife and Cultural Heritage.
5) Archive of Folk Culture. 2013. 3 October, 2013. <http://www.loc.gov/folklife/LP/CowboySongs_opt.pdf> Library of Congress.
# BEDLAM
By
Gregory Frost
The storm-tossed seas that night sank ships up and down the coast. Hulls smashed on reefs and rocks, crews dove or were flung headlong over rails, drowning unnamed and ungraced. Tangled in seaweed, bodies rocked in hard-pushed foam along beaches, Death being no kind nurse with gentle cradle for the loose-limbed and pale corpses. Tomorrow would bring new wailing as the dead and lost were claimed.
Lightning flashed and flashed again, tossed barred shadows down into the depths. Water sprayed like from a cannon's mouth.
In among the water devils and the churning dark, one prow shot up as if it had sprung from the deep, not ridden down trench and up wave. The tattered ship cut the night impossibly. It made straight for safe harbor and a quay that could have been no more than myth to those on board. The lighthouse on the point had been doused and darkened by the almost-sideways rain. What human hand could have guided any ship to so invisible a berth?
"Hard-a-lee!" shouted Tom, their captain. "Cling tight, you aloft!"
His crew was young as tyros on a maiden voyage, stripped naked or reduced to rags by the tempest, and bony as starving beggars, all. Their eyes pulsed a luminous blue as if sea creatures had slithered up into the sockets. They secured the lines and clung to the rails and up against the shrouds, the sails all reefed. He trusted in them, and they in him to bring them in across this wild, infernal sea.
And so came sailing into port a shape like a mound bearing three huge crossed spars as if Golgotha, ripped free of Jerusalem, had come scudding from ubiquitous darkness to a new destination.
Then the air seemed to shudder. A huge gust, and the storm had outpaced them, pushing on, ahead and over land. The final waves pitched and washed over them, slammed Tom against the wheel again. He heard mad laughter in it as if the sea were taunting him. He spit out spray and gripped the handles, but didn't so much steer as let the ship have its way.
Swells and lesser waves rolled beneath them, but moment by moment these, too, lost strength, until by the time they dropped anchor and it thumped the sand fathoms below, the surface had calmed as if no storm had ever passed.
Suddenly swallows winged across the sky, and in the black distance ahead appeared the small fairy lights delineating a town. Dripping wet, Tom followed the birds' flight past a crack in the clouds, his sharp-edged face presented to the burst of moonlight as the clouds pulled apart.
His sailors looked where he did with animal eyes, stung salt-red and wild, creatures not long out of the torrent, hardly longer out of hell. Doused in the faint light, they cheered him and laughed wildly. They had survived it all.
"Rum all around," Tom called. "But lower me a boat first." While most of them scurried for kegs, he scuttled across the foredeck and ducked under the bowsprit, where he leaned out beside the ship's figurehead, its lacquered arms bent and ending in a curl of claws that were ready to grab the wind, its grinning face stretched wide with sharp teeth and its eyes the yellow of a gorgon that might turn any opposing ship to stone. Its ample breasts peaked in nipples of iron, which had rusted over time so that they seemed to have bled in streams down the belly to where carved flesh became scales and joined the hull.
"I go in now," he whispered to her, his only confessor, "to fetch my love, my Maddie, who waits and has waited." He thought the gorgon smiled at his words. "She has waited," he insisted.
On deck below, the spindly boys dipped their cups into a cracked keg and then poured them back, empty of anything but spray and air. Yet they swayed as if affected by true spirits, pushing, wrestling one another for a place in line, a second or a third drink. Looking down upon them like a cleric from his pew, he said, "My bonny boys."
He climbed down and walked through them ignored, like a ghost himself, then climbed over the side and down the Jacob's ladder to where the lowered rowboat awaited, cast off the ropes, unshipped the oars, and began to row. He skimmed alongside the ship, and beneath the carved scroll bearing her name: Bedlam.
With each stroke of the oars, his moon-drenched body, barely whole at the start, seemed to take on mass and muscle. His ragged clothes knitted as though invisible silkworms were spinning their threads about him.
* * *
Along the rocks of the seawall he strode. The town above was mapped in his head. Where the plat had come from, he didn't know, didn't ask, didn't care.
As with each stroke of the oars, once he'd tied up at the quay, each step he took along the rocks put a finish upon him, blushed his face with color, oiled and curled his wild hair beneath a tricorner hat, drew sharp the line of his short, dark beard.
He started up one set of steps, but a tree had been uprooted in the storm and fallen across them. Above it he saw a man watching him, an odd bald fellow with eyeglasses, who waved to him as if they knew each other, gestured him to come up as though the huge tree didn't bisect their path. Tom raised his chin, seeming to assent, but he retreated down the steps, circled farther around and climbed up to the town along a dirt path so hard-packed that the rain had slid down it as if it were glass. As he climbed the slope, the wind picked up, gusting at him, sea spray or rain slapping him in the face. He ducked his head and let his hat take the brunt.
He entered the town now from its stern. The buildings cut off the wind again. Ribbons of his breath steamed into thick night mist.
At first he was alone, but even as he reached the wider paths, there were few people about and most of them drunk—who else would have braved this night but those whose need drowned common sense? They stumbled past, and he, like a darksome wind with purpose and destination, heard their jumbled, unvoiced dispatches, collected snippets of gossip out of the black air, gathered to him the capricious voice of the town. It told him everything.
The man with glasses was now down upon the rocks, calling out, seemingly seeking him where he no longer tread. "Village fool," he muttered. He chuckled to himself and pressed eagerly on up the hill.
A path of cobblestones appeared ahead. It glistened in the storm's aftermath like jewels and he fairly danced along now, higher up the hillside, past an inn and some shops, then in among individual hovels and homes. Time was of the essence.
He knew the place, his destination, though he had never seen it before this moment. A fine, small house. She'd married well, he thought, though he knew already the full story. Golden light fluttered in the front window—a fire burning in the warming hearth.
Awhile he stood outside in the darkness and just watched through the distorting glass as she bathed her two children in a tin pan with a sponge. Young boys, both, and he thumbed the empty scabbard at his hip where his long knife had previously dwelled. Its blade, lost to him, had tasted the meat of children before. "Pork pies" was what he'd called them. He flung the memory off. Two boys. They might, only a few years hence, join his crew. Some other night than this. A sly grin crossed his face but he wiped it away as the boys put on their nightshirts and she urged them ahead of her up into the loft. Then, from the top of the ladder, she climbed down, her skirt pulled high up, her legs and feet bare, palest thighs arresting his heartbeat for a moment, pulling him taut with honest yearning. He moaned softly with want, and her name slipped from his mouth swaddled in breath. In the ripple of the glass, she looked as young and beautiful as ever she had.
She stepped down and paused as if thinking what she had to do next, and as she did, she looked straight at him out in the cold night. He knew what she saw, lit gold by the fire of her own hearth and edged sharp in moonlight. He was a phantasm drawn from her memory.
Her fingertips pressed to her lips, peeling them apart. Such pearls, her teeth. He remembered.
Her eyes gleamed wet with joy and she ran to the door, flung it back and then herself into his arms.
"My Tom, oh, my Tommy! It can't be!" she cried.
"Maddie," he laughed. He caught her around the waist and swung her about. Hips thicker than he'd known, but not from bearing sons. Her smell—of soap, babies, and honest sweat—filled him up with lust he'd thought he would never feel again.
He set her down. She stood on tiptoe on the wet stones.
"I thought you dead. They told us—"
"They lied. To you, to everyone. They spoke from rumors, stories passed from ship to ship. We never sank—we don't know how. We were boarded, right enough, taken, yes. I was beaten plenty. They plundered us and would have done so again, but not with this crew, my Bedlam boys. I'm captain now and we sail on my command to anywhere you like."
"Anywhere I like. . ." She said it as if she'd misheard him. Then, "Why did you not send word? I waited forever."
He opened his arms. "This is it. I am my word. You were in my thoughts across the whole of the Atlantic. It's forever your face in darkness, in moonlight, when I close my eyes. I had to come—to collect you, to carry you off and give you the whole of the world."
"But, Tom, not knowing it, I married. I'm a wife." She gestured back at the house. "Two sons, and a husband who's away on—on business. Time has passed for you and me, though I swear, you look not one day older than I remember."
"No, we can't be severed. You are Proserpine and I've come to lead you out of the darkness of Hades that you don't know you're in. I never gave up in all this time. It's you and no one else, Maddie Maudlin. I know you've never stopped loving me. How else did I see you everywhere? I could feel you, even were my grave a hundred fathoms down. I saw it in your eyes through the window just then when you turned and I see it now, right here. Everything that ever existed 'tween us. You want to come with me." He held his hand out as if to take hers.
"But my boys—"
"They ain't yours, are they? They're his, from before. Nor's he your rightful helpmate." Then he laughed, his white teeth gleaming, as if none of it mattered. "I've a ship, a crew. We're invincible, can't be sunk by God nor nature. I'll give you the world, a thousand ports of call, if you'll only cast off this other life and join me."
She pressed a hand to her cheek. "How can I do that?" She glanced back through the open door. "My—the children. My husband."
His eyes gleamed sharp. "And where's he now, tonight?" She started to answer but he interrupted. "You stumbled in the telling. I'll tell you what he's away on. Riding his tender mistress in a room above a tavern the next town over, where they meet as often as he can steal away, and you pretend not to suspect so as to be able to go on." He clucked his tongue. "Let's go back inside. You stand for me in that tin pan, and I'll wash you clean, and we'll remark upon the bruises and the marks of his belt."
Horrified, she stepped back from him. "You only just arrived, how can you say—how can you know any such a thing?"
"One hears things in the wind and in the dark. Such as your new name, which I'll never call you by, nor where you reside." He opened his arms as if it was obvious. "How do you think I came here so directly?" When she didn't move, he added, "I saw the stripe on your thigh from his buckle as you got down, is how. Won't be the only mark. It never is." He clutched her hair, drew her face close. "You know I won't leave marks like that on your beautiful flesh." Her eyes were wet, but it was she who pressed her face, her lips, to his. Her hunger came to the fore now, drew him like an undertow inexorably away, and he let go and drowned with her.
He wiped a hand on his mouth, half-astonished by the intensity of her feelings. He turned quick, and pointed. "See, out there my waiting ship!"
It stood, still as something painted upon the night sky, sparkling with inviting lantern lights. They seemed to dot even the masts.
He licked his lips. Her taste lingered. "Let him have his boys and his belt. I have a whole ship full of boys who'll admire you—no—worship you. Worship you as I do." He offered his hand to her. It was flawless and strong. He curled his fingers, relishing the feeling. "Come with me, Maddie Maudlin. I've so much to share. Treasures and pleasures. But you must come now. Tonight, before midnight."
"Why's that?"
"Why, the tide of course. We must sail upon the tide before midnight, else find ourselves stranded here when the next storm hits."
She looked into his eyes then, and her expression shifted like a sail catching a favorable wind. He watched his promises flow into the curve of her lips. "Swear," she said, "you'll take me all the way to Italy, where the white lilies grow."
"Of course, of course. Anywhere."
She went back inside, lifted a poker and stirred the fire till it jumped up hot and bright again. She climbed the ladder a final time, stood at the top, her back to him. He watched, his will guiding her to sever each adopted tie. The resolve was his, though she would think it her own.
Down again, she snatched up her shawl and closed the door on the life lived in that house, and all its deceptions.
"None could ever hold a candle to you, my Tommy." She ran one slender hand over his taut, cold cheek. His eyes nearly closed at the sensation so long forgotten.
He took her by the hand and drew her through the curving, sloping streets of the town. He was the current now, pulling her out to sea, and she didn't resist.
"It's late," he said. "It's very late. Hurry."
Wind already was swirling in from the sea again. The spray stung him.
At the steps down to the seawall, the "village idiot" reappeared, closer now, running across the uneven dirt to catch up. The moon shone on his balding head. His eyes were like an owl's, magnified behind the lenses of his eyeglasses. "Thomas," the man called. "Stop now. Come back, for your soul's sake. Stay and ride it out with us."
"Get away, you." Tom quickly dragged Maudlin down the stone steps to the seawall.
"How does he know you?" she asked.
"He doesn't," he replied, and pressed on. "He never did."
"Thomas!" 'Twas as if the wind called from above.
They hurried along the wall, out the quay. In the distance, the glittering ship rose and fell now, the sea coming to life again as if agitated at his return.
He set her in the rowboat, clambered down and sat with his back to her as he hauled hard and fast on the oars. Facing the shrinking town, he stared at the man, who just stood there, leaning on the wall in a helpless way.
The water slapped Tom in the face. It seemed to leap over the gunwale to strike him. The whole sea wanted him back, didn't it? He craned his neck. Maddie sat with her shawl over her head, her face in its shadow, bowed as though in prayer. The ship rose and fell behind them, closer with each stroke. The gorgon watched their approach.
He saw his own hands losing their flawless shape. But he'd got her and no mistake. Not the ocean, not her careless husband nor any force of God she prayed to could stop him.
They glided alongside as he shipped the oars and made a grab for the rope ladder dangling where he'd left it.
With his head down so that she wouldn't see the change in him before she was on board, he urged, "Go up, my love, go up." She climbed ahead of him, barefoot still. The cold of the sea did not chill her. He only looked up when she was halfway and wouldn't see how his teeth showed through one cheek, and the sockets of his eyes glowed.
Boys' faces leered over the chain-wale. Their arms reached out for her.
Tom climbed along after. With each step he took, the Bedlam transformed around him, planks rotted, snapped, came unpegged. Moss grew up like a beard on either side.
Above, the sails unfurled without any hand tugging on any halyards. The sheets fluttered, tattered but swollen, pregnant with blue fire. The Bedlam boys' eyes burned with that same blue, and his own. "Yes," he mumbled. "Down and down, to hell we go, my Mad Maudlin. I'll show you where your white lilies grow before our final port of call, yes, I will." He slid over the side and onto the broken deck. "Oh, I've—"
"Got you now," she finished for him.
The naked boys smirked from around her skirts and under them, too. Her eyes gone the yellow of the gorgon's. Its polished face burnished her features.
"Mad!" he cried. "I swear! I'll show you—"
"No," she said. "Never can you show me a thing. That power was lost the day you drowned, my darling." She cocked her head. "Went mad did your Maudlin when you died. Off the sea wall she threw herself. Couldn't wait for your bones to wash up, so sent hers out to find you instead. An' here we lie, entangled below."
She turned to her crew, her boys. He was just one among them now. They all had ports of call and turns to take.
"Weigh anchor," she ordered. The sky cracked with jagged lightning. The wind bulged in the blue burning sails. It carried to his ears one final time that faded voice on the shore, a single shouted "Thomas!"
Then the sea rose up like a cobra, higher than the yardarm, and as he opened his mouth to scream, it slammed hard upon him, and them. She stood at the wheel, and the ship plunged downward to darkness.
His ears breached with the wail of a million souls drowned along with him.
* * *
Wailing echoed down the corridors. Lightning flashed and the thunder, right atop them, shook the very foundations of the hospital, and behind the terrified shrieks, one woman's voice screaming, "Tommy!"
The wild boy with the pail was ready to fling more icy water down on the patient tied to the iron frame below, but the physician raised his hand and called, "Enough!" He lumbered forward, removing his glasses and wiping at the misted lenses, at his own eyes. He leaned over and squinted dolefully at the soggy face.
Blue eyes stared wide, but somewhere far past and above him, all the way to the stars.
"Lost," said the physician. He put his glasses back on. "This time he's lost." To the two attendants he said, "Dry him off and take him back to his ward. And get your damned monkey down from that ladder before he strikes one of us with his pail. We won't have further need of him." He shook his nearly bald head, and walked over to the table in the corner where his notebook lay. He should have sat down and written up his notes immediately, but he couldn't face them. It was too great a burden tonight. He'd hoped. . . but he'd failed again. Even "hydro-therapy," as they were calling it, hadn't revived Thomas. If anything it had flung him out of this world altogether—not that his journey had been far to begin with.
The attendants—senior patients themselves—were cautious in wiping down Thomas. They didn't untie him until they were done. One still wore a bandage from the last time, when he'd been bitten by the savage madman. Rumor was the fiend had stabbed and flayed dozens of children before he'd been caught and locked up here. Another rumor had it that he was the doctor's only son, and that was why he was given every imaginable treatment from bloodletting to this water-dashing.
The attendants hauled him upright and dragged him out between them. They didn't bother to dress him. He wouldn't notice. Last time it had been a week before he'd come to what passed for his senses, claiming to be the survivor of some shipwreck. It looked this time like he wouldn't be coming back at all, and you didn't waste a gown on the mindless ones.
Lightning flashed again through the barred windows, making them jump as they left the chamber. Yowling filled the air.
Down the narrow corridor of Bethlem Hospital they staggered, though the patient hardly weighed more than a sack of bones. The near-naked boy who'd flung pails of water capered ahead of them as though leading a parade.
Wretched creatures ogled from cells on both sides, some grasping after them, but most laughing, howling at the sight of the naked man. Some threw their own stained nightshirts through the bars.
Across the walls obscene figures had been drawn—not all of them by the incarcerated—including a voluptuous female with fanged teeth and a snake's body. The word BEDLAM had been gouged out in huge letters beside her with a spoon.
At the sight of the procession, the woman stopped her cries and then began to sing:
"Tom o' Bedlam's home again
No escape in the deep.
Maudlin's mad with lust and sin,
All Bedlam boys do weep!"
The physician, following them, told her, "Hush up, Madeline."
She stepped back, curtsied, then raised her gown and pressed her sex against the bars at him. The Bedlam boys across the way yipped and strained to get a look.
The attendants pushed open the doors at the far end and a chill wind blasted down the corridor, snuffing out wall candles one by one. The hall went dark and the yowling recommenced. Then the doors shut, the bolts snapped, and one by one, they fell silent to listen to her soft chants, flowing like an undercurrent beneath their noise.
She was whispering singsong; it rose and fell. "I got you now, Tommy, got you, I got you." The susurrus bore them along, snaking through the cold darkness, drawing them from their cells. It might have been any of them she claimed—at least in their imaginations.
"Got you forever now, my love," she sang, "an' here we'll lie, together."
On...The Demon Lover
Known as 'The Demon Lover,' 'The Carpenter's Wife,' or 'The House Carpenter,' Child Ballad 243, also classified as Roud #14, was described by Child as "A Warning For Married Women, being an example of Mrs. Jane Reynolds (a West-country woman), born near Plymouth, who, having plighted her troth to a Seaman, was afterwards married to a Carpenter, and at last carried away by a Spirit, the manner how shall be presently recited. To a West-country tune called 'The Fair Maid of Bristol,' 'Bateman,' or 'John True.'" (1)
According to respected British folk singer and song collector A.L. Lloyd, the ballad was first printed as a broadside in the 17th century and before then was part of oral tradition.(2) There are many versions of this ballad—Child's description is a good general outline. The details change from version to version: the number of years between learning of her lover, often named Jamie Harris', death and her marriage, whether the one who comes to her is a spirit or the devil, the number of her children, how she dies and whether her husband simply mourns her or commits suicide.(3)
Shirley Jackson uses the name Jamie Harris throughout her short story collection THE LOTTERY AND OTHER STORIES. One of the stories, THE DAEMON LOVER, Jackson recorded in 1960 for Folkways Records and is housed by the Smithsonian Center for Folklife and Cultural Heritage.(4)
The many who have sung the ballad most recently include A.L. Lloyd, Bob Dylan, Joan Baez, Natalie Merchant, and Pete Seeger.
1) Child, Francis James. The English and Scottish Popular Ballads Vol. 4. Mineola: Dover, 2003.
2) Mainly Norfolk: English Folk and Other Good Music. 2013. 9 October, 2013. <http://mainlynorfolk.info/lloyd/songs/thedemonlover.html> English Folk Dance and Song Society.
3) Child, Francis James. The English and Scottish Popular Ballads Vol. 4. Mineola: Dover, 2003.
4) Smithsonian Folkways. 2013. 10 October, 2013. <http://www.folkways.si.edu/shirley-jackson/the-daemon-lover-and-the-lottery/prose/album/smithsonian>. Smithsonian Center for Folklife and Cultural Heritage.
# AWAKE
By
Jack Ketchum
In the dream he was lying on what could only have been a pebbled goat-path high above a whitewater sea pounding the rocks below, lying in the embrace of a beautiful young woman naked as he was, the path so narrow and so dangerous that the slightest move beneath her could send them both tumbling over the edge. He asked her to please let go and she said no, not this time, I don't think so and they began to fall.
He startled gasping up off the couch thinking what the hell was that all about? and in the flickering light reached first for the dregs of his scotch and then for a cigarette. On the flat screen in front of him a hard-faced cop stood by while his partner squatted in front of a distraught woman in a chair, her arm on the woman's shoulder.
He threw back the scotch and lit the smoke.
Right away he started coughing.
The cough took its usual course. Dry at first and insistent and repetitive as a single staccato note played over and over on the piano. Until finally it modulated into a wet cough and the ball of phlegm that released him from its grip.
Everybody knows they're going to die, he thought. It's another thing to have a schedule.
It was interesting.
His father had died of emphysema so he knew the drill. The progress of the thing.
He knew how it ended.
It was a nasty way to go, struggling for each breath. Until eventually your heart just threw in the towel. He gave himself a few more years. Maybe ten if he was lucky. But a simple head-cold could kill him too.
When the wet cough stopped happening he was in very deep shit.
His piss-hard-on was insistent too.
He took another drag on the Winston and stubbed it out. He shouldn't be smoking at all. But then there were a lot of things he shouldn't be doing.
The woman on the flat screen was quietly sobbing now. He muted her and hauled himself off the couch.
Maddie had left the light on for him in the hallway. That was good because the scotch was still working its dull magic.
It wouldn't be the first time he'd walked into a wall.
A dozen barefoot steps to the bathroom. A dozen more to their closed bedroom door and his daughter's open one, opposite. The usual. Lights off in both rooms.
In the bathroom he could hear himself wheezing.
The wheeze was roughly B flat. His piss in the bowl a clear A sharp.
He coughed again and the wheezing stopped.
He shook himself and zipped his jeans.
In the living room the cops were roaring silently toward a New York City brownstone. He tried for a moment to place the neighborhood. Couldn't. They shot the thing here — he'd see the crews on the street every now and then — and sometimes he could make out a landmark, a Grey's Papaya or a Love Cosmetics or, often, the Chrysler Building. But to him most all brownstones looked the same.
He lived on the eighteen floor of a decades-old high-rise. You couldn't miss it. He kept waiting for them to shoot there but it had never happened.
He reached for the bottle of MacPhail's and poured himself a short one. He'd need a clear head in the morning but he needed sleep too. You weighed one against the other and decided. Tonight it was sleep.
But an hour and two pours later he found himself grinding yet another Winston into the ashtray while the same cops in another episode grilled a wealthy matron about the supposed suicide of her husband and he was no closer to sleep than he was to Yellowstone National Park.
The sessions were eating at him. He knew they weren't right.
He, Lambert, Georgie and Kovelant were halfway through their fourth CD and the heads were fine, they caught the melodies nice and tight, the segues out of two-four and the ensemble turns were all fine, but his piano breaks just weren't making it. They were wooden, lacked the mix of fire and subtlety he was known for. They were competent. But they fucking bored him.
Nobody in the band was saying so but he'd been playing with them long enough to know what they were feeling. What's wrong with Fahner? And what the hell are we doing playing jazz takes on Appalachian folk ballads in the fucking first place?
It was his idea. Seemed like a good one at the time.
Still did. It was different. Dark, moody.
_Maybe_ , he thought, _it's the COPD. Maybe it's that good old death-sentence telling me I'm as good as croaked and might as well just lie down, already._
His first three CDs were solid hits. By jazz standards nowadays, big ones. At the moment he was the critics' darling. There was something to be said for quitting while you were ahead, he knew that. But he had bills to pay. And he wasn't quite ready for the piano-bar circuit, not yet.
He wasn't ready for that kind of applause.
Probably he should talk to somebody. Anybody. He hadn't. Not a soul.
Nobody in the band knew. During the sessions his Advair and Ventolin inhalers had the cough under control. Hell, they were all smokers. They all coughed, he no more than the rest of them. His producer didn't know. His agent didn't know. He'd hidden away the inhalers from Maddie well enough so far, usually in the piano stool, where neither she nor his daughter Leslie were likely to go.
The thought of his thirteen-year-old daughter digging around in his sheet music made him smile.
That would happen on the day that bears started reciting iambic pentameter and the menu at Le Bernardin offered up a side dish of Cheese Doodles.
Leslie used to love to listen to him play. She'd sit on the bench beside him. He taught her Chopsticks. Twinkle Twinkle Little Star. For Christmas, Deck the Halls. But that was years ago. She was hitting her teens now. And he was not Beyonce or Justin Bieber and certainly no Lady Gaga.
His little girl was growing up.
Last month she'd wanted a tattoo. Maddie was furious at the very thought. Actually screamed at her. He asked her what kind of tattoo. She said a rose. A red one. With thorns. She pointed to the back of her left shoulder. Right here, she said.
_Well_ , he said to Maddie, _at least she's not asking us permission to pierce her tongue_.
He thought he was being funny. Maddie missed the joke.
But basically he agreed. He was not about to let her deface her body.
No way.
When he considered it now though, he thought that her choice of said adornment had been appropriate, at least. A rose. A blossomed rose.
Because she was blossoming too, wasn't she? From skinny little kid to young woman. Her hips had begun defining themselves in graceful waves depending from her waist. Her breasts slowly building beneath her skin.
Another coughing fit set his nose to running and sent him headed back into the bathroom. Respiratory disease invaded not only the chest but the throat and nasal passages as well. He wondered if Maddie had noticed that they were going through a whole lot more toilet paper lately.
_Maybe I should talk to somebody_ , he thought. _A shrink. I'm pretty certain all this is depressing me_.
He sat down on the toilet and blew his nose, wadded up the paper and tossed it into the tank. Blew it again. He was quiet about it. He didn't want to wake Maddie. He sure wasn't ready to talk to her yet. He didn't know if he'd ever be.
There were problems with the marriage.
That's what Lambert and Georgie called them. Problems. Because they'd been having them too. Of the four of them only Kovelant seemed to have escaped the grim net of marital woes. But then Kovelant didn't fool around on the road or after sessions either.
Fahner wondered how he managed it. Jazz, when it was good, left you horny as a goddamn rabbit. It went with the territory. Jazz just...jazzed you up. The operative word being up.
But twice now Maddie had found out about it. Twice wasn't too many times, all things considered. But he guessed it was sufficient. A good day for them now was a day in which they were cordial to one another. Not openly hostile or aggressive.
But not exactly friendly either.
He didn't want a divorce. A divorce would devastate Leslie.
And he loved his daughter with all his heart. All his body and soul. He loved her as much as he loved his music and maybe even more.
He couldn't hurt her. Not in a million years.
In the hallway he gazed at her open door.
Maybe he should see somebody.
She lay awake thinking about what she was about to do.
There was a grim satisfaction to it, almost a sense of pride, now that it was in motion. Doing it this way. So many times she'd thought about it. Only she always seemed to falter, her nerves seemed to fail her at the last minute. But over the past few weeks she'd heard him noodling that goddamn song often enough, pointing the way.
And the idea had simply harpooned her. The song had harpooned her. Her anger had harpooned her and now she was finally pulling taut on the line.
It was one thing to screw around behind her back. There was a reason that she'd always considered the word groupie a diminutive. She could live with that. Had for many years now.
But this was another thing.
This was a dull muted rage in her that had been with her for so long she could barely remember when it began. She never spoke of it. She simply accommodated it as the Christian thing to do. She had her faith. Even if he never had his own.
Phil Fahner was a pagan through and through.
She'd loved that about him once.
She thought back to their first days together. The thoughts should have been sweet ones but they weren't. She'd been working as assistant to a painter, famous and prolific in the '60s but quietly losing it now that he was well past seventy — and sometimes losing it not so quietly at all.
It was an appointment-book and go-fetch job. Far beneath her talents and abilities. Get him into the town car on time. Take his strange, worried, sometimes furious calls at two or six in the morning. Pick up his paints and supplies at Lee's Paints on 57th Street. Deliver his paintings or sketches or lithographs.
Even shop for his wife for god sakes! Jesus! What she'd been through!
He was a rough, erratic taskmaker.
There were many, many times that she needed a drink after work and since she didn't drink alone — her father had, and look what happened to him — there was a quiet little bar on 68th Street not far from her apartment which was like those bars in Greenwich Village where you walked a few steps down off the street into dim lighting and some good quiet jazz and could sit among well-mannered upper-middle-class New Yorkers, and a woman could have a glass or two of Chardonnay and talk with the other customers without feeling like some tramp looking for a pickup.
He and his bassist Lambert and drummer Georgie did get to chatting her up, though. And they were smart and fun to talk to, so when the three of them got off their barstools to have a cigarette outside and asked if she smoked too — she did in those days — they invited her to join them. It was a warm spring night, just getting on to dark. At some point Lambert and Georgie faded back down into the bar while she and Phil Fahner — the Phil Fahner — lit another smoke and talked further about her job and her own painting and she felt this attraction. This magnetism. So that when he asked her out for dinner the following night she accepted.
Drinks and dinner, that was all it was for a couple of weeks and then it was drinks and dinner and bed and he was a very good lover, very attentive to her.
And she thought, he was attentive all right. He played her like a full-sized upright. All the keys.
So that when he asked in bed one hot sweaty August night why she didn't just dump the fucking job and marry him and do her own painting she said yes, and two months later they were at the City Clerk's Office.
And about a year after that she was pregnant.
Thirteen years ago, that was. Thirteen years was a long time. The good and then the bad. Then the very bad.
Thirteen years, to this.
There were far too many more years ahead of them for this family.
It had to end.
She could almost cry but she had stopped crying. She had always hidden the crying well, she thought. She had her makeup. Or else she had a cold, just the sniffles. But it finally stopped entirely only weeks ago when she heard him practicing for the session, playing that song over and over, and she recognized the lyrics within the melody from a record she'd owned in college, the harpoon deep into the body of the whale.
Tonight she would set the whale free.
Her daughter Leslie had awakened to her touch slowly, as though drugged. She'd always been a deep sleeper, a bed-wetter in fact from the ages of seven through nine.
And that had been her first clue. The bed-wetting beginning so late.
She should have seen it years ago.
If she had, they wouldn't have been here now. Here in this bed.
Her daughter rubbed her eyes as she woke and said whaaa? and she pressed two fingers to her lips. Lips warm with sleep.
I want you to go into our room, she whispered, and get into bed.
Huh? she said.
Don't ask me any questions, she said, just do as I say. I love you. We're trading places. You understand?
She watched as her daughter's eyes went gradually wide. Then after what seemed a very long time she nodded. Just the smallest nod of comprehension, of something that had passed between them. But it was enough. She slipped off the bed and started across the room.
Be sure to close the door in there, she said. And turn off the light.
She waited until she heard the door click shut and then reached for the stainless steel kitchen knife she'd honed early this morning which lay beneath her scarf on the night table beside her, lay down and pulled the covers over her and turned her back to the hall and the open door.
The song ran 'round and 'round in her head. An old Appalachian folk ballad. It was a comfort to her. She heard Joan Baez, her sweet thin soprano. She heard her girlhood.
"Don't sing love songs, you'll wake my mother.
She's sleeping here right by my side.
And in her right hand a silver dagger.
She says that I can't be your bride."
She didn't know exactly what she would do with the knife but breathed easily for the first time this long day and night and waited for him to come.
On...Silver Dagger or Katie Dear
As with so many beloved ballads, The Silver Dagger, Roud #711, has been shaped and changed by people as they sing and interact with it. Even though it's termed a Murder Ballad, the only death that takes place is a suicide. Like Romeo and Juliet, it deals with young love thwarted.
To look at three versions can give us a taste of how people change it to suit themselves. A version recorded by the Smithsonian has as its plot a 'comely youth' who courts 'a lady fair and bright.' Desperate at how her parents do their best to separate the lovers, the girl wanders out of the city, pulls out a silver dagger and stabs herself through 'her own true heart.' Dying, she calls on her act to be a warning never to part 'young true love.'
As she lies dying, her true love finds her. She bids him farewell, saying that they will meet on Mount Zion. Her lover then uses the dagger to kill himself. (1)
The theme of two daggers held by the girls' disapproving parents, silver for mom and gold for dad, is also common. In a ballad collected by Byron Arnold, Mrs. Hester—from whom he collected the ballad, couldn't bring herself to kill the young lovers. In her version of 'Katie Dear,' Mrs. Hester has the lover request that Katie ask her mother whether Katie can be his bride. Katie refuses, saying her mother has a silver dagger with which to stab "my true love's breast." The same request transferred to Dad is answered the same except his dagger is gold. Rather than end with a stabbing, Mrs. Hester rather confusingly has Katie say, "Oh, won't you be glad, my own true lover, When you and I become as one?" (2)
The Louvin Brothers version of Katie Dear includes both daggers and death. As with Mrs. Hester's version, the lover's request for Katie's hand in marriage is met with assurances that her parents keep daggers nearby for just such occasions. Heart-broken, he stabs himself with the golden dagger. After saying good-bye to her parents, Katie uses the same dagger to end her own life. (3)
Other than the Louvin Brothers, this ballad has been recorded by Joan Baez, Ian and Sylvia Tyson, Gillian Welch, and the Chieftains on the album Down The Old Plank Road, The Nashville Sessions.
1) Smithsonian Folkways. 2013. 10 October, 2013.
<http://media.smithsonianfolkways.org/liner_notes/folkways/FW03831.pdf> Smithsonian Center for Folklife and Cultural Heritage.
2) Arnold, Byron. An Alabama Songbook: Ballads, Folk Songs, and Spirituals. Tuscaloosa: The University of Alabama Press, 2004.
3) Bluegrass Lyrics. 2013. 9 October, 2013. <http://www.bluegrasslyrics.com/node/1665>.
# JOHN HENRY, THE STEEL DRIVIN' MAN
By
Jeff Strand
It happened in West Virginia, or maybe Alabama, around 1869, or maybe a decade later. This ain't a story about facts.
They say that no man alive could drive in steel like John Henry, and I believe 'em. With a hammer in his hand, he'd pound those spikes into the rock like you or I might stick a toothpick into freshly baked angel food cake.
Oh, he'd had muscles to spare when he was a slave, but he'd gotten even stronger after he was freed. He wasn't building the railroad all by himself—that would be crazy—but he was doing more work than any other steel-drivin' man, that's for damn sure. And believe me, the steel-drivin' men were not lazy people.
Too bad they all were gonna lose their jobs.
That's because some enterprising fool had invented a steam-powered hammer. How could a human being compete with such a machine? These poor workers and their families were gonna starve to death, all on account of "progress."
Well, John Henry, he put forth a challenge: he would race that mechanical hammer, and prove that a man could beat a godless contraption. And if he won, the workers would keep their jobs.
The race began, and oh, how the other workers cheered him on! Not to mention his wife, Polly Ann, who not only cheered louder than the steel-drivin' men but looked better doing it. John Henry's hammer, it came crashing down over and over, sparks a-flying, drivin' those steel spikes with the power of a god. Now, John Henry was a God-fearing man and would not have made that particular comparison himself, but to the outsiders watching the whole spectacle, it seemed appropriate.
Thing is, that steam-powered hammer was doing a mighty good job. I suspect that when John Henry put forth that challenge, he'd secretly hoped that the machine would break down after six or seven spikes and he'd win by default, but nope, it was pounding in those spikes at a rapid pace. Despite his muscles and his passion, John Henry was falling behind!
"Keep driving in that steel!" the workers shouted. "We believe in you! Don't let us lose our jobs!"
By now, John Henry had worked up a sweat of such quantity that more perspiration emerged from his pores than a normal man had of all body liquids combined. Oh, he was feeling the ache all the way down to his bones. His vision was starting to get kind of blurry at the edges, and that damn steam-powered hammer was generating so much dust his lungs burned with every breath.
But if you think John Henry gave up...well, you don't know John Henry.
He doubled his efforts. That's right, when any other man would have quit, John Henry hammered in those spikes even faster than before! I wish I'd been there to gape in amazement. He hammered and hammered, and though you might think that a couple of those spikes were crooked or not quite in all the way, you would be wrong. Every one of those spikes would have passed the railroad owner's inspection. John Henry was not a man to do slipshod work.
And then he caught up to that steam-powered hammer.
And then he passed it.
That's right, he passed it. Technological advancement was completely pointless when John Henry's hammer was at work. He was suffering, suffering bad, as if his arms might rip right off his torso at any moment, but John Henry was going to beat that infernal machine!
Yet with only three more spikes left to hammer, John Henry thought that he was going to die.
"Don't die!" shouted the other workers. "You've only got three spikes left!"
John Henry was so exhausted and he'd sucked in so much dust that for a moment, he wasn't sure that the voices of his co-workers were going to inspire him enough to finish the task. But then he heard the voice of his beloved Polly Ann making the same general point that the workers had made, and he knew that he could pound in those last three spikes.
Slam! Two spikes left.
Slam! One spike left.
John Henry, he raised his mighty hammer, and he let out the loudest grunt any human being had ever grunted up to that point in history, and he swung that hammer down and drove in that last piece of steel.
He'd won the challenge! He'd beat the machine! The workers were going to keep their jobs!
And then John Henry, with every ounce of energy in his body used up, dropped his hammer, fell on the ground, and died.
* * *
"John Henry, wake up!"
John Henry opened his eyes. "Huh?"
It was Polly Ann, crouching over him. Her beautiful brown eyes were filled with concern. "We think you may have been dead, but we resuscitated you!"
"I saw a bright light," said John Henry. "I was floating toward it, and some angels were beckoning, and then suddenly I was right back here. I think you did bring me back to life. Thank you, Polly Ann."
Charles, who was a steel-drivin' man just like John Henry but not as efficient, patted him on the shoulder. "We're glad you ain't dead, John Henry. Because we need you."
"Why?"
Charles pointed. "They've done invented an even bigger and faster steam-powered drill! They say it can do the work of twenty men! We're all gonna lose our jobs if you can't beat it!"
"I'm very tired," said John Henry.
Charles and Polly Ann took him by the hands and pulled him to his feet. "You're our only hope!" said Charles. "You've proven that a man can beat a machine once! Now we just need you to prove it one more time!"
"We can do this tomorrow, right?"
"No! The challenge is now! You've got to win, John Henry, or we're all going to lose our jobs worse than before!"
"That doesn't even make sense."
It took three men plus Polly Ann to lift the hammer, but they put it back in his hand. John Henry looked out at the faces of the workers and knew that he couldn't let them down.
Like I said, I wasn't there. But if I had been there, do you know what I would have seen in John Henry's eyes? Resolve. Resolve not to let down the other workers. Sure, he'd exhausted himself to the point where a medical professional would have declared him legally dead, but that was at a time when they didn't necessarily have the proper equipment to make such a declaration with complete accuracy. People used to get buried alive all the time.
An ugly thing, being buried alive. You may think there are worse ways to go, like when they chain each of your appendages to four different horses and then send those horses on their way in four different directions, but that's got nothing on the horror of waking up alone in a coffin, six feet under the cold ground.
But you know what? If that happened to John Henry, he would've busted his way right out of that grave, dusted himself off, and gotten right back to work. That's the kind of man he was.
Anyway, John Henry didn't get buried alive. He got to his feet, and he stared at that steam-powered hammer, which was all shiny and new, and he could feel the strength flowing back into his arms. He pointed at the hammer and said, "I'm sending you back to the scrap heap."
Well, everybody applauded and cheered, except for the driver of the steam-powered hammer, of course. He frowned a little.
And John Henry, he drove in those spikes like a man possessed. He'd been half-dead, and yet he worked like he'd spent the past week relaxing on the beach in a hammock, sipping drinks out of a hollowed-out pineapple. How many people do you know who could do that? I think you'll understand that I mean no disrespect when I say that, in similar circumstances, you probably would have just let those men lose their jobs. I know I would have. "No," I would have said. "Just let me die all the way in peace."
They weren't lying when they said that this steam-powered hammer was faster than the old one. Hell, that thing was twice as fast. It was so fast that a few of the workers admitted that though they didn't want to lose their jobs, they could see that the machine was indeed more efficient than human labor, with the added benefit that nobody had to suck dust into their lungs, and, yeah, they were still hoping that John Henry won the race, but they could understand the perspective of those in charge.
John Henry worked twice as fast as before without sacrificing quality. Slam! Slam! Slam! Slam! Slam! Inanimate objects or not, you almost had to feel sorry for those steel spikes.
The driver of the steam-powered hammer started to get kind of nervous. He was going to look like a real jackass if he lost to a half-dead man, and the financial implications of losing this challenge were dire. Railroad owners would cancel contracts all across the nation, and he'd have to lay off thousands of workers in his factories.
"John Henry's pulling ahead!" shouted Charles. Actually, John Henry had pulled ahead a couple of minutes ago, but Charles had been too flabbergasted by that fact to speak until now.
A big ol' cloud of dust had formed, so thick that the spectators couldn't see what was going on. But when the dust cleared, do you know who'd won the race?
That's right, John Henry.
Did you know that some railroad workers would swing their hammer so hard and so often that their intestines would come out? Yep, their intestines! Can you imagine that? But not John Henry. His torso was strong enough to keep those intestines inside where they belonged.
But he fell to the ground, closed his eyes, and everything went dark.
* * *
"John Henry...?"
"Go to hell."
"John Henry, wake up. It's me, Charles."
"I'm pretty sure I just asked you to go to hell."
"John Henry? It's me, Polly Ann."
"You can remarry after I'm gone. It's okay. I give you my blessing. Be happy."
"John Henry, open your eyes!"
John Henry didn't want to, but after some more coaxing, he finally opened his eyes. Charles and Polly Ann were crouched over him, looking concerned.
"Is everybody still employed?" he asked.
Charles and Polly Ann both nodded.
"Good. That's good."
"How are you feeling?" asked Polly Ann.
"Like somebody set my whole body on fire, and then took their sweet time in extinguishing me. I don't fear death. Death right now would be like a cold glass of lemonade on a hot summer day."
"Don't die," said Charles. "We need you."
"I can't help anyone."
"They say there's a man who can control the elements. A practitioner of the dark arts. By manipulating the earth, wind, water, and fire, he can drive in spikes faster than any steam-powered hammer! Maybe he doesn't use the water or fire. I'm not sure how it works, but unless you can beat him, we're all gonna lose our jobs!"
"My hammer's right there. Have fun."
"No, John Henry, you're the only one who can win the race!"
"If that's true, then maybe we need to accept the idea that progress isn't such a bad thing, even when there's collateral damage. You shouldn't continue to use outdated methods when a better option exists just to maintain the status quo."
"Come on, John Henry, you can't really believe that!"
"Do you want technological advancement to remain stagnant? This could be your chance to acquire some new skills."
"Please, John Henry! Don't let us down!"
John Henry looked into their eyes, and at that moment he knew that he had to accept this challenge. He had to show the world that supernatural abilities couldn't replace a man with a hammer.
The practitioner of the dark arts looked pretty much the way you'd expect a warlock to look. He wore a black cape, had a pointy mustache and pointy beard, and laughed a lot even when nobody told a joke.
John Henry lifted his hammer high above his head, and the challenge began.
John Henry was my father.
I kept trying to find a good place to insert that piece of information, but there really hasn't been one, so I apologize for just blurting it out like that. I assure you that you're getting an unbiased telling of the events, even though I'm his son.
Well, the warlock waved his arms, and cyclones appeared! Their winds were so strong that the other workers had to step back and shield their eyes, lest rock particles slam into their irises at a hundred and forty-five miles per hour. John Henry's eyeballs were more resilient and he kept his eyes wide open so he could see what he was doing.
With those cyclones, the warlock could lift spikes into the air and slam them down four or five at a time! He kept cackling with laughter the entire time. John Henry wanted to laugh right back at him, but he could hardly breathe.
Several of the workers shouted words of encouragement, trying to inform John Henry that they felt he was the superior competitor in this race, but their voices were lost in the swirling winds.
My guess is that a couple of the workers felt that it was worth sacrificing their wages to watch a warlock summon cyclones with his hands, but nobody ever admitted to it.
Some men, when faced with what seems to be an unwinnable challenge, drop into the fetal position and tremble. Well, John Henry trembled a bit, but he didn't drop into the fetal position even once. "I'm going to beat that warlock," he said, figuring that it didn't count as talking to himself if he couldn't hear his own voice, "and I'm going to save everybody's jobs. Then I'm going to take a nap."
He wondered if the cyclones would stop if he bashed in the warlock's face with the hammer. Then he decided that such a thing would not be true to the spirit of the challenge.
So John Henry, he began driving in that steel even faster than before! If you'd been there and been wearing protective eyewear, you would have gasped at that steel-drivin' man, I promise you that. I would never use language unbefitting a gentleman, but anybody standing there that day could be forgiven for taking the Lord's name in vain.
And when the cyclones dissipated, do you know who'd won the race?
Nope, it was John Henry!
The workers cheered and applauded even louder than before. "That John Henry, he's done it again!" they shouted.
The warlock, he was a sore loser, and he flung a lightning bolt at John Henry's head, intending to vaporize him. But John Henry, he held up his hammer at the last instant, and the lightning bolt bounced right off it, and that warlock became the one who got vaporized.
The workers stopped their cheering and applauding. Even if the victim is an evil warlock, you need to show respect after the loss of a human life.
Unfortunately for John Henry, you can't work that hard and just walk away whistling a merry tune. He got dizzy, collapsed, and then everything went dark.
* * *
When he opened his eyes, it was still dark.
And cold.
He lifted his hand. His fingers touched wood.
Dear God, they'd buried him alive.
Now, John Henry was braver than you or I, but don't let that fool you into thinking that he didn't let out a howl of primal anguish. Anybody else would have done the same, and there was no shame in it.
And then he cried.
I'll be honest—I wish he hadn't done that. It was a predicament to be sure, but that doesn't mean you need to go and blubber about it. Maybe I would have wept and maybe I wouldn't have; I just feel that, all things considered, a man should be a man when it comes to these matters.
I've never shed a tear in my life.
Perhaps there's something wrong with me. My wife thinks so. "It just ain't normal!" she wailed at our son's funeral. He drowned in the pond out back. I was supposed to be watching him. But that's not the story I'm here to tell.
Most men, upon waking up and discovering that they'd been buried alive, would claw at the underside of the coffin for a while, and then go back to sleep until their oxygen ran out. But not John Henry. They'd buried him with his beloved hammer. He picked up that hammer, and even though there wasn't much room to maneuver, he went and busted his way right out of that grave.
"John Henry!" Charles shouted. "Thank goodness you're out!"
Polly Ann gave him a great big hug and a kiss. "I love you so much, John Henry!"
"Why were you two standing around by my grave?"
"We were about ninety percent sure that you were dead," Charles explained. "It wasn't enough not to bury you, but it was enough that we felt we should keep watch for a while, just in case."
"I appreciate that."
"And now we need your help," said Charles. "They say there are dragons! With one tap of their enormous talons, they can drive in steel faster than any man alive! We're all gonna lose our jobs!"
"Why would you lose your jobs?" John Henry asked. "Surely they need workers to ride the dragons and make sure they don't fly off and kidnap maidens."
Charles shook his head. "They've got this new invention, this newfangled thing called hypnotherapy, and those dragons wouldn't touch a maiden even if she were rubbing up against their scaly tail!"
"There have to be other jobs out there."
"There aren't! We need you!"
"What about the poor dragons? Why should they be unemployed?"
"Please! Just one more race! That's all we ask!"
"My hands are covered with blisters," said John Henry, "and those blisters have even bigger blisters on the tips, and those blisters have even bigger blisters on the tips!"
"Exaggeration is not an admirable trait in a man," said Charles.
"You're right. I'll do it. I'll drive in that steel faster than that dragon!"
"Dragons. Plural. Four of 'em."
"Well...then...all of us workers will be racing as a team, right?"
Charles shook his head. "Nobody said it was a fair challenge."
Now, John Henry could have crawled right back down into that grave and nobody would have thought less of him. But that's not the kind of man he was. He stood up real tall, and he puffed out his chest, and he held his hammer up high and he vowed that he would beat those dragons, or die trying!
You may think you know how this story goes. "No man could beat a quartet of steel-drivin' dragons!" you're saying, "so clearly John Henry lost the challenge, and all of the workers lost their jobs, and everybody was sad."
Well, that's not how it happened.
He swung that hammer so fast that even the wings of a hummingbird had more visual clarity. And when the dust settled, it was a tie.
But you know what? One of those silly dragons had driven in a spike all crooked, so John Henry was declared the winner!
Hooray for John Henry!
Seriously, that was one hell of an impressive accomplishment. I don't care how jaded you are to superhuman feats of strength and endurance...that was impressive. It's difficult for my mind to even process what he did. He beat four dragons! Four! If he'd beat only one dragon, the world would be shouting "Oh my God! John Henry beat a dragon!" But he beat four of them! That just doesn't happen!
The dragons were taken away and put to death. All of the workers gave John Henry a great big pat on the back and told him what a fine job he'd done. He'd never shaken so many hands in his life.
He didn't even try to die this time. And before too long, sure enough, Charles hurried over to him, his eyes wide with panic.
"John Henry, we need you! They say they've got a man who can drive in steel faster than a steam-powered hammer, a more advanced steam-powered hammer, a warlock, and four dragons! We're all gonna lose our jobs!"
"Oh?"
Charles gave a frantic nod. "You have to beat him in a race! You're the only one who can..." He trailed off. "John Henry, you're gonna steal our jobs!"
"Or you could let me die."
"Yeah, I think maybe we'll do that."
And so, John Henry shook Charles' hand, and then he gave Polly Ann a hug, and he went off to die in peace. They say that late at night, if you're real quiet and you listen real close, you can hear the sounds of his hammer. Though I guess that means he's stuck doing this shit in the afterlife for all eternity, so it's not such a happy ending.
On...John Henry
Roud #790 the ballad, John Henry, is the most well-known and most-often recorded American folk song. (1) The English Folk Dance and Song Society list over 900 versions. (2) In the ballad, John Henry is a huge man who worked for the railroad (usually the song takes place in West Virginia, others place it in Alabama). (3) In order to build the railroad, tunnels had to be built through mountains and John Henry's job was to pound holes into rock where explosives would be placed. The bosses decided to bring in a steam-powered drill in hopes of speeding up progress. John Henry challenged the steam drill and its operator to a race to see who could drill faster. John Henry won thanks to his superhuman effort but it cost him his life.(4)
John Henry can be read in many ways: one is as a Luddite tall-tale with one man beating a steam machine that threatened to take away his job. As John Cephas, a blues musician from Virginia, states, "It was a story that was close to being true. It's like the underdog overcoming this powerful force. I mean even into today when you hear it (it) makes you take pride. I know especially for black people, and for other people from other ethnic groups, that a lot of people are for the underdog." (5)
Folklorist John Lomax writes how John Henry was often used as an inspiration by African Americans. While talking with a family on Fenwick Island, the father came in from work wet and exhausted. The family needed food that could only be acquired by rowing two miles to Bennett's Point. The father referred to the story of John Henry and trudged back out into the rain to row to the store. Later, another man rowing Lomax, in foul weather and against the tide, stated that if John Henry could beat the steam drill, he could row them back home. (6)
As one would expect, many people have sung this popular song. The Roud Index lists Woody Guthrie, Leadbelly, and Doc Watson (7), while Hugh Laurie and Johnny Cash can both be found on YouTube.
1) NPR.Org. 2002. 9 December, 2013.
<http://www.npr.org/programs/morning/features/patc/johnhenry/index.html>>. Present At The Creation.
2) Roud Folksong Index. 2013. 27 November, 2013.
<<http://www.efdss.org/library-and-archive>>. English Folk Dance and Song Society.
3) ibiblio. 2013. 4 December, 2013.
<<http://www.ibiblio.org/john_henry/>>. A collaboration of the School of Information and Library Science, the School of Journalism and Mass Communication and Information Technology Services at the University of North Carolina at Chapel Hill.
4) NPR.Org. 2002. 9 December, 2013.
<<http://www.npr.org/programs/morning/features/patc/johnhenry/index.html>>. Present At The Creation.
5) NPR.Org. 2002. 9 December, 2013.
<<http://www.npr.org/programs/morning/features/patc/johnhenry/index.html>>. Present At The Creation.
6) Lomax, John A. and Alan Lomax. American Ballads and Folk Songs. New York: Dover Publications, Inc., 1934.
7) Roud Folksong Index. 2013. 27 November, 2013. <<http://www.efdss.org/library-and-archive>>. English Folk Dance and Song Society.
# FISH OUT OF WATER
By
Keith R.A. DeCandido
I knew something was wrong when I saw the twenty-eight-foot Coast Guard boat coming toward us.
I cut back on the throttle to slow us down. It wasn't a collision course, but if the Coasties were coming this way, going the way they came from might not be such a hot idea. It was a Response Boat–Small, one of the new Defiant Class, and if an RBS was here, it probably meant something and/or someone went missing at sea. After double-checking the code painted on the other boat's hull, I grabbed the radio.
"USCG 25119, USCG 25119, this is Groucho on 16, over."
"Groucho, this is 119, over." I recognized the tinny voice that squawked over the radio as Boatswain's Mate 1st Class Cole Howard, who worked out of the USCG station located on Marathon Key.
"119, Groucho, wasn't expecting you guys out here. Something we can help with?"
"Groucho, 119, thanks very much. Don't suppose you've seen Soleado? It's a rec boat that usually launches from the Azucar Di—"
I interrupted before he could start describing a boat I already knew quite well. "Cole, it's Cassie—have those jackasses gone missing again?" Over Thanksgiving weekend, boats from the Azucar Dive Shop up on Big Pine Key had gone missing twice. In fact, Cole was the one who found them the second time.
"Groucho, 119, that's affirmative." Formal as ever, was Cole. "Last radio contact was at 1100 hours." Then he got less formal and more skeptical and sardonic when he added, "Said they saw a mermaid."
Last time they said it was the Loch Ness Monster, so at least they were getting more local. On the other hand, I numbered among my close friends and acquaintances the ghost of a wrecker captain, an immortal barfly, and four Norse gods, so I really wasn't in any position to poo-poo someone who said they sighted a mermaid. Or Nessie.
"119, Groucho, I've got six tourists who really want to dive Pickles Reef. Okay if I proceed there?"
"Groucho, 119, that is in our search pattern, but not for a bit. Proceed as planned, but be advised that we've received reports of an odd storm front over that way. Nothing on the radar, but a few boats radioed in tornado warnings."
That didn't make sense on any level. For starters, I'd checked the weather before we came out. And for another... "119, Groucho, those boats do know we're in the Florida Keys, right? We don't generally get tornadoes."
"Groucho, 119, wanted to mention it just in case. Good sailing."
"You too. Out." I thumbed the radio off and hit the throttle, zipping around the RBS.
The half-dozen tourists who had come to the Seaclipse Dive Shop on Stock Island and hired me to take them scuba diving were a bunch of martial artists from a dojo in Denver visiting Key West for the week. The woman who seemed to be the ringleader, a short, lithe Filipino named Isabel (all the others called her "Senpai Bel"), poked her head into Groucho's tiny bridge. "Everything okay?"
"Yeah, just touching base with the Coast Guard about a dive boat that's gone missing." I filled her in on what happened. "So good thing you went with us instead of Azucar."
Bel smiled and shook her head. "Our kaicho told us we should try Azucar, but it was too far up the Keys. We wanted somewhere closer to Old Town. Don't usually go against his wishes, but damn."
"Anyhow, we'll be at Pickles Reef in no time."
"Can I ask you something, Ms. Zukav? Kaicho told us the reef was named for the pickle barrels at the ocean floor. What he didn't tell us was why the barrels were left there."
My bosses at Seaclipse didn't warn me when I took the part-time job as dive-master last year that the job description included playing tour guide for the entirety of south Florida, but I quickly became quite the expert on the history and trivia of the Keys. "So many ships got wrecked on the reefs around here for centuries, there's no way to get everything back up above water. And these barrels are full of concrete, so hauling 'em up would be a major pain. No one's really sure where they came from—might be a wreck, might be a construction project gone bad. They used to think it was concrete to construct the forts they started to build during the Civil War, but they did an analysis a few years back that shows that it was a type of concrete made between 1890 and 1925. Might've been for the railroad they built through the Keys."
"What railroad?"
I smiled. "You guys drove down from Miami, right?"
Bel nodded. Flying into Key West International Airport was often more expensive than flying into Miami International Airport and renting a car. That had the added bonus of getting to drive on Route 1 through almost all of the Keys, which is some of the most scenic driving you'll ever see.
"The Overseas Highway is mostly on what used to be the railroad that went from Miami to Key West. It opened in 1912 and closed down after it got hammered by a hurricane in 1935. They built the road over the same right-of-way." I shrugged. "Anyhow, I'd rather they focused on cleaning up the actual litter that people throw down there. There's so much sh—so much crap down there..."
Smiling, she said, "It's okay, Ms. Zukav, you can say 'shit.'"
I laughed. "Sorry, first dive I ran, I made the mistake of saying 'fuck' in front of some Southern Baptists. Had to give 'em their money back. As a general—" I cut myself off when I caught sight of a boat in the water ahead of us. It wasn't on our exact course, but we'd pass pretty close. It was running a dive flag—the red flag with the white diagonal stripe that's become the universal sign for a boat filled with divers—but the motor was silent and it was bouncing around in the ocean as if it wasn't anchored.
Squinting at the boat, Bel asked, "Should we let that Coast Guard guy know?"
"Let's make sure they're okay, first," I said as I changed Groucho's course toward the dive boat, which looked more and more like Soleado the closer I got. "First time they went missing last weekend, it was another one of their boats that found 'em, but BM1 Howard found 'em the second time. He let ''em off once with just a warning, which means he's gonna rip several new—" I smiled at Bel, reveling in my newfound permission to speak freely. "—assholes this time. Probably close the shop for a major inspection, and whatever else he can do to make their lives a living hell."
Giving me a questioning look, Bel asked, "So you're gonna let them off?"
"There are about eighteen reasons why they might've gone quiet, starting with the radio breaking down."
"Cell phones?"
"Reception can be spotty out here. They also could be damaged in some other way that's not their fault. I'd rather not sic the Coasties on 'em until we know for sure they deserve it. Like the song says, we look after our own." I pulled up alongside the boat that had the word SOLEADO stenciled on its bow. "Mind giving me a hand?"
With the help of my very able-bodied tourists, I was able to tether Soleado to Groucho and then drop anchor so we'd all stay in one place for Cole to find us if we needed it. Groucho wasn't rated for towing, so if they were damaged, we'd still need help from the Coasties, but I wanted to know what was happening first.
Besides, it could've actually been a mermaid. Since moving to Key West, I'd dealt with a dragon, several different water fae, a half-dozen ghosts, what's left of the Norse pantheon, the spirit of the Calusa tribe, and a UFO on Dry Tortugas. I figured it'd be best to make sure it was something that the mundanes could handle before I called them in.
Okay, the Norse pantheon remnants I mentioned? I'm one of them. While I was "born of Midgard," as my friend Ginny, a.k.a. Sigyn, put it, I'm actually a Norse fate goddess—one of the Dísir. As a Dís, I tend to attract weird-ass shit, and sometimes I even can deal with the weird-ass shit. Cole was way too straight an arrow to handle an actual mermaid without his head exploding, so I needed to do triage on Soleado first.
Mind you, I had no idea if there actually were mermaids. I only found out I was a Dís seven months ago, and it didn't come with an instruction manual.
Once we secured the other boat, I jumped across to their deck. Soleado was a twenty-nine-footer just like ours, though it was made by a different company, so there were variations in design. However, the basics were the same: bridge fore, deck aft, galley and head below. I didn't see anyone in either of the former two places, so I climbed down belowdecks to find five people all standing crammed into the galley. They were all wearing some dive paraphernalia, but nobody was fully geared up—and nobody was fully un-geared up, either. One guy had a single fin on one foot, one wore a regulator but no mask, one had half his neoprene suit off—it was like they had all been interrupted in mid-prep and came down here.
I stared at the one person I knew, Al Martinez, one of the dive-masters at Azucar. He was the one wearing only a single fin. "Al, you okay? What happened?"
"I got a wife, back in Tampa, an' two kids. Ain't gonna be seein' her no more."
I frowned. "Al?"
The Asian couple crammed in on either side of Al then each said something in what I think was Korean; at the very least it sounded like how my Korean neighbors back in La Jolla always sounded when they chatted in their native tongue. After they said their peace, the other two—a couple of brunettes who looked related, probably sisters—spoke up.
"My fiancé's back in Ann Arbor. We'll never get to be married."
"My husband's waiting for me in Chicago. He'll be a widower."
I blinked. Even by my standards, this was weird.
Bel called from the deck—she had followed me across, apparently. "Everything okay, Ms. Zukav?"
"Lemme get back to you on that." I reached out to the dive-master. "Hey, Al, it's me, Cassie—from Seaclipse? Let's get you guys outta there, okay?"
But Al didn't move, he just said, "I got a wife, back in Tampa, an' two kids. Ain't gonna be seein' her no more."
That started the whole litany again, first the Asian couple, then the sisters.
Hesitantly, I grabbed 'Al's arm by the neoprene and tugged a bit, but he refused to budge.
I felt like there was something else I should do. This had all the earmarks of a spell cast over them. But my knowledge of spellcraft was limited to the fact that I was able to cast one if someone spent the better part of a day showing me how, and I only did that once. I needed to know a helluva lot more before I could try to cast a counterspell—and by "a helluva lot more," I meant "something."
I hopped back up onto the deck, asked Bel to keep an eye on the creepy quintet in the galley, and radioed Cole.
Once the RBS showed up, I untied Groucho and took the paying customers on to Pickles Reef, leaving Cole to handle Al and his clients. Bel and her friends paid for a reef dive, and they got a reef dive. Since there was an even number of divers, I stayed on the boat while they dove, since you always go down in pairs. (No certified diver would ever go down alone, and no reputable dive shop would ever let anyone go down alone.) Had I been in the mood, I could've been a third for one of the pairs, but these guys had some kind of teamwork bond stuff going on, so I let them do their thing.
I was fondling my smartphone when a call came in from BM1 Cole Howard's cell. I smiled and hit ANSWER. "Hey, Cole, what's up?"
"Just thought you'd want to know, we got Martinez and his customers out of the galley. They're on the RBS now. They just keep doing the same thing over and over, talking about who they left behind at home."
"You got a Korean speaker on board?"
"Yeah, me." I could hear Cole's smirk from here. "They were saying how they were going to leave their three kids orphans. And the boat was clean—no drugs, nothing untoward at all. All we found was a cooler full of non-alcoholic drinks, the first-aid stuff, dive equipment, and three iPads covered in waterproof casings."
I chuckled. One of Seaclipse's biggest sellers were the waterproof casings that allowed you to bring your smartphone or tablet underwater with you so you could take pictures with it. Those actually sold better than the disposable underwater cameras. As a veteran phone fondler, I totally got why people didn't want to be separated from their devices and wanted to have their underwater pics right there at their fingertips, but I never risked taking my phone down with me. Then again, I had a ridiculously expensive 14-megapixel underwater camera.
"We'll take them back to Marathon, drop them at the Fishermen's Hospital, let the docs there give them a full once-over."
"Okay. Hey, Cole, thanks for filling me in."
He snorted. "If I didn't, you'd have called me in an hour to bug me."
"Yeah, yeah." I tried and failed to sound offended. Cole knew me too well. "You ever gonna get your tight ass to Mayor Fred's?"
"Call me when they hire a blues band. Bye."
The dive went, you'll pardon the expression, swimmingly. My martial artists had a grand old time swimming around the reefs, the pickle barrels, and the pillar corals. They also frolicked with the blue angel fish and watched the spiny lobsters crawl on the barrels. I learned about all these things from the breathless descriptions provided by Bel's husband, aided by the pictures from Bel's tablet (nicely covered in a waterproof casing she bought at Seaclipse). Bel promised to e-mail me the link to the pictures when she uploaded them.
That night, as was my wont, I wandered over to Mayor Fred's Saloon. I lived and also worked part-time at the Bottroff House Bed and Breakfast, located on Eaton Street, just off Duval in Old Town. Mayor Fred's was a couple of blocks away on Greene Street, built around a big ficus tree (tourist web sites will tell you that it was Key West's hanging tree in the nineteenth century; they won't tell you that it's also a root of Yggdrasil, the world tree of Norse myth). I was always there to see my friends in the band 1812 play Thursday through Sunday. Of course, it was Wednesday, so the bar was less crowded, and also the music was provided, not by a four-piece band on the main stage in the back, but by an acoustic act over near the entrance.
I didn't recognize the person playing guitar, but I remembered Ihor, the bartender, telling me that the two guys with beards whose names I could never remember got a gig at a place up in Key Largo. The new guy was a painfully thin, absurdly pale, tall guy with a hooked nose and long, stringy brown hair. He also had a lovely voice—he was singing "Scarborough Fair" and doing it justice—and he played his battered old Yamaha acoustic guitar quite skillfully. The guitar had stickers from various cities on it, making it look like a tourist's suitcase from 1957.
I did recognize one of the people at the bar, though: Larry, who was the textbook definition of "regular." Every single day, from the moment Mayor Fred's opened at midday to the second it closed at four a.m., Larry was at that bar, guzzling coffee or a soda. He was the immortal barfly I listed among my acquaintances before, and he got that way by falling in love with a water elemental and then leaving her. As soon as he falls asleep, he'll die—hence the all-caffeine-all-the-time diet—but until he falls asleep, he'll continue to live. He's spent eternity to date at Mayor Fred's, going back to when Hemingway was a regular. I had no idea where he went from four to eleven in the morning, though I figured he must live somewhere. Someday I'd ask.
"What's the word, Cassie?" he asked me as I sat next to him.
I took a deep breath and then let it out in a single burst. "Weird" was the word I finally agreed on.
Larry laughed. I was the first person to answer Larry's rather old-fashioned greeting with an actual word, and it had become our thing. "I'd ask what's weird in your life, but I'm not convinced we'll have enough time for that."
"Yeah, well. I had a doozy on the afternoon dive."
After I told Larry the story of Al Martinez and Soleado, he got a faraway look in his eyes. "Wow. That takes me back."
This was interesting. "Back to what?"
He waved a hand around. "About a hundred years, give or take." He patted the pocket of his shirt to make sure his pack of cigarettes was in it. "C'mon outside, I need to smoke."
Florida state law actually allowed smoking in bars, but you still couldn't smoke in Mayor Fred's. Ihor, who was both the night bartender and the general manager, had asthma and reacted very badly to second-hand smoke, so he had to ban smoking in order to not collapse in a wheezing heap while doing his job. To Larry's credit, he remained loyal to Mayor Fred's even after the smoking ban went into effect and stuck around, rather than patronize one of the other gajillion bars on the island that would let him suck nicotine where he sat. Of course, that didn't stop him from complaining about it all the time...
I followed him out under the giant fish over the main entrance to the Greene Street sidewalk where he lit up. "Back around—oh, Jesus, Mary, and Joseph, I can't remember the damn dates, exactly, but it was definitely in the twenties, because I do remember that Harding was president. In any case, I used to spend my leisure hours at an exclusive cigar club down on Flagler. There was a fella named Ruben." He chuckled. "Ruben the Cuban, we called him. He sailed up from Havana and opened up one of the cigar factories. He and I would play cribbage over brandy and cigars."
Holding up a hand, I asked, "Wait, brandy? I thought Harding was president during Prohibition."
Larry just chuckled. "Letter of the law, sure, booze was illegal, but around here that didn't mean a 'hill'a beans. Treasury Department never really got down this far south, and when they did, they took the railroad, so we knew they were coming." At my confused look, he added, "Tickets had a note on them that said they were employees of the federal government, so if one boarded a train to Key West, the conductor would get on the wire and warn all the speaks to hide the booze."
"Didn't realize you were such an outlaw."
"Drinkin' brandy with my friend didn't make me an outlaw. It just made me another person on the island." He shook his head. "Now you've turned my head around with all this foolishness. What were we talking about?"
"Ruben the Cuban," I prompted.
"Right, so Ruben loved to fish, and kept needling me to come along with him. After what happened with Anne, I wasn't all that tickled to go out to sea, but he finally wore me down by making a wager out of it." Anne was the name his water fae ex used when she was in human form. He went on. "If I lost at cribbage, I'd fish with him. If I won, he'd give me a box of his factory's finest."
I smiled. "I take it you lost, or there wouldn't be a story."
He looked up at me as he took a puff. "You know that smart mouth of yours is gonna get you in Dutch one of these days."
"I think it's cute that you believe it hasn't already. So what happened on the fishing trip?"
Taking a longer drag on his cigarette, he inhaled, paused, and exhaled a puff of smoke before finally answering me. "All right, I don't wanna be unkind to Ruben, rest his soul, but that was one of the most boring afternoons of my life."
"Given how long your life has been, that's saying something."
"You said it." He shook his head. "After we whiled away most of the day without a single bite, we finally hauled anchor and went back home—and that's when we saw it. At first, we thought it was a sea lion, at least until, as God is my witness, I saw a woman's head. Looked like she was in trouble, so I dove into the water with one of the cork vests."
"No life-preserver?" I asked in surprise. We had ten inflatable donuts on each of the dive-shop boats.
"Not like what you're thinking—those got invented later. Back then, we just had a couple cork vests, and I swam out to put one on the lady so we could rescue her. Except she didn't need rescuing. She swam under me and came onto Ruben's boat. I was about fifty yards away when I realized she'd snuck past me, so I turned around to double back. I only caught a few glimpses, but—well, it sure as shootin' looked like a mermaid. Woman's head, long tail, and the body was scaly and strange. Reminded me a little of Anne's true form, to be honest—only got a gander of her like that once, but it was a doozy."
I nodded. As a Dís, I'm immune to disguises of any kind, so I only saw Larry's ex in her true, seaweed-encrusted, yucky form. He was lucky to only get the one gander.
"I did catch a glimpse of something peculiar. It was beautiful that day, sky blue as a marble, and not a cloud to be seen. But for a few minutes, the sky changed, went all green. Now, I spent some time in Kansas back in the day, and I've seen the sky turn that color right before a twister. But down here in Florida? Never seen the like, and never so sudden." He took a final puff, then dropped the cigarette onto the sidewalk and stepped on it. "By the time I got to the boat, though, the sky was normal and she was gone. And Ruben, he was just sitting there, carrying on to the nines about his brother back home in Cuba and how he wasn't gonna see him ever again. He wouldn't move, he barely blinked, he just sat around like a lump, and every time I tried to talk to him, he was just a broken record about his brother."
This was all sounding annoyingly familiar. Also, the tornado warning Cole told me about was sounding less far-fetched all of a sudden. "Then what happened?"
"I managed to steer the boat back to shore, but it wasn't easy since I hadn't been on a boat in decades. Ruben wasn't ever the same after that. Eventually, he became more himself, but we only spoke a few more times here and there. He lost interest in the club, in cribbage, in drinking, and eventually even in work. He sold the factory, moved back to Havana, and I never saw him again."
His cigarette done, Larry headed back inside under the giant fish. I followed him, and as I entered, I heard the guitar player start his next song with the words:
"One Friday morn as we'd set sail
And our ship not far from land
We there did espy a fair mermaid
With a comb and a glass in her hand..."
I did a double take as I was walking toward the bar, and stopped suddenly. Unfortunately, a tourist was walking toward the restroom from the bar, and crashed right into me, spilling his beer. I apologized and offered to buy him a new one, but he muttered that it was okay and kept going to the men's room.
As I sat down at the bar, where a pint of beer was waiting for me (it's good to be a regular), I caught the rest of the song, and was even more freaked out.
"Then up spoke the captain of our gallant ship
Who at once did our peril see
I have married a wife in fair London town,
And tonight she'll a widow be.'
And then up spoke the little cabin boy,
And a fair-haired boy was he.
I've a father and a mother in fair Portsmouth town,
And this night she will weep for me.'"
I wound up gulping down two-thirds of the pint at once.
After he finished the song, he said, "And now I take a pause. If you enjoyed what you heard, please make your pleasure known." With that, he held up the tip jar on the stool in front of him. Surprisingly, he had no CDs for sale, no cards listing his website, or anything like that.
I approached him as he put his Yamaha in its case. "What was that last song?"
"It's called 'The Wrecked Ship.' Based on an old sea shanty, I think. Always loved that one."
"Yeah."
A few days passed, and I didn't hear anything new about mermaids or boats going missing or tornado warnings or people muttering about the people they left back home. I did hear that Al Martinez quit Azucar to move back to Tampa. The owner of Azucar told me that right before he tried to poach me from Seaclipse, a request that I rejected, though it was flattering. If nothing else, it gave me leverage to bug my bosses for a raise.
Didn't know what happened to the four tourists who jammed into the galley with Al, but a little Google-fu revealed more about Ruben the Cuban. His real name was Ruben Hernandez Jr., owner of the prosaically named Hernandez Cigar Factory on Front Street. He sold it in 1922, when Warren Harding really was president, and he moved back to Cuba. The new owners of the factory kept the name, until it was badly damaged by the Labor Day Hurricane in 1935—same one that took out the railroad, actually—and it shut down for good.
Unsurprisingly, I found nothing online that mentioned that Hernandez saw a mermaid. Still, the details I could find matched Larry's story.
I also did a bit of research into mermaid legends, though there was a lot of stuff from a lot of different regions, and not much to match what I'd heard beyond the song the guy at Mayor Fred's did. I was amused to read that Christopher Columbus made references to seeing mermaids in his journals written during his infamous sea voyage of 1492, though some modern folk assume that he and his crew really saw manatees or rays and were just really hard up for sexual companionship.
None of my dives took me anywhere near Pickles Reef until the following Monday when the South Dakota Seafarers came back to the Keys. The SDS were a bunch of retirees who got together to do various water-related things, and every three months or so, a group of them came down here. This quarter, I got five of them, and they had their hearts set on diving the wreck of the Duane, a Coastie ship that was deliberately sunk back in the eighties to create an artificial reef. Wreck divers love the Duane, so I usually wind up going there at least twice a month.
We headed there in Chico this time, on a partly cloudy day with decent winds. Not enough to make the water choppy, thankfully.
Then the sky turned green.
For a second, I thought I imagined it. The sky just went from blue and white to an emerald-green instantly. And then it changed back just as fast.
Before I even had time to process this—or figure out how to answer the inevitable questions from a quintet of senior citizens as to what the hell was going on—I saw a hundred-and-twenty-five-foot gaff-rigged schooner hauling ass across our path. They were at full sail and pootling along at eight knots or so. Frowning, I double checked the charts and realized that they were heading straight for a shallow reef, and if they didn't change course in about thirty seconds, they were going to crash right into it.
Then I caught sight of the logo on the side, and realized it was the Lilly, a local schooner that did intimate little cruises for small parties in and around the Bahamas and the Keys. Her captain was Meg Michaels, and she wasn't usually batshit crazy. In fact, the last time I'd talked to her was about a month ago when she came back from a triumphant second-place finish in the Great Chesapeake Bay Schooner Race, and a bunch of us, including all twelve of her crew, got seriously drunk at Mayor Fred's to celebrate.
I snagged the radio. "Chico calling Schooner Lilly. Come about, you're gonna hit a reef. Meg? It's Cassie, you there?" I shook my head. "Fuck!" I hit the throttle, even though I knew I wasn't gonna make it in time.
I always used to make fun of TV shows and movies where they'd go into slow motion when something bad happens. After watching Lilly slam into the reef, I stopped doing that. It took maybe a second and a half for it to go from going full-bore through the ocean, to jutting at an angle up out of the water with a big-ass hole in the hull, but watching it felt like at least a full minute.
After sending out a general distress call, which would bring Cole and his buddies out here, as well as any other boats in the area, I set course for what was now a wreck.
Just like with Soleado, I pulled alongside Lilly, though I was careful to take it slow. At twenty-nine feet, Chico was able to handle the reef better than a schooner four times its length, but I didn't want to take any risks.
I tied us to Lilly's transom, which was the least damaged part of her, and then hopped on board.
Just like with Soleado, the crew was all bunched belowdecks, this time in the captain's cabin. (Which told me right there that supernatural forces were at work. Meg didn't let anyone in her cabin for any reason. In related news, the place was a mess and smelled kinda funky.) I got the same litany of family members they thought they'd never see again, from Meg's brother in Boston, to the chief mate's children in Norfolk, to the cook's husband and kids in New York, to the engineer's parents in San Francisco, and so on—just like Soleado, and just like that song.
Cole showed up soon enough, as did two other boats. He regarded me with concern. "This is some very bizarre stuff happening, Cassie."
"You ain't kiddin'."
That night, I returned to the Bottroff House, where I shared my room with the ghost of the wrecker captain who originally built the place, Captain Jeremiah Bottroff. As a Dís, I was the only person who could see or hear the captain, which meant I pretty much got stuck with his company, since 'I've been his only regular source of conversation for the past century and a half.
"I got to see how your job used to be today," I said as I climbed into bed.
"And how's that, exactly?" Bottroff asked.
I told him about Lilly—as well as Soleado, and I threw in Larry's fish story while I was at it—which prompted a derisive snort from Bottroff.
"If you didn't tow the vessel back to shore, if you didn't claim salvage of a percentage of its cargo in exchange for rescuing them, then what you did was nothing like my own travails as a wrecker." Back in the nineteenth century, wreckers like Bottroff would rescue boats that crashed on reefs and then do all that stuff he just said. When I first got here, I thought that meant he was a pirate, but it was actually all completely legal, and also seriously regulated. The practice faded away as boat construction improved to the point where those kinds of wrecks weren't everyday occurrences.
"Yeah, the Coasties handled the really fun parts."
"Actually," Bottroff said, "your story of the vessel Soleado does have a ring of familiarity to it, as does your immortal friend's tale of woe. I too encountered a vessel that claimed to have seen mermaids. They, too, babbled about their homes and family. At the time I dismissed them as madmen, but in light of what I've seen in the days since my death..."
"So what happened?"
"In truth, I only recall the incident, not due to the fatuous ramblings of the passengers and crew, but because I was forbidden from claiming salvage. The boat did not actually come upon a reef, nor was it damaged in any way. Rather it was adrift, and while we did indeed tow them back to shore, the judge denied our salvage claim. As you can imagine, my boys and I were rather upset."
That night, I dreamed of a mermaid swimming around one of Seaclipse's boats, with me and Larry and the guitar player on it, and I started wishing I could see my parents back in La Jolla again. Since I still hadn't seen the thing, my brain decided to make the mermaid look like the Disney character Ariel from The Little Mermaid.
The next day, I was working at the B&B all day, with no dives to run, and that night I headed to the open mic at Mayor Fred's. Larry was there, of course, as was Ginny. The drummer in 1812, as well as being a Norse god and the ex-wife of the trickster Loki, Ginny had been encouraged by the rest of the band to do some singing in addition to her excellent drum work. She wanted to try it at the open mic first, and if she was comfortable with it, she'd sing with the band onstage. Their already-impressive repertoire of covers (and harmonies) would be increased greatly by adding a fourth voice to the mix.
Because Ginny was there, a third familiar face sat at the bar: Loki, who was trying (and so far, to my delight, failing) to win Ginny back.
Loki greeted me with the supercilious smile he always used. "Larry informs me that you encountered a havsrå."
I blinked. Then I remembered my crash course in Norse myth that I imposed upon myself after learning I was a Dís eight months ago. "Right, that's what you guys called mermaids."
"In a manner of speaking. In truth, the only reason havsrå have been seen in Midgard or Asgard is because of a little joke I played on Thor once. You see, he had just married Sif, even though I kept telling her my cousin would have sexual congress with virtually any female he could get his hands on. To prove his lack of faithfulness, I summoned a havsrå from another of the Nine Worlds, and Thor did indeed attempt to ravish her."
I let out a long breath. "Let me guess, the sky is emerald-green on that particular world?"
"I believe so, yes." Loki grinned. "Sadly, in the end, no one was happy. The havsrå wished only to go home, Sif was furious at me for starting it all, and as for Thor, he faced certain—logistical difficulties."
I chuckled and shook my head. "Couldn't fit Tab A into Slot B?"
"Well, a havsrå has no, ah, 'Slot B' as such. Sif forced me to send the creature back, and I did. I suppose it's possible others of her kind have slipped between the worlds and arrived here."
"Just to board ships and make them act weird? It doesn't make sense."
Ginny was walking up to the mic now, and started singing an a cappella song called "Seven Bridges Road," which pretty much ended the conversation. Enraptured, Loki watched her sing the song in a lovely alto, and he clapped the loudest when she was done.
To be fair, I clapped the second loudest. She absolutely nailed it. When she came back to the bar to hearty congratulations from me, Larry, Loki, and the entire rest of the bar, I asked her, "Why didn't you sing before?"
"I must confess, Cassie, that, at this moment, I cannot imagine why."
Loki smiled. "I always knew you had a voice that could move the heavens themselves, Sigyn."
"Thank you."
I shook my head. On the one hand, Loki was a prime asshole. On the other hand, he really did love Ginny.
Two days later, I was in Harpo, taking three sisters back from a dive near some lovely coral, when I saw a sea lion swim by the boat. Scared the hell out of me at first, but I got a good look at it, and it was an actual sea lion. I found myself remembering the jokes about sea lions being mermaid dogs.
Just after I caught my breath and recovered, I saw another sea lion tail—but it was green. And then the head popped out, and it looked kind of like a person. The face was more or less feminine, with hair that extended down past her shoulders, but looking a lot like a pelt.
The tail was definitely that of a sea creature, and the head looked like a woman with funny hair, although her eyes were wide enough to qualify as an anime character; she didn't have a nose so much as two slits above her upper lip, and her lips were huge and full. But the weird part was her body. She had a flexible torso that didn't look like a sea lion or a woman—no boobs, no obvious thorax, just a stretch of skin and bone that linked the head to the tail. Two arms grew out of that torso, with no elbows, but ending with three-fingered hands, complete with opposable thumbs.
I also realized I couldn't see her skin directly. It was wet and matted and short, but her entire body was covered with the green pelt, making her appear a tiny bit like a green polar bear.
She was holding something in one hand that looked like a heavily serrated blade, and a glass ball in the other. I shook my head—there's your glass and comb from the song.
"Wow, she's beautiful." That was one of the sisters, who obviously had a different definition of "beautiful" than mine. Then again, the creature might've been using a glamour to appear more attractive to whoever's looking at her. After all, most of the mermaid stories, from Columbus to Captain Bottroff to Larry to the Walt Disney Company, have them looking all pretty, and if they saw what I was seeing, they wouldn't think that.
Anyhow, the mermaid—or whatever—burst through the surface just like a sea lion and flopped onto the deck of Harpo.
Then she just stared at us.
The sky turned emerald green—the same color as the mermaid's pelt.
The three sisters ran belowdecks and crammed themselves into the head.
I didn't move.
I felt—something in my head. A weird longing... and an overwhelming sense of wanting to be back in La Jolla.
No, that wasn't it. The images in my head were of the house I grew up in outside San Diego, but I wasn't thinking of the house specifically.
I was thinking about home.
I need to go home.
That wasn't me thinking that, even though the thought was in my head.
After a minute, I realized it was the mermaid who wanted to go home.
Fuck me.
"How can I get you home?"
You understand me? Finally, someone who understands me!
"I guess I do, yeah."
Are you of the Aesir?
I snorted. That was what Odin, Thor, Loki, and the other Norse gods liked to call themselves. "Sorta."
Their trickster kidnapped our sister once, and since then, the boundaries between the worlds have weakened. Sometimes we fall into this world and must find our way home to our sisters. I miss them so!
She slithered toward me on the deck, and I saved her some trouble by moving toward her, and as I approached, she held up the glass ball.
This talisman may return me home, but they do not function.
"May I?"
Of course.
I took the ball, but I had no idea what it was, really, or what it did. And me holding it had no effect on it. Sometimes, when I touched something, it acted weird. But not this time. Have I mentioned I didn't get an instruction manual?
How-some-ever, I knew someone who could help.
"My name is Cassie. I'm one of the Dísir. I'm—acquainted with the trickster who kidnapped your sister back in the day, and I can learn from him how to get you home. Can you meet me back here later tonight—say in six hours?"
Very well. And thank you, Dís.
"My pleasure." I handed her the ball back.
The mermaid—or, I guess, havsrå—took it and then dove back over the side of the boat. I ran down to the head, where the three women were just sitting there.
"My husband's back home in Philadelphia. I'll never see him again."
"My daughter's back home in Perth Amboy. I'll never see her again."
"My wife's back home in Tarrytown. I'll never see her again."
I sighed. I knew I forgot to ask the havsrå something.
That night, I went to Mayor Fred's, where 1812 was playing with the full band, including Ginny on drums, which meant Loki was in the audience. I walked in, grabbed his ear, and dragged him out onto Greene Street.
"Ow! What are you doing, little Dís?" He only called me that when he wanted to piss me off, but he'd already done that more than he realized.
"In April, when it snowed and the world was almost destroyed, that was your fault. Last month when the ghosts on the island got super-active and everyone could see and hear them, and one of them killed people, that was your fault. And now havsrå are causing shipwrecks and have been for centuries. Guess what? Your fault again. What the fuck is wrong with you?"
"How is this my fault, exactly?" he asked archly.
I told him what the havsrå told me.
His voice much more subdued with all the high dudgeon gone, he said, "Ah, I see. Sadly, Cassie, I cannot help you. While the spell itself is rather simple, it takes great power to bridge the gap between worlds to send the havsrå home. I am but a shadow of my former self. Even when the peoples of northern Europe worshipped the Aesir as gods, and my power was at its absolute height, it was difficult for me to manage it. Now, there is no chance of it."
"She has two talismans—a glass ball and a multi-bladed knife of some kind."
"I know not what the knife would do, but the glass ball is likely an orb, which can focus magic very much the way a lens focuses light. That will merely direct the spell, not create it."
"Fine, you said the spell was simple. Teach it to me. I know I've got the mojo." I grinned. "How do you think I stopped you?"
Raising a blond eyebrow, he then asked, "And what reason do I have to do you this kindness?"
I blinked. "I beg your pardon?"
"What boon shall you grant me in exchange for doing you this service?"
For a moment, I just stared at him. "My parents have a word for what you just did. It's chutzpah. And if you really want a boon, how about this? I won't have to tell Ginny that you were a total douchenozzle when I asked you to help me out. On the other hand, if you do help me, I could tell the woman you're trying to win back that you selflessly helped me get a havsrå back home."
He seemed to consider it, but I knew him well enough at this point to know damn well that he wouldn't do anything to jeopardize his chances with softening Ginny toward him. "Very well. Shall we retire to your dwelling?"
I sighed. The notion of Loki in my room did not appeal, but the havsrå needed to get home before somebody got hurt. So we went back to the Bottroff House and, over the strenuous objections of the captain, had a spellcasting tutoring session. The language of the spell was in the same unrecognizable tongue as the spell I used to stop Loki in April, which helped me pick up the cadence faster.
Once we were done, and I was sure I had the spell's words and proper pronunciation memorized, Loki said, "Excellent. Good luck, Cassie."
"Oh, I'm gonna have more than luck. You're coming with me."
"But—"
"But me no buts, butthead. I'm not taking any chances, and I need backup, even if it's just—" I shook my head, laughing at the irony even as I said it. "For moral support."
Loki grinned. "Not my usual type of support, I must say."
We drove out to Stock Island. I had already convinced my bosses at Seaclipse to let me take one of the boats out on my own, on the condition that I'd pay for the gas used on the trip. Luckily, there were only enough evening-dive signups for one boat, so they had two to spare.
Loki and I went out in Groucho. He looked really nauseated as we went, and by the time we reached the rendezvous spot, he was almost as green as the havsrå.
Speaking of whom, she was waiting for us in the same place as planned. She once again flopped onto the deck.
I am glad you returned, Dís. I was not sure you would. Our experiences with the Aesir are not ones that engender trust.
She was looking right at Loki. So was I. "You know who this is?" True, I'd said that I'd be consulting Loki, but he was a shapechanger, and I doubt he looked the same now as he did when he kidnapped her sister in order to tweak Thor and Sif.
We all are well aware of the trickster who ripped our sister from us.
"My apologies, sweet lady." Loki bowed and almost came close to sounding like he was trying to be sincere. "I was young and foolish, and was poor at thinking through the consequences of my actions."
"And you're using the past tense, why, exactly?"
Now Loki gave me a look. "Shall we begin?"
The havsrå handed me the orb. You will need this.
"Thank you." I took the orb from her and then started to chant the familiar words in the unfamiliar language. As I spoke, I felt something tug at my heart, and the orb started to glow from within.
The sky turned green, and a small whirlpool started forming in the water just off our port bow.
Thank you, Cassie of the Dísir. You have the eternal gratitude of the havsrå for what you have done today.
Then she turned to Loki and held up the knife that sea shanty writers had mistaken for a comb.
And thank you also for allowing us to at last have our vengeance.
With that, she slashed Loki's throat with the blades. Blood started spurting all over the deck, and Loki fell to his knees, a look of total surprise on his face. I grabbed him and eased him down onto the deck.
Turning to the havsrå, I cried, "What the fuck?"
Since the day our sister was returned to us from Asgard after being wretchedly ill-treated by the Aesir, we have travelled with two items in our possession. These were to be used if we fell through one of the many portals between worlds that this trickster created for his sport and for our agony. The orb could be used to return us home, though it does not always function as it should. The blade was to be used to avenge our sister for her mistreatment at this foul creature's hands, should we ever be fortunate enough to cross his path once again. Good-bye, Cassie of the Dísir. Fare you well. And good-bye Loki of the Aesir. Fare you poorly.
And then she dove over the railing and into the whirlpool, which closed up behind her, the water becoming still once again.
Loki stared upward. The sun was starting to set, painting the sky a spectacular orange and purple.
His voice a gurgling croak, he said, "Sigyn is—is back in Key West. I'll never see her beautiful face again."
Then he just faded away. Seriously, he went transparent and then disappeared. Even the blood he got all over Groucho's deck was gone (which, if nothing else, meant I wouldn't be stuck cleaning it up). Gods don't die the way we do—if they even really die at all.
But right now, Loki was gone. How the fuck was I gonna explain this to Ginny?
I sat completely alone on the deck, the orb rolling around next to me as Groucho bounced with the tide.
Whilst the raging seas do roar,
And the lofty winds to blow,
And we poor seamen do lie on the top
Whilst the landmen lies below.
On....The Mermaid
Child lists six versions of Ballad 289 while the Roud Folksong Index lists two hundred and sixty-two versions of #124, The Mermaid. When the crew of a ship on the ocean spies a mermaid, they know they are in trouble—mermaids were a portent of disaster. Even seeing one with a mirror and comb didn't forestall their troubles. In some versions, the Captain, Mate and Boatswain all mourn that soon their wives will be widows while the cabin boy bemoans his parents' grief. Whether the ship goes around three times in some versions or was en route to Greenland in another, they all testify to how important shipping was and how futile it was to fight the power of the sea. (1)
As shown on the YouTube uploads this is a popular ballad with amateur singers. Others such as Martain Carthy, Celtic Mayhem, The Pirates Of St. Piran, Celtic Stew, and The Sharecroppers have also recorded The Mermaid.
1) Child, Francis James. The English and Scottish Popular Ballads Vol. V. Mineola: Dover, 2003.
# MAKING MUSIC
By
Kelley Armstrong
Izzy sat in front of her computer, headphones on, eyes closed, listening to the latest top five rock songs, looping them over and over, struggling to find meaning in the lyrics. To find lyricism, poetry. Hell, at this point she'd settle for a clever rhyming couplet. She'd sampled the top country hits, but that wasn't really her forte. She loved good old rock-and-roll, not surprising, given that if she'd been a boy, her father had planned to name her Ozzy. An old-school roadie for a father. And an English lit professor for a mom. That particular mix was partly what landed her in this very place, at this desk, so deeply engrossed in her task that sweat had broken out along her hairline.
When the phone rang, she jumped, her headphones tumbling off, almost taking one dangling earring with them. She winced as she disentangled the earring with one hand and lifted her phone with the other.
"I have news," her agent trilled, somehow managing to get three syllables out of the last word.
"Let me guess," Izzy said. "There's a newly formed punk group out of Nowhere, Missouri, that wants to give me amazing exposure by covering 'The Unquiet Grave' free of charge." She paused. "No, wait. They're asking me to pay them, right?"
Monica sighed. "You are such a pessimist, Iz."
"No, I've just had that generous offer too many times. Sometimes brought to me by my agent, no less."
She swore she heard Monica's lacquered pageboy crinkle as she bristled. "That was a successful punk rock group from Kansas City. It would have been good exposure . . . before Grave. You are now a bona fide songwriting star, my girl. I head them off at the pass if there isn't money attached. Significant money."
Izzy eyed the bills stacked on her desk. "Actually, if there's any money attached, I'd like to hear—"
"Significant. Otherwise, you look desperate."
True, except for the fact that she was desperate. "The Unquiet Grave" was two years old now, and she was the writer, not the performer. It hadn't exactly made her rich.
"Do you want to hear my news?" One syllable for news now. Apparently, Izzy had not displayed the proper degree of gratitude.
"Sure."
"A major recording artist heard the NPR retrospective of your work and has contacted me directly. Directly. Not through his agent. Not through his manager. He wishes to speak to you about writing a large amount of his next album."
"Uh-huh."
The line echoed in silence. Then Monica said, "Did you hear me, Iz?"
"I did. That's why I said 'uh-huh.' I'm just waiting for a name attached to that 'major recording artist.'"
"I'm not at liberty to say."
Izzy shifted the phone to her other ear. "What?"
"He'd like me to keep his name out of it until you two meet in New York tomorrow—"
"New York? I'm expected to meet someone in New York tomorrow, at my expense, I presume, without even getting a name? Hell, I wouldn't drop everything and fly out for Bruce Springsteen."
"I should hope not. Have you seen the sales on his last album? I swear, girl, you are the oldest twenty-nine-year-old I have ever met. I'm actually glad I can't give his name because you'd probably say 'huh?' despite the fact his last three albums went plat-i-num."
"Give me a try."
Monica snorted. "Not likely. I made a deal. As for flying your ass out there, he's comping the ticket. First class. Have you ever even flown first class, Iz?"
"Sure, I got an upgrade once."
"Pack your bag. Gather your portfolio. You are going to New York at the crack of dawn, and when you find out who brought you there, just remember, I like Bollinger. Cristal is for thugs."
Izzy sat on the bank of Bank Rock Bay in Central Park, listening to the pound of feet jogging across the Oak Bridge. She closed her eyes and imagined them as drumbeats. There was music there, as there was everywhere. One only had to find it. She knew that better than anyone.
She fingered her portfolio of sheet music. Old-fashioned, Monica would sniff, but Izzy found music in the medium, in the crackle of paper, in the hand-scratched black lines. She'd brought an iPod, too, with recordings of her latest songs, but she preferred the sheets. An old-fashioned medium for an old-fashioned art. The crafting of music from poetry.
Even in rock, there was a place for poetry. For beauty. For balladry. She created both—turning old ballads into rousing rock anthems. It was, as one might expect, a very narrow specialty. Despite this, critically acclaimed artists weren't exactly clamoring to work with her. Even Whiskey Roar—who'd seen their first gold album based largely on the success of her re-imagining of "The Unquiet Grave"—had only contacted her once more for a song, buried deep on their second album, as if an act of charity.
"'Tis down in yonder garden green,
Love, where we used to walk.
The finest flower that ere was seen ..."
When Izzy heard the words, she thought she was still lost in her thoughts. Then she realized they were real and she jerked upright.
"Just don't ask me to recite more." The man's figure was almost hidden in the dark shade as he walked toward her. "That's the only part I memorized and only to impress you."
"But that stanza isn't in my song."
"Which makes it even more impressive, right?"
He stepped into the sun. A slender man in his mid-thirties. Dark hair falling into his eyes. A boyish grin. Glittering blue eyes. He was dressed like any other park-goer—in jeans, a T-shirt, and old sneakers—but the moment she saw him, her jaw dropped.
Shit. Holy fucking shit.
She made a mental note to send Monica her Bollinger, whether this meeting panned out or not. For once, her agent had earned it.
"Beau Wallace," she said.
"That's what my driver's license says. Or so I think. Haven't used it in a few years."
Before she could stand, he plunked himself down on the bank beside her without so much as a fastidious glance at the grass and dirt.
Beau Wallace. Monica might lament her client's old-school tastes, but Izzy sure as hell knew who this was. Former boy-band crooner turned mega-selling solo star. If she was being perfectly honest, she might admit to a Beau Wallace poster in her bedroom when she was twelve. These days, he got a little too much airplay on light-rock stations, but she still had a few of his hits on her playlist.
"I want to branch out," he said, as if reading her mind. "As flattering as it is to top the easy-listening charts, after a while"—he lowered his voice conspiratorially—"it's not so flattering, if you know what I mean. I want to go harder, roughen my edge. Maybe I'm kidding myself to think I have an edge to roughen but . . ." He shrugged. "I want to try."
"Sure." A lame answer, but it was all she could manage, her inner adolescent shrieking, "I'm talking to Beau Wallace!"
He glanced toward the path. A woman had appeared there. About Izzy's age, pretty but unsmiling, with a dark ponytail and darker shades. Dressed, like Beau, in jeans and sneakers, but wearing a denim jacket despite the warm late spring day.
"No, not a stalker fan," Beau said, nodding to the woman. He raised his voice. "Not a fan at all, are you, Jill?"
"I like your work just fine, sir." She tilted her head. "Some of it anyway."
Beau laughed, and though the woman—Jill—didn't crack a smile, Izzy got the feeling this was an old joke between them.
"Jill is my bodyguard," he said. "But right now, I don't think it needs guarding. Go amuse yourself, Jilly. Give me an hour."
The woman hesitated, but at a look from Beau, she dropped her chin in a nod and strode off.
"As I was saying," Beau continued, "I want a change and a challenge. But I don't want to forget my roots, either. I'm a balladeer, and that might chafe, but ballads have been good to me. What I want, then, is to acknowledge my roots and tweak them. If you know what I mean."
She did, better than he could imagine, but she only said, "You want a real ballad. With a heavymetal beat."
Another laugh, as easy as the one he'd given Jill. "Well, I wouldn't go that far. Heavy-metal isn't quite my scene or my audience. But I want rock. Solid rock."
"Old school."
"Exactly. More AC/DC, less Iron Maiden. More Zeppelin, less Sabbath. Can you do that?"
Izzy smiled. "With a little help from you, I sure can."
Most creative types were flakes. Flakes with money were worse. They'd think nothing of flying her out, wining and dining her, making vague pronouncements about their hopes and dreams, telling her how much they loved-loved-loved her work and were dying to collaborate with her...and then never contacting her again, their whims having drifted elsewhere, like a leaf floating downstream.
Beau Wallace was not one of those guys. He knew exactly what he wanted, and the minute she agreed to discuss it, he set to work. He'd brought a list of his favorite ballads, not only in order of preference but identifying the elements that spoke to him. They sat on that bank until Izzy's butt numbed. Jill came by half a dozen times, only to be waved away by Beau, too intent on their discussion to spare a word for her. The bodyguard didn't seem to mind. She brought coffees at the midway mark and then continued doing whatever she'd been doing, periodically checking in.
Then, after they were done talking business, he relaxed, seeming content to linger, looking out at the water and chatting. Chatting about her no less. Where did she get her interest in ballads? Which were her favorites? Had she ever considered folk music? They'd laughed about that. Folk was the obvious choice for ballads, but neither of them had any interest in it.
Soon the sun was dropping, the light playing on the water, and Izzy commented on that.
"It looks like music," she said.
He glanced over quizzically. "It does?"
She pointed out the sun's reflection on the ripples and started singing the notes as the sun bobbed between the ripple "lines."
Beau grinned. "Okay, I get it. Not a bad tune either. A little slow though. The wind needs to pick up."
He did get it. Few of his kind did.
She smiled. "It'll pick up tonight. And I bet the moon works just as well."
His grin turned wolfish. "Are you saying you want to see the moon on the water with me, Isabella?"
Her cheeks heated. "No, of course not. I just—"
"Damn."
The grin changed, simple boyishness now, putting her at ease again. They talked some more. Before they left, he told her where he was staying—the hotel and the alias. "In case you decide you do want to see the moon on the water." Another grin, one that said he was just kidding...unless she'd rather he wasn't. She'd blushed, said good night, and hurried off.
Izzy watched the moon play on the water. The bank in Central Park again, but not near the Oak Bridge. That was a little too public, even at this time of night.
She'd called Beau at his hotel just past midnight. Told him she had a song for him—the perfect song—and she wanted to show it to him in the perfect location—where they'd discussed it that morning. A bullshit story, of course, but he'd been the one to nudge-nudge-wink-wink about the moon on the water, so the moment she suggested it, he likely figured he was getting a rumble in the Ramble and hopped to it.
He was a bit of a fool, really, which was disappointing. All humans were, of course. Herself included, having joined their ranks nearly thirty years ago, after her rebirth. A regrettable but necessary transition. The world was no longer a safe place for fae. In the modern, wired-in world, people noticed when you didn't age. So she'd undergone the process of death and resurrection, brought back as a babe and exchanged with a human one, becoming human herself by feeding on her new mother's milk.
Some parts of her fae self remained, primarily her love of music. No, more than love. It was the stuff of life. She consumed it and was, in turn, consumed by it. Which led to a problem with only one solution.
Beau Wallace appeared at the stroke of two, like a shining faery prince. There was some fae blood in him—she'd seen that when he'd spoken of his music, and she could see it now, shimmering from his skin in the moonlight. She still felt that girlish flutter inside, seeing in the flesh the face that had once adorned her wall. She was, after all, human now. Mostly.
"So you have a song for me?" he said as he strolled to the bank.
"No, you have one for me."
His grin faltered for perhaps the first time since they'd met. She stepped forward and looked him in the eyes, calling forth every bit of fae charm she still possessed.
"You have a song for me," she said. "The sweetest, purest song I have ever heard." She took another step, her gaze fixed on his. "I need your song, Beau. I need it the way you need air to breathe. Your song feeds mine, and without mine, I would wither and die for wanting. I've taken seven songs before yours. Seven wonderful songs from seven wonderful men, and they live on, through me, through my music. That's what you want, isn't it? To live forever? Through music?"
Her lips went to his. It was easy. Always so easy. They looked into her eyes and they heard her words and they breathed their song—with their life—into her and—
He yanked back from the kiss. "What the hell?"
She reached for him, but he staggered out of her reach, his face screwed up. "No, seriously, what the fucking hell are you on, Isabella?"
Okay, maybe not so easy this time. Damn it. Those few drops of fae blood seemed to inoculate him to her charms.
She dropped her face into her hands. "Oh my God. I'm so embarrassed. You're right. I took something this guy offered me earlier, and I don't usually do that and . . ." She broke off on a sob.
"Okay, okay," he said. "Let's just get you out of here."
She cried louder, waiting for him to come over and comfort her.
"Enough of that," he said, and there was no sympathy in his voice, only annoyance. "Let's get you a cab. We'll—"
She sprang. She caught him by the throat, hands wrapping around it, cutting off his gasp. She pressed her lips to his again, kissing him and drawing out his—
He punched her in the stomach. Hard. She didn't let go, but squeezed his throat tighter, pulling out his breath with her kiss while choking it out with her hands. He went slack, finally giving in to her charm, kissing her back even, reluctantly at first, then picking up, kissing her hard, feeding her his song, his hands rising to wrap in her hair and—
He wrenched her away from him. She kept her grip, iron-tight, on his neck. When he opened his mouth to speak, she squeezed harder. Then his hands were around her neck.
"Let go," he wheezed. "Let go or—"
She kissed him again and when their lips touched, he flung her back, her grip tightening fast and hard as he began to struggle violently. He kicked, knocking her legs from under her, but she kept her grip and they went down, her on top of him, their hands still wrapped around each other's necks. Hers slipped just enough for him to gasp a few words.
"Stop. Damn it, I don't want to hurt—"
"Then don't," she managed. "Stop struggling. Let go. It'll be over soon."
He kicked at her again, and kept kicking, kneeing, scrabbling. She'd be a battered mess when this was over, but it would be over. She had the advantage of a little not-quite-human strength. Yet somehow it wasn't enough. The moment she loosened her grip, he found a better hold, pressing hard, gasping for her to stop, just stop, damn it, let go and he would, too. Only she couldn't. She was close—so damned close—and she needed this, needed it like she'd never needed anything before. This was music. He was music. Pure music. She could feel wisps of his song filling her, and she had to have it all, had to . . .
The world seemed to spin. He released her fast, grabbing her as she fell, his lips going to hers now, trying to breathe his life into her. Trying to bring her back, despite all she'd done to him. There was music in that. In that final act of kindness. Of goodness. She heard it, even as her life seeped away. The strains of death's music, so clear and perfect, and she breathed it into him. Her music, for him. Perhaps an apology. Perhaps, simply, because she did not need it anymore and because he could use it, and she could live on, in that small way, through him.
She gave it to him and she listened to those final strains and then . . .
Silence.
Ever since Beau's first hit single with the band, he'd been warned about parasites. "Folks will always be looking to take advantage of your talent, son. They don't have any of their own, so they'll steal yours." Which was true. It had happened many times. He looked down at Izzy's body. Just never quite so literally.
He had no fucking idea what just happened. Drugs, he could say—and would, when the police arrived, though he had a feeling they wouldn't find any in her system. He was just very, very happy that one of those early mentors had warned him to tape business conversations. He'd always heeded that advice—even tonight, because he had come here for business.
He'd had no intention of screwing around with Izzy. Jill would have his balls for breakfast if he cheated on her. Not that he would have anyway. He'd flirted with Izzy because, well, there were certain expectations that went with this career and he felt obligated to deliver. He would have just flirted and teased his way out of it as he always did. That, he expected. This...
He looked at her body again. Shit.
Footsteps pounded down the path. It was Jill, a damp coffee stain on her faded denim jacket, as if she'd literally dropped her cup when he texted.
"Apparently, she wanted more than a cuddle in the woods," he called as she ran over. Still feeling shaky, he rubbed the back of his neck with an unsteady hand. "Tried to strangle me. There was nothing I could do. She wouldn't stop, so I had to stop her, and I couldn't bring her back."
Jill stopped short and stared at him. "You stopped her?"
He chuckled, wincing as his throat hurt. "I'm a modern guy. I can take care of myself." He fingered the rising bruises on his throat. "Pretty much." He winced again and coughed softly. "But the next time I say I don't need you to stick close..."
"Ignore you?"
"Please."
Jill gave him a hug, quick and fierce. Then she placed the call. As she did, Beau stared down at Isabella. Why? Goddamn it, why?
For the music.
The answer seemed to whisper to him on the breeze and he looked out at the distant water, the moon playing on the ripples. Playing a song. Before, when he'd told her he could see it, he'd been lying. Humoring her. Now, he saw it. A song playing for Isabella. A tragic and terrible and haunting song, like all the best ballads.
He reached for his phone to record the tune. Then he stopped, rummaged in his jacket, pulled out a scrap of paper and a pen he carried for impromptu autographs. And he began to write.
On...Lady Isabel and the Elf Knight
Child wrote that Ballad 4 (Roud #21) might be the most circulated of all ballads. It's found in Southern as well as Northern Europe, the Netherlands, Poland, Germany, and the Scandinavian countries. There are many versions throughout North America as well. The earliest version Child lists is from 1560.
This ballad is wonderfully refreshing since in most versions Lady Isabel (referred in other variants as May, the King's daughter, Pretty Polly, etc.) gets herself out of trouble. The first ballad Child describes begins when an Elf knight blows his horn and inspires love-longing in Lady Isabel. He appears and convinces her to ride with him to the green wood. In the wood, he tells her to get down, she's come to the place where she's going to die. He's already killed seven kings' daughters here and she'll be the eighth. She convinces him to sit and put his head on her knee, then lulls him to sleep with a charm, ties him up with his own sword belt, and stabs him with his dagger, saying "If seven kings' daughters you have slain, lie here a husband to them all."
In another version, the elf knight brings her to a body of water with the same intent, but she drowns him. In some versions he tells her he will cut off her head. She suggests he take his coat off so her blood won't spurt all over it. As he does she cuts off his head. The Dutch version takes this further—she brings the head back with her to her father's house. There they celebrate by having a feast and put the head in the middle of the table. (1)
1) Child, Francis James. The English and Scottish Popular Ballads Vol. 1. Mineola: Dover, 2003.
# TAM LANE
By
Lisa Morton
The old newspaper building was haunted.
At least that was what Janet had always heard about May O'Greene's extraordinary 1910 masterpiece. Tucked in a forgotten corner of downtown L.A., wedged in somewhere between the Garment District and the endless cheap appliance stores, it had stood for over a century, long after the Daily Examiner had folded and the structure's original purpose had vanished. Three stories tall and occupying most of a block, its once glittering turrets and graceful, swirling archways were grimed over with decades of accumulated urban grit and neglect.
Still, it stood out enough that, as a youngster driving past it with her father, Janet had asked him about it. "I'll buy it for you one of these days," he'd told her.
With her father, that was no joke. Edward Carterhaugh III probably owned more of Los Angeles than any other mogul; he bought and sold properties, redeveloped and repurposed and renovated and demolished and rebuilt. The Carterhaughs' own home was a classic Greene and Greene craftsmen house; as a child growing up in it, Janet had gotten used to seeing strangers peering in through the fence, taking photos.
It was almost remarkable that eight years passed between the time eleven-year-old Janet had asked about May O'Greene's monument to journalism and her father's acquisition of same. During that time, Janet had developed an obsession for architecture that had transformed into a career choice. Now in the second year of her major, she had favorites (Gehry, Paul Williams), but none she admired so much as May O'Greene.
When her father mentioned at dinner one hot August night that his offer on the old Daily Examiner building had been accepted, Janet's eyes had widened and Carterhaugh laughed. "I thought that might pique your interest."
"I've always wanted to get into that building."
Edward wiped his mouth with a napkin, took another sip of his dirty martini, and said (with a slight smirk), "They say it's haunted, you know."
Janet returned his smile. "All the better."
Their housekeeper, Maria, placed plates before them. She squinted at Janet. "You don't eat enough. Too thin." It was an old joke between them. Maria had been with the family for a dozen years—she lived in a bedroom just off the kitchen—and Janet loved her like an aunt.
"It'll be your fault when I weigh three-hundred pounds," Janet said. Maria smiled and returned to the kitchen.
Edward picked up a knife and fork, resuming the conversation as he cut into his rare steak. "There's a film crew shooting on the ground floor right now—it's been used mainly for movies over the last thirty years, I guess, but only the first floor. The two upper floors are pretty useless."
"I want to see it."
Edward nodded. "I'll have Amy call in a pass for you tomorrow." He paused, a forkful of Maria's good beef halfway to his mouth, then lowered it and added, "I'm thinking of converting it to office lofts. Might be a great first job for a young architect..."
"Are you serious?"
He nodded. "You're welcome. Just don't embarrass me."
Janet knew this was probably where she should have risen and hugged him; that was how the scene would have gone in a movie, or even a lot of real households. But there'd been tension between them since Mother had died four years ago. It had been a miserable death—a month-long coma following too many years of too many prescription drugs, until finally the coma had ended in seizures and a lonely passing in a hospital bed. Janet secretly blamed her father for her mother's dependence on self-medication, and she'd wondered if he didn't feel the same about her.
At least he's trying. Of course he probably just knows I'll be the cheapest architect he can get.
Whatever the circumstances—whatever unearned privilege, whatever paternal guilt—Janet was still happy, the next day, to walk into the grand Daily Examiner building. She'd spent the rest of that night reading up again on May O'Greene's history. The Examiner had been her last major work; it had taken her nearly ten years as she'd overseen every detail, every tiny bit of tile and ornament and fixture. Many critics felt the Examiner lacked the sheer spectacle of some of May's earlier work, especially her Santa Monica amusement park Sealight, which had been completed in 1898 and torn down in 1942, its wood and metal used in the creation of war machinery instead. Sealight had been called "a fairy palace." The Examiner had been dismissed as "a wealthy Indian's brothel." It combined art nouveau and Orientalism in ways that hadn't always pleased architecture buffs; Janet, however, found it sensuous (as she thought O'Greene had likely intended). O'Greene's reputation should have been as great as Williams's, or certainly Julia Morgan's, but Janet doubted if most of the film crews that shot in its ground floor recognized the brilliance they stood within.
The guard at the building's entrance found her name on a list and admitted her. It was an uninteresting side entrance. Once inside, Janet found herself jostled by frantic assistant directors and grips hauling apple crates and coils of cable. She passed a row of glassed-in offices where she glimpsed cameras and actors (dressed in white lab coats). A few of the crewmembers eyed her curiously, but something about her—some indefinable aura of confidence, status—kept them from approaching her.
She moved past the crew, and the hallway she was in emptied out into the Examiner's front lobby. She stopped and gazed around, stunned; here, she could finally commune with O'Greene's glorious designs. The spacious interior went up three stories to skylights; curving flights of stairs flanked either side. Rich, original tile in burnished earth tones graced the floor; dual Moroccan-style turrets framed the carved oak front doors, locked for decades against the Los Angeles streets. The handrails on the stairs were rich mahogany, rubbed to a deep, nearly decadent finish from a century of use.
Janet stood unmoving, breathing it in. The sunlight that penetrated through the skylights and turrets was filtered by heavy dust (this building had never been designed for air conditioning); behind her, the sounds of the film crew had sunk to a constant low thrum.
It was easy to believe this palace was haunted.
Janet remembered, then, why she was here. Her father wanted her to lay waste to this space, to turn its offices and storage areas into the large single-room apartments he liked to refer to as "artists' lofts."
For a second the idea sickened her. She hated the job. She hated her father. It was a sacrilege.
But then she reconsidered. She could, after all, keep the lobby; it was functional as well as beautiful. It would provide pleasure to those tenants ("artists," she reminded herself) with enough soul to appreciate its art.
She was sweating as she climbed the stairs to the second floor; it was August in Southern California, and she was in a building that kept only part of its ground floor acceptably cooled. Here, on the second floor, the temperature rose sharply. Janet picked her way through ancient offices, most empty, a few strewn with trash. In one back room she found the remains of a party—a bag full of empty bottles of cheap red wine and a used condom. It was hard to tell how long they'd been there. She began to feel better about rededicating the Examiner to a purposeful existence.
A twisting back staircase led her up to a third floor that had plainly seen even fewer visitors in recent years. From her online study of the Examiner, she knew the third floor had once housed the executive offices; May O'Greene herself had even had a suite up here. Motes glittered in the air, turned opalescent by afternoon sunlight. Later, Janet would come back with a camera and tablet computer and begin planning; today was purely exploratory.
The heat on the third floor was stifling. Janet was wiping sweat from her brow when she poked her head into a room...and gasped.
She looked into a sitting room that could have stepped out of a silent film. Low tables, chairs, writing desk, vintage sofa, and cabinets all squatted, as clean as the items in a showroom. On the desk were a shaded lamp, a blotter, pens, and an elaborate Art Deco paperweight. On one gorgeous teak end table was something that looked for all the world like an original Tiffany lamp. Even Janet, who'd grown up with wealth, wondered what the lamp alone was worth.
She stepped into the space, running her fingers lightly along the fixtures, astonished by the lack of apparent age. Surely they were all reproductions, but...why? Who would install all of this in an unused office on the third floor of a nearly-abandoned building? Some eccentric film exec, maybe? She nearly pulled out her cell phone and called her father's office to ask if they'd forgotten to tell her that part of the third floor was rented out, but somehow she knew the answer would be "no," followed by questions she didn't want to deal with.
Was it possible that this room was just miraculously forgotten and preserved, a happy accident of time and environmental circumstance?
Janet reached the desk and picked up one of the items she found there. It was an antique fountain pen with a silver, scaled finish like a metallic lizard. Curious, she pulled the cap off and ran the nib across a sheet of cream-colored paper; it produced a clear, perfect dark line. Janet often made preliminary sketches in pen before rendering them in software; she somehow felt more connection to her work when she used ink instead of pixels. The idea of sketching out her designs for the Examiner's new artists' lofts in a pen she'd taken from it appealed to her, and so she replaced the cap and was lowering the pen into her purse when a voice sounded behind her.
"That's not yours."
Janet jumped and turned, trying to find the source of the words. It was a young man, seated in what looked like a genuine Louis XVI settee. White light streamed in through Venetian blinds behind the man, rendering him into a striped silhouette.
Janet struggled to make him out as her heart pounded. How could she not have seen him there before? Startled, nervous, she held the pen up, jiggling it. "Oh, I'm so sorry—is this yours?"
"Yes." The man rose now and stepped forward, and when his face caught the light, Janet nearly gasped.
He was the most beautiful man she'd ever seen. Light brown skin, glossy black hair that fell onto his forehead in charmingly unruly locks, golden eyes whose shape hinted at an Asian parent, dazzling white teeth that Janet didn't even think were capped. He was dressed in an oxford shirt that defined his sculpted, hard surfaces, and simple khaki pants.
There was also something familiar about him, something that tugged at the back of her mind, tickling. She caught herself before she used the hackneyed, "Have we met before?" Instead, she said, "I...didn't expect to find anyone up here."
"Well, I don't get many visitors. But the ones that I do are..." He reached out to take the pen from her, and when his fingertips brushed over her skin, Janet shivered—they were impossibly cold, icy. "...always special."
He stood there staring at her, a look that caused Janet's pulse to race.
Trying to cover her nervousness, she asked, "Just what are you doing here?"
"This place is mine."
Janet laughed, then said, "I'm sorry to disillusion you, but it's not. My father just bought it. It's in escrow already. We'll be turning it into artists' lofts."
"'We?'"
"Yes. I'm an architect." She felt like a fraud saying that, in this place, but it was too late to take it back.
"Like May O'Greene?"
"Yes," Janet said, before blurting out, "I mean, no! May O'Greene was a genius. I take it you're a fan...?"
The look that crossed his perfect face then was unexpected, full of fear and regret. "I can't really say I am."
"Oh...then, what...?"
Abruptly, he grinned again, apparently anxious to move past the mention of the architect. "My name's Tam—Tam Lane. And you are...?"
"Janet Carterhaugh," she said.
She placed her hand in his...and lost herself. His skin was chilled, but the sensation of it on hers ignited fire. He squeezed her hand, lightly, and it was enough to leave her weak-kneed and desperate. Janet was hardly virginal—her own aristocratic beauty had made many boys seek a place in her bed (when her father wasn't home), and she'd given it to a few of them—but she'd never experienced a need so all-devouring that it made her forget where, who, what she was.
"Tam," she breathed out.
"You know," he said, stepping closer to her (and that action alone flooded her with fresh arousal), "I did catch you trying to steal from me. I think I'm owed something."
"I already apologized."
He placed two fingers beneath her chin and tilted it up. "That's not what I had in mind."
He took his payment as a kiss. Janet returned it; even as she did so, feverishly, some small part of her whispered, Something's wrong here...walk away. No—RUN.
But she didn't listen to that part. All she heard were the soft, moist sounds of their lips and tongues, her own breath, his.
When, several moments later, he lowered her to the velvet couch as his fingers sought her buttons, she let him.
* * *
After, as they lay together, unclothed, her bathed in sweat, his skin only slightly warmed, she turned to him and asked, "Why do I feel like I know you?"
He answered, "I confess. In a former life, I was an actor. I starred in a television series called Thorn in the Roses –"
Janet cut him off. "Oh my God—I used to watch that show every week! It ran for—what, four years?"
"Five."
Pictures came together in Janet's mind, like two etched images lining up to form a moiré pattern. A beautiful boy, with dark skin and eyes the color of certain clouds at sunset. Every girl she'd known had had a crush on the star...Tommy Lynn, she remembered.
Tam Lane.
"But your name..." Janet said.
"The network made me change it."
"You were so young on that show..."
He nodded. "I was only sixteen when it ended."
"And what did you do then? I don't remember any movies, or..."
His expression darkened, and Janet instantly rued the question; she would, in fact, have rued anything that caused distress to crease that face. "I...well, I'll tell you some other time."
The last ray of light faded; the room plunged into shadow. For a second, Janet had the unsettling thought that all the other furniture had vanished, that the couch beneath her felt old and sprung...but then her eyes adjusted, she made out the shapes of the desk and tables and chairs, and velvet caressed her where Tam didn't. "I should be getting home," she said.
Tam said nothing.
Janet untangled herself from him, feeling suddenly modest as she sat up, gathering her discarded clothes. She dressed in silence and then turned to look down upon him. In the room's semi-darkness he looked unreal, like an out-of-focus photograph, or fog.
"Come with me," she said, even though she knew Daddy would throw a fit if she walked in with a man.
"I can't," he said.
"Oh." The thought struck her: She'd just had unprotected sex with a man she didn't know. She nearly turned and ran, so strong was her urge to be home, safe, away from...whatever was so wrong here.
She had nearly reached the doorway when he called after her, "Wait."
She turned, saw him rise and walk to the desk, where he picked something up. Even in the room's darkness, growing stronger with each passing minute, his beauty—freed from clothing now—shone, paralyzing her.
"Here." He held something out to her, and, numb, she reached a hand up. He placed the object in her palm. "Your pen."
"Will I see you again?"
"I truthfully don't know."
The last of the day vanished. Janet stumbled out, feeling her way blindly along the walls, shivering as her fingers scraped away chips of paint; something sticky clung to her like a poisonous spider's web. The floorboards groaned beneath her; she felt their rot and wondered if they'd hold her. The silence was otherwise complete.
Janet forced herself to remain calm, to find her exit. She had a good memory for layout and located the staircase; she stumbled twice negotiating its creaking turns in the dark, but soon she was on the ground floor, filing out with the last of the film crew. Some of them cast side glances at her, and Janet wondered what they saw: A disheveled woman with a shell-shocked look, perhaps?
When she reached the sanctuary of her car, she cried.
After ten minutes she stopped. She wondered, with a turn so complete she couldn't begin to understand, why she'd been sobbing.
She started the car and drove home.
* * *
On the way, she stopped at a pharmacy and bought two items: A bottle of juice, and a pill that would prevent pregnancy.
She swallowed the pill with the juice and disposed of the box in the drug store parking lot.
When she got home, she took a forty-minute shower, let Maria feed her, then fell into bed. Her sleep was black, dreamless.
* * *
The next day she asked her father's secretary, Amy, to find out if anyone was renting part of the Examiner's third floor. Amy checked. The answer, of course, was no.
She Googled Tommy Lane. It turned out that his show, Thorn in the Roses, had been canceled because he'd disappeared, not the other way around. His parents claimed they knew where he was and that he was in good hands. Speculation was that he'd run afoul of drugs, been shipped off to a very quiet rehab somewhere. After a time, mentions of him stopped altogether.
Amy arranged a permanent pass into the building; after all, she was now lead architect. The next day she returned to the Examiner, her heart in her throat as she climbed the spiral staircase to the third floor. She was shaking by the time she reached Tam's corner; she half-expected to look in and see nothing at all, no furnishings, not even a trace.
The furnishings were still there. But Tam was not.
She called his name. She sat on the couch, quivering as she remembered the feel of his icy fingertips on her shoulders, her breasts, between her legs. She jumped to her feet, trying to ward off arousal, calling again, again. "Tam! Are you here? Tam?"
He didn't appear.
As night fell, she left.
Over the next month, she visited the room as often as she could, working her trips in around classes. When a good-looking PoliSci major named Matt asked her out, she politely declined. The next day she sat on the velvet cushions in Tam's room and talked to thin air. She told him about Matt. She told him she'd turned down the invitation.
She told him she wanted him.
He didn't appear.
A month passed, and the memory of their lovemaking became more precious to her, not less. August spilled into September, and one morning Janet woke up sick. As she knelt over the toilet, one hand instinctively clutched at her belly. It was too soon, of course, to show anything there but she knew.
October arrived, and the sickness continued. Janet tried to be quiet, but she'd never been able to hide anything from Maria. "You sick a lot lately," the housekeeper said, looking away.
Janet had no answer.
Maria came out of the bathroom with a load of towels for the laundry. She stopped before Janet. "You know you're like family to me, Miss Janet..."
"Of course. You, too."
"So you'd tell Maria if anything was wrong? If you needed help?"
Janet's throat was dry, her tongue useless.
Maria kissed her gently on a cheek and left the room.
* * *
A day later, Janet sat on the couch in Tam's room and cried. "Tam, goddamnit, I really need to see you. I...I think I'm..."
She couldn't say the words, not to empty air. And emptiness was all there was.
* * *
A home pregnancy test confirmed what she already knew. It shouldn't have been possible, but it had happened.
She finally gave up going back to Tam's room. She grew distracted and missed classes. She wondered what Maria's offer for help meant. She was afraid of the answer.
The boy at school, Matt, asked her out again, this time to a Halloween party. Was it really October already? She politely declined. As she walked away from him, she laughed.
Halloween party? Sure...I could go as Stupid Pregnant Girl. I won't even need a costume.
One morning she awoke to a knock on her bedroom door. She was sprawled on her bed, looking at old pictures of a handsome, young television star named Tommy Lane on her tablet. "Yes?"
The door opened and her father entered. Janet thumbed a tab on the screen to switch to another website, then set the computer aside. "Oh, hi, Daddy."
"Hi, sweetie. Listen, I've got some bad news. The Examiner failed inspection, and I mean as in epic fail."
Dread crashed down on Janet, nearly suffocating her. "So what does that mean?"
"I'm sorry, I know you've put a lot of work into this, and I really liked your ideas, but...the lofts aren't gonna happen. The building's just too far gone—it would cost more to bring it up to code than it's worth."
"So...?"
"It's going to have to come down."
"When?"
"Demolition starts tomorrow—"
"You can't!" Janet leapt from the bed, face flushed, fingers clawed.
"There's really no choice."
Of course there's a choice, she wanted to say. Spend the money to bring it up to code. Honor the building.
Love your daughter more than business.
Instead, Janet pleaded, "Can't you at least postpone it?"
"Why? What difference will that make? There's nothing left to get out of that building, Janet."
Janet strode past him, pausing only long enough to grab a jacket. "Yes, there is."
He reached out as she ran by. "Don't go down there—"
But she was gone.
* * *
The side door she always used was secured with a heavy chain that had been added since the last time she'd been here three days ago. She wouldn't be getting in that way.
But she knew the Examiner now; it had become an old and dear friend. She knew its secrets, the hidden ways in and out. She knew about the rickety door next to a loading dock in the back that gave easily when she applied pressure.
The ground floor was dark, but she'd brought a mag lite. She turned it on, heading for a back stairwell.
She shivered.
It wasn't cold; October in L.A. never was. But something in the Examiner had changed. It was no longer the friend she'd come to know intimately. It felt charged with fear, electrified with dread.
It felt haunted.
Janet remembered what day it was and forced herself to smile. Right...it's Halloween and I'm in a haunted house. All I'm missing is a jack-o'-lantern full of candy. But the thought didn't cheer her; it didn't dispel the Examiner's newly acquired aura.
She was relieved when she reached the third floor and sunlight spilled in through a few windows, wan but enough to reassure her that there was a world outside where kids were putting on costumes, laughing, looking forward to their night of mock terrors.
Tam's room looked the same. She knew crews had toured every room in the Examiner, and she wondered what they'd seen here. A Tiffany lamp? Antique fountain pens? She guessed not.
"Tam...goddamnit, Tam, you need to hear me! The building's coming down tomorrow!"
Nothing.
She paced the carpeted floor of his room, desperate. She'd called him once before, hadn't she? She tried to recall exactly what she'd done on that sweltering August afternoon. Of course she'd tried this before, dozens of times, but she had to try again. She'd come in...she'd gaped...she'd admired the lamp...she'd picked up a pen...
The pen.
It had to be the pen. But she'd tried this before, on so many other days. She'd picked up one of the other pens (there were still half-a-dozen on the desk), even waved it around...but nothing. No Tam.
There had to be something else—
Wait—I tried to steal it.
Janet grabbed one of the pens and shoved it into a pocket of her jacket.
"I've been waiting for you to figure that out."
She turned as Tam rushed forward to embrace her. "God, Tam, I've got so much to tell you..." she began, her voice husky, but she broke off as his mouth found hers. They were ravenous for each other, and need to speak gave way to other, more urgent desires. She fell back over the desk, he moved between her legs, and two months of separation vanished.
It was over quickly, and then Janet's rush of words came out. "Tam, you have to leave here, tonight. My father's tearing the Examiner down tomorrow and I can't stop him—"
He put two fingers to her lips, gently. "Just slow down..."
She looked into his eyes, and she saw compassion, a hint of mischief, love...and a trapped animal. She lifted herself from the rough wood of the desk, pulled clothing back into place. "I'm sorry. It's just that I have so many questions, and with the baby and all –"
"Baby?"
The ability to make coherent words left her. She looked at him uncertainly. Would he mock her? Question if it was his? Dismiss her?
Instead, he pulled her to him. "Janet...my Janet..."
She let him hold her for a few seconds before pulling away. "Tam...what exactly are you? Why are you here?"
Tam sighed, a slight expression but enough that she felt his breath on her skin. "I'm something both more and less than human. Remember that show, Thorn in the Roses? We shot the last episode here. I was a stupid kid, sixteen. I got bored once while I waited for the cameras to set up, and I wandered off to the third floor. This was where I met her."
"Who?"
"Who else? May O'Greene."
Janet's mind raced back to the biography she'd read months ago. "But...May O'Greene died in something like 1912, not long after she finished the Examiner."
Tam shook his lovely head. "No. She can't die, because she's not human."
"What...what is she?"
"She's had many names. Titania... Oona... Mab... Maeve... May... Queen of the Fairies."
Janet couldn't suppress a sharp laugh. "Fairies? Now I'm supposed to believe in fairies?"
Tam took her hands, kissed her fingers (sending another ripple through her), and said, "Nowadays people would call us ghosts. We're the ones who live on the outside...or, maybe more accurately, tucked into the edges of your reality. We never die, but we don't really live, either."
"But you...?"
"Ahh, yes...the boy who wandered away from the film crew. Well, it was a Halloween—like today—and May came through the veil between the worlds and took me back with her. She laid magicks on me first and then on my parents, so they lied about knowing where I was. And she kept me prisoner, or rather—collected, a pretty trinket she can look at when she's bored."
"But if the hotel is torn down..."
Tam's look contained almost inexpressible weariness. "I'll go with it."
"How do we get you out of here, then?"
"There's one way, but..."
He turned away from her, and Janet walked to his front, forcing him to look at her. "I don't care. What do we do?"
"Do you know about Halloween? I mean, it's real meaning beyond silly costumes and pumpkins? It's the night when fall becomes winter, when sun's life turns to night's death, when the veil between worlds is thinnest. You sense it, don't you?"
May thought about the walk from the first floor, how she could almost feel a vibration in the air itself. "Yes."
"At midnight, that veil will be the thinnest and you might be able to pull me through, but May will try to stop you."
"Stop me? How?"
"By tricking you. She'll tell you that you can have me...if you can hold me. Then she'll try to frighten you into letting go. She'll change me three times, and each transformation will be worse than the last."
Janet's hands moved to his face, his cold skin. "I don't care. If I get you out..."
He encircled her wrists with his hands. "You can. But don't make the mistake of underestimating her, Janet. It will be the hardest thing you've ever done. Just remember: Don't let go. When I return to my true form, wrap me in your jacket and it'll be over."
She nodded, too overwhelmed to speak. He smiled warmly at her. "You can do this."
She wished she shared his confidence.
* * *
They spent the rest of that day talking, holding each other. Tam asked her questions about what had happened in the ten years that he'd been May's prisoner. Janet talked about her life, her plans, how she both loved and hated her father.
As the sun set, the Tiffany lamp glowed softly, casting the room in the soft shades of its colored glass squares. The light would have been pleasant under any other circumstance, but as the time drew closer to midnight, the oppressive atmosphere intensified. Janet felt it like a low frequency hum in her midsection, setting her on edge, making her want to flee.
But she didn't. She stayed with Tam as he spoke to her, soothing her, telling her that May had treated him well (for a prisoner), and that he believed she loved him.
"Who wouldn't?" Janet asked.
Janet glanced at her phone, and saw the time was just after 8 p.m. "Four hours to go," she said, setting the phone up where she could see it.
She glanced out the window, and her heart skipped a beat as she saw eyes looking in at her. Yellow eyes that didn't blink, that weren't shaped like any animal she could name.
"Tam!"
He followed her wide gaze before pulling her close. "It's starting. The veil is lifting."
"But it's too early, it's only—" She looked at the phone.
It read 11:58.
"That's not possible. It was just eight o'clock—"
The building began to rumble as if a small earthquake were happening. The Tiffany lamp blinked out so that the only light in the room came from the glowing eyes of whatever watched them from outside. Janet clutched at Tam, her breath quickening; the temperature plummeted, and steam puffed from her with each exhalation.
The other side of the room began to glow, softly, a bluish light that might once have been seen above a dank graveyard or in the heart of a darting will-o'-the-wisp.
"Tam...?"
"It's her."
Something moved in front of the glow—a feminine shape. Tall, dressed in some sort of full-length dress or gown, hair flowing... "Well, now, Tam, what's this?" asked a husky feminine voice, with a hint of old world accent.
The glow moved to reveal her face, and Janet gaped—the photo of May O'Greene she'd found hadn't begun to capture her extraordinary face. Her features were too sharp—almost feral—to be truly beautiful, but her eyes glinted with both youthful passion and the madness of great age.
Janet saw that even Tam was unnerved by May. "This is Janet, and we're leaving."
May laughed, a sound that was every Halloween witch's shriek mixed together. "Oh, dear Tam, are we actually going to do this?"
Tam ignored her and turned to Janet. "Are you ready?"
She nodded and put her arms around him. He whispered into her ear, "Remember—don't let go."
Whatever Janet expected—a snake, a giant spider, a flame that might scald her—she didn't anticipate Tam trying to pull away from her.
Except it wasn't Tam's voice that said, "What the...? Janet? Where am I?"
She pulled back as far as she could without releasing him, and saw: a college boy who'd asked her out. "Matt...?"
Even in the dim light, she made out the terror on his face. Her first instinct was to release him, tell him to run, this wasn't his fight... "No." She held him tighter.
May said, "That's one..."
Janet closed her eyes. Maybe if she just kept them shut, if she didn't look at whatever came next...
"Janet, look at me," said her father.
Her eyes snapped open involuntarily, and Janet gasped and shrank back at what she saw. It was her father, but his features had altered, become savage, almost demonic. He leered at her and pulled her to him; she felt the bulge in his crotch and cried out wordlessly. "There's been nobody but you since your mother died," he said, and he ran his tongue down her cheek.
Janet willed her knees to hold her up, her arms to keep their circle around him. It's a phantom...It's not my father...
"You've disappointed me in so many ways," he said, and she hated the tear that she felt trace down her cheek, "but you could still make it right, if you're good to me..."
He ground against her. She swallowed back a scream, turned her head, but finally forced out, "I won't let go!"
"Oh, really, child?" May asked. "How about now?"
Her father's face melted away, the figure in her arms grew smaller, more compact, until Janet saw that she held—
Herself.
She stared in confusion. It was like looking into a mirror, except she felt the body solidly in her arms—her body. How was this supposed to scare her? It was strange, yes, but hardly fearful enough to make her relinquish her hold—
The face began to shift. To age.
Twenty...twenty-one, an adult now, and doubt etched itself in her features...twenty-three, twenty-six, and she saw anxiety, premature lines...twenty-eight, thirty, thirty-three, and failure was deeply graven, the eyes half-lidded and sunken, the mouth turning down forever...
Janet sobbed as she realized—this could be her life with Tam. Joyless and careworn, burdened with a child at too early an age—
Thirty-eight, forty, forty-five...
She looked like her mother. Her unfocused eyes and sallow cheeks were testament to prescription drug abuse, and she knew she'd be dead soon, dead like her mother, worn away by life...
But she could have life back if she let go. The next thirty years didn't have to go like this.
Let go...
"Noo!" It was a shout of desperation, but it worked. It roused her to realization: a trick.
She tightened her grip.
And she felt bare, warm skin beneath. She held Tam again. He looked drained, barely conscious. Janet reached one arm to the desk, found the jacket she'd placed there, and draped it around him. He fell against her. She lowered him to the floor, kneeling with him, keeping the jacket in place.
May O'Greene didn't shriek or storm. Instead she sobbed.
"You've won him, girl," she managed between cries. "Be good to him."
She blinked out. The world changed. The eyes at the windows were gone, the lovely furnishings vanished, the smothering sense of wrongness replaced by the ordinary air of a world that held a future for her again.
"Is it over?" Tam muttered, trying to re-gather energy he hadn't possessed in ten years.
"It's over," Janet said as she bent to kiss his head.
Or, she thought as she felt a lifetime's worth of happiness wash over her, it's just beginning.
On...Tam Lin
The ballad of Tam Lin resonates throughout our culture. It's the 39th Child ballad (Roud #35) - a "transformation story." Tam Lin is a lusty young man, loved by the ladies, human and fairy. When Janet (or Margaret or Jenny) finds she is pregnant by Tam Lin, she goes to pick an herb that will cause her to abort. There Tam Lin himself appears and explains that he has been taken by the fairies and is going to be given to hell that Halloween as a tithe. He tells her to wait by the road until the fairy procession rides by. She must then pull him off his horse and hold him, no matter what he transforms into. Janet is brave enough to hold on as Tam Lin transforms into a myriad of creatures. Some listed include an adder, a bear, a lion, a toad, an eel, or a black dog—creatures no one would want their arms around. In some versions he then becomes his normal comely self, in others he must be submerged in either milk or water to become human. Many versions end with the fairy Queen saying if she had known Tam Lin would be taken from her, she would have replaced his heart with a stone or stored his heart in a tree. (1)
Janet does much better than a husband in a similar story, but with the roles reversed. When he tries to save his wife in the same way, he lets her go in fright, thus condemning himself to current and eternal unhappiness. (2)
The Ballad of Tam Lin has inspired many writers including Susan Cooper, Diana Wynne Jones, Charles Vess, and Holly Black. Musicians love the ballad and a small selection includes The Decemberists, Enter The Haggis, Fairport Convention, and Frankie Armstrong.
1) Child, Francis James. The English and Scottish Popular Ballads Vol. 1. Mineola: Dover, 2003.
2) Child, Francis James. The English and Scottish Popular Ballads Vol. 1. Mineola: Dover, 2003.
# JOHN BARLEYCORN MUST DIE
By
Marsheila Rockwell and Jeffrey J. Mariotte
The sign read "Bacchanal Brewing—Ale Fit for the Gods," and not only was it the last place in town I wanted to be, it was also the last story in all of this news-challenged state that I wanted to be covering. And considering my other choices were batteries made from cowpies and allegations of fraud at the local dog show, that was saying something.
This piece would have been a natural for Paul Hendricks, who worshipped microbrew as if those aforementioned gods had wafted down from paradise to personally hand-deliver it to us lowly mortals. So of course Paul was out with a flu that came out of nowhere, hit him hard, and looked like it would keep him down for the count. Which meant Jayne had assigned the story to me, and it was about as welcome as a punch in the kidneys.
Send a dry drunk, an alcoholic with fourteen years, nine months, and twenty-two days of sobriety under his belt, into an up-and-coming microbrewery. What could possibly go wrong, right?
I checked my notes again, not wanting to turn the car off and step out onto the newly paved and striped asphalt. Not that I needed the refresher—I'd committed the words to memory when I had first written them down, and the neat block letters weren't suddenly going to divulge any information about the place or its owners that I didn't already know. Still, it gave me a chance to steady my nerves and a few more minutes to wait for a text from my sponsor—something I wasn't sure I had the guts to enter the brewery without.
Bacchanal was owned by three women who'd gone to college together at Southern Montana State—Anna Reeves, the business major; Carrie Reeves, her sister and the trio's public relations guru (though I wasn't clear on what exactly her major at SMSU had been, or if she'd even had one—she didn't appear to have graduated); and Honey Harrelson, the nerdy one with degrees in both botany and agricultural engineering.
Both Anna and Honey had graduated last year, Honey with honors. Honey's picture was a fixture in the Big Sky Standard (or the BS Standard, as those of us who had to work there liked to call it), for everything from winning scholarships to charity work around town. In the space of a year, she and the Reeves sisters had gotten Bacchanal up and running with money from . . . well, that wasn't exactly clear. Definitely one of the things I was going to try to discover in the course of our interview.
As I was flipping through the articles on Honey—a pretty blonde with big green eyes and a Marilyn Monroe figure—my smartphone chimed.
Finally.
"Sorry, John, been AFK. No way 2 get there 2day. Reschedule?"
Damn it.
"No," I typed back quickly. "This is the only day the owners would agree to." I hated the lazy shortcuts texters used and refused to indulge in their wholesale slaughter of the English language. Not that any of the five people I exchanged texts with ever noticed.
"U can do this. Just another interview. Treat it that way."
Easier said than done.
When I didn't immediately reply, Eddie—who'd just earned his twenty-five year chip—sent a quick torrent of follow-up texts, making my phone sound like a doorbell being abused by an impatient and probably not very personable salesman.
"U CAN do this, John."
"I got faith in U. Have some in URself."
"But . . . maybe do the interview outside?"
"Good luck!"
With that, I knew he was gone—my only life preserver in what was sure to prove a perilous ocean of pale ale and dark lager sharks. Sighing, I put the phone on vibrate and stuck it in my pocket. I stuffed my notes back into their file folder, pulled the key from the ignition, and with a wordless plea to my higher power, climbed out into the late June heat to meet my demons.
* * *
The brewery's interior was cool and dim and smelled of dank hops, making my palms sweat and my mouth go dry even as my nose crinkled in response to the sour aroma. How I'd ever found this scent—vaguely reminiscent of cheap weed—to be appealing, let alone tempting, was beyond me. And yet, here I was, swallowing hard for sudden want of it.
The place seemed deserted, and a quick glance at my phone showed that I was a few minutes early, so I turned the flash on and started taking pictures to fill the time. Considering the Standard didn't have the budget for more than one full-time photographer—who was busy covering the scandal-plagued dog show, lucky bastard—whatever I got with my phone's camera would have to do.
I'd expected a Greek motif, what with the name and the reference to gods, but there were no marble columns, replica Venus de Milos, or bunches of grapes to be seen. Instead, drying herbs hung from the foyer's low rafters, green candles burned in holders on every available sill and shelf, metal moons in various phases graced the walls, and a pair of wooden-handled brooms hung crossed over the entryway, like props from cheap Halloween costumes.
"They're called 'besoms,'" a soft voice said from behind me as I snapped a photo. Turning, I saw Honey Harrelson standing there, dressed in a black and gold V-necked Bacchanal Brewing T-shirt, jeans, and tennis shoes. A black cord wound its way about her neck and underneath her shirt, so I couldn't see what pendant might be nestled in her generous cleavage, but I found myself jealous of it all the same. A feeling I quickly smothered; an alcoholic getting involved with a woman who owned a brewery could only end badly, no matter who was writing the story. "An old tradition to keep negativity from entering a space."
Didn't stop me, I thought, but didn't say. Instead, I pocketed my phone, pulled out my notebook, and dutifully wrote this tidbit down, wondering what ever happened to my big-city, Pulitzer Prize-winning dreams.
As I did, two other women stepped into the foyer, both brunettes and obviously related, though one was several inches taller and had a cleft in her chin the other lacked. They, too, wore low-cut Bacchanal T-shirts, and if they weren't Monroe-caliber beauties, they still had a way of moving that drew and held the eye. I suddenly didn't have as many questions about how they'd acquired the capital to open this place—a smile and a wink from any of them was probably all it would take for me to empty my wallet on the floor at their feet, though the most they'd be able to buy with the contents would be some real brooms to go with the decorative ones over the door.
"Mr. Woodward," the taller siren said, sticking out a well-manicured hand. "So glad you could join us today for a tour . . . and perhaps a tasting?" I wasn't sure if I imagined the sardonic twist to her grin or not.
As I shook her hand, hoping she wouldn't notice how much sweatier my palm had become at her words, the other woman—Carrie—spoke.
"Don't mind Anne, Mr. Woodward," she said, casting her sibling a warning look. "She thinks because it's her birthday, she can take liberties."
"Oh? Happy birthday," I said, wondering why they'd chosen to schedule an interview on a day when they should be downing their product instead of showing it off.
"Thank you," Anne said, her smile still bordering on mischievous. "We share that, though, don't we?"
"I'm sorry?"
"I was born on the summer solstice, and you were born on the winter. We're both children of the solstice." At my surprised look, her grin widened, and she winked. "We do our homework."
Not very well, I thought, but again didn't say. Though she wasn't the first person to make that mistake. My ex had been into astrology, and made a big deal about how compatible our charts said we were—her an Aquarius and me a Sagittarius. Until she found out that since I'd been born "on the cusp" in Arizona, a place that didn't observe Daylight Savings Time, I was really one of those hard-headed Capricorns who only wanted to stifle her airy creativity.
It was all bullshit, of course, but if she wanted to blame the stars for our breakup instead of my drinking, I was more than willing to let her.
"Anyway," Carrie said, the glance at her sister sharp enough to cut this time, "while we'd normally offer you the opportunity to taste some of our award-winning brews for your piece, your editor made us aware of your particular . . . sensitivities . . . in that regard, so we'll skip it this time. We'll take you out to the fields instead, so you can see how our products go from barn to bottle without ever leaving town. A fact we're very proud of, and one we believe has greatly contributed to our success."
"That—that'd be fine," I answered, taking her hand in turn, using the movement to cover my surprise at both her bluntness and the fact that it seemed like she'd read my mind. The sooner we could move this show outside and into the light the happier I'd be.
Carrie smiled broadly.
"Excellent," she replied. "Shall we begin?"
* * *
"The angle Jayne McClure—my editor—wants," I said, "is how three young women, fresh out of school, have managed to create one of the most popular and profitable microbreweries in the state in such a short time. You attribute your success to your locally grown hops or whatever, so explain that to me. That sort of thing. I'm not writing a hit piece. Think of it as free advertising."
"How much do you know about beer, Mr. Woodward?" Honey asked.
"How it tastes," I answered. "How it feels going down." I swallowed, hard. Here, enveloped by the smell of it, those things felt fresh, as if my last drink had been fourteen minutes ago instead of fourteen-plus years.
I also remembered what it did to my head. I remembered the paychecks that had never made it home, the lost days and nights, the time I'd forgotten a laptop in a bar. A laptop owned by the Boston Globe, containing the only copy of an investigative piece I was writing on Russian gangs—former Soviet thugs, KGB types—taking over the drug and hooker businesses in the Combat Zone. Those had been the days of my Pulitzer dreams. But the farther away those dreams got, the more I drank, and the more I drank, the faster those dreams raced away. The lost laptop had ended my stint in New England and my marriage. When I heard that one of my confidential sources had been found in the Charles River with two bullet holes in the back of his head and his tongue cut out, presumably because in my notes on the laptop his name hadn't been confidential, it nearly ended my stint on the planet.
It turned out I couldn't drink enough to kill myself because I always passed out first. Didn't stop me from trying, though—first in Massachusetts, then back in the arid wastelands of my birth, and finally here, in the Nothing-But-Big-Sky State. Still, it was touch-and-go a few times, until I finally hit bottom and a drunk named Eddie Kemp reached down for me, pulled me up, and introduced me to his friend Bill W. Eddie'd had to kick my ass a few times to keep me in line, but eventually, it took.
Yet here I was, inside a brewery, with the aroma digging around under my skin like some kind of invasive parasite.
I realized Honey had been talking the whole time I'd been inside my own head, and I'd been wandering around half-blind as the women showed me their brewing process. I snatched a couple of words from the atmosphere, where they'd hung while I was barely listening—grain mill, rollers, stuck mash, wort. I blinked, and saw that we'd stopped in front of a shiny silver tank with hoses attached here and there, and various gauges and dials on top.
". . . the mash/lauter tun is where we hydrate the grains," Honey said. "We don't want any dry spots in the mash. We wet every bit of it down, to draw out all the sweetness, and mix the grains with the liquor. We say liquor, but at this point, it's just water. Hot water; we go with a steady one-fifty Fahrenheit. We use water temperature to regulate the rate at which the enzymes in the malt convert the starches to sugars. Sweetness is good, but as with most things in life, I've found, it's balance that's critical."
"Can you spell that?" I asked, making my first note since we had left the foyer.
"C-R-I-T—" Anne started.
"No, the machine. The tank. It's a what?"
"A lauter tun," Carrie said. "Ours is a mash/lauter tun." She spelled it out for me, slowly. I appreciated the effort. "We're a small operation, and we want to keep every batch we brew just like the one before. Bigger outfits use bigger lauter tuns, with automated rakes, but we use stirring blades." She laughed. "I'm glad you asked how to spell it, instead of just assuming, Mr. Woodward. I was a philosophy major, for a time. Words and their meanings—their true meanings—are important to me. I guess we're both seekers of truth, you and me."
"Please, call me John," I said. "Just because I'm old enough to be your father doesn't mean I want to be reminded of it."
They all laughed at that, and then continued the quick tour and lesson: sparging to remove the sugars from the mash, boiling the mash in a brew kettle and adding hops to make the wort, spinning out the solids and cooling the wort so yeast could be added. That was when the fermentation began. Bacchanal Brewing had two 2,400-gallon fermentation tanks, where yeast converted the glucose in the wort to ethyl alcohol and carbon dioxide, which was continually vented until near the end of the process. At that time, the vent was capped so the CO2 would be forced into the beer, partially carbonating it.
By the time we had walked through the process, I was desperate for a taste. I swallowed again and again, my mouth as dry as the Sonoran desert in the month before the rains came. My mind was starting to wander once more when Honey grabbed my arm. I'd felt worse sensations.
"Perhaps a drink?"
I'm not exactly sure what my expression was on hearing those words—deer in the headlights of an oncoming Peterbilt, probably. Whatever it was, it made her laugh, a sound that sent a not-unwelcome thrill through me.
"Of water, silly. Infused with herbs to offset the taste from the tap."
She waited while I took the proffered glass and downed half of it, then grabbed my arm again, this time linking hers through it.
"Enough of this," Honey said. "The real story is outside, in the fields. Let's go, before we lose the light."
* * *
I had seen the field behind the brewery as I drove up, but paid it scant attention. You didn't have to live long in Arizona (and I hadn't, either time, thankfully) to see enough dried out, sun-yellowed grass to last a lifetime, especially in early summer when the whole world was as parched as a drunk . . . well, anytime his hand was empty. Same was true of Montana, most years. But stepping out Bacchanal's back door, Honey leading the way down two wooden steps and across a small gravel lot at the edge of which the field pressed, as if impatient to march on the building, Carrie and Anne a few steps behind me but present in the sound of their feet crunching across the small stones, the rhythm of their breathing—and it had never before struck me that you could hear a smile in somebody's breath—with the late afternoon sun sending slanting golden beams across the bristled tufts waving at the ends of long, slender stalks, I knew this was no vacant lot gone to seed, no rancher's grazing land.
"Is that . . .?" I began.
Honey let go of me and stepped into it like a beachgoer venturing into the surf. On the far side of the field was a silo, and beyond that a straight line of pines that might have indicated a road. "It's barley!" she exclaimed, running her hands across the spiked heads—awns, I think they were called. I had done a little prep for the piece. "Beautiful, beautiful barley!"
"It is," I said. "Beautiful, I mean." I wasn't sure we were talking about the same thing.
Honey whirled, arms outstretched, a human crop circle. "Isn't it? It's two-row barley. Old World barley, which is part of what sets Bacchanal ales apart. Lower protein gives it more fermentable sugars, sweetens the malt. This particular variety is called Summer Isle. Sadly, the crop this year doesn't look to be as robust as in years past." She stopped and looked back the way we'd come, toward the brewery. "Though we think we've found a remedy for that."
"You'll have to forgive Honey," Anne said quickly from somewhere over my left shoulder. "She's a little in love with barley."
I flashed on a tall glass of golden nirvana, foam sliding down the outside. "Who isn't?"
"I apologize, John," Carrie said, coming around to my left side, while Anne stood on my right. "We're not trying to tempt you, here. We're enthused about our process and our product, but we know it isn't for everyone."
"It's okay," I said. "I mean, that's a gorgeous—" I tugged my gaze away from Honey's lush form. "—plant. It's a grass, right?"
"Yes," Anne said. "It's also one of the healthiest foods you can find on the goddess's green earth. The FDA even allows barley-based foods to claim they aid heart health. It lowers blood pressure and can help control blood sugar."
I followed the path Honey had carved into the barley. The spikes were sharp-edged, snagging at my jeans and the discount-store dress shirt I could never quite keep tucked in. "I never knew."
"There's probably a lot you don't know," Honey said. She was walking back toward me now, her gaze fixed on my face. I wondered if there was something between my teeth, or a bug on my cheek.
"I have no doubt," I replied.
She snapped off a piece, just beneath the head, and handed it to me. "See?" she said. "The long ones are spikes, the little ones spikelets."
I glanced at the double row she indicated. "It looks braided," I said.
"Take it, John."
I reached for it, but as soon as my fingers started to close around the awn, she yanked it free. "Ouch!" I said. I opened my hand. A drop of blood beaded on the end of my index finger—my best typing finger.
"Oh, I'm so sorry," she said. She took my hand and lifted it toward her. For a brief, foolish instant, I thought she would press it to her breast, or place it in that delectable mouth to suck the blood away, but instead she brushed my fingertip lightly with the barley.
"There."
I looked at my hand. The blood was gone. "Let me see that." I snatched the barley from her hand before she could react. I expected to see a bloodstained spike; maybe a couple of them, the one that had cut me and the one she had wiped away the blood with.
It was clean. Golden. Not a trace of red.
"It wasn't a lot of blood," Honey said.
"No. Still."
"We told you," Carrie said. "It's a miracle grain."
"And it knows its own, and approves," Honey added with a strange, almost hungry smile. "As do I."
Was it possible to be sexy and a little creepy at the same time? Because she sure seemed to be pulling it off.
"Can you show me the rest now, please? You grow your own hops, right?" I looked at the field as I asked the question, not at any of the women. I suspected my cheeks were flushed, with embarrassment and not a little lust. But I could feel Honey's gaze, boring into me.
God help me, I didn't mind it a bit.
* * *
"We do," Anne said, answering my question as we walked deeper into the field. "But we're more interested in showing you the harvesting process. If you don't mind?"
"No, of course not," I said, blinking against the light, which was suddenly making my head hurt. My eyes were watering now, too, and my vision seemed to be blurring a little. Allergies? Seemed a little late in the year for them, but then I didn't normally spend a lot of time traipsing through amber waves of grain.
Honey had moved ahead of us, toward something that looked like a tractor with the blade reel from an oversized push mower attached to the front. I watched her climb into the cab and heard the splutter and roar as she started it up.
"Is that a . . .?" I began, but found that I couldn't think of the word. I couldn't think of much of anything but Honey's body and the smooth taste of beer sliding down my throat. I wasn't sure which one I wanted more.
"It's a combine harvester," Anne replied, seemingly oblivious to my mental lapse. "In the old days, all the steps in harvesting barley were done separately, by hand. And while we prefer the old ways for most things, in some cases, the benefits of modern technology are too great to ignore. The harvester reaps, threshes, and winnows, all in a single process, and in much less time than it would take to do each task individually. It's really quite a sight to behold."
It looked like I was going to get to behold it, though, and up close, too, because Anne and Carrie had each taken me by one arm and were guiding me straight toward the thing.
"Shouldn't . . . shouldn't we . . .?" I tried to say, searching foggily for the last words. Move? Out? Of? The? Way?
"Hush, now, John Barleycorn. It will all be over soon."
The sound of the harvester was so loud now I couldn't tell which of the sisters had spoken, so didn't know who to correct. That's . . . not my name.
Was it?
I was so confused, and the rhythmic slicing of the harvester blades through the rows of bowing barley were almost like a lullaby, soothing me, making me want to lie down in front of the approaching machine, like an offering.
Wait.
Like a what?
The utter foreignness of the thought pulled me out of what I realized must have been some sort of drug-induced stupor. What kind of herbs had Honey put in that water, anyway?
And, more to the point, why?
I yanked my arm from Carrie's grasp, but Anne was bigger, stronger. She held on.
"What the hell are you doing?" I demanded, shouting to be heard over the whirring blades.
"Hell has nothing to do with the Craft, John," Anne said grimly, still lecturing, even now. She grabbed onto me with both arms, a grip I fought to break, especially with Carrie clawing at my head and back to regain her own hold.
As I struggled, I heard Carrie start to speak in a strange, almost sing-song meter. After a moment, Anne joined in, matching her word for word. "Seeker of truth," they said. "Lover of barley. Child of the solstice."
I thought my swimming head was playing tricks on me, making me hear them wrong. But they repeated it, in the same tones, the same pattern. "Seeker of truth. Lover of barley. Child of the solstice."
The harvester was coming closer, its engine tearing apart the calm silence that had engulfed the field at first. I didn't know what their nonsensical patter was about, but I was no solstice child. I kept fighting, trying to twist out of their hands.
"Just give in, damn you," Carrie snarled in my ear. "It's an honor to be chosen to bring back the harvest. And you won't even feel anything, thanks to Honey. More mercy than your Puritan kind deserves."
She was right. I could feel her nails tearing at my neck, but whatever herbs had dulled my wits also seemed to be doing the same to my other senses, blocking any associated pain.
Which is probably how I was able to shake her off, plant my foot, and spin around, catching Anne off guard. Pulled off her feet, she slammed into Carrie, releasing me as both women went down in a heap.
Right in front of the harvester.
I didn't bother to see if either of them got up again.
Instead, I turned and ran.
* * *
They were between me and the parking lot, so I went the other way, cutting through the field toward the silo and the road I hoped was beyond it. My feet felt heavy, as if I'd waded through wet cement, and I was aware that I was stumbling in more of a zigzag pattern than charging straight across. The rasp of the barley was loud, as was my breathing, and I could hear my pulse in my ears. None of it, though, was loud enough to mask the roar of the harvester, that sickening squelching sound, or those horrific screams, which ended as abruptly as the engine noise did.
I didn't look back. I was pretty sure that trying would ensure that I fell on my face. Instead, I kept going, my gaze fixed on the tree line. I hoped the harvester had reaped, threshed, and winnowed both sisters. But whatever luck had allowed me to escape before the harvester reached me had since fled; I heard someone behind me, racing through the field.
Moving faster than me, from the sound of it.
I'd never make the trees. And I still couldn't tell if there was a road on the other side; if not, reaching them would do me no good.
I veered toward the silo. It was closer, and I could hide in there, or perhaps find something to use as a weapon. Make a stand, though I could barely remain upright without the world tilting crazily beneath my feet.
Somehow, I made it, but whoever was coming behind me was catching up in a hurry. I'd been in a grain silo once before, reporting on a story about the dangers faced by children working on family farms. I hadn't spent much time inside; the air was thick and foul, and the owner's warnings of methane gas buildup made me more than a little nervous.
But I remembered some of the basics. I unlatched the lower access door and climbed inside, pulling it shut behind me. It didn't seem very full—this year's harvest had yet to take place—but barley was crusted against the walls, as far up as I could see. Fiberglass panels set at intervals in the upper reaches of the tall structure let in just enough light to deepen the shadows. I stepped carefully over a piece of equipment on the floor, a kind of spiral-bladed, pivoting arm, then remembered its name: sweep auger. It whisked grain from the sides of the silo and toward a conveyor system in the floor. It was powered off, and I didn't think it went that fast to begin with, but I didn't want to take any chances. Leaning against the far wall was a shovel. I had just taken a last staggering step toward it when the door opened.
Anne stepped through. In the dim light I could see her face and neck were spattered with blood, presumably her sister's. She held a steel rod that looked to be at least six feet long. I recognized it as the kind workers used to dislodge stuck grain from silo walls. Suddenly my shovel seemed an inadequate weapon—by the time I could get close enough to hit her with it, she would already have skewered me.
I had the feeling that if I swung the thing, I would only wind up dumping myself on my ass. But I had to try. Her gaze was fixed on me, her hands tight on the rod. Everything about her spoke of menace.
I took a step to my right, her left. She turned with me and came closer. I took another step, and she reached the auger. As she stepped over it, she thrust the rod toward me. I swung the shovel, batting it away, but the motion made the whole place swim and I fell back against the wall. Dust rained down on me.
She charged. I lunged to the right just in time to dodge the rod. It slammed into the silo wall, sending up a reverberation that shook the whole structure with a deep, echoing hum I could feel in my teeth. I tried to catch it with my left hand, hoping to snatch it from her grasp, but I snagged only empty air.
The rod came at me again, and this time I didn't move fast enough. The end scraped my ribs, ripping the shirt that by now was completely untucked and filthy. The absurd thought that the Standard didn't pay me enough to replace my clothes flitted through my head in the instant before pain flared through me. The drugs must have been starting to wear off; my side felt like I had walked into the path of a blowtorch.
I took two more lurching steps to the right before she thrust the length of steel at me again. I swatted it with the shovel. I couldn't keep that up forever, though. All she needed was to hit me dead center once, impale me, and I was done.
The next time she tried to spear me, I hurled the shovel. She dodged it easily, but lost her footing as she did. She crashed into one wall, while the shovel clanged off another.
More dust fell, and grain now, too, striking the floor like hailstones. I stumbled toward the door as an ominous vibration shivered through the steel walls.
Anne scrambled to her feet, but her rod had slid beneath the auger and she was trying to wrench it free. My last glimpse, just before I fell outside into the dirt, was of her just as she yanked it loose. But doing so slammed its back end against the wall, and the vibration grew more intense. I heard bigger chunks of crusted barley hitting the floor, and Anne gave a cry as—from the sound of it—another hit her, knocking her once again off her feet.
I made it to my hands and knees and saw the power box right in front of me. Throwing it open, I jammed my finger down onto the switch. The motor engaged instantly. Inside, Anne started screaming. "Shut it off!"
Judging by the racket, she was still trying to regain her footing. Larger quantities of grain were falling with a noise like an entire marching band full of drummers, and her cries were interrupted by grunts of pain. "Come back!" she shrieked. "Come back here, John Barleycorn!"
I stayed where I was, breathing hard, dizzy, seconds away from puking. Anne's shrieks became wails of agony, and I thought maybe I could hear the sound of the sweep augur chewing through her legs, but the falling grain had turned into a rumble. The vibrations as it hit the floor spread through the silo, working more off, and the rumble became thunder. Barley dust puffed out through the open door, followed by a spill of grain. I sat there, a dozen feet away, until the din stopped. Almost reluctantly—sitting felt so good—I found my feet and went back to the door.
All I could see was barley, mounded up high on the floor.
* * *
The sun was on the last leg of its slow descent, its rays catching the harvester still sitting in the field. I didn't want to go near it. I could have called somebody—911, sheriffs, anybody—but I had just participated in the deaths of two local businesswomen and I was under the influence of some unknown drug. I didn't want to put myself into that kind of meat-grinder until I was free of its effects. I didn't see Honey anywhere, but I couldn't imagine her calling the authorities. Not after what she and her partners had tried to do to me.
I made my way across the field again, toward the brewery and my car parked in front of it. As I walked, the awns poked at me and I batted them wearily away. If I never saw barley again, it would still be too soon. For once, I didn't even crave a beer.
* * *
I'd intended to circle around the brick building and make straight for my car, but as I neared the closest corner, I thought I saw a flash of movement in the shadows gathering there.
Honey?
I couldn't afford to take the chance. If she was at the corner, going through the building would get me to my car faster than trying to go around the other way.
I changed direction, heading for the brewery's service entrance, the one we'd all walked out of so cheerfully just a short while ago, a dry drunk improbably surrounded by a trio of beautiful, attentive women, like something out of a dream.
Or, in this case, a nightmare.
Either Carrie or Anne had left the door wedged open with a cinder block, which I shoved inside before pulling the metal door closed behind me. I hoped the click I heard was it locking and that the fact that it had been propped open in the first place meant none of the three women had bothered to take a key. Not that I planned on sticking around long enough to find out.
As I hurried through the darkened brewery, past the two fermentation tanks, I thought I heard footsteps echoing behind me. I tensed and whirled, expecting Honey.
No one.
Christ, John. _Get a grip_.
I turned back around, picking up my pace. The echoes increased in response, and this time I thought I heard whispers, too.
"Seeker of truth."
I spun so fast I made myself momentarily dizzy.
Nothing but shadows.
I peered into the darkness, trying to see, but it was impossible. If Honey was hiding there, the only way I'd know was if I walked back and bumped into her, and that was not happening.
I resolutely turned my back and headed past the long bar for the foyer, just able to make out the flickering light from the candles still burning there. I was running now.
"Lover of barley."
I refused to look, kept my eyes on the arched entrance, ignoring the impossible echoes of many feet, the whispers, the movements at the edges of my vision.
"Child of the solstice."
I fetched up hard at the arch, not because of some unseen wall—though with all the other strangeness tonight, that wouldn't really have surprised me—but because of what I could see, just beyond, in the foyer.
Honey knelt there, naked, chanting words I couldn't understand in the center of a circle inlaid in the tiles. Thick black candles stood at five points along its circumference, white lines of . . . salt? . . . connecting them across the circle's interior, forming a star.
A pentagram.
No. A pentacle. I remembered them from my ex's Tarot cards, recalling the distinction only because of her frequent pontification on the topic, something I never thought I'd be grateful for. I guess I owed her an apology.
Assuming I ever made it out of here.
So, a pentacle. And if Honey was kneeling naked in the middle of one, that could only mean she was a . . . I couldn't quite bring myself to think the word, despite what I'd already seen and heard. Been through.
Because I could handle three psychotic women luring me here, drugging me, and trying to kill me for some insane reason that only made sense to them—unfortunately, we lived in that kind of world, and I'd covered worse and more bizarre stories back in New England.
But three . . . witches—because that's what they had to be, right?—choosing me to be their . . . harvest sacrifice?
I'd had booze-fueled fever dreams that made more sense.
But I couldn't deny the reality—the awns stuck in my pants, the burning pain across my torso, the lovely woman before me.
"You shouldn't have run, John," Honey said softly, her chant finished. She looked up at me with those incredible green eyes, her hair falling about her shoulders in a way that drew my gaze downward . . . I jerked my head back up, met her come-hither look with a glare of my own. "Running only ever makes things worse."
"Only if you get caught."
Honey laughed aloud at that, a sound that still thrilled me, even though I knew she wanted me dead and would do anything in her power to make that happen.
And that was the question, of course.
What exactly was in her power?
"Oh, I'd say you've been caught. Quite handily, too." Honey rose as she spoke, and it was all I could do to keep my eyes on her face and not lose myself in the sight of her curves. "The doors are locked, windows barred. And you won't find me as unprepared as you did the others."
She gestured with her hand, taking in the circle and the candles. Behind me, there was a rush of sound, as if a thousand birds had taken flight at once, all of them intent on a single target.
Me.
I winced, bracing myself for an impact that never came.
Honey laughed again, then held both arms out, as if in welcome. A rattling noise sounded on the walls to my left and right, and then suddenly the metal disks flew off the walls and into her waiting hands. In the candlelight, I could see now that the edges of the moons—one half, one a crescent—had been sharpened.
"Are you ready to die now, John Barleycorn?"
In reply, I turned and ran back the way I'd come. As I did, candles I hadn't noticed behind the bar and on the tables flared to life, like lights on a runway, leading me to something I was pretty sure wasn't safety. But what choice did I have? I could hear her bare feet slapping against the wooden floor behind me.
And then I could hear the whoosh of something slicing through the air. I couldn't know if she'd aimed left or right, high or low, so I ducked behind the nearest table and heard the thunk of metal sinking into wood, right about where the small of my back had been just moments before.
Not quite center mass, but enough to get the job done, if it had connected.
I had little doubt the next one would.
I tried to keep low, working my way through the tables and toward the back of the brewery, where the fermentation tanks were. I thought there were bathrooms there, too, and if any of the windows had escaped the rebar treatment—used in actual urban environments to keep the bad guys out, but here in small-town Montana, as a form of hipster ornamentation—it would be those.
The tables slowed me, but they also kept Honey from getting a clear shot with her Xena Warrior Princess chakram, or whatever the hell those things were supposed to be.
Of course, I had no idea if those were the only weapons at her disposal. I had a bad feeling they weren't.
On cue, the echoes started back up, masking her footsteps.
And the whispers, too, though they had a few new refrains this time.
"Are you ready to die, John Barleycorn?"
"Just give in."
And the last, most insidious one, repeating over and over.
"Is your life really so worth living?"
Maybe not, but I'd made it this far, despite myself. I wasn't about to give up now.
And then I was out of tables, and there was nothing between me and the door to the men's room but a long stretch of open floor and the twin silver cylinders of the fermentation tanks.
I'd never make it without something to distract Honey, slow her down.
I looked at the tanks again, considering.
Judging the distance.
Then, with a wordless prayer to my higher power, I took off.
* * *
I made it to the tanks and had just managed to squeeze myself between them when the higher-pitched thunk of metal piercing metal rang in my ears; she'd been going for my head that time, and she'd barely missed.
I had no idea if she was out of throwing moons now or not—she could have pulled the other one out of the table, or even summoned more off the walls as she walked. I had to assume she was still armed and dangerous.
So I did the only thing I could think to do.
I reached around, felt for the valve, and opened it up to full. Then I did the same with the second tank.
The heady scent of beer flooded my nostrils as the cool liquid poured from the tanks in a frothy rush. Hoping that wading through a river of lager would slow her down some, I peeked around the curved edge of the tank to look for Honey. She was standing back near the tables, half moon in hand, ready to throw as soon as I stepped out of my hiding place, unmindful of the beer already sloshing around her ankles.
Damn it. Now what?
As if in answer, the tank that she'd hit with the crescent moon groaned, shook. Weakened by the puncture and with thousands of gallons flowing out from it unchecked, the cylinder was losing its structural stability. Seeing my chance, I wedged myself in between it and the wall, and heaved.
At first, nothing happened, but after a handful of eternities, I felt it give.
Redoubling my efforts, my back and legs braced against the brick wall, clothes ripping and skin tearing, I pushed as hard as I could.
And slowly, like a tree unwilling to admit it had been felled by a logger's axe, the tank toppled.
As it started to fall, I slid out from my hiding place. I watched as it crashed down on the floor and part of the bar, the top splitting open on impact, the remainder of the golden liquid within bursting out across the tables like a flash flood. I saw Honey go down as tables collided with each other. Candles fell from their perches, igniting the alcohol-soaked wood, and flames flared to life across the brewery, turning it into a virtual lake of fire. I watched Honey struggling to rise, her hair brushing up against a burning table, bursting into a fiery corona. She screamed as I turned and ran again, slamming into the men's room door and rushing over to the window above the urinals, chased by fire and ale and a dying witch's curses.
The window was small, and as suspected, unbarred. But I could squeeze through it.
I had to.
Climbing precariously up on two of the urinals, I used my elbow to break the glass, and my forearm to clear the shards from the pane. I was halfway out when the contents of the second tank caught fire and exploded. The blast propelled me from the building onto the packed gravel outside, and the last thing I saw as the world faded from view was the edge of the red solstice sun sinking below the horizon.
* * *
The inquest raised a lot of questions, but no answers that satisfied anybody. Fortunately for me, there was also no proof beyond the proverbial reasonable doubt that the deaths weren't a series of horrible accidents. I was plenty beat up myself, so it wasn't like I had come out of it unscathed. I got a lot of sideways looks, and for a while heard people whispering until I came within earshot, after which they clammed up and pretended not to stare.
Once it was finished, I tried not to think about it. I buried myself in work, focused on sobriety, met a woman and lost her in the same month. I never went back to the neighborhood where Bacchanal Brewing had been located, and I found that sometimes an entire day could pass without me thinking about Honey, Carrie, and Anne.
Then one day in the latter half of June I was out in that general direction, reporting a story about a survivalist compound being turned into an adventure park, with paintball and zip lines and an obstacle course. On my way back into town, near the day's end, I was approaching the turnoff to the brewery, and almost before I had consciously decided, I was braking and turning the wheel. A few minutes later I was parked on the side of the road, eyeing the remains.
The building was still charred rubble. The winter had been a hard one, and most of what had been left standing had collapsed under the weight of the snow. The silo was gone—dismantled and moved, or razed, I couldn't tell.
But the barley field was lush, the stalks tall, shining like spun gold in the afternoon light. I had found it impressive the first time I'd seen it, but that was nothing like this. I had thought that it would be fallow, abandoned, left to rot. Instead, it looked healthy and rich.
My phone rang once. Glancing at the screen, I realized that I had come on the summer solstice. No number appeared, and it didn't ring again. I stopped, eyeing the crop, knowing what fine beer it would produce, were there anyone around to brew it.
Seeker of truth.
Lover of barley.
Child of the solstice.
Three witches, each fitting one of those descriptions as well as I did—no, better—had died on this ground.
An appropriate sacrifice? Maybe. Or maybe it was nothing but winter snow and late spring sunshine working on soil fertilized by ash.
Sitting there in the car, gazing at the shimmering golden grass, I felt a pull like nothing I had experienced since those first dry days, when every bar's OPEN sign flashed just for me, every bottle promised salvation, every beer commercial taunted me with what I could no longer have.
I wanted to wade into the barley. I wanted to cup it in my hands, to brew it into ale, to sip, then guzzle, until the field was nothing but bare earth.
Instead, I turned the key, gunned the engine, and left it all behind me, where it belonged. All the way home, I avoided the rear-view, and kept my eyes on the road ahead.
On...John Barleycorn Must Die
"John Barleycorn Must Die" (Roud #164), also called simply "John Barleycorn" or "Sir John Barleycorn," is a song that often worries scholars and folklorists. It's too perfect. As A.L. Lloyd comments: "The song is related to the ancient idea of the Corn King. Perhaps too neatly so, hence the suspicion that it may not be a genuine piece of primitive folklore. It is old (it was already in print c.1635) and has been passed on by generations of country singers. The tune is a variant of Dives and Lazarus."
In the most common beginning three men come from elsewhere (west, east, north) and make a solemn pact that John Barleycorn should die.(1) From there the song describes in detail what happens when a cereal crop is planted. There's plowing and harrowing, planting, the crop absorbs the rain, grows and ripens, is harvested and the grain allowed to cure before [winnowing] removes the grain from the stalk. Then the grain is ground. The song describes this all as if it were inflicted on a writhing man.
In some versions, John Barleycorn now extracts his revenge: "Here's little Sir John in the nut-brown bowl[ale], And here's brandy in the glass, And little Sir John in the nut-brown bowl, Proved the strongest man at last." Without him man cannot hunt or work, proving that no matter what they did to him, he is still the stronger. As Martin Carthy wrote: "Forget the academic stuff about death and rebirth, fertility symbols and corn gods! The reason that this is one of the best known and most popular of all ballads—and one which has crossed a great many musical thresholds—is that it's actually about that other activity which most commonly accompanies the singing of traditional songs—drinking!"
1) Roud Folksong Index. 2014. 22 June, 2014. < http://www.vwml.org/roudnumber/164 >. English Folk Dance and Song Society.
2) Mainly Norfolk: English Folk and Other Good Music. 2014. 22 June, 2014.
<http://mainlynorfolk.info/lloyd/songs/johnbarleycorn.html >
English Folk Dance and Song Society.
# IN ARKHAM TOWN, WHERE I WAS BOUND
By
Nancy Holder
It was all on a ghastly, gloomy day that at last I reached the outskirts of Arkham, Massachusetts. After a carriage ride lasting innumerable, endless rainy days and bitterly cold nights, I climbed down at last with time-worn satchel and patched valise, and a chill washed over me with the fog. The moon shone down on a misshapen street crowned with gambrel roofs, and the familiar panic seized my heart as I contemplated why I was there. I walked into the station house on shaking legs in fear for my dear Virginia—Sissy, as I called my little wife. I had failed her again.
The magazine that had employed me had gone bankrupt, and I had no funds. Sissy and I had nothing but molasses and bread to eat, and Sissy—I can admit it now!—was overdue for medical attention. I had not wanted to admit that she had consumption—I had denied it for far too long!—and as a result of my neglect, she was dying, a death made all the harder by our poverty.
As you may well know, my father, David Poe, abandoned my mother, brother, sister, and me before I even knew him, and when I was but two years of age, my mother died. We children were farmed out, and I became the ward (but never son, never that) of John and Frances Allan. My foster mother suffered pitiable heartbreak over the unfaithfulness of my foster father, and I quarreled with him bitterly for her sweet sake. Claiming to find me sulky and ungrateful, he cast me out, and when he died, I was disinherited. He bequeathed his second wife and their children a fortune but for me, not a penny. Owning no property and having been expelled from West Point, I had no employable skills except that I had been raised a gentleman and I was facile with words, and so I determined to make my living with my pen.
I sought through the years to remain gainfully employed as an editor and critic, and to publish my verses. Fame came my way, but not fortune, as had come to others who, I confess, I still consider my literary inferiors (Longfellow comes to mind).
I became shameless in my pursuit of relief. Inquiries on my part revealed that a branch of the Allan family made their home in Arkham. This limb of the Allan family tree had split off two centuries before and spelled their name "Allen." Thus knowledge of them among "my" Allans (though of course they were not mine at all) had been utterly lost.
The patriarch of the Arkham Allens was named Mr. Demeter Allen. I had no claim on him and he and I both knew it, but as I had achieved some repute (others might say notoriety) for my literary work, he agreed that we should meet, and invited me to stay at his home, which was a small distance from the town itself.
Now I waited for him in the milky, thick fog; quatrains sprang into my head as I paced to stay warm. The refrain was ever the same: my love cannot die. She must not, would not; she had endured so much for love of me.
Presently an old woman limped along the cobbles. She was bent over, her face gray and lined, and her clothes tattered. She was wearing no coat. She extended a raggedy glove toward me and said, "A penny, sir? Anything? I'll say a prayer for you and yours."
I gave my head a rueful shake. "I'm sorry for your trouble, missus," I said, "but I barely have a cent to my name."
She wrinkled her brow and when she sighed, it was as if she exhaled a ghost. She shook her head and said, "Woe to you, sir, for my prayers have weight."
"I'm sure of it," I said, and impetuously, I was in a mind to offer her my coat when a fine carriage clopped down the lane, and she vanished into the darkness.
The carriage wheeled to a stop in front of me and a fine, handsome gentleman emerged with hand extended in welcome. Whereas the Allans of Virginia are fair, Demeter Allen's hair, eyebrows, and beard were raven-black, and his eyes were so dark I could discern no color in them as he smiled at me and we shook hands.
"Mr. Poe, so very pleased to meet you," he said warmly. Then he turned to peer into the interior of the carriage and said, "Barbara, say hello to your cousin Edgar from Virginia."
He did me great service, for we were not cousins at all, much less on a familiar, first-name basis, and while it was true that I hailed from Virginia, I did not currently live there. Still, Richmond was the home of the Allans, and I felt that he was attempting to emphasize to me that he felt in some way connected to my life. This gave me hope that at last I might have found sympathetic friends.
After a moment, I heard the rustle of silk, and then a very beautiful young woman appeared in the door of the carriage. Her hair was as black as her father's, and she was fine-boned and quite dainty in appearance. She was brilliant with joy, and I was somewhat taken aback, as I could not imagine her broad smile and flashing eyes were a result of making my acquaintance. Indeed, she barely seemed to notice me.
As she alighted, her father said to the coachman, "Eustace, put Mr. Poe's belongings in the carriage." Then he said to me, "What a cold night this is, cousin. Let us repair to the tavern to warm our blood before we embark on the journey to the house."
I was deeply touched, but all my senses sprang to alert as we walked across the street toward a golden-hued bay window blossoming with shadowed movement. Echoes of laughter and conversation greeted my ears, and I began to worry about how I should manage to pay for refreshments for my two new companions, which was the least hospitality my Southern upbringing required. And in all truth, I was afraid that I should forget my own vow to drink spirits in moderation, and humiliate myself. When upset, I overindulge, even to this day.
My "cousin" Barbara practically bolted ahead of us, as eager to enter the inn as I was hesitant. Soon we were wreathed in a steamy crowd of warm breath and the scent of mulled wine, and Mr. Allen began the introductions. Out of the corner of my eye, I spied Barbara making a quick way toward a tall, handsome youth. She had no eyes for anything but him. But he, on the other hand, was striding past her toward me, his hand extended.
"Mr. Edgar Allan Poe!" he cried. "Oh, sir! I have read everything you have ever written!"
He pumped my hand and tears welled in his eyes. I was at once charmed, for I admit that I have never shrunk from public acclaim, and my poor, distressed heart was grateful for this man's eagerness to meet me.
"Flip!" he called.
This was a beverage of hot rum, I knew, such as they drink in New England. A buxom barmaid arrived with a tray of four steaming pewter mugs. The young man handed one to me, one to Mr. Allan, and took a third off the tray. There was one mug left, and Barbara rustled forward to claim it. But before she could do so, the young man raised his mug and cried, "A toast to the greatest writer who ever lived!"
"Thank you," I said, pausing in hopes that he would give the young lady the fourth mug, but he did not. "Mr...?"
"Jemmy Grove," he answered. "Drink up, I pray you, sir!"
Young Mr. Grove, Mr. Allen, and I drank, and Barbara's face blazed bright red. I wondered if Mr. Grove was ignoring her on purpose—if, perhaps, she entertained false hopes and he was attempting to dash them—and I winced inwardly as she touched an anxious, gloved hand to her hair and adjusted her jet-encrusted shawl around her shoulders. Her uncertainty of her beauty made me think of my own sweet Sissy, who had wondered aloud what I saw in her, as lovelier, grander women had sought my company.
After resolving to take only this one mug of flip, I finally drank, and the pungent rum punch spread throughout my chilled limbs. I found myself surrounded by a ring of faces eagerly calling for me to recite "The Raven," and after quaffing more of my refreshment, I felt eager to oblige.
I was ushered to a small stage at one end of the tavern and as I climbed onto it, I spied Barbara Allen rushing in tears out the tavern door and into the night. Planted firmly in the public center of attention, I could see no way to alert her father without embarrassing the lady, and so I kept my peace and hoped she would come to no harm.
I recited my poem, and then was asked for another. I complied. I was in a fine mood by then, having accepted a second cup of flip, and basking, I do confess, in the adulation. More verses were requested; more, given.
After an interval, Mr. Allen decreed that it was time for us to depart. As if on cue, at that very moment the tavern door opened and Barbara slipped in unnoticed by either her father or Mr. Grove. Her hair was slightly mussed and her exquisite jet shawl was gone.
She and I traded looks; hers said, I beg of you, do not betray me, and I dipped my head ever so slightly in reply. She appeared much relieved.
It was announced that we were to depart, and there was much commotion with fetching "the poet's hat and coat." I made a half-hearted attempt to settle the bill with Mr. Allen, but he would not have it—to my shamed relief.
At long last, Mr. Grove seemed to realize that he had neglected Barbara all evening. He hurried toward her and extended a hand. But she gazed at him with hard, angry eyes, lifted her chin, and showed him her back.
He came up behind her and attempted to place his hand on her shoulder. She shrugged it off. He called her name: "Miss Allen?" and she pretended not to hear.
Then, "Barbara?" And she turned her head in his direction and sneered at him like a queen confronted with a beggar. Mr. Grove was utterly crestfallen, but her features were set as if made of porcelain, and she placed her hand on my arm, clearly preferring my company.
Her father saw none of this, but notice was taken by those in the tavern, and a few knowing grins were exchanged. I smiled faintly as well.
I knew a lovers' quarrel when I saw one.
* * *
"Arkham's not like other towns. It would suit you, cousin," said Mr. Allen as we headed home. Then he began to recite many of the ghastly legends attending the town—of monsters from other worlds, and curses, and madness, and hideous beings that rise from the sea to mate with the daughters of men. Of witches. I was quite astonished that he spoke so freely of such peculiarities in front of his daughter. Divested of her cloak, she was huddled under the carriage blanket, shivering with cold, and he didn't seem to notice this, either.
As he unfolded tales of horror and more horror, I lifted the carriage curtain and peered outside. Beneath the moonlight, the trees were blasted as if struck by lightning, then twisted into poses resembling grotesque human hunchbacks. Fog congealed into human faces that scowled at me, then dissipated. At a crossroads, I thought I saw a crooked figure in a shroud. It raised a hand in greeting, and I realized it was the beggar woman who had offered to pray for me and mine, wearing Barbara's cloak. I felt a twinge as I ticked my gaze toward Barbara, who clearly had sacrificed her outer garment on this woman's behalf. Miss Allen stared back at the lady, then furtively made the sign of the cross and nervously licked her lips.
* * *
The Allen home was an imposing country manor house of the same gambreled roof design I saw everywhere in Arkham, its stern gray windows glaring down at us as we alighted. It was not welcoming, but it was large, and I blush to admit that it raised hopes in me of financial help. Even a loan would be welcome.
Lightning flashed as the front door opened and Mr. Allen's man took my coat and hat. I was tired and had imbibed perhaps a bit too much, but Mr. Allen insisted that I accompany him to the parlor, there to meet Mrs. Allen.
Barbara excused herself and went on to bed. We two men entered a poorly lit room lined with oil portraits of dark-haired men and women resembling Demeter Allen. Then, seated before the fire with an embroidery hoop resting in her lap, sat the image of Barbara Allen perhaps some twenty years hence, although somewhat drawn and haggard. Her eyes were focused on the fire; at the sound of our footsteps, she seemed to struggle to blink her eyes and look at us.
"My dear, look. It is dear Mr. Edgar Allan Poe from Richmond," said Mr. Allen, but I noticed that he did not draw near to her.
For an instant, fury glittered in her eyes. I was quite taken aback, but I realized that she was still not looking at me. Mr. Allen was the object of her ire. The exchange was a mirroring of what had transpired between Jemmy Grove and their own daughter, and I wondered at the cause.
"What a pleasure to meet you," she said to me without much enthusiasm.
I inclined my head and made my response as I took the seat across from her. Her husband perched on a settee somewhat more distant from the mantel. Then she called for warm rum and I did not protest. We drank to her health and then to mine.
"Tell me, Mr. Poe, how do you like this wicked place?" she asked me.
"Begging no disrespect, ma'am, but as a gracious lady such as yourself lives in this place, surely it cannot be too wicked."
She smiled at my pretty gallantry and said, "Let us not forget that Eve turned the Garden into Hell."
"Adam found paradise with her," I rejoined, even though that wasn't entirely accurate. But I am a writer, after all, and writers often varnish the truth. And, I confess, the imp of the perverse within me goaded me to see what her response might be.
But she only smiled neutrally and called a servant to refill my goblet. I was weary, and I knew I was drinking too much. When at last I was released from the obligations of a visitor and shown to my room, I could hear the rain pounding on the roof. The wind wailed against the glass of my window, in a room of dark wood and burgundy velvet. Lightning crashed.
I wandered to the window, drew open the curtains, and started at the sight of a figure down below in the carriage yard. It wore a greatcoat and a top hat, and stood utterly still with its head tilted back. When lightning illuminated its features, I saw that it was Jemmy Grove, Miss Allen's sweetheart.
His mouth was moving; I unlatched the window and tilted my head in an effort to catch his words through the whistling tempest.
"Barbara," he pleaded. "Barbara Allen." He wailed as if suffering the greatest of torments.
I wasn't sure where her room was situated, and I considered whether I should alert her to the fact that her beau—if beau he still was—was serenading her like an Irish banshee. For some moments I debated and had just made up my mind to seek her out when Mr. Grove dejectedly strode to a horse tied up at a gate. Thunder rolled and the miserable, sodden horse reared. The young man was obviously an accomplished horseman, for he gentled the steed with some ease, then mounted. He cast one last longing look at the house, and then he cantered away.
* * *
I slept heavily, for I was exhausted and half-drunk. The sun had barely risen when there was a knock on my door. I expected a servant, but after putting on my dressing gown, I opened the door to Barbara Allen. High color gave her a rosy glow, and I saw that she was fully dressed in a coat and bonnet.
She said, "Papa has gone. There's a fire at the mill. And I beseech you, Cousin Edgar, will you escort me to Mr. Grove's? I've had a message that he is quite ill."
I was much astonished by her request, but not at all surprised that the young man had suffered from standing in the storm. I was about to say as much when she reached out and squeezed one of my hands with both of hers.
"I beg of you, please take me to him. Mama, well, you see, sir, she has a condition and in order to sleep she must take...It will be hard to rouse her and I need to go now!"
It was utterly imprudent of me to assent. In the first instance, it was most improper to go at all, but for me to whisk away the daughter of the house without consulting her parents? Unpardonable. I had come with hat in hand to look for money from her father. My wife was starving, dying.
No, my love cannot die.
But in my life, I have often done the one thing I should not. I have sought out the dramatic in situations that other, wiser folk shun. So I told her I would escort her, dressed, and had a quick breakfast and very strong coffee.
The day was chill but we took two horses rather than a carriage. She pushed her mare and I had trouble keeping up as we charged through a blasted, black-and-white landscape so dank and dreary that I wished with all my heart to be back in New York. Then we came upon a cheery country home surrounded by pines, such a contrast to its surroundings, that my heart lightened upon seeing it.
True love shall win the day, I thought as a boy came forward to gather our horses and assist Miss Allen. She scarcely waited for me as she hurried to a large red wooden door set between two white columns. A fine black carriage with matching horses sat in the drive. I surmised that a physician had come.
Her gloved hand was on the knocker when the door opened and a young lady stopped on the transom. She was as fair as Barbara Allen was dark.
Miss Allen's eyes widened and I saw the rage of the mother reflected in the daughter; the other woman raised her chin, and said, "He is asleep. It would be better if you did not disturb him."
The fair young lady looked at me haughtily, then walked past us both to the carriage. Barbara Allen's fury did not abate and she stomped into the house without invitation. I followed hesitantly behind.
A maid appeared and curtseyed. I prepared to give her my coat and hat but Miss Allen walked right past her and started up a staircase. I raised a brow and the maid curtseyed again. Tears were streaming down her face.
"You may as well go up, sir, if you wish to say good-bye to our boy," she said, and then she fell to weeping.
I wondered where the rest of the household was. A well-mannered gentleman would have waited in the foyer. But I followed Barbara Allen up, then trailed behind her as she ran down a hall and pushed open the door at the end of it.
I knew it was a sickroom before I was one step inside. The odor made my heart clench. I thought of my love, my beautiful Sissy.
"Barbara," said a voice from the bed. It was young Jemmy Grove, blankets up to his chin. I was shocked at his appearance. His sunken cheeks and eyes gave him the aspect of an elderly man, older even than the beggar woman of the night before. His rheumy eyes ticked toward me, and he smiled with thin, bluish lips. "Mr. Poe. You do me such an honor."
"Why was Jennet Swanson here?" Barbara Allen demanded, and I caught my breath, astonished at the depth of her jealousy in the face of Mr. Grove's grievous condition. "You told me she meant nothing to you."
Evidently he means something to her, I thought, but did not say. Instead, I drew a bit away and thought to quietly walk back out the door in order to give the man his dignity. He fell to coughing so violently that I saw in my mind's eye droplets of blood upon a handkerchief, and the world whirled around me. What was I doing here? Oh, why had I come to Arkham? For all I knew, my love lay in similar agony, forsaken by me because of this foolish quest!
"I went to your window. I called for you," he said. "Forgive me, love, I was so thrilled that Mr. Poe had come to Arkham—"
"Why was she here?" Barbara demanded. "You are all alike! You men, you faithless devils!" This last she cried in a scream like that of a spoiled child who had been refused a plaything. "I have laid a curse on you, Jemmy! Death to a faithless lover!"
She hissed at him like a cat and whirled on her heel. I was so shocked I stood rooted to the spot. She did not wait for me. I heard her dash down the stairs, and still I did not go. I went to the young man, who was doubled over in coughing. Tears and sweat were pouring down his face.
"I was never...I am not faithless," he managed to grind out, through it took him some time. I heard Barbara Allen galloping away. "She fears it most because her father...Forgive me, Mr. Poe, I am a gentleman." And then he fell back against the pillows, gasping.
"Help! Mr. Grove needs his physician!" I cried, and started for the door, but Jemmy Grove grabbed my forearm, and his grip was uncommonly strong.
"Tell her I loved her. I always loved her," he pleaded, and then I was firmly moved out of the way by a gray-haired gentleman, who was indeed his physician.
I went downstairs and the little maid saw me. She threw her arms around me and sobbed as if her heart would break.
"Please, send any news, any change, to the Allen home," I asked her. "I—we would like to know."
She shook her head against my chest. "It will be the bell, sir. In our little chapel. When you hear it toll, you will know he's gone." She burst into fresh tears.
I stayed until she composed herself, and then I quitted the house, grateful to be gone, but sorry for the lad inside it. I was angry with Barbara, so very angry, until I remembered what had been said about her father, and I thought of my own faithless foster father. My foster mother had had no recourse but to endure John Allan's mistresses and bastards. She had died a broken woman. Perhaps it was the same with Demeter Allen and his wife. I knew the kind of special hell that brought to battered hearts.
The melancholy sight of the Allen house presented itself to me just as the sound of a bell tolled in my ear. I drew my horse up short and cocked my head to make sure I heard it true, and not simply in my poetic imagination. It rang. Dolefully, mournfully, endlessly. Jemmy Grove was dead then. I murmured, "Requiescat in pace." Rest in peace.
A scream echoed like thunder across the yard where Jemmy Grove had caught his death. Barbara Allen came flying out of the house, and limping after her was the old woman, Barbara's jet-encrusted shawl wrapped around her.
I could not hear their conversation, but their voices were raised. Then Mrs. Allen appeared in the doorway, and as I dismounted and ran to the trio of women, she saw me and swayed, tumbling to her knees.
"Demeter is dead!" Mrs. Allen cried. "In the mill fire!"
I approached. The old woman looked my way, and across her face spread the most evil, malevolent smile I have ever seen and hope never to see again. It was inhuman. Her eyes glistened like the jet of the shawl.
"He was unfaithful," the hag decreed. "A curse was laid."
Mrs. Allen seemed not to hear or else to not comprehend. She was lost in her misery.
Barbara Allen clutched her bosom as if her heart would burst from her chest. "Jemmy, Jemmy," she moaned. "Could you not be true?" And then she fell to hard, heavy sobbing in cadence with her mother.
"He was true," the woman said. Her hideous smile bore down on Barbara Allen. "A true love stands in the rain to beg forgiveness. A true love uses the last of his strength to tell his beloved that he loves her. But tell me, girl, what kind of lover asks for a death curse on a young man like that? What kind of faith does she show? No faith."
The woman pointed a gnarled finger at the distraught girl and said, "And a curse has been laid that the faithless would die. This curse must run its course."
Barbara Allen's sobs turned to gasps. Her eyes went wide. Then I heard the rattle in her throat as she fought to draw breath. She clamped her hand around her neck and reached out to her mother, who roused from her frenzy to rush to her daughter.
"Barbara! Barbara!" Mrs. Allen screamed. Then, as her daughter collapsed in her arms, she shouted at the woman, "What are you doing to her, you old witch? Stop it, for the love of God!"
"She did it to herself," the woman replied. "And not for any sort of love at all." And she burst into merry peals of laughter.
* * *
You know, of course, that I did not return to New York with money. The mill was not properly insured, and Mrs. Allen was ruined. It is not true that Jemmy Grove and Barbara Allen were buried near each other, but it is true that flowers blossomed on their graves mere days after their coffins were lowered into the earth.
On his bloomed a calla lily, for innocence, and rich, green grass, and a weeping willow tree.
And on hers, deadly nightshade, and nothing else, ever—no grass, nor nettle, nor weed.
Jennet Swanson married her fiancé a fortnight after Jemmy Grove's death. Mr. Grove was to have been her fiancé's best man. The old woman was never seen again in Arkham town.
And I wish down to my immortal soul that I had asked her to pray for Sissy.
On...Barbara Allen
One of the best known of all ballads is Child #84 (Roud 54), Barbara Allen. It is also known as "Bonny Barbara Allan," "Sir John Grehme and Barbara Allan," and "Barbara Allens' Cruelty." This ballad can be traced back at least three and a half centuries—Samuel Pepys writes in his diary on January second, 1666, that at a gathering Mrs. Knipp, an actress, sang the "little Scotch song of Barbary Allen."
In the ballad a man, sometimes named Sir John Graeme, (and sometimes, Jemmy Grove) is dying. In some versions he's dying for the love of Barbara Allen, in others, it's of an undisclosed ailment. He sends for Barbara Allen who eventually arrives. In one version she's angry at him for slighting her, in others she is simply unmoved by his condition. He turns his face to the wall, she leaves and he dies. Upon hearing of his death, Barbara Allen feels remorse for her treatment of him and in all but one version death takes her as well. (1)
Recently Barbara Allen has been sung by Bob Dylan, on the 2nd Gaslight Tape in late 1962, as well as Jean Ritchie, Shirley Collins, Joan Baez, and Pete Seeger.
1) Child, Francis James. The English and Scottish Popular Ballads Vol. 2. Mineola: Dover, 2003.
# DRIVING JENNY HOME
By
Seanan McGuire
NOVEMBER
Jenny can you hear me, I am driving in the rain,
I am looking for you darling, I am calling out your name,
For I'll do as much for my true love as any lover known—
Let the rain fall on the highway. I'll be taking Jenny home.
* * *
It only took three days for them to let me out of the hospital, but that was long enough. Jenny was already in the ground. So I stayed in my room, "recuperating," until my parents told me that I was going back to school whether I liked it or not. "Homecoming was a month ago, honey," they said, and "Leigh, this isn't healthy," they said, and I didn't have a choice.
I told them there was something I had to do before I could go back, and while they didn't like it much, at least they understood. I missed the funeral, after all.
It's raining when I walk into the cemetery where they buried my girl. That's right. That's exactly right. It should be raining because Jenny is dead, and it's all my fault. It should keep on raining forever.
Walking between the rows of graves is like something out of a nightmare or a dream—a bad dream, the kind that escapes nightmare territory by dancing on the razor's edge between surrealism and insanity. The kind where the walls bleed lobsters and the sky burns to ash when it touches the horizon. But the flowers I've brought to place on Jenny's grave are just flowers. They don't sing or whisper prophecy or turn into butterflies and fly away. When I finally find her tombstone—a simple granite rectangle that can't possibly summarize everything she was and should have been—and lay them down beneath the line that lists her date of birth and death; they don't bring Jenny back to me.
"God, Jenny," I say, sinking to my knees in the grass. "This is like some sick joke, only no one's telling me the punch line, and nobody's laughing. I'd take it back if I could. I'd take it all back if it would bring you home."
Jenny doesn't answer me. Jenny's never going to answer me again.
I stay where I am for what feels like an hour or more, the rain running down the back of my jacket and soaking my hair as I bow my head and cry. This can't be true. This can't be my life. It feels like just yesterday that I was picking her up for homecoming. She wore that dress the color of moonlight on the snow, and she fit in my arms like the missing piece of a puzzle. She's not fitting into anyone's arms now. She's dead and gone and she's not coming back to me.
They lock the cemetery gates at sunset. I don't want to spend the night here among the graves, and so when the light starts to fade, I force myself to move, wiping my tears away and leaning forward to press a kiss against the cold granite of Jenny's headstone.
"I love you," I say. "I'll see you soon."
Then I stand, wet clothes sloshing with every step, and start toward the distant gates. The rain is slacking off a bit, which is good since the setting sun is making it hard to see, and I'm starting to feel like a teenage emo cliché: sad lesbian in the graveyard in the rain. Jenny wouldn't like seeing me like this.
Maybe that's not the best train of thought because Jenny's never going to see me again, not like this, and not like anything else.
I'm crying again by the time I reach my car, a 1978 Volvo Bertone I inherited from my father when I turned sixteen and Mom made him dig it out of the garage. It handles like a tank and guzzles gas like nobody's business, but the collision that killed Jenny and put me in the hospital didn't even bend its frame. Jenny would have been fine if she'd been wearing her seatbelt, instead of twisted around and rummaging through the back seat. I guess I should hate the car for taking her away from me, but I like being able to get to school, and Jenny always loved the Bertone. She said it was just the right combination of old-fashioned and ugly as hell, and that she appreciated the contrast.
"That's my girl," I said, putting a hand gently on the door. That's when I see the ticket fluttering against the windshield like a trapped bird in the process of beating itself to death. It hasn't been there long; it's not even soaked through. I grimace. "Fucking cops," I mutter, and reach for it, only to freeze when another hand snatches it up before I can get there. A hand with long, slender fingers, the nails painted a shade of perfect moonlight gold that didn't come out of any bottle. Jenny always mixed her own nail polish. She said wearing anything off-the-rack, from clothes to cosmetics, showed a lack of commitment.
I lift my head. Jenny is standing next to the car in her moonlight dress, her homecoming corsage still tied around her wrist. She smiles, and she looks so sad that it makes me want to die, even though the sight of her is making me want to live like nothing has in weeks.
"Hi, Leigh," she says, and oh, God, her voice is the same as it always was. Nothing has changed. Everything has changed. "Can I get a ride home?"
What is there for me to say?
"Yes," I say, and unlock the car.
But she doesn't get in. Jenny doesn't get in. She just looks at me, her smile fading away a little bit at a time, until she finally says, "There are rules."
"Rules," I echo dumbly. My dead girlfriend is standing in front of me, and she's telling me there are rules. I don't understand why there should be since I'm clearly losing my mind.
"You have to drive me straight home," she says. "No stops—we're not going to the movies or going to Taco Bell or anything like that. You can't ask me why I'm here. And you can't give me a kiss good night."
It's an odd set of restrictions to get from a hallucination. I frown at her for a moment, trying to figure out what my subconscious is telling me—and then I decide that I honestly don't care. Jenny's here. Whether she's a hallucination or a ghost (and the fact that she's still holding the parking ticket is a vote for "ghost," unless I'm hallucinating that, too) doesn't matter. All that really matters now is driving Jenny home.
"Okay," I say, and we both get in the car, and I drive us away from the cemetery, and everything's okay again, if only for a little while.
DECEMBER
Jenny was gone when I turned onto her street, disappearing in the moments between checking my mirror and looking back at her. She left the parking ticket in the seat where she had been. She was always thoughtful that way.
That was a month ago. The ticket's been paid, and the shock waves have mostly finished washing through our school. I'm off the cheerleading squad, of course—even if the rest of the girls had sort of known about Jenny and me back when I was a base and she was a flyer, even though they'd been cool about the fact that we were lesbians, I was responsible for the death of one of our own. They couldn't look at me without seeing her, and I couldn't look at them without seeing her, and we were all tired of being haunted. So I quit the squad, and now no one's trying to use Jenny to define me. They're too busy using her to define everything else.
Jenny's dress at homecoming was moonlight satin? Fine, then, the theme for prom is going to be "Love Under the Moon," and everyone already knows Jenny will be awarded the coveted crown—the other girls who should be talking about campaigning by this point lower their eyes and murmur her name whenever the position of Prom Queen is mentioned, like wanting it would mean wanting what Jenny got: a brutal death and an early grave. They're canonizing her with memory, wearing away her hard edges bit by bit until all that's left is something perfect that they can tell their own children about someday when they're telling them not to drink and drive.
Jenny hadn't been drinking. Neither had I. Tyler did all the drinking for both of us, big muscle-headed football asshole who thought he was so great—thought Jenny was his for the taking—but learned he was so beneath her notice when he saw her dancing with me on the far side of the gymnasium, the fat girl in the tuxedo pressed up against the high school fantasy. We were both cheerleaders, but she was the one everyone remembered. I was just the one who made sure she didn't fall. I had one job. I couldn't even accomplish that.
But God, she was so beautiful that night. She was like the goddess of the moon, come down to hold hands with a mortal girl until the sun came up and the fantasy burned to ash. She was a fairy tale, she was a fiction, and I still felt like I had mud under my fingernails, despite the hours I'd spent scrubbing them clean. I guess that was why I let Tyler and his asshole friends get to me when they cornered me by the punchbowl. "Be the bigger person" and "just ignore them" is all well and good for people whose own high school days are fading in the rearview mirror of adulthood, but in the moment, when you're wearing teenage skin...
I didn't throw the first punch. I didn't throw the last one, either. I would probably have been beaten to a pulp if Jenny hadn't thrown herself into the middle of things, shouting at Tyler for being a Neanderthal idiot and shouting at me for letting myself be baited. I said I was sorry. I said I was stupid. I said I wouldn't let it happen again.
She said, "Take me home."
I drove home angry. That's the worst part. That's the part I'm never going to forgive myself for. Jenny wasn't speaking to me, just flipping through the radio and sulking, right up until she decided she was cold and undid her belt so that she could rummage through the backseat for my discarded tuxedo jacket. That was when Tyler came around the corner behind us, drunk and angry and driving a Prius that didn't stand a chance against my ancient bruiser of a Volvo. He slammed into my bumper, crumpling his car like a tin can and putting Jenny through the windshield.
The impact was enough to knock me out, even though I was wearing my seatbelt. By the time I woke up, it was too late. Jenny was gone.
How could I have been so stupid?
It's been two months since Jenny died and a month since I saw her. I've parked next to the cemetery gates every goddamn day since then, listening to a playlist of her favorite songs—too much country, too many teenage death ballads from the 1950s, and not nearly enough good old-fashioned rock and roll—and waiting to see her again. Even if it's just a glimpse, I need to know I'm not losing my mind. I need to know I wasn't hallucinating when I saw her before.
But the sun is going down, and there's no sign of Jenny. The grassy lawn is empty; the cemetery gates are closed and locked, as impassable as the wall to some forbidden city. I sigh. She's not coming. I'm losing my mind. I've had a month to think about what happened here, what I heard and saw and smelled. There's no way it was real; things like that don't happen. I close my eyes and lean back in my seat, breathing in through my nose and out from my mouth, and wait for the urge to wait for her to pass away.
It doesn't pass.
But someone's knocking on the window.
The sound is enough to make me yelp, twisting in my seat...and there's Jenny in her dress like moonlight with her hair pinned up just so, long gold curls anchored in an elaborate updo by little pins with glittering silver stars on them. My fingers itch to plunge into that hair and unwind it strand by strand, until the smell of her shampoo fills the world. I don't move. I can't move. Jenny frowns and knocks on the glass again.
I don't answer. Instead, I open the car door and get out, my eyes fixed on her face the whole time. I can see her—fine, the mind plays tricks. I can hear her, I can smell her, and that doesn't prove anything, that doesn't prove anything except that I miss her so badly I can taste it, and every day that I wake up and Jenny's not there is like another needle in my heart. I keep thinking it'll run out of room, and then the morning comes and proves me wrong. She's not real, she can't be real, and I'm going to prove it to myself here and now. I'm going to—
My hand catches her wrist. Her skin is cool, like she's been standing outside without a coat for too long, but it's still her skin, soft and familiar and real. My breath catches in my throat. I try to speak. Nothing happens, and Jenny daintily pulls herself free.
"You haven't answered," she says.
"Jenny, God, what is going on here?" I grab for her again, but she's too quick for me—she steps back and out of the range of my questing fingers, leaving them to close on empty air. "Why are you doing this? Do your parents know that you're not dead? Why are you hiding?"
"Because all I'm doing is asking for a ride home, and I'm only doing it because you keep coming here and wanting me to," she replies. She looks at me, eyes wide and sad and pleading, and I know the truth. There's no other explanation. "I died, Leigh. I died, and you didn't get me home. Even though you promised. You didn't get me home."
"I'll get you home, I promise." I'm making promises I can't keep—I'm making promises to a dead girl, and I've seen enough horror movies to know that this is a terrible idea, but fuck, what do I care? A week ago I was thinking about killing myself. At least this way I'll do something useful with what's left of my life. Any promise in the world is worth making, if I'm making it to Jenny.
"You can try," she says, and disappears, leaving the scent of vanilla in her wake. Vanilla, and something darker, something wet and green and old, like the moss that grows on gravestones.
Somehow, I'm not surprised when I turn around and find her sitting in the car, belt already buckled, hands folded primly in her lap. This is a dream, I think, and I'm opening the car door, I'm sliding back into my seat, and Jenny is there, Jenny with her corsage on her wrist and a sad, distant look on her face. This is a dream, and I never want to wake up again.
"Drive, Leigh," she says, and there's a sudden tension in her voice, like she wants to say so much more, but she doesn't know how. "You have to drive, or I can't stay."
"What?"
"Drive."
There's no denying the urgency in the word, and so I start the engine and hit the gas, and we're rolling, moving away from the cemetery and starting on the long road back to Jenny's house. She lives—she lived, she's dead, and I can't let myself forget that, not even with her sitting next to me, the smell of her vanilla perfume rolling through the cab like a storm front—on the other side of town. We have twenty miles to go, and that's if I take the short way.
She never said I had to do that. I turn left when I should have turned right, going for the route we always used to take when we were thinking that maybe a brief stop by the side of a wooded road would be a good way to spend a little of the afternoon.
Jenny is silent for the first part of the drive. I glance her way, but her attention is on the window, watching as the housing developments that ring the town like mushrooms melt away into forest, the semi-untouched wood that still owns this part of the state. If it weren't for her perfume, I'd think I was hallucinating. Maybe I could still be hallucinating. Do hallucinations have a scent?
I'm mulling that over when Jenny says, in a small, wounded voice, "You were supposed to drive me home. That was the deal, when my parents said that I could go to the dance with you. You told my father you would get me home by midnight. Remember? You promised."
I glance at her, startled, and startle myself all over again when she fills my field of vision, Jenny in her moonlight dress, Jenny with her golden hair, Jenny not under the ground and filling the bellies of a million worms. "I didn't mean to have an accident. You should have been wearing your belt," I say. The words are mulish, sullen; they fall into the space between us like clots of earth onto her grave.
"It scared me when I saw Tyler hitting you," she says and turns away from me, looking out the window. "He was so much bigger than you were, and he'd been drinking, and I just wanted to get away before somebody got hurt. I knew it would piss him off, seeing us there, and I didn't care, because I loved you. I should have cared. All-American boys get the All-American girls, right? That's what the Founding Fathers died for." Scorn drips from her voice. I flinch from it. I can't help myself. "I knew it was stupid. I knew I should have yelled for one of the chaperones and told them that we wanted to stay, and that he needed to go. But I was so mad. You made me so mad, Leigh. Why couldn't you keep your temper for just one night? It was supposed to be our night."
"I'm sorry," I say, and it's small, and it's stupid, and it's not enough.
"So am I," says Jenny.
She doesn't say another word during the drive. I have to look away from her when I turn onto her street—it's an unprotected left, if I watched my dead girlfriend, I'd be joining her in the ground—and when I look back, Jenny's gone, leaving only the faint scent of vanilla and the feeling that I've lost her all over again.
I drive by her house without slowing down. There's nothing for me there.
Nothing at all.
JANUARY
It's apparently been long enough since the funeral for public opinion to have shifted. I come back from Christmas break and the shrines to Jenny are gone, and the plans for the prom theme to match her homecoming dress have all been forgotten, replaced by something cliché about Greek gods and the beauty of Olympus. I should be happy. I'm not being reminded of Jenny every time I turn a corner or go into the school office. Instead, rage paces and snarls under my sternum like a captive animal. How dare they forget about her? How dare they go on with their lives like nothing has changed? Jenny is dead. She's not coming back to school tomorrow, or the next day, or ever, and it's not right for them to let go of her like this.
The school counselor says this is healthy. Says they're "moving on" and "coming to terms," and that maybe it's time my parents start paying for some independent counseling services for me since it's pretty clear she's not going to be able to help me properly through my grief—not with me refusing to let go of Jenny's memory. I call her a bitch and get thrown out of her office with a week's detention and a letter to take home to my mother. I should probably feel bad about that. I can't find the energy. I'm walking through a school that was haunted by Jenny only a few weeks ago, and now seems content to go on as if she'd never existed.
Three months: that's apparently the lifespan of teenage grief. That's how long our fickle hearts are meant to hold on to someone who's not there anymore. Anything more than that is cause for concern.
Poor, absent Tyler is the new darling of the student body. Tyler, the football hero who may never play again; Tyler, who was just trying to have a good time when he got drunk and crashed into my car, killing my girlfriend almost instantly; Tyler, who was the sort of guy every one of us should aspire to be.
I'm starting to think about murder.
I'm thinking about murder when I park at the curb in front of the cemetery, my hands resting lightly on the wheel and my eyes fixed on the middle distance, visions of Tyler's tortured face dancing like sugarplums through my daydreams. There's no rap on the window, but there's the smell of vanilla, and the sudden, definite feeling that I am not alone in the car. I start the engine before I turn to flash a smile at Jenny, and say, "I'm happy to give you a ride home."
"This isn't healthy," says Jenny, a frown on her face and a curl of golden hair hanging across her forehead like a banner. "You shouldn't still be showing up here."
"Why wouldn't I?" I pull away from the curb, turning my attention back to the road at the last possible moment. I don't want to take my eyes away from her, but I'm coming to learn the rules of our monthly encounters: I have to drive. That's what matters more than anything else. "You need a ride home. I promised."
"Leigh..."
"How could I live with myself if I knew you weren't at peace because I broke my word to you?"
"But I'm not at peace, Leigh." She sounds like she's in pain. I start to take my foot off the gas, automatically turning to reach for her and try to hug that pain away. I see her face when she recoils, when she cries, "Drive! You have to drive!"
My foot presses down almost of its own accord. The car lurches forward, my heart pounding against my ribs, and for a moment, I think this is it: this is how I die.
But the moment passes, and the car is back under my control, and we're rolling easy down the road as Jenny says, "I can't rest in peace. You won't let me. Everyone else is starting to let go, they're starting to ease up on my memory—even my parents. And then there's you."
"I miss you." The words are small and stupid and big enough to encompass the entire world. I miss her. That's all that I'm capable of doing anymore.
"You have to let me go."
"Or what?"
"Or this keeps happening over and over again," she says. "I keep showing up. You keep driving me home. I keep disappearing. Over and over."
"For how long?"
"I don't know. Forever, I guess."
I think about that as I drive, the streets melting away around us. The air in the cab smells like vanilla. At some point Jenny realizes I'm not going to say anything else, and she turns on the radio, spinning through the stations until she finds a channel that's playing the kind of music she likes, all soft country ballads and too much auto-tune. One of her favorite bands is on, performing a song that hadn't been released yet when she died. She makes a small, wordless sound of delight, and that's it: that seals the deal. I love her and I want her to rest easy, but that doesn't mean I can let her go. Not yet. Maybe not ever.
"I'll have their new album when I pick you up next month," I say, as we turn onto her street. "You can listen to it during the drive."
"Leigh—"
"I'm sorry you're not resting in peace. But I'm not resting in peace either, so at least we can not rest in peace together for a little while longer. If that's selfish, I don't really care. I miss you too much. I can't just stop."
Her hand touches my cheek, fingers cool. I don't turn. I don't want to see her disappear.
"I love you," she says.
"I love you too," I answer, and she's gone, and I drive past her house without slowing down or stopping. Forever is a long time. I'm not sure it would be long enough.
FEBRUARY
Now they say that love is ended by betrayal or the grave,
And they tell me to give up on her, the one I couldn't save,
But I'll do as much for my true love as any lover known—
I will know no rest or solace 'til I'm taking Jenny home.
* * *
Valentine's Day without Jenny is another word for Hell. I stay home sick, choosing another black mark on my attendance record over the school halls festooned with paper hearts and filled with girls giggling over their discount chocolates and wilting roses. Two weeks later, when Jenny appears in front of the cemetery, there's a bouquet of daisies—her favorite flower—waiting for her on the dashboard, along with the CD I promised her.
She's still my Valentine. Death doesn't change that.
Death doesn't really change anything.
MARCH
It's been four months since Jenny died, and I'm starting to think about the mechanics of suicide. It can't be that hard to kill somebody, can it? Tyler managed it, and Tyler's a dumb jock with more muscles than brains. Or he was, anyway; they still don't know whether he's ever going to wake up, and even if he does, there's no way of knowing whether he'll ever walk again. His football career is over, buried alongside Jenny's body, and that might be a comfort to me if I didn't miss her so goddamn much, if his teammates didn't glare at me when they pass me in the halls, like I was the one who suggested he try to put the make on my girlfriend at the homecoming dance. None of this was my idea, I want to scream. None of this is the way that I wanted my junior year of high school to go. But try telling them that. Too many words, too many syllables, too many concepts for their atrophied little brains. Tyler, Jenny, and I wound up in a weird sort of triangle on the night of the dance, two of us competing for one girl, and now Jenny's dead and Tyler's in a coma and I'm still here, which makes me the perfect target.
I guess if I were as suicidal as I feel I'd bait them and let them beat me to death behind the school. At least that way they'd get punished for it, and I'd finally get to stop living in a world that doesn't have Jenny in it anymore.
...but I don't really live in that world, do I? I just exist there. It's four months since Jenny's funeral, and here I am again, parked in front of the cemetery, waiting.
The sun reaches the horizon and again, Jenny—Jenny in her moonlight gown, Jenny with her golden hair, and her corsage still fresh on her wrist. Mine is just so many fallen petals now, sitting in a dish next to my bed where the smell can chase me down into my dreams. Jenny, looking at me with fond exasperation, one satin-toed foot tapping on the grassy knoll.
"Again, Leigh?" she asks, and before I can answer her, she chases the question with a question, asking, "Can I get a ride home?"
"Always," I say, and then she's in the car, appearing like a miracle, and my foot is on the gas, and everything is right with the world. As long as I'm with Jenny, everything is fine.
Again, I take the long way back to her house, choosing more time with her over the expediency of city streets. It's better this way, I tell myself, and I realize I actually mean it: if this is a dream, I know it'll end when I make the final turn onto her block, and if it's not a dream—if my dead girlfriend is really riding in my front seat like this sort of thing happens every day—then the last thing I want is for someone to see us. They might react the same way I did, with confusion and disbelief and denial. Or they might decide I didn't deserve this, and give someone else the task of driving Jenny home.
"Tyler's still in a coma, you know," I say, because I have to say something; I have to fill the silence between us with words, or it's going to drown me. "They're not sure whether he's ever going to wake up again."
"Good," she spits, with such venom that it startles me. "He deserves to be caught that way. Not living, not dead. Just lost, for as long as their machines can keep him there."
I worry my lip between my teeth before asking, "You really mean that?"
"Yes," she says, and "no," she says, and "I would be here with you if it weren't for him. I would be here with my family. Really here, I mean, with skin and bones and a heartbeat, not drifting through every time I need someone to drive me home. I'd be getting older. You know, I read this book once, about a unicorn who'd been turned into a human? And she hated it. She said that she could feel her body dying all around her. I know what she meant now, Leigh, and I miss it. I miss the feeling of my body dying, because it meant that I had a body that could die. It meant that I was real. More than just a memory. It meant I belonged to the world. I'm never going to have that again, and it's all Tyler's fault."
There are so many words that it's almost overwhelming. It takes me a moment to process them all, and we're moving the whole time, getting closer and closer to the point where she leaves me again. I want to ask the best question in the world, I want to stun her with how well I understand, but when I open my mouth, what comes out is, "So stay."
"I can't," she says. "I'm dead, remember?"
There's laughter in her voice, and pain too, like she likes remembering what she is as little as I like being reminded of it. I don't say anything, but I hit the gas just a little harder, and neither one of us says anything for the rest of the drive.
It's just like before. I turn onto her street, and the smell of vanilla fills the car, and when I turn to look at her, she's gone like she was never there. I may as well have been driving a hallucination across the city. I pull up to a stop sign and lean over to touch the seat. It's warm. She was there enough to warm up the seat with the body that she doesn't have.
Maybe that means something.
I hold that thought firmly as I drive myself back home. Jenny could still warm a seat, and maybe that means something...but what, I just don't know.
APRIL
Jenny's already standing on the curb when I pull up, her arms clasped tight around herself like she's cold. I find myself wishing she'd died wearing something warmer, which turns quickly to wishing she'd never died at all, and that's dangerous; that's the kind of thinking I can chase down the rabbit holes of my mind all night long. So I just stop the car and roll down the window and wait for her to ask the question.
"Can I get a ride?"
"Always," I say, and she disappears, leaving me alone. This time, my heart doesn't stop. I've learned enough to know what comes next. I turn, and there she is in the passenger seat, her seatbelt already fastened, a smile on her perfect lips. Trying to be casual, I say, "You look good today."
"One good thing about death: no more bad hair days," she says with a laugh that isn't a laugh at all, more a close cousin to a sob. Kissing cousin, even, the two so tangled together that I couldn't pry them apart if I tried. "I'm really glad I like this dress."
"I wonder if that's what you'd be wearing if they'd buried you in something else." The words are thoughtless. I cringe.
Jenny doesn't seem to mind. If anything, she looks relieved. She's been dead for six months, and this is the first time I've talked to her like that mattered at all, like it changed anything about our relationship, apart from how often we get to see each other. "I think so," she says. "It's different for everybody, but most of the ghosts I've met have been wearing something that really mattered to them when they were alive. Lots of wedding gowns, tuxedos, graduation robes...this was the prettiest dress I ever owned. It only makes sense that I'd be wearing it now."
"I'd say that wearing it made you the prettiest girl in the world, but you didn't need a dress for that."
Jenny laughs without the sob this time, and leans forward to turn on the radio. "Just drive," she says.
So I do, and everything is perfect, and I could go on like this forever, just me and Jenny and our monthly date, her in her homecoming dress, me in whatever I threw on that day, driving into eternity.
MAY
Now they say that love is something you'd be lucky to forget,
But I say that I was lucky on the day that we first met,
And I'll do as much for my true love as any lover known—
I will roam the lonesome highways 'til I'm bringing Jenny home.
* * *
The halls are buzzing with the news that Tyler—elevated in his absence to young god, deified like Jenny was, but without the absence of flesh to allow his memory to erode—has started responding to treatment. Why, he opened his eyes yesterday, which is nothing short of a miracle as far as his legion of adoring fans is concerned. If this continues, he could actually wake up soon! Imagine! Tyler, All-American high school god, walking among us mere mortals like we have the right to glory in his physical presence and breathe his rarified air!
It makes me want to vomit, or punch someone, or scream. But all these things are anti-social, and the school counselors are still watching me more closely than I like, since apparently my inability to "move on" from the death of the first girl I ever loved means that I'm a potential suicide risk or school shooter or something. I'm not really clear on what the problem is, and no one else seems to be either. They just know I'm not fitting easy into their pre-fab high school mold anymore, and so they watch me, and they wait for me to make a mistake.
Tyler's name is on everyone's lips today, even the teachers, who urge us to focus on our studies because "that's what Tyler would want us to do." By seventh period, I've had enough, and when my history teacher invokes Tyler's name in an effort to quiet us down, I stick my hand in the air and ask, "Do you think Tyler was quiet when he was committing vehicular manslaughter? Or do you think he had time to scream at the sight of Jenny's corpse before his brain damage kicked in?"
I'm sent to the principal's office for my trouble, a red detention slip clutched in my hand, and it's not until I see the sun setting through the study hall window that I realize what this means—that I won't reach the cemetery until hours after I usually arrive. I bolt to my feet.
"Sit down, Miss Winslow!" barks the Vice-Principal, and my knees buckle, years of trained obedience ordering me back into my seat before my conscious mind is invited to the party. I shoot another panicked glance at the window, but it's too late, it's too late; the sun is dipping down below the horizon. I don't know much about the strange dance that I've been locked in since homecoming, and still something tells me that this is a line that should never have been crossed. She appears at sunset. Well, the sun has set, and when I reach the cemetery, Jenny won't be waiting.
Even knowing that, I run for my car as soon as we're released, breaking speed laws all the way down to the cemetery gates. But Jenny isn't there. I stay until midnight, listening to her favorite CD over and over again and praying to a god I don't entirely believe in, and Jenny never comes. I broke the rules. I broke the chain.
What if Jenny never comes again? What am I going to do then?
JUNE
I skip school on the day Jenny's due to appear, even knowing she won't be there until sunset. I missed her once. I can't run the risk of missing her again. Not when we only get one night a month, and that night is limited to however long it takes to drive from the cemetery to her house—not exactly the kind of dates I used to lay awake dreaming about. I haven't kissed her since the night she died. I ache to hold her in my arms, kiss her cheek, and tell her how much I've missed her, how much I miss her every single day. But if I can't do that, I can at least do this; I can be here, I can wait for her until she comes.
I have to move the car three times when the security guard comes by and gives me the hairy eyeball, suspicion written plainly on his face. What's a chubby teenage girl in a beat-up Volvo doing parked in front of their cemetery? What mischief am I planning?
No mischief, sir, I want to say, but I can't imagine he'd take well to being told that I was just here to pick up my girlfriend, who's been dead for seven months—don't worry, she looks just fine, on account of how she doesn't have a body anymore. I'd be lucky if he called the cops. It's more likely he'd call the local loony bin, and I'd be hauled away by men in white coats, screaming for my ghost girlfriend all the while. No. Nuh-uh. I don't have time to be committed, and so I move the car again and again, wasting gas and wasting time as I wait for the magic moment when Jenny will appear.
Then I'm coming around the corner and there she is, blazing up in the headlights like a fairy tale princess, all moonlight gold and helpless longing. I stop the car, roll down the window, look out at her, and smile.
"Hey," I say. "You need a ride?"
I'm trying to sound cool and smooth and like the sort of person who belongs in a place like this. All I really manage is sounding like my dorky self. Jenny still smiles as she walks toward the car. That's all the validation I need.
"Where were you last month, Leigh?" she asks. "I thought maybe you were getting over me."
"That's never going to happen," I say. "I had detention." She disappears, and then she's in the car with me, and my heart hurts from the reality of her. I know that what I have to say will hurt her, but I have to say it anyway. "Tyler's waking up."
Jenny goes still, and it's not until that moment that I realize she isn't breathing, she's never been breathing, not any of the times I've seen her or any of the nights when I've driven her home. The realization changes something. It's like for the first time I can't pretend that she's not dead, not on any level. There is only one living body in this car, only one person who's getting older, only one unicorn trapped in a human form. The other person, the Jenny-shaped person...she's dead and gone, she's dwindling into dust underground, and she's never coming back to me. This is the closest we're ever going to get, these strange moments of stolen time in my car.
"He shouldn't get to wake up," she says finally. There's a bitterness in her voice that runs all the way down to the bone, the kind of dark, resentful hatred that used to be reserved for people who abused animals or argued against gay marriage. I bite my lip and keep driving, not wanting to see the look on her face as she continues, "I didn't get to wake up. Why should he?"
"He may never walk again."
"Well isn't that a shame—oh, wait. I'm definitely never going to walk again because I'm dead, and it's his fault, and I can't even rest easy in my grave and forget about all this, because you won't let me go. So what do you want me to do, Leigh? Be happy for him? Oh, hooray, the man who killed me is waking up, and maybe his life's been changed forever, but it's still a life. He still gets to have a life. He gets to grow up and get old and I get to keep asking you to drive me home. How is that fair?"
"It's not," I say quietly.
"Then what are you going to do about it?" There's a challenge in her voice that I don't know quite how to answer, and so I don't. I just drive her home.
What else am I supposed to do?
JULY
Well, I'm supposed to commit murder, for one thing.
It's a little surprising that I didn't think of it sooner, but as the days stretch out and the school year winds to an end, the thought preys on me more and more often. How hard could it be, really, to kill someone who's bedridden, slipping in and out of consciousness, and incapable of fighting back? Not that hard. Getting to him is going to be the difficult part...and wouldn't it make Jenny happy to know that I loved her enough to kill for her? I want to see her smile, really smile, just one more time. And not just from her memorial page in the yearbook, where they've used a picture of her in her homecoming dress—the only dress she has anymore—and her corsage. None of the pictures in her memorial collage have me in them. Without Jenny and the cheerleading squad to drag me into the school's social limelight, I'm fading from everyone's memory, another high school weirdo worthy only of dismissal.
Maybe that's a good thing, given what I'm planning to do.
It wasn't a plan at first, just an idle thought, a continuation of that half-conversation that I hadn't been able to finish with Jenny. But it's grown, bit by bit, into something bigger and more powerful than it was when it began. Tyler killed Jenny. Tyler doesn't deserve to have any kind of a life, not when Jenny doesn't get to. Tyler needs to die, and if I want to show her that I'm still a good girlfriend, I need to be the one who kills him. It's as simple as that.
I don't say anything when I go to pick her up; I just smile, and hand her a copy of the yearbook so she can see all the nice things people said about her after she was gone and in the ground, and then I drive her home. It's the least that I can do, all things considered.
AUGUST
Jenny, I was foolish, I was selfish, I'm ashamed,
And I'm praying you'll forgive me, though I know I should be blamed,
For I'll do as much for my true love as any lover known—
I will never know salvation 'til I'm taking Jenny home.
* * *
Tyler has his own room—naturally he does, his parents have money and they've never been shy about spending it on their only beloved son. That makes things a little easier. Finding it is easy; our classmates have sent offerings of flowers and stuffed toys in such great numbers that when I walk up to the admission desk with a bouquet of roses in my hands, the nurse barely even looks up from her romance novel before spitting out his room number, three little digits that don't seem like nearly enough information to lead me to murder. But there they are, and there I go, walking down the hall unquestioned, the roses in my hands somehow serving as an all-access pass.
I've heard they don't allow flowers in the ICU, but apparently, Tyler's far enough along the road to recovery that they've moved him into a lower security grade. That, too, works in my favor.
His door isn't locked. Three strikes, Tyler, you're out.
The room is dim and quiet, save for the soft, steady rasp of the machines that keep him alive. He's a wasted skeleton of a man, all that football muscle melted away to reveal the scarecrow that was sleeping for so long inside his skin. I stop at the foot of the bed, looking at him. Maybe this isn't about unicorns at all; maybe this is some kind of strange Wizard of Oz parable with Jenny just trying to get home and Tyler trapped in the echoing cavern of his own mind. I can't decide whether that makes me the Lion or the Tin Man. Am I looking for courage or am I wishing for a heart?
Now that I'm here, I don't know what to do. I don't have a syringe; I can't inject air bubbles into his arm like they do on Dad's crime shows, and I wouldn't know how to do it even if I could. I'd put a pillow over his face, but he has machines to do his breathing for him, and I'm pretty sure they'd start beeping like crazy if I pulled them loose.
The door opens and closes behind me. "More flowers?" asks an unseen woman. "Well, put them with the rest. Poor boy's lucky he doesn't have allergies. That would just be one more problem on top of a mountain of them."
"Is...is he going to be okay?" I'm not really concerned about Tyler's welfare—I'm not—but I'd expected him to look more like, well, himself. A great big bear of a teenage boy, briefly bedridden, gathering the strength to jump right back into his life. Not this wasted bundle of bones and sickness.
"That depends on how you measure 'okay,' I suppose." The owner of the voice is a middle aged woman in pale pink scrubs. She takes the flowers from my hands, apparently able to sense that I can't bring myself to move. "Is he going to live? At this point, the prognosis is good. His folks have paid for the very best care, and he's got a strong will. He's stubborn. Stubbornness counts for a lot in cases like these. But is he ever going to walk again? Throw a pass or kick a ball or anything like that? No, I don't think that's likely."
"Oh." I pause then, frowning. "Should you be telling me this?"
"A pretty little thing like you only sneaks into a hospital with a bunch of flowers for two reasons: love or hate. I read the papers. I know which one it is." The nurse looks over her shoulder at me as she sets my roses down amongst the rest. "She won't rest any easier if you kill him, and neither will you. Are you sticking to the rules? Do you drive her home when she asks you to?"
My throat is a thin straw through which only air can pass. I squeak a few times, trying to speak, and finally settle for a nod.
"That's good. Have you tried to kiss her?"
I've dreamt about it. That isn't the same thing. I shake my head.
The nurse nods approvingly. "That's good. That's real good. It's not safe for a girl like you to be kissing a girl like her. There are consequences, when the living love the dead. Now I'm going to ask you one more question, and I'll thank you to find your voice, since your answer is going to determine whether or not I call for security. I'm assuming you came here to kill this boy. You can go ahead and correct me if I'm wrong, but that's not my question. Are you still planning to try?"
"N-no, ma'am," I manage. "He shouldn't be alive when Jenny's not, but it's not my place to change that. I guess the worst thing I can do to him is leave him alive."
"Good girl." She smiles at me, and then glances meaningfully to the clock above the door. "You'd better hurry if you want to pick her up on time."
I want to stay and ask her what she knows about ghosts, how she understands my arrangement with Jenny...who she's been driving home. But she's right; the sun will be down soon, and if I want to make it to the cemetery, I need to go, and I don't really want to be here, in this room where time is rotting on the vine. So I turn and run, leaving the hospital, leaving Tyler to his own living hell, and it's not until Jenny is sliding into her place beside me that I really wonder what that nurse knew, or why she asked if I'd been trying to kiss my girl.
I go back to the hospital the next day. The nurse I met in Tyler's room isn't there.
Somehow, that's not really a surprise.
SEPTEMBER
School starts up again in an explosion of cheerleaders and football players wearing orange and green uniforms and smiles like they just won the universal lottery. It's freshmen slinking in the halls and sophomores seeming suddenly easy in their skins, it's juniors with their chests puffed up with upperclassman pride and seniors smiling beatifically, rulers of their small, time-delineated kingdom. It's all so stupid, and there was a time when I would have loved it. I would have been walking in formation with the rest of the cheerleaders, Jenny by my side, and we would have been queens of the world. Homecoming last year was supposed to be our grand declaration of love, and by now, everyone would have been used to the idea that we were together. We could have had one perfect year of high school, me and Jenny, Jenny and me.
Instead, I'm just another nobody at the edge of the crowd, wondering why this ever seemed to matter. All the memorials to Jenny are gone, and there's a whole new class of freshmen who never knew her, and will never know that they're supposed to miss her. Summer has restored the campus, sweeping its ghosts away, and now I'm the only one who's haunted.
I don't know how long I can live like this.
I'm still dwelling on that when it comes time to pick Jenny up again; she slides into the car, all vanilla and moonlight, and asks, "Well? How's senior year?"
She still loves the idea of the future we crafted for ourselves, back when we were lying on our backs behind the tumbling mats, hands entangled, and both of us were breathing. I shake my head and start the car. "Same shit, different semester," I say. And then, because the question is burning me, I ask, "Why am I not allowed to kiss you?"
Jenny is silent.
"I mean, I get to drive you home. I know you're solid. I've touched your hands and seen you hold things. So why can't I kiss you? What makes it so wrong to want to kiss my girlfriend?"
"I'm dead, Leigh." Her voice is the whispering of wind among the gravestones, barely audible, impossible to ignore.
"So what? You're still my girl. I'm not giving up on you."
"I'm dead, and you're not."
I've thought about changing that so many times. "And?"
"And if I kissed you, it would..." Jenny sighs. "Everything dies, Leigh. Everything. But some things make you die faster."
"Like kissing dead girls?"
"Like kissing dead girls." Jenny reaches over and touches my hand, gentle as a promise, cruel as a prayer. "Homecoming is next month. I've been buried for almost a year. Don't you think it's time you let me go?"
I don't answer her because that question is every answer I've been seeking for the last eleven months. "I think it's time that something changed, yeah," I say, and we drive on.
OCTOBER
My tuxedo still fits. Grief either bulks you up or slims you down, and I guess I've gone for the latter—too many days when I couldn't bring myself to eat, too many nights spent crying until I threw up. It was a little snug when I wore it the first time and now it's a little loose, but I look okay. I think Jenny will appreciate it.
"Are you going to homecoming?" asks Mom when she sees me coming down the stairs. She sounds surprised and maybe a little hopeful, like this is a sign that things are changing for the better.
"Yeah." Homecoming isn't tonight—it's always on a Saturday, and it's only Monday now—but this is the anniversary of Jenny's death, and I don't feel like correcting my mother. I finish descending the stairs and press a kiss against her cheek, sweeter than a note left on my dresser, crueler than an explanation. I don't think there's any way to explain what I've decided to do. "Don't wait up, okay?"
"I won't," she says, relief plain on her face. "Who's the lucky girl?"
"It's a surprise." My new boutonniere is already pinned to the front of my tuxedo. I figure Jenny doesn't need a corsage—she still has hers from last year—but I bought her daisies, just to make the symbolism clear.
"Have fun tonight."
"I will." I should feel bad, I know I should, but I can't. There's nothing for me here, and I still have to keep my promise. I have to finish driving Jenny home.
I pull up to the cemetery just as the sun is starting to go down, and Jenny's there, my Jenny, in her dress like moonlight. She gasps a little when I get out of the car and she sees me in my tuxedo. That's what I was hoping for, that moment of shock, when she's too busy staring to react as I stride toward her, the daisies held out in my hand.
"These are for you," I say, pressing them into her hands. She takes them—she's solid, she's solid, because she's holding the flowers—and is still looking at them when I lean forward and press my lips to hers.
She still tastes like vanilla lip gloss, but there's something else there, something dark and sad and dry as dust. She pulls back, eyes wide and filled with dismay. "Do you know what you've done?" she demands, and I do know, I do know what I've done.
So I smile and say, "Too late now," and this time when I kiss her, she doesn't pull away. We'll get into the car soon; I'll drive her home, and this time, when whatever happens, happens, I won't survive. That's all right. That's what I wanted.
I made her a promise, after all. I promised that I'd get her all the way home.
* * *
Jenny, darling Jenny, there is no need to explain,
I have seen your lonely graveside, I have waited in the rain,
And I'll do as much for my true love as any lover known—
Though my family will grieve me, I'll be driving Jenny home.
On...The Unquiet Grave
The Unquiet Grave is Child Ballad 78, Roud #51. Child wrote that, like Sweet William's Ghost, it sprang from the belief that excessive grieving for the dead would disturb their repose.
Child quotes many traditions that believed too many tears or too much mourning by the living would hurt the dead. In Scotland there is a story where a sister cried night after night for her brother. He finally appeared and reprimanded her for the "extravagance" of her sorrow and its "rebellion" against the decrees of Providence. He told her that every tear made him cold and weighed him down. Child also cites a story in the Grimm records of a child coming to its mother and asking her to stop her tears because they make him wet and he cannot sleep. She stops and he later returns to inform her he's now dry and at peace.
In what is the strictest ban against grief, Child wrote that the ancient Persians forbid weeping for the dead. It was against the express command of the Almighty and as such was a heinous sin and caused the dead to be trapped in a river, where they constantly experienced the "agony" of drowning.
The plot of the ballad has someone (male or female) mourn on the grave of their love for a year and a day. At the end of that time, the dead love appears and asks why they are not being allowed to rest. The mourner requests a kiss good-bye to which the corpse replies that their breath smells foul and if you kiss me you will soon join me. (1)
The Unquiet Grave has been sung by the British folk band Lau, by Ween, as well as by Joan Baez, Kate Rusby, Bobby McMillon, and The Dubliners. Composer Ralph Vaughan Williams wrote several arrangements for the song.
1) Child, Francis James. The English and Scottish Popular Ballads Vol. 2. Mineola: Dover, 2003.
# HOLLOW IS THE HEART
By
Simon R. Green
Bradford-on-Avon is an old town, in an old country. Sick and feverish with centuries of history. And some things older than history. Older, and more foul.
My name is Jason Grant, and if there were any justice in this world, you'd already know my name. My books would be everywhere, my name on everyone's lips, and my face on all the chat shows. Instead, I make a precarious living researching dubious articles for partwork magazines, and generally hacking it out for pitiful returns. I grub about for work wherever I can find it, cranking it out by the yard to pay the bills. It's been a long time since I've written anything just to satisfy my soul.
I did write a whole bunch of novels and screenplays, but nobody wanted them. I was a journalist, once, doing my bit for a small but respected local paper, the Wiltshire Record and News. But that was then, and this is now. I finally have a story worth the telling. A story to prise open the eyes of the world, and make them see things in a whole new way if only I dared submit it. It has been made very clear to me that silence and obscurity are the price of my survival. But I'm not sure I care anymore.
It all started when I made one last attempt to get my old job back.
I sat in the outer office of the Wiltshire Record and News, waiting to see the editor. Being calm and quiet and not making any trouble, playing the part of the prodigal son and the penitent return. Returned to the scene of my crime, like a dog to its vomit, to beg a few crumbs from the editor's table because my cupboard was bare, and I was getting hungry. My old boss, Samantha Walsh: editor, publisher and Conscience in Chief of the local weekly rag. There was no way in hell she was ever going to give me my old job back, I knew that. But if I could just get my foot in the door, hold her attention with some fast talking... I might yet walk out of here with a story assignment. Do a good enough job, and it could lead to regular work. And I wanted that.
Not a staff position, obviously, but a local stringer with local connections is always going to be a useful asset. The odds were stacked against me. I couldn't have blotted my copybook more thoroughly the last time I was here if I'd pissed in the printer's ink.
I sat politely on my fiendishly uncomfortable visitor's chair and glowered at the clock on the wall. The editor was deliberately keeping me waiting, to make sure I understood my place in the scheme of things. That I was not needed, or even welcome. Everywhere I looked in the outer office, something old and familiar looked back at me. It was as though I'd never been away. The same dreary old fittings and furnishings, cheap but durable. The carpet worn thin from people pacing up and down as they waited to be summoned into the inner sanctum to learn their fate. Dusty plastic venetian blinds at the windows, cutting the sunshine into strips. The same framed front pages proudly displayed on the walls; names and faces and news from a time when people actually read the paper to learn what they needed to know. Significant stories and excited headlines, forgotten moments from the county's past, going all the way back to the First World War
I sat forward in my chair and stared at the floor. So I wouldn't have to look at anything else. There had been a time... when this had felt like home.
There's nothing like visiting an old haunt to make you feel old and unwanted. The surroundings might not have changed, but I had. I look back at the person I was then, and I barely recognise him. I looked at my reflection in the long mirror on the other side of the room. A man in his late thirties who looked older. Thinning hair and a hard-used face, more than a little scruffy. I should have shaved before I came out or at least found the time to force a comb through my hair. But it had been a long time since anyone cared what I looked like, including me.
The door to the inner office opened suddenly, and the editor glared at me. As though I was the one who'd kept her waiting. She nodded briefly as though she didn't trust herself to speak, then turned around and stomped back to sit behind her editor's desk. Her place of power. Leaving me to trail into the inner office after her and shut the door very quietly and politely behind me. Being careful not to slouch. She always used to yell at me when she caught me slouching. There was a time I could have got away with it, but not now. I sat down on the bare wooden chair set out before her desk piled high, as always, with papers overflowing her In and Out trays. She just sat there and waited for me to speak. Because, after all, I was the one who wanted something from her.
Samantha Walsh was middle-aged with prematurely grey hair and deep lines etched around her eyes and mouth. She dressed neatly and conservatively, in a way that didn't so much ignore fashion as bypass it completely. The ultimate authority figure, the iron hand in the iron glove. Who'd spent so much time occupying the moral high ground, it was a wonder she didn't get nose bleeds from the altitude. I was amazed she could even see us poor mortals down below. The editor fixed me with her usual steely gaze, and I gave her my best respectful smile.
"Hello, Sam. Been a while, hasn't it...?"
"It's Ms. Walsh to you, Grant, and don't you forget it. You have no friends here. And sit up straight. You make the place look untidy."
So that was how it was going to be. I sat up straighter, squared my shoulders, and did my best to look like a professional. The editor sniffed as though reluctantly giving me credit for trying.
"All right, Grant. I read your e-mail. You haven't forgotten how to grab a reader's attention, I'll give you that. So against all my better judgment, I'll admit I'm intrigued. Hit me with your proposal, but make it quick and succinct. I've got a paper to run and a deadline to meet. Just because we're a weekly, it doesn't mean I've got time to waste on the likes of you."
"You'll like this one," I said, doing my best to sound confident. "I've got a new local take on a very old legend. A story that used to be on everyone's lips, in this town and around, that no one has talked about in centuries. I stumbled across this particular piece of local history while looking for something else, which is always the way. I was doing research on a story for Hidden Worldz magazine. A story that will not be appearing because the magazine had the bad manners to fold before I could hand it in. Or get paid. Do you remember the old story of the Hollow Women?"
"Refresh my memory," said the editor. Which was her way of saying that she didn't, but was prepared to listen.
"Women who can only be seen, and only appear to have substance, from the front. If you look at them from the back, they're just an empty, hollow husk. A shell of a woman, no depth to her at all. The old legend tells how they prey on young, unattached men. They win the men's hearts and then break them, seduce them and make a child with them to continue their own kind... and then disappear. Always girl children... never been a Hollow Man. These women were predators, giving every appearance of being human. But inside they were empty, emotionless, inhuman.
"Obviously, this was designed as a moral warning for young men back then. Don't go off with strange young women or there might be unfortunate consequences. Avoid shallow types, stick with a real woman and make a commitment to home and family."
"Funny how, in these old stories, it's always the men who have to be warned, and the women who are presented as the villains of the piece," said the editor.
"Well, quite," I said. "The point is, I have uncovered evidence that strongly suggests this old legend had its roots and beginnings right here, in the town. A basis in truth and real history. I started with an old folk song from 1815. 'The Foggy Foggy Dew.' I went jumping from link to link across the Net and ended up with a series of stories coming out of that old disreputable part of Bradford-on-Avon, back when we had real slums. The Hollows. I think... this all goes back to women from the Hollows."
The editor sniffed loudly, but I knew I had her. Sam does love the old folk stories, thinks they're part of what shapes local character. Add a local connection, and she was hooked.
"You might have something there," she conceded. "Not enough for me to offer any advance money, or even a guarantee of publication when the story's completed, but... I am interested. If you do good work on this, turn in something good enough to demand publication... I might be able to do something for you."
"Can't say fairer than that," I said.
"What do you want from me?" said the editor. "Access to the paper's archives?"
"I've already been through the old editions," I said tactfully. "It's all online these days. No, what I need from you is to be able to say I represent the Wiltshire Record and News. People will talk to the paper, where they wouldn't talk to me."
"Agreed," said the editor. "On one condition. You work on this story with an assistant I will provide."
I looked at her. I hadn't seen that one coming. "What?"
"Emma Tee. Girl reporter, new to the paper, young and enthusiastic and on her way up. Like you used to be. Work with her. If she can survive you, she'll make a great reporter. And just maybe some of her youthful integrity will rub off on you. But, James, listen to me. The story might be based on a legend, but I expect you to keep your prose tight and factual. No flights of fancy."
"Understood," I said.
"You'd better," said the editor. "Go on. Get out of here. Emma's waiting in the outer office. And be nice to her! Don't frighten her off. Bright, young reporters are getting hard to come by."
She really was waiting for me, sitting in the chair I'd just vacated, reading last week's edition of the paper. I could barely see any of her, behind the Wiltshire Record and News. Because she was such a small thing, and the paper remained an old-fashioned broadsheet, despite financial pressures. The editor still believed that readers still believe you can't trust anything you read in a tabloid. And we, she was fond of saying, are the local paper of record. If we say it happened, it happened.
The paper lowered abruptly, revealing a fresh, young face with a big, beaming smile. The kind that would probably have been irresistible to anyone else. Emma Tee was barely out of her teens with fluffed-out blond hair, a cheerful, young face without even a trace of makeup, shining blue eyes and a sweet demeanour. She was so full of youth and energy I felt old and tired just looking at her.
"Hi!" she said brightly, folding up the paper and tossing it casually to one side. "I'm Emma Tee, and you must be Jason Grant. Don't worry. I've already been warned about you by practically everybody, so let's just take that as read and move on."
She bounced up out of her chair and extended a small hand for me to shake. I did so solemnly. She still hadn't stopped smiling.
"So!" she said. "What are we, as journalists of record and reporters of fact, doing investigating an old fairy tale?"
"The things we choose to believe," I said carefully, "the stories we cherish and preserve, tell us who and what we really are. That's what makes old folk tales so important. We are going to investigate which local people and conditions gave birth to this particular legend, of the Hollow Women."
"Marvellous!" said Emma. "Where do we start? General search engine or something more specific?"
"I've already tried that," I said. "And beyond the basics... there's nothing there."
"But that's not possible!" said Emma.
"Not on its own," I said. "Which is what started me thinking. Someone seems to have gone to a lot of trouble to erase all but the original story of the Hollow Women. And I want to know why. Some old scandal perhaps? Featuring, or maybe even implicating, some old established families in the town?"
Emma grinned happily. "We can but hope. Nothing like a good local scandal to sell the local paper! How far back do we need to look to get to the beginning of this legend?"
"If I'm right, the eighteenth century," I said. "And for that we need access to the old records, the original sources. The books and papers that make up the church and parish records. The kind of thing you can't erase or delete. I already approached the local church, but the vicar wouldn't talk to me, let alone allow me access to his precious historical archives. Not as long as I was just a local hack. But since we are now official representatives of a respected local paper..."
"Does it have to be the church?" said Emma, her sunny face suddenly clouded. She'd finally stopped smiling. "Don't like churches. They give me the creeps."
"If you want to report the news," I said solemnly, "you have to go where the news is. Or in this case, where the news was."
On the way to the church, I filled Emma in on what I'd already turned up. A series of stories in the local press, about that most disreputable area, the Hollows. Stories from the eighteenth century, of drunkenness, debauchery and bad behaviour in the streets. Nothing in the least supernatural or fantastical. Just... warnings for men of good character to stay away from the bad women of the Hollows.
The Saint Laurence Church was mostly blocky Norman architecture with later Gothic flourishes, and a handful of stone gargoyles up by the guttering, showing their bare stone arses to the world. The church was surrounded by an old graveyard so packed full of the eternally resting there was no room left for new arrivals. Stones and crosses and monuments were jammed so close together there was hardly any room to pass between them. Wild flowers blossomed in profusion, where they weren't being choked by weeds. I led the way down the narrow gravel path with Emma hanging back and scowling mutinously in the rear.
I couldn't see what the problem was. The sun was shining brightly, and as graveyards went, this one seemed open and cheerful. A pleasant enough setting in which to contemplate eternity. I've always liked graveyards. Always a good place to go teenage drinking, late at night, with a few convivial friends. Secure in the knowledge no one would come barging in to bother you.
The vicar emerged abruptly from among the headstones, and came bustling forward to meet us. Oliver Markham had to be in his late seventies, but he still had a great mane of grey hair, and a bristling grey beard. It gave him something of the air of an Old Testament prophet, somewhat undermined by his cheerful smile and vague eyes. A pleasant enough sort in a dotty and distracted kind of way. He kicked his way through the last few weeds and stepped out onto the gravel path. He remembered meeting me before, but had to be reminded of my name. And he made enough of a fuss over meeting Emma that she quite forgot she didn't want to be there. He went to shake my hand, and only then realised he was still holding the trowel he'd been using for a bit of weeding. He tossed the trowel casually away and made a point of giving me a good hearty handshake. And a more careful one for Emma.
"Well, well, Mister Jason Grant," he said finally. "Back again! Yes, yes... The local archives, isn't it? I'm glad someone's taking an interest in them. They're all stored away in the church basement. Because no one else wants them. I keep hoping the local historical society will take the damned things off my hands and spend the money it will take to preserve them properly. I don't have the budget, you see! No, no... Sorry I had to drive you away, earlier, Mister... Grant! Yes! But I needed to be sure you represented the right sort of people. The archive records are very old, very valuable... and very fragile. So I have to be careful about who gets to see them. Oh yes! As long as they're in the Church they're in my care, you see...."
And yet, all the while he was saying this, he seemed to have trouble concentrating on me. His gaze kept sliding away, to Emma. Which was only natural in that she was a great deal prettier than me, but still...
The vicar finally stopped talking not long after he ran out of things to say and produced a large ring of old-fashioned keys, great solid metal things. He sorted carefully through them, muttering cheerfully to himself until finally he separated out one particular key and presented it to me. Slapping the heavy thing into my palm with enough emphasis to make me wince.
"There you go!" he said happily. "All yours! I'll leave you to it, if you don't mind. All that dust in the basement does terrible things to my sinuses. And I have work to do... work that needs doing... Where did I put my trowel?"
The basement under the church turned out to be a dank and gloomy place with no windows and just the one bare light bulb to push back the heavy shadows. All four walls were covered with shelves, packed full to bursting with old books and folders of even older documents. Some of the folders were labelled or dated, most weren't. More books tottered in piles across the stone floor. Dust and cobwebs to all sides suggested it had been some time since anyone had been down there. Emma didn't like the look or feel of the place, and I didn't blame her. So much history in one place has an oppressive weight.
It took us hours to locate the necessary volumes of town history, written out in longhand in a series of over-sized leather-bound books. Emma and I piled them up on the single reading desk, and then I sat down on the only chair (as the senior partner in this team) and worked my way through the volumes. With Emma standing right behind me, peering over my shoulder, and getting just a bit agitated when I didn't finish reading a page fast enough to suit her. I worked steadily through the old records, making notes where necessary. After a while, Emma started to fidget.
"If someone did go to all the trouble of removing knowledge of the Hollow Women from the Net, why didn't they destroy these old archives as well?"
"Because that might have drawn attention to them?" I said, scowling at the handwritten pages. My eyes ached. "Any attack on local records might make people think there was something important in them... Okay, this is it. A whole series of incidents in the town from the late seventeen hundreds onwards. Reports of certain unruly women from the Hollows preying on unfortunate young men. Taking their innocence and their valuables and sometimes sending them home in just the clothes they stood up in. This is the source of the legend! These terrible predatory women from the Hollows. The Hollow Women!"
"Well, yes," said Emma. "But isn't there anything more recent? You know how Ms. Walsh always wants to tie stories to modern settings and people. Makes it more accessible for today's reader."
To keep her happy (and because she was right, the Editor would want that), I skimmed through the more recent volumes. There were any number of incidents in the Hollows, everything from public drunkenness to open riot... but the stories of the Hollow Women just seemed to fade out. And no matter where I looked, I couldn't turn up any actual names, addresses, or anything that would serve as hard evidence.
Nothing to tie a scandal to any local family name. Unfortunately...
I slammed the final volume shut, sat back in my chair, and stretched my aching back.
"I think we've done all that can be reasonably asked of us," I said. "We've connected the dots and made a reasonable connection. Enough to put together a solid story for our beloved editor."
"It's a good, strong story as far as it goes," Emma said carefully. "But we still need to show a link to the town today. I think we need to pay a visit to what's left of the Hollows. The last mention of the Hollow Women was in the nineteen twenties. There might still be some people living there who heard the stories firsthand from their grandparents. I think we should check this out if only because..."
"Because if we don't, Ms. bloody Walsh will ask why we didn't," I said. "All right, then. To the Hollows it is. I wonder if there's time to buy a Kevlar jacket and update my immunisation shots?"
Of course, the Hollows as such didn't exist anymore. The slums of old were pulled down long ago, replaced by a series of run-down Council houses. Entering the Hollow Estates was like crossing a line into new and dangerous territory. Overgrown lawns with old refrigerators and other large objects just dumped in the gardens. Ugly graffiti on every wall and lots of peeling paint. No attempt to smarten the place up because nobody cared. Small groups of youths lurking around in hoodies, waiting for something to happen. And ready to start something if it didn't. Emma and I were careful to stick to the main roads, and took it in turns to brave the awful gardens, knock on doors, and talk as charmingly as we could to whoever answered. No one wanted to talk to us. They were all suspicious of strangers, particularly snooping strangers. We might be the law or social services or looking for money. We got a lot of doors slammed in our faces, and I would have given up if Emma hadn't been there. But finally, we struck gold in the form of an old woman called Alicia Tiley.
A very old woman who lived alone in a crumbling wreck of a house with far too many cats. She scowled all through Emma's cheery and engaging questions until she realised we were only interested in stories about the Hollows women, and then her head came up and she fixed me with a sharp look before stepping suddenly back, and inviting us in.
The narrow entrance hall smelled of damp. And cats. And damp cats. Dozens of them hurried back and forth, excited by the arrival of strangers, darting between our legs and jumping from one high spot to another. Alicia Tiley led us through into her pokey little parlour, stepping over and around the cats without looking while they did their best to trip us up. The parlour was crowded with all kinds of colourful junk and tatt. It looked like Alicia hadn't thrown away anything in years. She bustled around, making us a cup of tea, while advising us to just turf the cats out of any chair we fancied. The first cat I approached bared its teeth and hissed at me, but Emma chased the animals out of two chairs with effortless efficiency.
I sat down gingerly. The chair smelt very strongly of cats and not in a good way. To take my mind off that, I studied Alicia surreptitiously. She had clearly been a tall woman once, but age and presumably infirmity had bent her right over. She was large-boned, but still slender to the point of scrawny, her hard-edged face more full of character than anything else. She wore her thin grey hair scraped back in a tight bun. Her hands were bent almost into claws by arthritis, but she still managed the tea things easily enough. She moved slowly and steadily, pacing herself, so her strength would still be there when it was needed.
She put an old-fashioned china tea service down on the table before us. I took one look at the state of the cups and decided immediately there was no way I was drinking anything that went in them. Even if it did involve boiling water. Alicia finally finished pouring out the tea, thrust a cup into my hand and Emma's and then lowered herself carefully into a chair facing us.
"The Hollow Women," she said, harshly. "The ruiners of men. Seducers and betrayers. And murderers too, sometimes. If there was a thing that needed keeping quiet. Or men who should have known enough not to go back after them. The Hollow Women could make any man love them and give up their hearts just so they could have the fun of breaking them. Was a time, everyone around here knew, to beware of the Hollow Women. But people forget...
"I lost my dear Jack to them, long and long ago. He was never the same, afterwards. Oh yes... I was young once, and a young man loved me. Until one of them got him... If you want to know the truth, you need to talk to the nuns. They know."
"I'm sorry," I said. "Nuns? Which nuns are these?"
"The Holy Sisters of Saint Baphomet," Alicia said sharply. "You know the ones. They all live together at Barrow Farm, down by the river."
"Oh... yes," I said. "An order of reclusive nuns. They bought Barrow Farm and moved in... how long ago? Must be years..."
"More than twenty years," said Emma. "I'm not surprised you forgot about them. Most people have. They don't get out much."
"I sort of got that, from reclusive," I said.
"They keep themselves to themselves," said Alicia, sipping loudly at her tea. "But there's no denying they know things."
"Why would nuns know anything about the Hollow Women?" said Emma. "One of the few things we know for sure from the old legend was that these women had a violent antipathy for all things religious. And apparently, vice versa. It was always the church who spoke out most strongly against the sinful practices of the Hollows women."
"They know things," Alicia said darkly. "Know thy enemy and all that."
"I still don't think we should go barging in on an order of reclusive nuns," said Emma.
"We're reporters," I said sternly. "And that means we go where the story is."
"Then you can go on your own," said Emma. "Churches are spooky enough. I'm not doing anything that might get a whole bunch of nuns mad at me."
At first, I thought she was joking, but she just sat there stubbornly and refused even to discuss the matter. Alicia looked on, quietly enjoying the argument. So in the end, I got up and left Emma there to see if she could get any useful information out of Alicia. I hated to do it, not least because the editor had made it very clear I was supposed to work this story with Emma, but it wasn't my fault if the little girl reporter couldn't keep up. You have to go where the story leads you.
Barrow Farm was a sprawling old stone building, right on the bank of the River Avon, where it cuts through the centre of the town. No telling how old the place was, but the local creamy grey stone was deeply discoloured from the ravages of time and weather, and the tiled roof looked like it could use some serious repairs. There was no bell at the front door, just a large, black iron knocker in the shape of a wolf's head, the ring hanging from its snarling mouth. Not exactly the most welcoming first impression from a company of nuns. I looked around for signs of life, but there didn't seem to be any. All the windows were covered by heavy wooden shutters as though the nuns felt they were under siege from the modern world and were determined to keep it out.
I banged the iron knocker heartily. It raised a hell of a din, but there was still a really long pause before the door finally opened just enough for a single nun to stare out at me with a cold and entirely unwelcoming gaze. The black robes and starched white wimple gave her a nun's usual anonymity. Her face could have been any age, and the only expression I could read was open disapproval. I nodded and smiled politely, introduced myself, and explained why I was there. The nun showed no interest at all, until I mentioned the Hollow Women. She fixed me with a firm stare and then opened the door wider.
"I am Sister Joan. I know the story of the Hollow Women. We all do. We are the Holy Sisters of Saint Baphomet and sin is our business." She smiled briefly, and I realised that was meant to be a joke. "You'd better come in, Mister Grant. And we will discuss the matter further. I should make it clear, none of us are at all interested in publicity."
I assured her it was the Hollow Women who were the story, not the sisters, and she stood back to allow me to enter. She locked and bolted the door very carefully and then led me through a series of narrow rooms that finally opened out onto a large hall. Sunlight fell in through a number of tall narrow windows, but still it seemed to me that the room had too many shadows for my liking. For all its size, the hall felt... isolated, cut off, not part of the world. A very private and very secure place. A long wooden table took up the middle of the room, and around it sat a great many nuns in full regalia. All of them looking at me with cold eyes and tightly pursed mouths. None of them got up to greet me.
Sister Joan explained who I was and why I was there and not one of the sisters even nodded to me. Sister Joan pulled out a chair for me at the head of the table and I sat down. The presence of so many staring eyes would probably have been intimidating to anyone else. I just smiled politely back at them while Sister Joan sat down beside me.
She then proceeded to interrogate me on the subject of the Hollow Women, hitting me with question after question, drawing out everything I knew. She didn't challenge or correct anything I said. I got the impression she was checking what I had discovered against what she already knew. The other nuns remained silent throughout, never taking their eyes off me.
More and more, the great open hall made me feel uneasy. It was all very clean, nothing out of place, but it was just so... characterless. The nuns had been here for twenty years and more, but they'd made no impression on their surroundings. No religious paintings or texts on the walls, not even a single crucifix. This had to be a really austere order.
In the name of self-defense, I interrupted Sister Joan's questions to ask a few of my own, including the lack of religious items on show. Sister Joan smiled tightly.
"Our order does not believe in idolatry or the need for religious paraphernalia. Our belief is pure without distractions. Let the world go its own way and we shall go ours."
"I've told you everything I know," I said. "Now it's your turn. What can you tell me about the Hollow Women? And why are you so interested? I thought the Hollow Women couldn't abide religious people and vice versa."
"It's all about faith," said Sister Joan. "So lacking in modern times. The legend of the Hollow Women is old... They have existed alongside civilisation under many names. Before this town was a town, there were Hollow Women preying on the men. Before there were people, there were Hollow Women. They learned to look like people, the better to prey on them. Perhaps these days they have learned to look like something else. It's hard to be sure of anything where the Hollow Women are concerned. They are very secretive. They've had to be to survive so long. The church has tried to stamp them out many times."
"Which church?" I said.
"All of them, Mister Grant! Perhaps because only those of true faith can see through the illusions that hide the Hollow Women from the eyes of the world. It is a war, Mister Grant. Make no mistake. There can be no forgiveness for things that prey on men."
"You've clearly amassed a great deal of information during your researches," I said carefully. "Would it be possible for me to take a look at what you've discovered?"
Sister Joan was already shaking her head, even before I finished speaking. "No, Mister Grant, it will not be possible."
"May I ask why not? My story wouldn't have to quote you or mention the Sisters in any way if that's what's worrying you."
"Information is ammunition, Mister Grant. And as I said, there is a war on. We guard what we know most jealously for when it might be needed."
"You seem convinced these Hollow Women of legend still exist," I said. "Do you see them as supernatural creatures? Like vampires or ghosts?"
"Those are dead things, Mister Grant. The Hollow Women are as real, as natural, as you. Every species has its predator."
"But you do believe they still exist, here in the town?"
"Oh yes, Mister Grant, we know they do. Hiding in plain sight. Only emerging to prey on the weak and then disappearing again. Any woman could be a Hollow Woman. That's the point. And be warned, Mister Grant. If you go looking for them, you can be sure they will come looking for you."
I looked up and down the table to see if the other nuns were taking this as seriously, and everywhere I looked, cold eyes and cold faces stared implacably back at me. There's nothing scarier than a faith backed up with utter certainty. They believed. Sister Joan stood up and indicated it was time for me to leave. And I couldn't get out of there fast enough.
The door closed firmly behind me. I heard the lock turn and bolts slamming into place as Sister Joan sealed Barrow Farm off from the intruding world again. I breathed in deeply and shook my head to clear it. Sometimes intense beliefs can be... catching. I had to remind myself I only got into this story to prove the mythical Hollow Women had a real world source in the Hollows women. The Holy Sisters of Saint Baphomet had been locked up together for too long. Stewing in their own conspiracy theories and the need for someone who needed punishing. I suppose, if you believe in devils and possessions and miracles, it's not too big a leap of faith to believe in women who can only be seen from the front.
I shuddered suddenly despite myself. When faith turns inwards, it becomes unhealthy. I did not believe in anything supernatural. I'd spent enough years writing and researching the weird shit to know it was all just bullshit and wish fulfilment. Whatever the Holy Sisters knew, or thought they knew, I didn't need to know it. They were just a dead end. I needed to put them behind me and press on with my research into historical records. The church archives had been a good start, but where next? The vicar had mentioned a local historical society...
I turned my back on Barrow Farm and strode determinedly away, not looking back once.
I reached for my phone to call Emma and bring her up to date only to realise she hadn't given me her number. So the editor couldn't blame me if her precious new reporter wasn't a big contributor to what was, after all, my story.
I walked back into the middle of town and headed straight for my favourite watering hole, the Dandy Lion, for a quick drink and a think. It's always been my experience that the two go well together as long as you don't overdo either of them. The Dandy is a cosy and comfortable drinking establishment with traditional fixtures and fittings and absolutely no piped music. I can usually find someone worth talking and drinking with. But I really wasn't expecting that when I walked through the doors, the first person I found waiting for me was Emma Tee.
She was sat by a table right by the door with a drink in front of her that she'd barely touched. She smiled winningly at me. I looked briefly past her to where a group of old friends were sitting round a table farther in, but I had promised the editor I would work with Emma. And Sam Walsh was perfectly capable of spiking my story out of hand if I didn't. So I got myself a pint of good cheer from the bar and sat down opposite Emma. She gave me her best happy smile, backed up by bright shining eyes... and it was hard to stay mad at her.
"How did you get on with the Holy Sisters?" said Emma, smiling perhaps just a little mischievously.
"Don't ask," I said. "I'm sorry about just going off and leaving you to cope with the mad old cat lady."
"Oh no, I should apologise to you!" Emma said immediately. "For not following the story. You were completely right. The story must come first. I just didn't want to meet the nuns. Nuns are creepy. Even more than old churches. So you didn't get anything useful from them?"
"Not a thing," I said. "Except that they seem convinced the Hollow Women of legend are still a real and present danger."
"Let them think what they like," Emma said firmly. "Our story will prove the Hollow Women are just an urban legend, mistranslated and misunderstood down the years. That's what reporting is supposed to be about, isn't it? Shining a light into dark places and uncovering the truth."
"Yes," I said. "That is what it's supposed to be about."
We sat and talked, and drank our drinks, and talked some more. She was very easy to talk to. And somewhat to my surprise, I found we were getting on really well. She had an endless interest in all things journalistic and was fascinated by my tales of researching weird stuff for strange magazines. And it helped a lot that she thought my jokes were funny. All my cynicism and world-weariness seemed to just evaporate in the face of her youthful enthusiasm. I'd forgotten how it felt to get properly excited about a story. But then it had been a long time since I had a story worth getting excited about.
Emma was quite open about why she wanted to become a journalist. She'd left her home, and her family, to make her own way in the world. I got the impression this had been very much against her family's wishes. That they were very strict, very traditional, and apparently believed they had a right and a duty to map out her life for her. And Emma wasn't having any of it. She wanted to be a journalist so she could tell the truth about things, things that mattered. Because her family had tried so hard to hide the truth about the world from her because it conflicted with what they believed. Emma wanted to know everything there was to know about the world. So she could tell everyone else. My heart went out to her. Looking at Emma was a lot like looking at my younger self.
"My parents never wanted me to be a writer," I said. "No money in it, that was what they said. Get a proper job with prospects. So I sort of drifted sideways into journalism. I did quite well for a while."
"What happened?" said Emma. "I know something happened. Ms Walsh said... some things when she told me I'd be working with you. What went wrong, Jason?"
"I did," I said. "I had my chance, and I blew it because I couldn't stand the hard discipline of real journalism. I decided it was more important to tell a good story than sticking to the facts. So if facts got in the way, I just changed or suppressed them to make the story more sensational. I wrote some really great stories—they just weren't entirely true. On a modern daily tabloid, that wouldn't have been a problem. That would have been business as usual. But here, in the local paper of record..."
"Ms. Walsh fired you."
"Hell yes. More in sorrow than anger, I like to think. But it was definitely 'Go and never darken my doors again.' It's taken me years to get this opportunity. And years to understand that she was right. People need to be able to believe what they're told is the truth."
"Even when a little white lie can be so much more comforting?"
"Perhaps especially then. You can't base decisions that matter on someone telling you what you want to hear. No other local paper would touch me after word got out as to why I was fired, and without a good local history, the dailies didn't want to know. And that's how I ended up hacking it out and phoning it in for any rag that would have me." I smiled, briefly. "It does feel good to be working on a real story at last."
"I did get some more information out of Alicia Tiley after you left," said Emma. "After a little encouragement and open pleading..."
"You didn't actually drink that tea, did you?"
She winced. "Please. Don't remind me. And one of her cats pissed on my shoes. Anyway, Alicia remembered a part of the legend of the Hollow Women that was new to me. Apparently they only emerge, only reveal themselves as their true selves, at night. And only when the fog rises to blur reality... and hide them from prying eyes. That's when they go forth to prey on unattached young men. And strike down their enemies, anyone who might be getting too close to the truth about them."
"You mean they kill people?"
"Oh yes," said Emma. "They kill people."
"You're right," I said. "That is a new twist. Makes sense, I suppose. The women from the Hollows were probably professional women, plying their trade away from the light of day. And because what they were doing was illegal, they or their protectors would kill anyone who threatened their livelihood, or their territory. The Holy Sisters said the Hollow Women would come after me if I went after them."
"They actually said that to you?"
"Yes. Very sternly."
Emma looked at me for a long moment. "Do you really think you might be in danger, Jason?"
"From a supernatural myth?" I said, grinning despite myself. "Hardly. You mustn't take any of this too seriously, Emma. It's just an old moral fable that's outlived its significance. Don't let the material spook you."
She forced a smile. "As long as you don't go hanging around the Hollow Estates at dawn."
"Don't worry," I said. "I never get up that early. Another drink?"
"Don't mind if I do," said Emma.
Time passed in a pleasant fashion. When Emma and I finally left the Dandy Lion, leaning on each other in a companionable sort of way and giggling a bit, it was well into the evening. A fog was slowly forming on the air, a pearly grey haze, swallowing up the distance and spreading milky halos around the street lights. It seemed to thicken slowly even as I looked at it. There was no one else about, not even any traffic passing. It was like staring off the edge of the world. Everything seemed vague and uncertain. As though if I went walking off into the fog, the places I expected wouldn't be there anymore.
It was all very quiet, the fog soaking up sound. I put an arm around Emma, protectively, and looked about me. Suddenly feeling a hell of a lot more sober. The foggy evening seemed the perfect setting for some old legend to come walking back into the world. I glared into the curling mists. I was damned if I'd let my own story get to me. I looked at Emma, and she was staring into the fog with wide, worried eyes. She turned suddenly to look at me, and she seemed genuinely scared.
"Don't worry," I said. "We're a long way from dawn."
She didn't smile. Not even a little bit. "You don't understand," she said.
"Come on," I said. "I think you've had a few too many. I'll walk you home. You've nothing to worry about as long as I'm with you. And tomorrow, when the fog's all gone, you'll see how silly you were. It's just a story!"
"It's a long way to where I live," said Emma. She looked at me. "Where's your place, James? Is it near?"
"Yes," I said. "Just a few streets away."
"Can I stay with you tonight?" said Emma, her wide eyes fixed on mine. "That's what I want, James. I want to stay with you, tonight. Can I?"
And I said yes.
I took her back to my place. Nothing special, just a reasonably comfortable flat above a newsagent's. It wasn't until I unlocked my door and ushered her in that I realised how much of a mess I'd let the place get into. I was a man who lived alone and let things lie where they fell. I made a token effort to clear some of it up, while Emma looked around her, not commenting.
She was nervous. I could tell. I stopped what I was doing and went to her.
"You haven't done this before, have you?" I said.
"No," she said.
"It's been a while for me. Emma... you don't have to do anything you don't want to."
"I want to do this, Jason."
"There's a spare bed. I just need to sort out some sheets for it..."
"I want you, Jason."
I put my arms around her. And she put her arms around me. Our faces were so close now; I could feel her breath on my mouth.
"You're so much younger than me," I said. "And so beautiful. You deserve better than me..."
"Hush, Jason. You deserve me. I'm here for you."
She hugged me tightly, pressing the side of her face against my shoulder. The smell from her hair filled my head. She held me as tightly as she could, as though afraid someone might drag her away.
"Hey," I said. "It's all right. Really. Everything's going to be all right, Emma."
"Yes," she said. "It is." She looked up at me, and smiled. "Take me to bed, Jason. Take me to bed and love me so we can forget everything except us. That's what I want."
And that's what I did.
Sometime later, I lay on my back in bed, the sweat drying on my bare skin, stretched out and relaxed, feeling more at peace with myself than I had in a long time. Emma was sat up beside me, her back against the headboard, staring out across the room. I couldn't read the expression on her face. She suddenly swung her legs over the side of the bed, and padded silently across the bedroom, entirely naked, to stand before the window. She opened the curtains just a little, and looked out.
Nice arse, I thought.
"What is it?" I said.
"I thought I heard something." She didn't look back at me.
"Come back to bed," I said. "It's nothing. Just the night. There are always noises, at night."
"Yes," she said. "There's always something happening in the night."
I glanced at my alarm clock, on the bedside table. "Getting on for ten o'clock. The night's barely started. Come back to bed, Emma."
"In a minute."
She was still staring out through the crack in the curtains at the street below. I rolled over onto my side, thinking vaguely about getting out of bed to join her and that was when I saw her back reflected in the wardrobe mirror. She had no back. Seen from behind, in the mirror's reflection, there was just a hollowed out shell. A concave depth, all ridges and whorls. As though something had reached in and scooped out everything that made her human. It was like looking into the husk of a dead insect or the hollow trunk of a diseased tree with all the insides eaten away.
I cried out. I couldn't help myself. And she spun round to look at me. She saw the truth in my face and I saw the truth in hers.
She looked at my wardrobe mirror and then back at me. For a moment she seemed to shrink in on herself, and then she drew herself up again and faced me squarely. She seemed entirely human. As long as I looked at her from the front. But I couldn't forget what I had seen. And what I had just done with something that only pretended to be human. She started towards me. I sat up sharply and put my back against the headboard. She stopped at the foot of the bed.
"I'm sorry, Jason," she said. "I'm so sorry. I was so happy I let my concentration slip, just for a moment. I never meant for you to know."
"You're real," I said. "They're real. The Hollow Women. The ones who prey on men."
"Yes, Jason."
She reached out a hand to me, and I flinched back. She looked at me sadly and let her hand drop again.
"I could have loved you, Jason. Don't you know that?"
"How long...?"
"All my life. Hollow Women are born, not made. Just like you. I had no choice in the matter. That's why I left home. Left my family and my own kind because I didn't want to be like them. I wanted to be what I wanted to be." She smiled, briefly. "The name they gave me, that I never thought to change, should have been a clue. Emma Tee. Empty. I tried so hard, Jason! Trying to live as a human, among humans. I do care for you in my way."
"You can't stay here," I said.
She looked at the curtained window behind her, and then back at me. She seemed scared.
"Please, Jason. Don't make me go. Don't throw me out. It's night, and there's a fog, and I'm so scared about what might happen..."
"Scared of what?" I said. "Why were you so determined to spend the night here with me?"
"Because they're out there. Looking for me. I know it. You've been asking too many questions, Jason. Getting too close to the truth. Despite everything I could do to distract you."
"What is the truth, Emma? Really?"
"If I tell you everything, will you let me stay?"
"Tell me everything," I said. "Tell me all about the Hollow Women."
She turned her back on me and got dressed. It looked like a perfectly ordinary human back now. I got dressed too. And then we sat down on chairs a respectful distance apart, facing each other. And she told me what I needed to know in a calm, emotionless voice.
"I saw something moving, down in the street," she said. "Something in the fog. A human shape that didn't move like anything human. In the fog, in the night, the only time when Hollow Women appear as themselves."
"How do they... pass, normally?" I said. "How can they, how can you walk among us, and not be seen for what you are?"
"A glamour. A broadcast telepathic illusion. It can be undermined, seen through, if someone has more faith in their religion than they do in the illusions of the world. That's why I was so nervous at the church, earlier. The vicar kept getting glimpses of me out of the corners of his eyes. That's why he was so jumpy. Only his refusal to believe what he was seeing with his own eyes protected me."
"Where are you from?" I said. "I mean, you said you left your home and your family. A family of Hollow Women. Where are they?"
"You don't need to know, Jason. It's safer for you if you don't know."
"I need to know! This isn't my story anymore. It's my life!"
"They'll kill you to keep their secret safe."
"It was the old woman, wasn't it?" I said. "Alicia Tiley. She was a Hollow Woman!"
"No, you fool," said Emma. "She was just an old woman. She didn't know anything. I already knew everything. All the Hollow Women, all that are left, live together in one place now because they're not human and when they're alone, they don't have to act human. You've already met them, Jason. At Barrow Farm. The Holy Sisters. Hole-y. Get it? What better disguise..."
"Where do you come from originally?" I said. My throat was tight. It was getting hard to breathe. "I mean, if you're not human, what are you? Mutations? Aliens? Supernatural? What?"
"I don't know," said Emma. "If the sisters ever knew, they forgot long ago. We're just predators. That's all you need to know."
"How is it that you're so... human?"
"Television," Emma said simply. "I'm the first generation of Hollow Women to be exposed to television. It really is a window on the world. A better world, a better way of living. And I wanted it." She glanced back at the window. "They'll know I've talked. They'll come for me. And for you. I was only allowed to stay away as long as I kept my head down. Didn't get noticed. I was doing so well. And then you came to Ms. Walsh with your idea for a story. I was the one who convinced her to go for it with me attached. So I could watch over you, steer you away from the truth. Towards a nice, safe historical interpretation that would help hide us. You can't defend yourself from something you don't believe in. But you wanted this so much and I..."
"You need to leave town," I said. I got to my feet. "Come on. You need to get away. Start over, somewhere else. I'll see you safely away, and then later, I can come and join you. With both of us gone maybe the sisters won't feel so threatened."
"You'd come with me?" said Emma, getting to her feet. She looked at me wonderingly. "You'd do that, for me? After... everything?"
"Of course," I said. "I care for you in my way."
I put out my hands to her, and she grasped them tightly, like a drowning woman.
"It's the human thing to do, Emma. I'm sorry I freaked out at first. It's just... I never had one of my stories turn on me before."
"I'm sorry too," said Emma. She didn't say for what, though I didn't realise that till later.
"We can't go by car," I said. "They'll be expecting that. Looking for that. No—I'll take you to the railway station. It's not far. There's still time to catch the last train out of here and you'll be safe among the other passengers. Just... keep going, keep changing trains until you're far away. You can hide yourself properly in a big city. They'll never find you."
"Will you come and find me if I call for you?" said Emma.
"Do you want me to?"
"Yes. More than everything."
"Good. Because that's what I want too."
We held each other for a long moment. In the end, she pushed me away.
"I have to go, Jason. It's not safe here. For either of us."
Outside in the street, the fog had come down hard. Thick, grey walls surrounded us on every side, cutting us off from the rest of the world. There was no one about. Not even a single passing car. As though everyone somehow knew it wasn't safe to be out and about this night. Emma held tightly to my arm, staring frantically about her.
"I've never seen a fog this thick," I said.
"I have," said Emma. "It's them. They're here."
"It's all right!" I said roughly. "I'll get you to the station."
I set out confidently enough, but in the fog, all the streets looked the same, and without landmarks to guide me, I soon lost my way and all sense of direction. I kept going anyway, striding out, Emma clattering along beside me, still hanging tight to my arm. And then, one by one, they appeared. Just dark shadows at first, appearing and disappearing in the mists around me like sharks circling silently in murky waters. Bursting out of the mists in front of me just long enough to turn me aside, guiding me, herding me, closing in from all sides. I kept going, even broke into a run, but it did no good. They were everywhere. Moving faster than I could because they didn't have human limitations. They never made a sound, just appearing and disappearing, until finally they all came out of the fog at once, forming a great circle around me and Emma. The Holy Sisters. The Hollow Women.
The black and white of their disguising robes stood out starkly against the grey mists. They stood very still, inhumanly still, watching me with their cold, empty faces. Sister Joan loomed up suddenly before me. I struck out at her, but she was gone before the blow arrived, vanished back into the mists. She reappeared while I was still off balance, and her fist came flying at me impossibly fast. She hit me once, clubbing me down with sudden, vicious strength. I hit the ground hard, driven to my knees by the force of the blow, all the strength knocked out of me. I cried out in shock and pain. When I looked up, she was standing over me, studying me with cold predator's eyes.
Dark shapes rushed in from every side, and just like that they were all over me. Lashing out with large, hard fists driven by more than human strength. Blood flew from my battered face. I tried to defend myself, but I couldn't even touch them. I ended up curled in a ball on the ground, hurting all over. And suddenly, they stopped. I slowly uncurled and looked up. Sister Joan was standing over me, her face entirely unmoved, unconcerned. She wasn't even breathing hard.
"Forget your story," she said. "Forget her. Forget any of this ever happened. And we will let you live as a cautionary example."
"All right," I said shakily, blood spilling from my mouth. "All right..."
"He lies," said another voice, from one of the Hollow Women looking on. Others took it up. He lies, he lies...
"You can't forget us because you won't forget her," said Sister Joan. "Such a pity... Say your prayers, Jason Grant. Your story has come to an end."
"No!"
I turned my head, slowly, painfully, and there was Emma. Standing beside me, her hands clenched into fists, glaring defiantly at Sister Joan.
"If you kill him, I'll never forgive you! Never!"
"You forget yourself, child," said Sister Joan. "You forget what you are. When he is gone..."
"No. I won't let you kill him."
Sister Joan considered her thoughtfully. "How will you stop us?"
"By giving you what you want. If you'll let him live, I'll come home again. I promise. I'll come back to you, and I'll never try to leave again. That's what you want, isn't it?"
"Come home?" said Sister Joan. "No more arguments, no more running away?"
"Yes," said Emma. She didn't look at me. "That's what I want."
It was the bravest thing I ever saw. A Hollow Woman, demonstrating her humanity. Giving up her life for mine.
Sister Joan looked at me, and then at Emma. "Did you...?"
"Yes," Emma said steadily. "I slept with him. And made a child with him."
I looked at her speechlessly. I had no doubt she was telling the truth. That she knew. It was, after all, what Hollow Women did.
"Then come home, child," said Sister Joan. "Your place is waiting for you."
Emma walked away from me, into the ranks of the waiting Hollow Women. And together, they turned sideways and disappeared, back into the fog. Only Sister Joan remained.
"I could still tell the world all about you," I said.
"But you won't," said Sister Joan. "Not if you want Emma to stay safe. Silence and obscurity are the price of your survival, and hers." She smiled, very briefly. "And anyway, who would believe you? A hack writer of so many wild stories? No one believes in the things you write. The world only likes its legends in stories these days. We have what we came for. Nothing else matters."
She walked away and left me. I caught a last brief glimpse of Emma standing alone in the mists. She looked at me and didn't smile or wave good-bye. She took one last look at me and then walked away forever. As she turned away, I saw her back was empty. Just a hollow shell.
And I was left alone, in the fog, and the night. Alone, with my hollowed out heart.
On...The Foggy, Foggy Dew
Roud #558 denotes the British ballad The Foggy, Foggy Dew (not to be confused with the
Irish ballad The Foggy Dew). Reading through the nearly fifty late nineteenth/early twentieth century versions in the Vaughan Williams Memorial Library as posted by The English Folk Dance and Song Society one finds that the plot of this ballad tends to go one of four ways.
1) At night a girl is frightened by "the foggy dew" or a "bugaboo" and comes into a young man's bed for comfort.
2) A young man, a weaver, courted a girl who, frightened of the "foggy dew," came into his bed. Part of the night they "sport and play." In the morning she worries about being undone but he says she is not to worry since the foggy dew is gone. From then on whenever she gives him a wink or a smile he thinks of the foggy dew.
3) This version is similar to #2 except later in life the now older man is with his son, with the inference being that the son was the result.
4) Again this is similar to #2, except that afterwards the girl asks, what if they should have a child? They marry and are happy and forever more when he looks at her he thinks of the foggy, foggy dew. (1)
Alan Lomax mentions on the sleeve notes for Shirley Collins' 1958/1959 recording that this is one of the few "frankly erotic songs" that made its passage from Southern England to America more or less uncensored. (2)
"The Foggy, Foggy Dew" has been sung by Burl Ives, Shirley Collins, Martin Carthy, A.L. Lloyd, and in countless pubs.
1) Roud Folksong Index. 2013. 6 November, 2013. <http://www.efdss.org/efdss-the-full- english>. English Folk Dance and Song Society.
2) Mainly Norfolk: English Folk and Other Good Music. 2013. 6 November, 2013.
<http:// mainlynorfolk.info/lloyd/songs/thefoggydew.html> English Folk Dance and Song Society.
# THE CONTRIBUTORS
CHRISTOPHER GOLDEN is the award-winning, bestselling author of such novels as The Myth Hunters, Wildwood Road, The Boys Are Back in Town, The Ferryman, Strangewood, Of Saints and Shadows, and (with Tim Lebbon) The Map of Moments. He has also written books for teens and young adults, including Poison Ink, Soulless, and the thriller series Body of Evidence, honored by the New York Public Library and chosen as one of YALSA's Best Books for Young Readers. Upcoming teen novels include a new series of hardcover YA fantasy novels co-authored with Tim Lebbon and entitled The Secret Journeys of Jack London. A lifelong fan of the "team-up," Golden frequently collaborates with other writers on books, comics, and scripts. In addition to his recent work with Tim Lebbon, he co-wrote the lavishly illustrated novel Baltimore, or, The Steadfast Tin Soldier and the Vampire with Mike Mignola. With Thomas E. Sniegoski, he is the co-author of multiple novels, as well as comic book miniseries such as Talent and The Sisterhood, both currently in development as feature films. With Amber Benson, Golden co-created the online animated series Ghosts of Albion and co-wrote the book series of the same name. As an editor, he has worked on the short story anthologies The New Dead and British Invasion, among others, and has also written and co-written comic books, video games, screenplays, the online animated series Ghosts of Albion (with Amber Benson) and a network television pilot. The author is also known for his many media tie-in works, including novels, comics, and video games, in the worlds of Buffy the Vampire Slayer, Hellboy, Angel, and X-Men, among others.
DAVID LISS is the author of eight novels, most recently The Day of Atonement. His previous bestselling books include The Coffee Trader and The Ethical Assassin, both of which are being developed as films, and A Conspiracy of Paper, which is now being developed for television. Liss is the author of numerous comics, including Mystery Men, Sherlock Holmes: Moriarty Lives and Angelica Tomorrow.
DEL HOWISON Along with his wife Sue, Del owns Dark Delicacies, "America's Home of Horror." He is a Bram Stoker Award winning editor and multi-nominee as well as having been nominated for a Shirley Jackson and the Black Quill awards. His short story "The Lost Herd" was retitled to Sacrifice and released as the premiere episode of the horror television series Fear Itself on NBC.
GARY BRAUNBECK is the prolific author of 25 books, as well as nearly 250 short stories. He is a multiple Bram Stoker Award winner and as won the International Horror Guild Award. He is a past-president of the Horror Writers Association. His fiction has been translated into Japanese, French, Italian, Russian and German.
GREGORY FROST is a writer of dark fantasy, SF, Young Adult, and historical thriller fiction. He has been a finalist for every major fantasy genre award. His latest novel-length work is the YA-crossover "Shadowbridge" duology; voted "one of the four best fantasy novels of the year" by the ALA. His historical thriller, Fitcher's Brides, was a Best Novel finalist for both World Fantasy and International Horror Guild Awards. Other Frost short stories appear in Ellen Datlow's Supernatural Noir anthology, and in V-Wars, edited by Jonathan Maberry. He directs the fiction writing program at Swarthmore College. He is the co-founder of the Liars Club.
JACK KETCHUM is an American author. He is the recipient of four Bram Stoker Awards and three further nominations. Many of his novels have been adapted to film, including The Girl Next Door and Red.
JONATHAN MABERRY is a NY Times bestselling author, multiple Bram Stoker Award winner, and comic book writer for Marvel, Dark Horse and IDW. His novels include _Code Zero_ , _Rot & Ruin, Fall of Night, Ghost Road Blues, Patient Zero, The Wolfman_, and many others. Nonfiction books include _Ultimate Jujutsu_ , _The Cryptopedia, Zombie CSU_ , and others. Several of Jonathan's novels are in development for movies or TV including _V-Wars, Extinction Machine, Rot & Ruin_ and _Dead of Night_ He's the editor/co-author of _V‐Wars_ , a vampire‐themed anthology, and is editing a series of all original _X-Files_ anthologies. He was a featured expert on The History Channel special _Zombies: A Living History_. Since 1978, he's sold more than 1200 magazine feature articles, 3000 columns, two plays, greeting cards, song lyrics, and poetry. His comics include _V-Wars, Rot & Ruin, Captain America: Hail Hydra, Bad Blood, Marvel Zombies Return_ and _Marvel Universe vs The Avengers_. He lives in Del Mar, California with his wife, Sara Jo and their dog, Rosie. www.jonathanmaberry.com
JEFFREY J. MARIOTTE is the award-winning author of more than fifty novels, including supernatural thrillers Season of the Wolf, Missing White Girl, River Runs Red, and Cold Black Hearts, horror epic The Slab, the Dark Vengeance teen horror quartet, and others. He also writes comic books, including the long-running horror/Western comic book series Desperadoes and original graphic novel Zombie Cop. With writing partner Marsheila Rockwell, he has published short fiction and is working on more. He has worked in virtually every aspect of the book business, as a writer, editor, marketing executive, and bookseller. He lives in southeastern Arizona. Visit him at www.jeffmariotte.com.
JEFF STRAND is the four-time Bram Stoker Award nominated author of such books as _Pressure, Dweller, A Bad Day For Voodoo_ , and _Wolf Hunt_. He lives in Tampa, Florida, and because he cares about your ears, he lets others do the singing where folk ballads are concerned.
KEITH R.A. DeCANDIDO is the international, best-selling, award-winning author of more than 50 novels as well as dozens of short stories, novellas, comic books, and blog entries. His many works of fiction in media universes such as Star Trek, Supernatural, World of Warcraft, Doctor Who, Spider-Man, Buffy the Vampire Slayer, Farscape, Leverage, and more won him a Lifetime Achievement Award from the International Association of Media Tie-In Writers in 2009. Recent and upcoming work ranges from the Sleepy Hollow novel Children of the Revolution to the acclaimed "Precinct" series of fantasy police procedurals (Dragon Precinct, Unicorn Precinct, Goblin Precinct, Gryphon Precinct and Tales from Dragon Precinct) to the Star Trek coffee table book The Klingon Art of War to the short story collection Ragnarok & Roll: Tales of Cassie Zukav, Weirdness Magnet (which features the protagonist of "Fish Out of Water"). He has also contributed to the anthologies Bad-Ass Faeries: It's Elemental, More Tales of Zorro, Stargate: Far Horizons, V-Wars Volumes 1 & 3, and The X-Files: The Truth is Out There, and he's also a regular blogger for Tor.com, doing a twice-weekly rewatch of Star Trek: Deep Space Nine. Find out more at his web site at DeCandido.net.
KELLEY ARMSTRONG is a #1 New York Times bestseller. She has published eighteen fantasy novels to date, set in the world of the Women of the Otherworld and the Darkest Powers series, also two crime novels in 2007 and 2009.
LISA MORTON is a screenwriter, author of non-fiction books, award-winning prose writer, and Halloween expert whose work was described by the American Library Association's Readers' Advisory Guide to Horror as "consistently dark, unsettling, and frightening." Her most recent releases include the novella By Insanity of Reason (co-authored with John R. Little) and the novel Zombie Apocalypse: Washington Deceased. She lives in North Hollywood, and can be found online at www.lisamorton.com.
MARSHEILA (MARCY) ROCKWELL is the author of The Shard Axe series, the only official novels that tie into the popular MMORPG, Dungeons & Dragons Online. She has two collections out now (Tales of Sand and Sorcery and Bridges of Longing and Other Strange Passageways), and is currently hard at work on the second book in a trilogy based on Neil Gaiman's Lady Justice comic books. "John Barleycorn Must Die" is her second published collaboration with writing partner Jeffrey J. Mariotte. The first, "A Soul in the Hand," can be found in the Neverland's Library anthology from Ragnarok Publications. Learn more here: http://www.marsheilarockwell.com/.
NANCY HOLDER is a New York Times bestselling author (the dark fantasy series Wicked) who has written over seventy novels, and two hundred short stories, essays, and articles, many of which have appeared in "Best of" anthologies. She has received five Bram Stoker awards and a Scribe award for her supernatural fiction, as well as a Pioneer award from Romantic Times for her young adult fiction. She also received a Special Sales Award from amazon.com. She is well known for her work on such fantasy properties as Buffy the Vampire Slayer, MTV Teen Wolf, Saving Grace, Hellboy, Hulk, Highlander, and many others. She has written two retold fairy tales for Simon and Schuster's Once Upon a Time series, and a nursery rhyme retelling ("The Lion and the Unicorn") for Month 9 Books. Many of her MFA students have explored dark fantasy retellings in their third semester projects. She will be the Author Guest of Honor at the 2014 World Horror Convention.
SEANAN McGUIRE is the New York Times Bestselling fantasy author of the October Daye series, and the InCryptid series, both published by DAW Books. She won the John W. Campbell Award in 2010, and was the first person to be nominated for the Hugo Awards five times in a single year. Seanan majored in folklore and mythology at the University of California Berkeley (go Bears!), and periodically vanishes into haunted corn mazes for days at a time. She lives in Northern California, where she writes stuff. She also writes as Mira Grant, author of the Newsflesh series, and talks about horrible things at the dinner table.
SIMON GREEN was born in Bradford-on-Avon, Wiltshire, England (where he still resides), in 1955. He has obtained an M.A. in Modern English and American Literature from Leicester University and he also studied history and has a combined Humanities degree. His writing career started in 1973, when he was a student in London. His first actual sale was a story titled "Manslayer," back in 1976, but it didn't appear till much later; Awake, Awake.... was his first sale to a professional editor in 1979. Furthermore, he sold some six or seven stories to semi-pro magazines before that market disappeared practically overnight. After years of publishers' rejection letters, he sold an incredible seven novels in 1988, just two days after he started working at Bilbo's bookshop in Bath (this after three and a half years of being unemployed!). This was followed in 1989 by two more, and a commission to write the bestselling novelization of the Kevin Costner film Robin Hood: Prince of Thieves, which has sold more than 370,000 copies.
NANCY KEIM-COMLEY has a degree in English and a Master's in Folklore from Western Kentucky University. She has written about and been published on diverse topics such as the death, burial and funerary rites of an African American community and a storyteller in Tennessee.
| {
"redpajama_set_name": "RedPajamaBook"
} | 1,660 |
Q: xPath inside xPath? This xml below retrieve results from a query and have an embedded XML document of responseText element.
<resultsResponse>
<errors/>
<responseformat>xml</responseformat>
<responseText><![CDATA[<?xml version="1.0" encoding="UTF-8"?><records><record><slice><![CDATA[40224]]]]>><![CDATA[</slice></record><record><slice><![CDATA[40224]]]]>><![CDATA[</slice></record><record><slice><![CDATA[40224]]]]>><![CDATA[</slice></record></records>]]>
</responseText>
</resultsResponse>
I want to get slice attribute, but i don't know exactly how can do that without make two xpaths separately.
Any chance to get slice information using only one xpath query? Like turn responseText into a node tree?
What I'm doing now: use a xPath Query to get the responseText attribute: //responseText. It returns:
<?xml version="1.0" encoding="UTF-8"?> <records><record><slice><![CDATA[40224]]></slice></record><record><slice><![CDATA[40224]]></slice></record><record><slice><![CDATA[40224]]></slice></record></records>
after i have to keep this new xml inside of a variable and apply another xPath Query to get slice elements: //slice
40224
40224
40224
The problem is: Is mandatory to use xPath query just one time and without code (just using xPath expressions).
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,818 |
The Lt. Dan Band is an American cover band originally formed in Chicago in 2004 by Gary Sinise and Kimo Williams to perform at USO shows, entertain troops, and raise money for disabled veterans. The band is named after the character Lieutenant Dan Taylor, whom Sinise portrayed in the film Forrest Gump. Sinise has said in interviews that many people know him by sight as "Lieutenant Dan" rather than by his real name, hence the band's name. The concept came about when Sinise asked for permission to bring musicians on his USO tours. The group was initially known as "Gary Sinise and the Lt. Dan Band".
The Lt. Dan Band has grown from the occasional jam session and Chicago-area gigs to performing for charities and non-profit organizations including the USO and Operation Iraqi Children, the latter of which was co-founded by Sinise in March 2004. In 2011, a documentary was released regarding the band and Sinese's work to benefit veterans. They frequently visit military bases in the United States and abroad, they have played over 400 concerts.
Sinise was involved in building a memorial to America's three million, living, disabled veterans. Completed in 2014, the American Veterans Disabled for Life Memorial was built due to the efforts of Sinise and the band championing the cause of the Disabled American Veterans.
Awards
For his humanitarian work, Sinise has received the Bob Hope Award for Excellence in Entertainment, the Spirit of the USO Award, Sylvanus Thayer Award, Doughboy Award, Dwight D. Eisenhower Award, and the Ellis Island Medal of Honor. He was given the Presidential Citizens Medal in 2008.
Gallery
References
External links
Lt. Dan Band Official Site
Cover bands | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,351 |
María Isabel Gea Ortigas (1956) es una escritora y periodista española. Licenciada en Ciencias de la Información por la Universidad Complutense de Madrid.
Autora prolífica de estudios, recopilaciones, reediciones y anecdotarios sobre diversos aspectos relacionados con Madrid, publicó en 1989 su primer libro, Casas, casos y cosas de Madrid.
Obras
De entre su abundante obra pueden destacarse un Diccionario Enciclopédico de Madrid, una Guía del Patrimonio Artístico de Madrid, sus monografías sobre la Historia de los distritos de Madrid, una Guía del Plano de Texeira (1656) y su aportación al Madrid desaparecido.
Notas
Referencias
Enlaces externos
Escritores de España del siglo XX
Escritores de España del siglo XXI
Estudiosos de Madrid
Alumnos de Ciencias de la Información de la Universidad Complutense de Madrid | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 376 |
\section{Introduction}
\label{sec:introduction}
\IEEEPARstart{P}{ower and energy} systems have evolved tremendously over the last few decades, as pushed by various parallel (and inter-related) processes towards decarbonisation, digitalisation and liberalisation. On that last point, there has been a clear trend in many Western countries (e.g., in Europe, North America and Australia/New Zealand) among others, to design and deploy markets for the exchange of electric energy primarily, but also to support power system operations through the procurement of ancillary services, as well as investment in generation assets through capacity and auction mechanisms. Today, in regions of the world like Scandinavia and the U.K., there have been wholesale and retail electricity markets operating for nearly 30 years. Many academics, practitioners in industry and policy-makers continuously underlined over that period that this approach to the liberalisation of electric power systems was an undeniable success. At the same time, over the last 10-15 years, existing electricity markets have been challenged by various evolutions of power and energy systems (decentralisation, increase in renewable energy penetration, etc.), and more recently by, e.g., tensions in the procurement of gas as well as availability of the nuclear generator fleet in Europe. In a way, we already got many hints of the fact that existing electricity markets may not be suitable for our current situation with a strong push for an energy transition, as well as for future power and energy systems that would be dominated by carbon-neutral and renewable-based power generation. For instance, in the context of power systems without fuels, Ref.~\cite{Taylor2016} insisted on the fact existing electricity markets will be fundamentally challenged.
A key question then arises: ``what may (or what should) future electricity markets look like?" This question has attracted substantial attention recently -- see, e.g., \cite{Newbery2018, Sorknaes2020, Pollitt2022}. We could surely stick to the status quo and regularly find a few patches to add to existing markets hoping to contain their limitations. Though, alternatively, we could consider their necessary and profound evolution towards mechanisms that are fit for purpose. Power and energy networks are complex socio-techno-economic systems that are of utmost important to industries, economies and societies, while being at the core of the necessary ongoing energy transition. Obviously, the answer to this question may be different if taking the perspective of an economist, of an engineer, of a policy-maker or of a social scientist. The necessity to accommodate different perspectives to such complex system was for instance recently discussed \emph{(i)} in the context of the energy transition generally \cite{Cherp2018}, and \emph{(ii)} for the specific case of electricity distribution systems \cite{Wang2022}. More holistically, taking a designer's perspective, one should recognize that markets are an object (or a mechanism, following the terminology within economics and applied mathematics) with a purpose, for that complex socio-techno-economic system. There are certainly laws of physics that cannot be avoided (e.g., Kirchoff laws) and economic principles that cannot be circumvented (e.g., marginal pricing of energy, as discussed by \cite{Conejo2023} in this same issue). But then, there are actually a myriad of options that could be considered in terms of market organization and operations, depending on the purpose we seek for electricity markets.
Since we cannot start from a blank page, we argue that it is important to appraise the reasons why electricity markets have been designed and operated in such a way so far, to understand how they may be to evolve in different directions in the future. Therefore, Section~\ref{sec:past} gives a succinct overview of the key aspects of the development of electricity markets from both academic and actual implementation points of view. Subsequently in Section~\ref{sec:purpose}, we will consider alternative perspectives (economical, engineering and related to environmental, societal and governance -- ESG, aspects), allowing us to ask a number of key questions about the purpose and relevant drivers for the design of current and future electricity markets. Eventually in Section~\ref{sec:features}, we will then describe some of the potential features we foresee for future electricity markets, for them to be able to meet their purpose. Finally, the paper will close with Section~\ref{sec:concl}, gathering a set of conclusions and perspectives.
\section{Understanding and Learning from the Past}
\label{sec:past}
Even though many countries are only recently engaging in developing and deploying electricity markets, these are not fundamentally new mechanisms as they have been around for decades in certain areas of the world. Their existence originates from the convergence of ideas born in academia that are at the basis for such markets, combined with a will from policy-makers and industry to restructure electricity markets, with a view towards more transparent and efficient operations of power systems. Today, still, we expect that future evolutions of electricity markets should be grounded within rigorous and well-motivated academic considerations, at the interface between relevant disciplines, e.g., power system engineering and operations research, economics and social sciences more generally. We therefore concentrate in the following on both regulatory and academic sides of the early development of electricity markets. Readers aiming to have an exhaustive overview of current electricity markets and their specific challenges are referred to \cite{Glachant2023, Sioshansi2013}.
\subsection{On the Regulatory and Real-world Deployment Side}
The large-scale deployment of electric power infrastructure, in parallel to the electrification of our industries and societies, made access to electricity a primary need for many, if not a basic human right \cite{Bradbrook2006}. Today, looking at the most important requirements for developing and fast-growing countries, electrification stands high on the list, while generally, high reliability in the delivery of electricity is seen as of utmost importance in many advanced economies and their industry. In principle, when considering such large, expensive, complex and strategic infrastructures, it makes sense for those to be handled by governments or government-controlled organizations since forming natural monopolies. This is has been the case for a long period for electric power system infrastructures throughout the world, following the so-called vertically integrated structure \cite{Kirschen2004}. Such an approach has allowed many countries to develop and maintain large infrastructures, support ambitious generation development plans (as for the case of nuclear energy in France), while providing electricity to citizens with high reliability levels. Even though vertically integrated and highly linked to governments, these were already seen as electricity markets, since also involving monetary transactions all the way from final consumers to the generation side.
Following the wave of liberal thinking of the 1980s in, e.g., the U.K. and the U.S.A., it was envisaged to restructure such electricity markets by liberalizing some of its components, and more generally to consider alternative ways to look at the organization of generation, transmission and distribution, as well as retail of electric energy. A gentle overview of the various approaches to electricity market organization are presented in \cite{Kirschen2004}, while more advanced discussion of organizational aspects of electricity markets can be found in \cite{Cramton2017, Joskow2019} -- we do not aim here to go into detail with such matters. The key point of this restructuring process, similarly to other strategic infrastructure and industry, was to find a way to separate activities that would form a natural monopoly like the management of transmission and distribution, to activities where competition could be introduced, as for the case of generation and retail. In principle, this then means that there are two sides to thoroughly analyse within liberalized electricity markets: \emph{(i)} the generation side and the related wholesale electricity markets, and \emph{(ii)} the retail side and the advent of retail markets. We will mostly focus on the former one, since being the side where most of today's challenges are most visible and pressing. We will see later on that we may also see a stronger convergence of such markets in the future, as supported by digitalisation and decentralisation of power and energy systems.
Eventually, three countries acted as front runners for the liberalization of electricity markets, and for the deployment of wholesale electricity markets, which are Chile, the U.K. and Norway. For an exhaustive overview of the Chilean process and experience, the reader is referred to \cite{Serra2022}, while a broad coverage of the U.K. and Norway, as well as subsequent developments over the whole European area, is provided by \cite{Bolton2021}. At the time, liberalized electricity markets appeared as the natural way to more efficiently operate power networks and to eventually make electricity cheaper for final consumers. Even though the development of electricity markets in different regions of the world has placed varying focus levels on accommodating operational constraints and on the way to meet desirable market properties, the main and core ideas are very similar and supported by the strong academic contributions to the design and functioning of modern electricity markets.
\subsection{On the Academic Side}
Many consider that the fundamental methodological concepts underlying the development of modern electricity markets can be traced back to 1988 and the seminal book of Schweppe and co-authors about spot pricing of electrical energy \cite{Schweppe1988}. The concept of spot pricing is there proposed to accommodate the temporally and spatially varying costs of electricity, accounting for both operational and capital costs, while also accommodating operational constraints related to power system operation (e.g., network constraints). Arguably, this also builds on earlier results related to networked resource allocation and economics based on the foundational works of Samuelson \cite{Samuelson1952}. Since then, academia has continuously supported the developments of electricity markets through direct interaction with regulators and market operators, as well as by proposing novel ideas that would allow to improve the design and functioning of electricity markets.
At stages, the academic contributions to the functioning of electricity markets first consisted in observing and documenting outcomes of electricity markets that may be seen as counter-intuitive or unintended. This is often complemented by simulations and scenario-based analysis to foresee how electricity markets may behave in the future. A clear example is that of negative electricity prices, which have been increasingly observed in electricity markets like Nord Pool for instance (see, e.g., \cite{Skytte2018}). In that market, it was decided in 2009 that supply offers with negative prices were accepted by the market, hence opening to the possibility of market-clearing prices to be negative. While this appears to be a natural consequence of the balance with supply and demand when supply is plentiful, this may also have major consequences, e.g., for the pricing of derivatives, and by not sending the right signal for investment in future capacities.
In parallel, the role of academics is also to help with specific challenges that emerge in electricity markets, owing to their evolution. As an example, the increasing penetration of renewable energy generation from wind and solar power sources brought inherent change in terms of variability and limited predictability, which have to be accommodated within the current realm of electricity markets. This has led to the rethinking of the sizing of reserves to be procured through markets \cite{Matos2010}, the definition of new market products, e.g., ramp products (recently reviewed in \cite{Sreekumar2022}), etc.
Most importantly, academics have continuously taken a more exploratory and visionary role in their consideration of electricity markets. To start with, fundamentally, electricity markets were originally seen as a way to coordinate and remunerate supply to meet demand. Though, early works, e.g., in \cite{Kirschen2003}, hinted at the fact that demand had a role to play in electricity markets through their flexibility and elasticity. This has now opened up to the strong emphasis placed on demand-side flexibility. Pushing it further, one may see this leading to a change of paradigm where, instead of supply following demand, it may be that demand should follow supply (if mostly coming from non-dispatchable renewable energy generation). Similarly, electricity markets were designed based on the idea that supply was dispatchable (up to a reliability-related uncertainty), hence making that supply offers in the market should be seen as deterministic. The limited predictability of renewable energy generation has challenged this idea, while leading to various proposals towards using stochastic optimization approaches to the clearing of electricity markets \cite{Pritchard2010} and more generally novel approaches to pricing energy in wholesale electricity markets \cite{Morales2012}.
On a more exploratory note, different works are pointing at the fact it is not new products, clearing and pricing approaches that are necessary, but more fundamentally new views of electricity markets. For instance in a future of power systems without fuels, pricing may negligibly rely on the marginal cost of producing renewable energy generation, but relying on strategic behaviour of producers and consumers to recover capital investment costs instead \cite{Taylor2016b}. Alternatively, as we see that energy systems are increasingly interconnected (i.e., among gas, heat and electricity) and that such inter-dependencies may further support the integration of renewables, alternative approaches are put forward for markets that would accommodate energy systems altogether \cite{Sorknaes2020}.
\section{The Purpose of Future Electricity Markets}
\label{sec:purpose}
Most often, researchers and practitioners concentrate on the actual functioning of electricity markets, while aiming to accommodate their evolving context, e.g., with increasing penetration of renewable energy sources, the need for additional flexibility, etc. However, more fundamentally, it may not just be the functioning of electricity markets that ought to be rethought, but their very purpose instead. This was for instance recently illustrated by a report produced by Energinet, the Transmission System Operator (TSO) in Denmark \cite{Energinet2022}, which challenged some common thoughts about what markets are for.
Let us explore in the following what the purpose of future electricity markets may be, by considering three different perspectives: the economic one, the engineering one, and the Environmental, Societal and Governance (ESG) one. For all of those, we additionally describe important challenges ahead.
\subsection{The Economic Perspective}
Conventionally, the term ``market" is given an economic meaning, as it relates to the organization of buyers and sellers, for the exchange and pricing of a given or multiple commodities. For electricity, the market principally deals with electric energy as a commodity, even though we usually consider the extension to the procurement of ancillary services (i.e., all services that system operators may require for the safe and efficient operation of the power system) as part of electricity markets. This is since ancillary services are commonly provided by the same agents who buy and sell electric energy. The purpose of electricity markets, from the economic perspective, is \emph{(i)} to optimally allocate resource, \emph{(ii)} to reveal prices as a basis for payment and revenues, and \emph{(iii)} to provide the right signals to support investment in relevant assets.
Within the market, the way buyers and sellers interact can be through direct contracts (for a given quantity, time period and price). This situation is for futures contracts and over-the-counter (OTC) trading. However, at the day-ahead stage, it is more common today to see wholesale markets as an exchange or a pool. There, the interaction between buyers and sellers is standardized, for instance by having set the lead times, periods, product types, etc., while using a common software platform. Principles from mechanism design are employed to define these products (e.g., quantity-price bids, offer curves, etc.) and the market-clearing algorithm eventually yields the dispatch (quantities to be supplied on the supply side, quantities to be consumed on the demand side), while allowing for price discovery.
The principle of price discovery follows a social welfare maximization process, which, given stated constraints (to be discussed in the following section), aims at maximizing both supplier and consumer surpluses. A stylized representation of these principles is given in Figure~\ref{fig:socialwelfare} for a given market time unit (say, an hour of the day). Market participants on the supply side place offers expressed in terms of quantity and price. These are to be interpreted as the maximum amount of energy they can produce at that time and the minimum price they are willing to accept to be paid for. These offers are ordered with increasing price and yield the supply curve depicted. On the demand side, market participants place similar offers. Inversely though, these are to be interpreted as the maximum amount of energy they can consume and the maximum price they are ready to pay. The demand offers are ordered with decreasing price and form the demand curve that can be seen in the figure.
\begin{figure}[!ht]
\begin{tikzpicture}
\draw[black, thick, -stealth] (1,1) -- (1,7) node[above=3pt]{price};
\draw[black, thick, -stealth] (1,1) -- (9,1) node[below=3pt]{quantity};
\draw[fill=gray!10] (1,1) -- (2,1) -- (2,1.2) -- (2.5,1.2) -- (2.5,1.5) -- (3.2,1.5) -- (3.2,2.1) -- (4,2.1) -- (4,2.5) -- (4.8,2.5) -- (4.8,3) -- (5.7,3) -- (5.7,4) -- (4.5,4) -- (4.5,4.5) -- (4,4.5) -- (4,5.5) -- (3,5.5) -- (3,6.5) -- (1,6.5) -- (1,1);
\draw[red] (1,1) -- (2,1);
\draw[red] (2,1) -- (2,1.2);
\draw[red] (2,1.2) -- (2.5,1.2);
\draw[red] (2.5,1.2) -- (2.5,1.5);
\draw[red] (2.5,1.5) -- (3.2,1.5);
\draw[red] (3.2,1.5) -- (3.2,2.1);
\draw[red] (3.2,2.1) -- (4,2.1);
\draw[red] (4,2.1) -- (4,2.5);
\draw[red] (4,2.5) -- (4.8,2.5);
\draw[red] (4.8,2.5) -- (4.8,3);
\draw[red] (4.8,3) -- (5.7,3);
\draw[red] (5.7,3) -- (5.7,4.1);
\draw[red] (5.7,4.1) -- (6.2,4.1);
\draw[red] (6.2,4.1) -- (6.2,4.4);
\draw[red] (6.2,4.4) -- (7.5,4.4);
\draw[red] (7.5,4.4) -- (7.5,4.9);
\draw[red] (7.5,4.9) -- (8.3,4.9) node[above=3pt, red]{supply curve};
\draw[blue] (1,6.5) -- (3,6.5);
\draw[blue] (3,6.5) -- (3,5.5);
\draw[blue] (3,5.5) -- (4,5.5);
\draw[blue] (4,5.5) -- (4,4.5);
\draw[blue] (4,4.5) -- (4.5,4.5);
\draw[blue] (4.5,4.5) -- (4.5,4);
\draw[blue] (4.5,4) -- (6,4);
\draw[blue] (6,4) -- (6,3.5);
\draw[blue] (6,3.5) -- (6.5,3.5);
\draw[blue] (6.5,3.5) -- (6.5,1.5);
\draw[blue] (6.5,1.5) -- (7,1.5);
\draw[blue] (7,1.5) -- (7,1.2);
\draw[blue] (7,1.2) -- (7.7,1.2) node[above=8pt, blue]{demand curve};
\fill[black] (5.7,4) circle (3pt) node[right=12pt, black]{equilibrium price};
\draw[gray!80, dashed] (1,4) -- node[below=8pt, midway, gray!85]{social welfare} (5.7,4) ;
\end{tikzpicture}
\caption{Stylized representation of the approach to clearing of wholesale electricity markets based on a social welfare maximization principle. Market clearing yields both dispatch (i.e., production and consumption quantities for all participants) and equilibrium price.}
\label{fig:socialwelfare}
\end{figure}
Doing so graphically, for such a stylized setup, is equivalent to employing the optimization-based approach used to clear wholesale electricity markets, where the aim is to maximize social welfare. Social welfare is formally defined as the signed area below the demand curve and above the supply curve (in light gray in Figure~\ref{fig:socialwelfare}). It is signed in the sense it is positive if the demand curve is above the supply one, and negative if demand is below supply. The crossing point between the two curves reveal the equilibrium price, which is used as a basis for pricing electricity. All offers (both demand and supply) on the left of that point are accepted, while those on the right are not. If using the equilibrium price for both buyers and sellers, consumers whose offers are accepted pay less than they were willing to, while sellers receive a price higher than what they required.
The design and clearing of electricity markets rely on relevant principles from optimization and game theory (mechanism design, more precisely), allowing them to enjoy certain properties that are crucial for its functioning. As we aim at keeping the exposition of this paper concise, we will not go through all these properties individually. For instance, one would expect that market participants do not enter markets in a loss-making position (individual rationality), are incentivized to participate in an honest manner (incentive compatibility), and that the sum of revenues is equal to the sum of payments (budget balance). While it is fairly straightforward to get these properties in a very simplified environment, real world constraints (e.g., congestion on power networks, start-up/shut-down of individual assets) challenge these properties and potentially require adaptations (e.g., uplift payments). More importantly, there are expectations such that markets cleared based on the social welfare maximization principle also allow for the suppliers to recover investments (as discussed by, e.g., \cite{Conejo2023, PerezArriaga1997}).
\subsection{The Engineering Perspective}
While many will give a purely economic meaning to the concept of market, the reality of electricity markets is that they must also have a strong technical component\footnote{We refer to this technical component as the ``engineering" perspective, since it is mainly engineers who focus on such technical aspects.}, since they deal with the operation of the largest and most complex engineering system ever built by humans, i.e., the electric power infrastructure. This comes with a number of constraints related to of how power systems are to be operated, as well as constraints related to the power production and consumption assets themselves. These constraints ought to be accommodated in one way or another by the market. Eventually though, there are limits to what can be handled through markets \cite{Energinet2022}. Maybe the biggest challenge from an engineering perspective is to find the fine line between what can (and should) be handled through markets, and what can (and should) not. And, for the operational constraints and aspects that are to be handled through market, one should find a way to embed those in the market design (possibly through the definition of market products) and in the market clearing algorithms (which needs to be kept tractable).
There are variations in the way constraints stemming from power system operation are accommodated. For instance, in terms of network constraints and flows, the historical model in Europe has been to use markets zones and an import-export representation of cross-zonal exchanges, while a detailed power flow-based network-constrained representation of the network has been employed in the USA. Even if employing the more advanced approach of embedding power flows and their constraints in market clearing, these are simplified and linearised (following a DC linearisation -- see \cite{Taylor2015} for clarifications). Such simplification and linearisation still yield a gap between the outcome of market clearing and actual power system operation since, basically, DC power flow outcomes are not AC feasible \cite{Baker2021}.
Besides network-related aspects, substantial emphasis is to be placed on constraints to ensure the secure operations of the power system. This is done by ensuring the availability of relevant resources in case of contingencies, e.g., through the provision of ancillary services. However, this has to be done in a way that incentivize market participants to provide services, while also coordinating markets for various products to get consistent and non-conflicting outcomes (typically, if a given share of capacity of a production asset is booked to provide reserves, it should not be used to produce energy). This has then led to the proposal and implementation of security-constrained market-clearing approaches \cite{Alvey1998}, and to varied proposals for the joint clearing and pricing of energy and ancillary services -- see \cite{Arroyo2005, Galiana2005} among others.
While some of the aspects from power system operation are accommodated within market design and clearing procedures, others are internalized by the market participants when offering in electricity markets. Indeed, some of the characteristics of the assets of market participants may not be well accommodated in market clearing procedures and pricing approaches that rely on linear programming. This is the case of non-convexities of these assets, for instance related to commitment decisions. Alternative pricing principles, e.g., convex hull pricing \cite{Hua2017, Andrianesis2022}, were proposed to deal with such asset-based operational constraints. Recently, some even advocated for a necessary shift from a linear to a conic programming paradigm \cite{Ratha2022}, which would allow to better accommodate more complex operational constraints and uncertainty in market clearing procedures and related pricing approaches.
The same goes with asset flexibility: as we want to have more flexible assets, and more specifically energy storage assets, in the energy system landscape, emphasis must be placed on how to optimally accommodate those in electricity markets. As of now, storage is mainly seen as a merchant asset, i.e., participating in electricity markets as for other production and consumption assets (and focused on short-term incentives). However, there is a path towards having non-merchant storage assets in electricity markets, allowing to perform temporal arbitrage, the same way that transmission is often seen as non-merchant asset allowing for spatial arbitrage. Accommodating storage in electricity markets comes with challenges, e.g., related to locational pricing \cite{Weibelzahl2018} and more generally impact in social welfare \cite{Sioshansi2014}, though they are also novel instruments generalizing financial transmission rights \cite{Taylor2014, MunozAlvarez2017} that comprise promising approaches.
\subsection{The ESG Perspective}
Over the last decade, we witnessed a substantially increased focus on Sustainable Development Goals (SDGs), leading to an Environment, Societal and Governance (ESG) perspective to investment and operations of our industries. In many ways, the power and energy system is at the core of this transition, with a clear shift towards renewable energy sources, but not only. Indeed, while the restructuring and liberalization of electricity markets has meant a shift towards wholesale and retail markets as we know them today in many Western countries, this focus on ESG could be one of the drivers of how we rethink the organization and purpose of electricity markets. An obvious example is the push towards various forms of community-based and peer-to-peer electricity markets, along with the associated novel business models \cite{Parag2016}.
So, what would be the purpose of electricity markets from an ESG perspective? For one, markets should support the decarbonization of power and energy systems, in line with national and international objectives. So far, the actual functioning of markets, as well as additional measures and subsidies, have amply supported the deployment of renewable energy generation. As of today, however, it is not only investment in actual renewable energy generation capacities that is required, but a general transformation of the infrastructures and practices. Integration of renewable energy generation in existing power and energy systems requires additional temporal and spatial arbitrage flexibility, e.g., provided by power networks, storage, as well as various forms of demand response and energy conversion.
More than the decarbonization of power and energy systems, the design of electricity markets reflects the way we envisage how we should produce, exchange and consume electric energy. Indeed, with a transition towards more distributed energy sources and for which the marginal cost of producing electricity is close to 0, many potential paradigm changes are in sight. Instead of having supply following demand, it may just be that part of the demand (i.e., except for crucial uses and infrastructures) will have to follow supply. Another change of paradigm relate to the fact the ownership of power production units will also be more distributed (think of solar panels on houses, car parks, small businesses, etc.) and they may want to have a say about how the surplus energy they produce is consumed. This social component of future electricity markets is for instance illustrated by the approach of \cite{Morstyn2018} and the concept of federated power plants, in which some of the participants may have an altruistic behaviour by expressing their wish to share their energy with others potentially in need. This idea was recently pushed further based on the concept of smart energy neighborhood \cite{Savelli2021}, where the agents involved in a local energy system and market are seen as community with shared interested and objectives. Electricity consumers are increasingly interested in expressing preferences in the way they source (in terms of location, energy type, etc.) and use electricity. Certain forms of peer-to-peer electricity markets allow for such heterogenous preferences \cite{Sorin2019}. This may substantially change the way all think of electricity as a commodity: even if being ubiquitous when sourced from any socket being available, the fact that consumers express their views on the sourcing of their electric energy is a form of empowerment that could have substantial consequences, e.g., on investment in generation, storage and energy conversion assets\footnote{A relevant recent example is that of Ripple Energy in the UK (\url{www.rippleenergy.com}), offering collaborative investment in wind generation capacities, with direct impact on subsequent energy procurement costs.}.
Besides the traditional economic and engineering perspectives to electricity markets, we expect that this ESG perspective will allow to embrace important concepts from social science to rethink the purpose of energy infrastructures and electricity markets in the coming century. Historically, emphasis has been placed on giving access to plentiful amounts of energy to support industry growth and the comfort of the population. Many other considerations are emerging and now gaining importance, e.g. focusing on energy poverty, fairness in terms of access and costs of electric energy, etc. Several academics have been pushing to revisit some of the basic economic principles in our society, see e.g. \cite{Raworth2017, Lavie2023} towards a better use of resources and a cooperative approach. Such concepts are directly relevant for design of future electricity markets, for instance to be thought of within the energy justice framework discussed in \cite{Sovacool2016}.
\section{Expected Features}
\label{sec:features}
In view of the purpose of future electricity markets and the various perspectives (economical, engineering and ESG-related ones) discussed in the above, we introduce here some of the resulting expected features of future electricity markets.
\subsection{Variability, Uncertainty and Flexibility}
Future power and energy systems will heavily rely on renewable energy generation sources, with their variability and limited predictability. Energy uses are also changing, with increasing electrification and the deployment of new types of electricity consumption assets, e.g., heat pumps and electric vehicles. All in all, this means more variability and uncertainty in electricity markets. This also calls for more flexibility, as for instance underlined in the new strategy of Energinet, a TSO that deals with one of the power systems with the highest penetration of renewable energy generation \cite{Energinet2022b}\footnote{Actually, it is also the case that Denmark has the advantage to have a highly interconnected power system (see presentation and discussion in \cite{Pinson2017} for instance). Hence, penetration of renewable energy sources should also be seen as relative to interconnection capacity, making countries like Ireland and Spain/Portugal facing possibly bigger challenges at lower penetration levels already.}.
When modern electricity markets were first designed and implemented, such variability and uncertainty was not so prominent and the main source of uncertainty was on the electric load side (see, for instance, the pioneering works in load forecasting in \cite{Gross1987}). Hence, electricity markets were thought off in a deterministic setup, with the possibility to handle the consequences of uncertain demand at the balancing stage (and based on adequate reserves procured a priori). However, increasing uncertainty may justify rethinking electricity markets in a stochastic framework instead, e.g., based on stochastic programming \cite{Pritchard2010} or chance-constrained optimization \cite{Dvorkin2020}. We may generally refer to these markets as stochastic electricity markets. And, since most suppliers and consumers should be seen as uncertain, a potential approach is for markets to allow for offers expressed in a probabilistic manner \cite{Tang2015}, for instance in the form of distributions or prediction intervals.
Since stochastic electricity markets may be difficult to implement in practice, an alternative is to find ways to define new products that would allow to better cope with uncertainty and variability on the one hand, and bring some desired additional flexibility on the other hand. For the latter case, relevant examples are that of \emph{(i)} policy-based reserves \cite{Warrington2013}, since reserves should not be thought only in terms of capacity, but in terms of how they can react to how uncertainty unfolds, and \emph{(ii)} price-region bids \cite{Bobo2021}, which allow to naturally accommodate flexibility characteristics for assets at the interface between multiple energy systems (e.g., combined heat and power plants at the interface between electricity and heat energy systems).
A consequence of the profound changes to electricity markets driven by variability, uncertainty and flexibility is that it will eventually have an impact on our approach to pricing. Today, pricing is mainly related to the energy commodity itself and to capacity if providing ancillary services. In the future, pricing may also be driven by reliability and security of supply concepts, in terms of firmness of supply on the production side, and flexibility in consumption on the demand side. In that direction, an interesting example is that of risky power markets introduced by \cite{Zhao2014}.
\subsection{Distributed and Coordinated}
Electricity markets have always been thought of as a way to coordinate the operation of power systems in a somewhat decentralized manner since allowing for all agents involved to be in control of their decision-making process \cite{Pollitt2022}. Even though the decision-making is decentralized, one still relies on a central marketplace that serves as an interface to all agents involved. In contrast, novel approaches involving transactive energy, community-based and peer-to-peer electricity markets, also aim at decentralizing the marketplace itself. Eventually this may lead to a convergence between wholesale and retail markets by having a direct connection between producers and consumers \cite{Sousa2019}.
Future electricity markets are expected to adapt their level of decentralization and coordination to the fact that assets are increasingly distributed, while additionally allowing for small consumer and prosumers to actively participate in such markets. There are obviously both benefits and caveats if going for more or less decentralization \cite{Ahlqvist2022}. As an example, in a centralized pool setup, the minimum bid size (in the order of a MWh) in electricity markets has traditionally been a barrier to entry for small agents. However, as increasing flexibility is sought after, flexibility which can be provided by smaller assets in power and energy systems (e.g., at the consumer level), there is a trend towards lowering the minimum bid size, while also supporting the emergence of so-called aggregators that would coordinate groups of smaller assets \cite{Burger2017}. Again, this participates to an increase in decentralization and coordination. Ideally in peer-to-peer and community-based electricity markets, there should be a virtually no minimum bid size, reflecting setups with very low transaction costs.
Coordination is also motivated by the necessary interplay between various energy systems (electricity, gas and heat) in the future, to optimally support the integration of renewable energy sources. A substantial challenge though is that these energy systems have different operational constraints driven by their underlying physics (e.g., gas and water flows are not the same as power flows), as well as established operational and market practices. Aligning such practices for these interconnected and complementary energy systems will already be a big step towards their improved coordination. Eventually, the design of optimal interfaces between these energy markets, e.g., through co-optimization under a leader-follower setup \cite{Ordoudis2019} or information exchange \cite{Chen2022}, and alternatively joint energy markets \cite{Sorknaes2020}, will yield an agile coordination approach to these energy systems.
\subsection{Data-driven and Fair}
Digitalization has already had a fundamental impact on electricity markets, with an increasing role of forecasting and data-driven decision-making. Most importantly, data and data-driven techniques have been enablers for many of the current and foreseen evolution paths of electricity markets. For instance, peer-to-peer electricity markets \cite{Sousa2019} and the business models of aggregators \cite{Ostergaard2021} necessarily rely on \emph{(i)} the availability of data at the level of consumers (and possibly at the detailed level of their assets), possibly with high temporal resolution, and \emph{(ii)} on advanced analytical approaches to get value from such data.
At some point though, with increased decentralization and digitalization, comes the scalability challenge. While it is possible to clear a pool-based markets with 1000s of participants, and generally not prohibitevely computationally expensive, clearing a peer-to-peer market with the same number of participants requires a lot of communication among peers, it is computationally expensive and it may not even be feasible. Therefore, new computational approaches are necessary if aiming to further decentralize electricity markets. Going beyond decomposition and distributed optimization, this is where AI-based approaches (more precisely, based on machine learning) to market clearing may become very relevant, since such approaches could learn from the past and clear market based on contextual information (e.g., about the weather, the state of the power system, etc.). AI and machine learning are becoming more prominent in power system operations anyway, hence possibly supporting their potential role in electricity markets too. Another advantage from AI-based approaches is that they may allow to rethink electricity markets by bringing some relevant ideas from theoretical computer sciences and economics. Here, we mainly think of the concept of fairness. Indeed, one of the regular complaints against electricity markets (at both wholesale and retail levels) is about their lack of fairness. Maximizing social welfare is a key principle, it could be complemented by fairness objectives and or constraints.
When the operation of power and energy systems, as well as electricity markets, is to crucially rely on data, we are reaching a point where the value of data is non-negligible. One could even argue that in a future where renewable energy comes with a marginal cost that is close to 0, the value of data will be more than the value of energy itself. In practice though, data is collected and owned by many distributed agents, e.g. small consumers, retailers, power producers, system operators. These agents have a very low willingness to share their data in principle, since they believe that this may yield a loss in privacy, the loss of a competitive advantage, or potential exposure of critical asset information. To unleash the value of distributed data, it is difficult to envisage future electricity markets without data sharing and data monetization platforms, also with privacy-preserving components.
\section{Conclusions and Perspectives}
\label{sec:concl}
Current times have witnessed an increased focus on electricity markets in most countries that have had a long experience with seemingly well-functioning electricity markets. While the debate was mainly contained to a limited community gathering academics, system operators and policy makers until recently, the recent increase in prices in wholesale electricity markets have broad intense scrutiny to the functioning and very nature of electricity markets. While some technical and economical basics of electricity markets are not negotiable, we have aimed to show here that many aspects of modern electricity markets may (and even ought to) to be rethought. Today's context is not what is was when electricity markets were first designed and implemented. Electricity markets are to be fit for purpose -- and, their purpose will necessarily change with time.
If revisiting the previous paragraphs, we have explained that electricity markets should go from deterministic to stochastic, that reserves should not be seen as just capacity but procured based on control policies, that markets will be more decentralized, that data may be more valuable than energy and that AI could be used to clear electricity markets. To this, we could actually add that since renewable-dominated power and energy systems mark a transition from low-capital investment and high operational costs to high-capital investment and low operational costs, there is a shift from wholesale electricity markets to auctions and other capacity remuneration mechanisms to insure that market participants recover their investment. Overall, that means that a part of the very role of wholesale electricity markets is fundamentally changing. Similarly, it will impact the retail side of electricity markets. This opens the door to a wealth of new business models, some of which we already see emerging. Some of these new business models take a more cooperative view for instance by allowing to co-invest in renewable energy generation capacities, to share surplus solar power production with others, to pay a subscription for having a storage unit at one's house, etc.
Today, we see strong needs for supporting the energy transition, insuring the reliability and resilience of power and energy systems, and for making electricity markets fair for all. The context has also changed in the sense that the status quo within the underlying science, as well as the availability of data and computational power, allows to rethink electricity markets in ways that could not be envisaged a few decades ago. These are exciting and challenging times for electricity markets, requiring a multi-disciplinary approach and broad expertise (in, e.g., mathematics, economics, computer science, power system engineering, social sciences) to re-design electricity markets that fully acknowledge the socio-techno-economic nature of power and energy systems.
\section*{Acknowledgements}
Acknowledgements are due to many colleagues in academia and industry for many regular discussions about current and future electricity markets, but also to numerous funding bodies and agencies for supporting my research on that topic over the years. The author is also grateful for the detailed feedback provided by Antonio Conejo and Antonio G{\'o}mez-Exp{\'o}sito, which allowed to improve the manuscript originally submitted.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,543 |
\section{Introduction}
Observable macroscopic forces produced by quantum vacuum fluctuations of the electromagnetic field have attracted great attention in theoretical and experimental studies. The most renowned are perhaps the attractive Casimir force between two neutral bodies \cite{Casimir,Milonni,Dalvit-Milonni-Roberts-Rosa,Woods-et-al} and the Casimir-Polder force between an atom and a neutral body \cite{Casimir-Polder}. The Casimir effect, for example, has found various applications in macroscopic physics, cosmology, hadron physics, supersymmetry and supergravitation. Since the Casimir effect (CE) has been verified with an astonishing high precision \cite{Lamoreaux,Mohideen,Roy,Klimchitskaya}, it has become a testing ground for the predictions of new fundamental physical theories.
Outside the paradigmatic CE between two parallel conductive plates, many theoretical and experimental researches have been conducted regarding the calculation of the Casimir force for different geometries and spacetime topologies \cite{Deutsch-Candelas,DeRaad-Milton,Dowker, Aliev,Huang}. In particular, the CE in the cylindrical and spherical geometries have attracted the most attention due to its simplicity and important applications. Interestingly, the former leads to an attractive (although small) force \cite{DeRaad-Milton} while the latter produces a repulsive force \cite{Boyer}. Regarding the applications, the CE for a sphere has proved to be useful in the understanding of other phenomena. For example, the van der Waals force is equivalent to the CE for a dielectric sphere \cite{Milton 1, Milton's book}, and it has also been considered as the responsible for the production of visible-light photons in the bubble collapse which occurs in sonoluminescence \cite{Milton 2}. It has also turned out to be strikingly important in hadron physics, particularly in the context of the bag model and chiral bag model \cite{Johnson}. In these systems, quarks and gluons are absolutely confined inside the bag which is bounding a hadron, and hence the Casimir energy of these fields must be incorporated in the total energy of a bag in hadron properties calculations.
The analysis of the CE and other physical systems in $D$ spatial dimensions have been found useful. In quantum field theory, for example, spacetime dimensions have been employed as a perturbative parameter, thus allowing to provide analytical solutions that are nonperturbative in the coupling constant. On the other hand, the CE for a $D$-dimensional sphere has also been used in hadron physics. As shown in Ref. \cite{F-G-K}, the zero-point energy in the flux-tube connecting a heavy quark-antiquark system can be regarded as the zero-point energy of an hypersphere with dimension $D=1$. Also, in the same context, it has been applied to study the quantum stabilization of $1+1$-dimensional static solids \cite{Graham 1} and the quantum energies of interfaces \cite{Graham 2, Graham 3}. These studies exemplify the usefulness of the CE in $D$ spatial dimensions in the context of hadron physics, and suggest the need for additional research in this direction. This is precisely the main goal of this work, but within the context of a Lorentz-violating (LV) scalar field theory.
Lorentz violation is currently a topic of great interest in particle physics. This is motivated from the fact that the standard model, although phenomenologically successful, suffers from some theoretical inconsistencies and long-standing unresolved problems. Nowadays, investigations concerning Lorentz violation are mostly carried out under the framework of the Standard-Model Extension \cite{SME2,Kostelecky 3}, which contains translation-invariant but Lorentz-violating corrections to the standard model parameterized by small tensor-valued background fields. No evidence for a deviation from Lorentz invariance has been found, but experimental Lorentz tests are constantly being refined. Due to its potential scope, the CE stands as a good arena to test Lorentz-violating field theories. This has motivated the study of the CE in Lorentz-violating scenarios in the classical geometry of two parallel plates \cite{Cruz 1, Cruz 2, Escobar-Medel-Martin, PLB}. This, together with the above discussed applications of the CE for a sphere, motivate the present work, where we study how Lorentz symmetry violation manifests on the CE for a real scalar field when it is confined to a $D$-dimensional spherical shell.
The outline of this paper is as follows. Section \ref{LV Sec} introduces the theoretical model for a real massive scalar quantum field $\phi$ in the presence of Lorentz violation. This consists of the Klein-Gordon Lagrangian density supplemented with the Lorentz-violating term $\lambda\left(u\cdot\partial\phi\right)^2$, where $\lambda$ is a parameter and $u^{\mu}=(u^0,\vec{u})$ is a constant vector which control Lorentz symmetry breaking \cite{Gomes-Petrov}. As required by the local approach to the CE, in Sec. \ref{GFsection} we derive the Green's function for the two cases we consider in the present study: the timelike case for which $u ^{\mu} = (1,0, ... , 0)$ and the radial spacelike one where $u ^{\mu} = (0,1,0 ... , 0)$. In Sec. \ref{CEsection} we derive analytical expressions for the Casimir stress in each case. Afterwards, in Sec. \ref{Numerical Section} we follow the procedure reported in Ref. \cite{Bender-Milton} to evaluate numerically the Casimir force in the massless case and we present the results in Sec. \ref{NumResultsSec}. Finally, Sec. \ref{ConcluSection} summarizes our results and gives further concluding remarks. Throughout the paper, natural units are assumed $\hbar = c = 1$.
\section{General settings} \label{LV Sec}
Let us consider the LV Lagrangian density for a massive real scalar field \cite{Cruz 1}
\begin{align}
\mathcal{L} = \frac{1}{2} \left[ \partial _{\mu} \phi \, \partial ^{\mu} \phi + \lambda \left( u ^{\mu} \, \partial _{\mu} \phi \right) ^{2} - m ^{2} \, \phi ^{2} \right], \quad \quad \quad \mu=0,1,2,...,D \; ,
\label{Lagrangian1}
\end{align}
where $u ^{\mu} \equiv (u _{0} , \vec{u} \, )$ denotes a non-zero $(D+1)$-dimensional constant vector and $ \vert \lambda \vert < 1$. The second term in Eq. (\ref{Lagrangian1}) mimics a background field that specifies privileged directions on the spacetime and encodes the Lorentz symmetry violation.
In this work we assume that the scalar field $\phi(x)$ is confined in a $D$-dimensional sphere of radius $R$. Formally, one
can think $D$ as a continuous variable that ranges from $0$ to $\infty$. In the present study we restrict ourselves to the case $D>2$ only. The Lorentz-invariant case corresponds to $\lambda=0$. It is worth mentioning that the LV term $\lambda \left( u ^{\mu} \partial _{\mu} \phi \right) ^{2}$ was originally conceived within the scalar sector of the Standard-Model Extension (SME) \cite{Kostelecky 3,SME2}. In the SME context, it is expected that physical relevant systems correspond to $\vert \lambda u ^{\mu} u ^{\nu} \vert \ll 1$ for any $\mu$ and $\nu$. However, there are scenarios in which Lorentz symmetry is naturally broken, such as condensed matter systems, where Lorentz-violating coefficients are not much smaller than one. Examples of this are properly discussed in Sec. \ref{NumResultsSec}. These Lorentz-violating models also appear in connection to Riemann-Finsler spacetimes \cite{Finsler} where the notion of distance is controlled by additional quantities beyond the Riemann metric, which can intuitively play the role of Lorentz-violating coefficients \cite{KFinsler}. The model in the present work is a particular case of those presented in Ref. \cite{Finsler} through the identification $(\hat{k}_c)^{\mu\nu}=\lambda u^\mu\, u^\nu$.
From the Lagrangian (\ref{Lagrangian1}) it follows the modified Klein-Gordon equation
\begin{align}
\left[ \, \Box \ + \ \lambda \left( u _{\mu} \, \partial ^{\mu} \right) ^{2} \ + \ m ^{2} \right] \,\phi (x) \ = \ 0 , \label{EQLV}
\end{align}
where the symbol $\Box = \partial _{\mu} \partial ^{\mu}$ stands for the standard D'Alembert operator in $(D+1)$-dimensions. We will consider solutions of the equation of motion (\ref{EQLV}) that satisfy Dirichlet boundary conditions (BCs) on the sphere. The corresponding stress-energy tensor takes the form
\begin{align}
T ^{\mu \nu} \ = \ (\partial ^{\mu} \phi ) (\partial ^{\nu} \phi ) \ + \ \lambda u ^{\mu} \, (\partial ^{\nu} \phi ) (u _{\sigma} \partial ^{\sigma} \phi ) \ - \ \eta ^{\mu \nu} \mathcal{L}\ , \label{Tmunu}
\end{align}
where $\eta ^{\mu \nu} = \textrm{diag} (1,-1,-1,...,-1)$ is the usual Minkowski flat spacetime metric in $(D+1)$-dimensions. It can be checked that the stress-energy tensor (\ref{Tmunu}) is conserved, i.e. $\partial _{\mu} T ^{\mu \nu } = 0$. However, it is not traceless $T ^{\mu} _{\phantom{\mu} \mu} \neq 0$ and, unlike most of the cases where Lorentz symmetry is preserved, it cannot be symmetrized \cite{Cruz 1}.
Since the spherical shell divides the space into two regions, the interior and exterior of the sphere, it is natural to use hyperspherical coordinates for the spacelike components of the stress-energy tensor (\ref{Tmunu}). In these coordinates, the Casimir force per unit area $F/A$ on the sphere is obtained from the discontinuity of the radial-radial component of the vacuum expectation value of the stress-energy tensor \cite{Milton's book}
\begin{align}
\frac{F}{A} = \langle \, 0 \, \vert \, T ^{rr} _{\mbox{\scriptsize in}} - T ^{rr} _{\mbox{\scriptsize out}} \, \vert \, 0 \, \rangle \big\vert_{\| \vec{x}\|\,=\,R} \, , \label{F1a}
\end{align}
where the subindices in/out indicate the region where the vacuum expectation value of $T ^{rr}$ has to be evaluated.
In the present study two particular cases will be analyzed in detail, namely
\begin{itemize}
\item[(I)] the radial spacelike case $u ^{\mu} = (0,1,0, \cdots ,0)$, where all the coordinates of $u^\mu$ are zero except the radial spacelike coordinate, and
\item[(II)] the timelike case for which $u ^{\mu} = (1,0, \cdots ,0 )$, where the timelike component $u _{0}$ is different from zero only.
\end{itemize}
Equivalently, in terms of the two-point Green's function (GF) defined as the vacuum expectation value of the time-ordered product of two fields \cite{Peskin}
\begin{align}
G(x , x ^{\prime} ) = - i \,\langle\, 0 \, \vert \, \hat{\mathcal{T}} \phi(x) \phi( x ^\prime) \, \vert 0 \, \rangle , \label{Grel}
\end{align}
the radial Casimir force (\ref{F1a}) can be rewritten as
\begin{align}
\frac{F}{A} = - \frac{i}{2} \, \Lambda \lim _{x ^{\prime} \rightarrow {x}} \bigg[ \frac{\partial }{\partial r}\frac{\partial}{\partial r ^{\prime}} G _{\mbox{\scriptsize in}} (x,x ^{\prime}) - \frac{\partial }{\partial r} \frac{\partial}{\partial r ^{\prime}} G _{\mbox{\scriptsize out}} (x,x ^{\prime}) \bigg] \bigg\vert _{\| \vec{x}\|\,=\,R} ,
\label{FAbyG}
\end{align}
where $G _{\mbox{\scriptsize in}}$ and $G _{\mbox{\scriptsize out}}$ are the GF for the interior ($r<R$) and exterior ($r>R$) of the sphere, satisfying Dirichlet BC on the surface ($r = R$). Further, $\Lambda = 1 - \lambda$ for case I while $\Lambda = 1$ for case II. In general, there are other terms not included in Eq. (\ref{FAbyG}) which depend on angular derivatives of the GFs that vanishes by virtue of the Dirichlet BC on the sphere.
\section{Green's function}\label{GFsection}
In this section we derive, for the cases I and II, the GF appearing in Eq. (\ref{FAbyG}). As mentioned above, we assume that the GF obeys the Dirichlet BC at the surface of the sphere
\begin{align}
G( x , x ^\prime ) \, \big| _{\| \vec{x}\| = R} = 0 , \label{Dirichlet}
\end{align}
and finiteness at the origin
\begin{align}
G(x,x^\prime) \, \big| _{\| \vec{x}\| = 0} < \infty . \label{Finiteness}
\end{align}
From Eq. (\ref{EQLV}) follows that the corresponding GF satisfies the equation
\begin{align}
\big[ \, \Box + \lambda \left( u _{\mu} \, \partial ^{\mu} \right) ^{2} + m ^{2} \, \big] \, G( x , x ^{\prime})\ = \ - \delta ^{(D+1)} (x - x ^{\prime}) . \label{GF eq}
\end{align}
Now, since the GF is translationally invariant in time let us take the time Fourier transform of $G( x , x ^{\prime} )$:
\begin{align}
G _{\omega} ( \vec{x} , \vec{x} ^{\, \prime} ) = \int _{- \infty} ^{\infty} \, dt \, e ^{-i \,\omega (t-t ^{\prime})}\, G( x , x ^{\prime})\ . \label{GFT}
\end{align}
Substituting Eq. (\ref{GFT}) into Eq. (\ref{GF eq}) we find that the reduced GF $G _{\omega} ( \vec{x} , \vec{x} ^{\, \prime} )$ satisfies the differential equation
\begin{align}
\big[ \, \omega ^{2} + \vec{\nabla} ^{2} - \lambda \, (i u _{0} \omega - \vec{u} \cdot \vec{\nabla} ) ^{2} - m ^{2} \, \big] \, G _{\omega} ( \vec{x} , \vec{x} ^{\, \prime} ) = \delta^{(D)} (\vec{x} - \vec{x} ^{\, \prime} ) . \label{GF eq omega}
\end{align}
Following the same nomenclature used in Eq. (\ref{FAbyG}), we denote the reduced GFs in the interior and exterior of the sphere as $G _{\omega} ^{(\mbox{\scriptsize in})}$ and $G _{\omega} ^{(\mbox{\scriptsize out})}$, respectively. Also, note that the expression for the Casimir pressure (\ref{FAbyG}) requires the GFs in spacetime coordinates, $G _{\mbox{\scriptsize in/out}} (x,x ^{\prime})$, and this can be obtained by inverse Fourier transforming the reduced GFs $G _{\omega} ^{(\mbox{\scriptsize in/out})} ( \vec{x} , \vec{x} ^{\, \prime} ) $, i.e.
\begin{align}
G _{\mbox{\scriptsize in/out}} (x,x ^{\prime}) = \int _{- \infty} ^{\infty} \, \frac{d\omega}{2 \pi} \, e ^{i \,\omega (t-t ^{\prime})} \, G _{\omega} ^{(\mbox{\scriptsize in/out})} ( \vec{x} , \vec{x} ^{\, \prime} ) . \label{GFT-inv}
\end{align}
Therefore, the problem now consists in determining the reduced GFs.
\subsection{Green's function: radial spacelike case}
\label{GFRa}
First, let us consider the radial spacelike case $u ^{\mu} = (0,1,0,\cdots,0)$. To solve Eq. (\ref{GF eq omega}) it is convenient to introduce polar variables $(r,\theta)$, with $\| \vec{x} \| = r$ and $\theta = \angle(\vec{x},\vec{x} ^{\prime})$ \cite{Bender-Milton}. In terms of these variables, Eq. (\ref{GF eq omega}) becomes
\begin{align}
\bigg[\omega ^{2} - m ^{2} + \Lambda_R \frac{\partial ^{2}}{\partial r ^{2}} + \frac{D-1}{r} \frac{\partial}{\partial r} + \frac{\sin ^{2-D} \theta}{r ^{2}} \frac{\partial}{\partial \theta} \left( \sin ^{D-2} \theta \frac{\partial}{\partial \theta} \right) \bigg] G _{\omega} (r,r ^{\prime} , \theta) = \frac{\delta (r-r ^{\prime}) \delta (\theta) \Gamma \left( \frac{D-1}{2} \right)}{2 \pi ^{\frac{D-1}{2}} r ^{D-1} \sin ^{D-2} \theta } , \label{GFLV ur}
\end{align}
where
\begin{align}
\Lambda _{R} \equiv 1 - \lambda > 0 \ ,
\end{align}
is the effective parameter for Lorentz violation.
As in the Lorentz invariant case ($\Lambda_R=1$), Eq. (\ref{GFLV ur}) admits separation of variables. Its solution can be factorized as
\begin{align}
\label{Godef}
G _{\omega} (r,r ^{\prime} , \theta) = g ( r ,\, r ^{\prime} ) \, \Theta ( \theta ) .
\end{align}
Eventually, we arrive to the following equations
\begin{align}
\left[ -\Lambda_R\, \frac{d ^{2}}{dr ^{2}}\, -\, \frac{D-1}{r} \frac{d}{dr} \,+\, \frac{n(n+D-2)}{r ^{2}} \,-\, \omega ^{2}\, +\, m ^{2} \right]\, g(r,r ^{\prime}) = 0 , \quad (r \neq r^{\prime}) \label{radial1a}
\end{align}
and
\begin{align}
\left[ \sin ^{2-D} \theta \frac{\partial}{\partial \theta} \left( \sin ^{D-2} \theta \frac{\partial}{\partial \theta} \right) + n(n+D-2) \right] \Theta(\theta) = 0 , \label{Angular eq}
\end{align}
where $n$ is a non-negative integer parameter and $n(n+D-2)$ plays the role of the separation constant. We observe that the differential equation (\ref{radial1a}) for the radial part in (\ref{Godef}) depends on the coefficient $\Lambda_R$ for Lorentz violation, while the angular part (\ref{Angular eq}) does not depend on it explicitly. The following remarks are in order:
\begin{itemize}
\item Only for $D=1$ with $n=0,1$ the second and third terms in the l.h.s of (\ref{radial1a}) vanish simultaneously. In this case, making the substitutions $\omega\rightarrow \omega\,\sqrt{\Lambda_R}$ and $m\rightarrow m\,\sqrt{\Lambda_R}$ in Eq. (\ref{radial1a}), all $\lambda$-dependence is removed and we arrive to the corresponding Lorentz symmetric equation ($\Lambda_R=1$). This special case will be treated separately later on.
\item In Eq. (\ref{radial1a}), the singular term $\propto r^{-2}$ vanishes at $n=0$ independently of $D$. For any value of $n$ and $D>2$, this term is positive whereas for $n=1$ and $0<D<1$ it becomes negative. This fact, will be relevant in the construction of the corresponding Green function (see below).
\end{itemize}
Now, the solution to Eq. (\ref{radial1a}) that is well-behaved at the origin is given by
\begin{align}
g _{n} (r,r ^{\prime}) = r ^{\alpha} \left[ \,a _{n} \, J _{s _{n}} (\Omega r) + b _{n} \, Y _{s _{n}} (\Omega r) \,\right] , \label{gnylv}
\end{align}
where $J_{s _{n}}$ and $Y _{s _{n}}$ are the Bessel functions of the first and second kind \cite{Abramowitz}, respectively, and
\begin{align}
\alpha & = \frac{\Lambda _{R} + 1 - D}{2 \,\Lambda _{R}} , \quad \Omega = \sqrt{\frac{\omega ^{2} - m ^{2}}{\Lambda _{R}}} , \quad s _{n} = \frac{1}{2 \, \Lambda _{R}} \sqrt{ (\Lambda _{R} - D + 1 ) ^{2} + 4 n \Lambda _{R} (n + D -2) } , \label{Coef1}
\end{align}
where we have assumed that $D>2$. This condition will be justified below.
Next, making the change of variable $z=\cos\theta$ in Eq. (\ref{Angular eq}) we obtain
\begin{align}
\left[\, (1-z ^{2}) \frac{d ^{2}}{dz ^{2}} - z \,(D-1)\, \frac{d}{dz} + n(n+D-2) \,\right] \Theta(z) = 0 ,
\end{align}
which can be identified as the Gegenbauer equation \cite{Abramowitz}. Its regular solutions for $|z|=1$ are the ultraspherical or Gegenbauer polynomials
\begin{align}
\Theta _{n} (z) = C _{n} ^{(-1+D/2)} (z) , \qquad n \in \mathbb{Z} ^{+} \cup\{0\} \ . \label{Sap}
\end{align}
Therefore, the most general solution for (\ref{GFLV ur}) can be expressed as a linear superposition of the separated-variable solutions
\begin{align}
G _{\omega} (r,r ^{\prime} , z) = \sum _{n = 0} ^{\infty} g _{n} (r,r ^{\prime}) \, C _{n} ^{(-1+D/2)} (z) \, . \label{G1omega}
\end{align}
Now, we proceed to determine the GF in the inner and outer regions of the sphere. To this end, we recall that the reduced GF $g _{n} (r,r ^{\prime})$ must satisfy the Dirichlet BC (\ref{Dirichlet}) on the sphere, finiteness at the origin (\ref{Finiteness}), continuity at $r = r ^{\prime}$
\begin{align}
\lim _{\epsilon \rightarrow 0 ^{+}} \, g _{n} (r,r ^{\prime}) \big| ^{r = r ^{\prime} + \epsilon} _{ r = r ^{\prime} - \epsilon} = 0 \, ,
\end{align}
as well as the discontinuity in the first derivative
\begin{align}
\lim _{\epsilon \rightarrow 0 ^{+}} \, \frac{d g _{n} (r,r ^{\prime})}{dr} \bigg| _{r = r ^{\prime} - \epsilon} ^{r = r ^{\prime} + \epsilon} = \frac{(2n + D - 2)}{4 \Lambda _{R} \pi ^{D/2} \, r ^{\prime \, D-1} } \Gamma \left( \frac{D-2}{2} \right) \, ,
\end{align}
which follows from integration of Eq. (\ref{GFLV ur}) in the vicinity of $r ^{\prime}$. Also, its behavior at $r \to \infty$ should decay appropriately. For the interior region of the sphere the general solution of Eq. (\ref{gnylv}) can be written as
\begin{align}
g _{n} ^{\mbox{\scriptsize in}} (r,r ^{\prime}) &= \ \left\lbrace \begin{array}{l} r ^{\alpha} a _{n} \, J _{s _{n}} (\Omega r) , \\[7pt] r ^{\alpha} \left[ b _{n} \, J _{s _{n}} (\Omega r) + c _{n} \, Y _{s _{n}} (\Omega r) \right] , \end{array} \begin{array}{c} r<r ^{\prime} < R \\[7pt] r ^{\prime} < r < R \end{array} \right. \,, \label{gnin}
\end{align}
where $a _{n},b_{n}$ and $c_{n}$ are arbitrary parameters. In Eq. (\ref{gnin}), for $r<r ^{\prime} < R$, we have eliminated the linearly independent solution $r ^{\alpha} \,Y_{s _{n}} (\Omega r)$, which is singular at $r = 0$. This becomes clear when considering the asymptotic form of $Y _{s _{n}} (\Omega r)$ for small argument. In that case, the leading contribution in $r ^{\alpha} \,Y_{s _{n}} (\Omega r)$ is of the form $r ^{\alpha - s _{n}}$, which diverges at the origin provided that $s _{n} > \alpha$. The latter condition yields $D>2$ for all $n$, which is exactly the same condition obtained in the Lorentz symmetric case. This is why we stick to the case $D>2$. The special case $D=1$ will be considered separately. The BCs lead to the system of equations
\begin{align}
b _{n} \, J _{s _{n}} (\Omega R)\ + \ c _{n} \, Y _{s _{n}} (\Omega R) &\ = \ 0 , \notag \\ b _{n} \, J _{s _{n}} (\Omega r ^{\prime})\ + \ c _{n} \, Y _{s _{n}} (\Omega r ^{\prime})\ - \ a _{n} \, J _{s _{n}} (\Omega r ^{\prime}) &\ = \ 0 , \notag \\ b _{n} \, J _{s _{n}} ^{\prime} (\Omega r ^{\prime})\ + \ c _{n} \, Y _{s _{n}} ^{\prime} (\Omega r ^{\prime})\ -\ a _{n} \, J _{s _{n}} ^{\prime} (\Omega r ^{\prime}) &\ = \frac{(2n + D - 2)}{4 \Lambda_R \Omega \pi ^{D/2} r ^{\prime \, \alpha + D-1} } \Gamma \left( \frac{D-2}{2} \right)\ ,
\end{align}
which determine the coefficients $a _{n}$, $b _{n}$ and $c _{n}$. In that way, the reduced GF for the interior region takes the form
\begin{align}
g _{n} ^{\mbox{\scriptsize in}} (r,r ^{\prime}) \ =\ \frac{(2n + D - 2) \Gamma (-1+D/2) }{8 \Lambda_R \pi ^{\frac{D-2}{2}} } \frac{r ^{\alpha}}{ r ^{\prime \, \alpha + D-2}} \frac{J _{s _{n}} (\Omega r _{<})}{J _{s _{n}} (\Omega R)} \left[ J _{s _{n}} (\Omega R) Y _{s _{n}} (\Omega r _{>}) \,- \,J _{s _{n}} (\Omega r _{>}) Y _{s _{n}} (\Omega R) \right] , \label{Green}
\end{align}
where $r _{>}$ ($r _{<}$) is the greater (lesser) between $r$ and $r ^{\prime}$. In Fig. \ref{gin} we plot the reduced GF of Eq. (\ref{Green}) for a massless scalar field in 3 dimensions as a function of the dimensionless radius $r/R$ for $r ^{\prime} / R = 0.4$ and different values of $n$ and $\lambda$.
\begin{figure}[H]
\centering
\includegraphics[scale=0.6]{Fig5.pdf}
\caption{The reduced GF $R \, g _{n} ^{\mbox{\scriptsize in}}$ (\ref{Green}) for a massless scalar field (in a $3-$dimensional sphere) as a function of $r/R$ with the values $r ^{\prime} = 0.4\, R$ and $\omega R = 1$.} \label{gin}
\end{figure}
For the outer region, the general solution of Eq. (\ref{gnylv}) can be expressed in terms of the Hankel functions of the first and second kind:
\begin{align}
g _{n} ^{\mbox{\scriptsize out}} (r,r ^{\prime}) = \left\lbrace \begin{array}{l} r ^{\alpha} \left[ d _{n} \, H _{s _{n}} ^{(1)} (\Omega r) + e _{n} \, H _{s _{n}} ^{(2)} (\Omega r) \right] , \\[7pt] r ^{\alpha} f _{n} \, H _{s _{n}} ^{(1)} (\Omega r) , \end{array} \begin{array}{c} R < r<r ^{\prime} \\[7pt] R< r ^{\prime} < r \end{array} \right. \, . \label{gnout}
\end{align}
The single Hankel function in the region $R<r ^{\prime} < r$ is required by the BC that $g _{n} ^{\mbox{\scriptsize out}}$ remains finite for $r \to \infty$ and has the asymptotic form of an outgoing plane wave. Again, the BCs lead to a set of equations for the coefficients $d _{n}$, $e _{n}$ and $f _{n}$, namely
\begin{align}
d _{n} \, H _{s _{n}} ^{(1)} (\Omega R) + e _{n} \, H _{s _{n}} ^{(2)} (\Omega R) &= 0 , \notag \\ f _{n} \, H _{s _{n}} ^{(1)} (\Omega r ^{\prime}) - d _{n} \, H _{s _{n}} ^{(1)} (\Omega r ^{\prime}) - e _{n} \, H _{s _{n}} ^{(2)} (\Omega r ^{\prime}) &= 0 , \notag \\ f _{n} \, H _{s _{n}} ^{(1) \, \prime} (\Omega r ^{\prime}) - d _{n} \, H _{s _{n}} ^{(1) \, \prime} (\Omega r ^{\prime}) - e _{n} \, H _{s _{n}} ^{(2) \, \prime} (\Omega r ^{\prime}) & = \frac{(2n + D - 2) \, \Gamma (-1 + D/2)}{4 \Omega \Lambda_R \pi ^{D/2} r ^{\prime \, \alpha + D-1} } .
\end{align}
By solving the above system of equations, we arrive to the solution
\begin{align}
g _{n} ^{\mbox{\scriptsize out}} (r,r ^{\prime}) (r,r ^{\prime}) = \frac{(2n + D - 2) \, \Gamma (-1 + D/2)}{16 i \Lambda_R \pi ^{-1+ D/2} } \frac{r ^{\alpha}}{r ^{\prime \, \alpha + D - 2} } \frac{H _{s _{n}} ^{(1)} (\Omega r _{>})}{H _{s _{n}} ^{(1)} (\Omega R)} \left[ H _{s _{n}} ^{(1)} (\Omega R) H _{s _{n}} ^{(2)} (\Omega r _{<}) - H _{s _{n}} ^{(1)} (\Omega r _{<}) H _{s _{n}} ^{(2)} (\Omega R) \right]\ , \label{Green2}
\end{align}
where, as before, $r _{>}$ ($r _{<}$) is the greater (lesser) between $r$ and $r ^{\prime}$. In Fig. \ref{gout} we plot the real part of the reduced GF of Eq. (\ref{Green2}) for a massless scalar field in 3 dimensions as a function of the dimensionless radius $r/R$ for $r ^{\prime} / R = 1.4$ and different values of $n$ and $\lambda$. With the above GFs, we are in position to evaluate the Casimir force (\ref{FAbyG}) for the radial spacelike case.
\begin{figure}[H]
\centering
\includegraphics[scale=0.6]{Fig6.pdf}
\caption{The real part of the reduced GF $ R\, g _{n} ^{\mbox{\scriptsize out}}$ (\ref{Green2}) for massless scalar field (in a $3-$dimensional sphere) as a function of $r/R$ with the values $r ^{\prime} = 1.4 R$ and $\omega R = 1$.}
\label{gout}
\end{figure}
\subsection{Green's function: timelike case} \label{GFT1a2}
Our objective now is to derive the GF appearing in Eq. (\ref{GF eq omega}) for the timelike case $u ^{\mu} = (1,0,0,\cdots,0)$. In the polar variables ($r,\theta$) the function $G_{\omega} (r,r ^{\prime} , \theta)$ (\ref{GFT}) satisfies
\begin{align}
\bigg[ \Lambda _{T} \, \omega ^{2} - m ^{2} + \frac{\partial ^{2}}{\partial r ^{2}} + \frac{D-1}{r} \frac{\partial}{\partial r} + \frac{\sin ^{2-D} \theta}{r ^{2}} \frac{\partial}{\partial \theta} \left( \sin ^{D-2} \theta \frac{\partial}{\partial \theta} \right) \bigg] G _{\omega} (r,r ^{\prime} , \theta) = \frac{\delta (r-r ^{\prime}) \delta (\theta) \Gamma \left( \frac{D-1}{2} \right)}{2 \pi ^{\frac{D-1}{2}} r ^{D-1} \sin ^{D-2} \theta } , \label{GFLVUT}
\end{align}
where $\Lambda _{T} = 1 + \lambda$. We observe that, unlike the radial spacelike case where Lorentz violation enters into the GF equation (\ref{GFLV ur}) in a nontrivial fashion, in the timelike case the LV-dependence can be absorbed into the frequency through the redefinition $\sqrt{\Lambda _{T}} \, \omega \rightarrow \omega$, thus leaving us with the Lorentz-symmetric GF equation. Therefore, we safely take for granted the reduced GFs for the timelike case. Since the spherical symmetry is untouched in both cases, the angular part is still given by Eq. (\ref{Sap}). However, for the inner region the reduced GF $\tilde g _{n} (r,r ^{\prime})$ is
\begin{align}
\tilde{g} _{n} ^{\mbox{\scriptsize in}} (r,r ^{\prime}) = \frac{(2n + D - 2) \Gamma (-1+D/2) }{8 \pi ^{\frac{D-2}{2}} } \frac{1}{ (r\, r ^{\prime} )^{ \, -1+\frac{D}{2}} } \frac{J _{\nu} (\Omega ^{\prime} r _{<})}{J _{\nu} (\Omega ^{\prime} R)} \left[ J _{\nu} (\Omega ^{\prime} R) Y _{\nu} (\Omega ^{\prime} r _{>}) - J _{\nu} (\Omega ^{\prime} r _{>}) Y _{\nu} (\Omega ^{\prime} R) \right] ,
\label{GreenIn}
\end{align}
while for the outer region we obtain the solution
\begin{align}
\tilde{g} _{n} ^{\mbox{\scriptsize out}} (r,r ^{\prime}) = \frac{(2n + D - 2) \, \Gamma (-1 + D/2)}{16 i \pi ^{-1+ D/2} } \frac{1}{ (r \, r ^{\prime} )^{ \, -1+\frac{D}{2}} } \frac{H _{\nu} ^{(1)} (\Omega ^{\prime} r _{>})}{H _{\nu} ^{(1)} (\Omega ^{\prime} R)} \left[ H _{\nu} ^{(1)} (\Omega ^{\prime} R) H _{\nu} ^{(2)} (\Omega ^{\prime} r _{<}) - H _{\nu} ^{(1)} (\Omega ^{\prime} r _{<}) H _{\nu} ^{(2)} (\Omega ^{\prime} R) \right] , \label{Green2In}
\end{align}
where $\nu = n-1+D/2$, $\Omega ^{\prime} = \sqrt{\Lambda_{T} \, \omega ^{2} - m ^{2}}$, and $r _{>}$ ($r _{<}$) is the greater (lesser) between $r$ and $r ^{\prime}$. As we shall see later, the fact that in the present case the parameter $\Lambda _{T}$ for Lorentz violation enters in a simple manner into the GF will be reflected in the expression for the Casimir pressure as a global factor.
\section{Casimir effect}\label{CEsection}
In this section we present for the cases I and II the formal expression for the Casimir force (\ref{FAbyG}), with $D>2$, explicitly. Substituting the Fourier representation of the GF (\ref{GFT-inv}) into Eq. (\ref{FAbyG}) we obtain an expression for the Casimir pressure in terms of the frequency-dependent GF:
\begin{align}
\frac{F}{A} = - \frac{i}{2} \, \Lambda \int _{- \infty} ^{\infty} \, \frac{d \omega}{2 \pi} \, \lim _{\vec{x} ^{\, \prime} \rightarrow \vec{x}} \bigg[ \frac{\partial }{\partial r}\frac{\partial}{\partial r ^{\prime}} G _{\omega}^{(\mbox{\scriptsize in})}( \vec{x} , \vec{x} ^{\, \prime} ) - \frac{\partial }{\partial r} \frac{\partial}{\partial r ^{\prime}} G _{\omega}^{(\mbox{\scriptsize out})}( \vec{x} , \vec{x} ^{\, \prime} ) \bigg] \bigg\vert _{\| \vec{x}\|\,=\,R} .
\end{align}
This can be further simplified by using the expansion (\ref{G1omega}). Upon substitution we find
\begin{align}
\frac{F}{A} = - \frac{i}{4 \pi} \, \Lambda \sum _{n = 0} ^{\infty} \frac{\Gamma (n +D - 2 )}{n! \, \Gamma ( D - 2 )} \int _{- \infty} ^{\infty} \, d\omega \, \lim _{r ^{\prime} \rightarrow r} \bigg[ \frac{\partial }{\partial r}\frac{\partial}{\partial r ^{\prime}} g _{n} ^{\mbox{\scriptsize in}} (r,r ^{\prime}) - \frac{\partial }{\partial r} \frac{\partial}{\partial r ^{\prime}} g _{n} ^{\mbox{\scriptsize out}} (r,r ^{\prime}) \bigg] \bigg\vert _{r = R} , \label{CasPressureGeneral}
\end{align}
where we have used the value of the ultraspherical polynomials at $z = 1$ (which corresponds to $\theta = 0$),
\begin{align}
C _{n} ^{( \alpha )} (1) = \frac{\Gamma (n+ 2 \alpha )}{n! \, \Gamma (2 \alpha )} .
\end{align}
Having determined the above general expression for the Casimir pressure in terms of the reduced GF, we can now evaluate it explicitly for each case.
\subsection{Case I: radial spacelike case}
From the reduced GFs $g ^{\mbox{\scriptsize in/out}} (r,r ^{\prime})$, given by Eqs. (\ref{Green}) and (\ref{Green2}), one can directly compute the limit appearing in the integrand of Eq. (\ref{CasPressureGeneral}). For the interior region, without loss of generality we take $r _{<} = r ^{\prime}$ and $r _{>} = r$ in Eq. (\ref{Green}), wherefrom we obtain
\begin{align}
\lim _{r ^{\prime} \rightarrow r} \frac{\partial }{\partial r}\frac{\partial}{\partial r ^{\prime}} g _{n} ^{\mbox{\scriptsize in}} (r,r ^{\prime}) \bigg\vert _{r = R} = \frac{(2n + D - 2) \Gamma (-1+D/2) }{4 \Lambda_R \pi ^{D/2 } R ^{D}} \left[ \Omega R \frac{J _{s _{n}} ^{\prime} (\Omega R)}{J _{s _{n}} (\Omega R)} - (\alpha + D-2) \right] , \label{partialrr}
\end{align}
where we have used that the Wronskian of $J _{\nu} (z)$ and $Y _{\nu} (z)$ has the value $W [ J _{\nu} (z) , Y _{\nu} (z) ] = 2/ ( \pi z )$, and the prime in the Bessel function denotes derivative with respect to its argument. The converse choice between $r _{>}$ and $r _{<}$ produces, at the end of the calculations, the same final result for the Casimir force. A similar procedure for the exterior region (for which we take $r _{>} = r ^{\prime}$ and $r _{<} = r$) yields
\begin{align}
\lim _{r ^{\prime} \rightarrow r} \frac{\partial }{\partial r}\frac{\partial}{\partial r ^{\prime}} g _{n} ^{\mbox{\scriptsize out}} (r,r ^{\prime}) \bigg\vert _{r = R} = - \frac{(2n + D - 2) \, \Gamma (-1 + D/2)}{4 \Lambda_R \pi ^{ D/2} R ^{D} } \left[ \Omega R \frac{H _{s _{n}} ^{(1) \, \prime} (\Omega R)}{H _{s _{n}} ^{(1)} ( \Omega R)} - (\alpha + D - 2) \right] ,
\end{align}
where we have employed that the Wronskian of $H _{\nu} ^{(1)} (z)$ and $H _{\nu} ^{(2)} (z)$ is $W [ H _{\nu} ^{(1)} (z) , H _{\nu} ^{(2)} (z) ] = 4 / (i \pi z)$. Finally, inserting these results into Eq. (\ref{CasPressureGeneral}) we obtain the following expression for the Casimir force
\begin{align}
\frac{F_r}{A} = - \sum _{n = 0} ^{\infty} i \,\frac{(n - 1+ D/ 2) \,\Gamma (n+D-2)}{ (2R) ^{D} \pi ^{\frac{D+1}{2}} n ! \, \Gamma \left( \frac{D-1}{2} \right)} \int _{- \infty} ^{\infty} d \omega \left[ \Omega R \frac{J _{s _{n}} ^{\prime} (\Omega R)}{J _{s _{n}} (\Omega R)} + \Omega R \frac{H _{s _{n}} ^{(1) \, \prime} (\Omega R)}{H _{s _{n}} ^{(1)} ( \Omega R)} - 2 (\alpha + D - 2) \right] .
\end{align}
where we have used the duplication formula $\Gamma (2 \alpha ) = 2 ^{2 \alpha -1} \Gamma (\alpha ) \Gamma (\alpha + 1/2) / \sqrt{\pi}$. For the massless case $m = 0$ ($\Omega = \omega / \sqrt{\Lambda _{R}}$) the rotation of $\pi /2$ in the complex-$\omega$ plane, namely $x = i \, \omega\, R / \sqrt{\Lambda _{R}}$, transforms the above expression into
\begin{align}
\frac{F_r}{A} = - \sqrt{ \Lambda_R} \sum _{n = 0} ^{\infty} \frac{(n - 1+ D/ 2) \Gamma (n+D-2)}{2 ^{D-1} R ^{D+1} \pi ^{\frac{D+1}{2}} n ! \, \Gamma \left( \frac{D-1}{2} \right)} \int _{0} ^{\infty} d x \left[ x \frac{I _{s _{n}} ^{\prime} (x)}{I _{s _{n}} (x)} + x \frac{K _{s _{n}} ^{\prime} (x)}{K _{s _{n}} ( x)} - 2 (\alpha + D - 2) \right] . \label{FLVR}
\end{align}
In the limit $\lambda \to 0$ we obtain $s _{n} \to n-1+\frac{D}{2}$ and $\alpha\rightarrow 1-D/2$, which corresponds to the Lorentz-invariant result provided that $D>2$ \cite{Bender-Milton}. Hereafter, we will restrict ourselves to the massless case $m=0$.
\subsubsection*{Special case $D=1$ }
Formally, Eq. (\ref{FLVR}) was derived under the assumption $D>2$. However, in the particular case $D=1$ the expression (\ref{FLVR}) is well defined and can be evaluated analytically. At $D=1$, the series appearing in Eq. (\ref{FLVR}) truncates after two terms. This happens because of the identity
\begin{align}
\lim _{D \rightarrow 1} \frac{\Gamma(n+D-2)}{\Gamma(\frac{D-1}{2})}\ = \ -\frac{1}{2}\delta_{n0}\ + \ \frac{1}{2}\delta_{n1} \ . \label{identi1}
\end{align}
Hence, only the terms with $n=0$ and $n=1$ will give contribution. As mentioned in Sec. \ref{GFRa}, this implies that the Eq. (\ref{radial1a}), up to a redefinition of the frequency $\omega\rightarrow \omega\,\sqrt{\Lambda_R}$ (equivalently, $x \rightarrow x \,\sqrt{\Lambda_R}$), can be cast in the form of the Lorentz invariant case for which $\alpha\to1/2$ and the Bessel function order is $\nu \to n-1/2$, see Ref. \cite{Bender-Milton}. Therefore, the Casimir force in the presence of Lorentz violation is given by
\begin{align}
\frac{F_{r} ^{(D=1)}}{A} &= \sqrt{\Lambda_R}\,\times \,\frac{-1}{4 \pi R ^{2}} \int _{0} ^{ \infty} d y \left[ y \frac{I _{-\frac{1}{2}} ^{\prime} (y)}{I _{-\frac{1}{2}} (y)}+y \frac{I _{\frac{1}{2}} ^{\prime} (y)}{I _{\frac{1}{2}} (y)} + y \frac{K _{-\frac{1}{2}} ^{\prime} (y)}{K _{-\frac{1}{2}} (y)}+ y \frac{K _{\frac{1}{2}} ^{\prime} (y)}{K _{\frac{1}{2}} (y)} +2 \right]\, . \label{F2R}
\end{align}
where the second factor is just the Lorentz invariant result derived in \cite{Bender-Milton}. We emphasize that this situation occurs only for $D=1$. Now, using the explicit forms of the Bessel functions required by Eq. (\ref{F2R}) and evaluating the integral over $y$ we get
\begin{align}
\frac{F _{r} ^{(D=1)}}{A} &= - \sqrt{\Lambda_R}\,\frac{\pi}{96\, R^2} . \label{FR1a}
\end{align}
Thus, in the case $D=1$, the Casimir stress (\ref{FR1a}) can be greater or smaller than the Lorentz-symmetric case depending on the sign of $\lambda$. In particular, recalling that $\vert \lambda \vert < 1$, $\lambda < 0$ tends to increase the force, while $\lambda > 0$ produces the opposite effect.
\subsection{Case II: timelike case}
In this case we have to evaluate Eq. (\ref{CasPressureGeneral}) with the reduced GFs $\tilde{g} ^{\mbox{\scriptsize in/out}} (r,r ^{\prime})$ of Eqs. (\ref{GreenIn}) and (\ref{Green2In}). As discussed in Sec. \ref{GFT1a2}, the timelike case behaves exactly as the Lorentz symmetric case with a simple redefinition of the frequency. Taking similar steps as in the previous section, after some algebra we find that for the timelike case the Casimir force takes the form
\begin{align}
\frac{F_{t}}{A} = - \frac{1}{\sqrt{ \Lambda} _{T}} \sum _{n = 0} ^{\infty} \frac{(n - 1+ D/ 2) \Gamma (n+D-2)}{2 ^{D-1} R ^{D+1} \pi ^{\frac{D+1}{2}} n ! \, \Gamma \left( \frac{D-1}{2} \right)} \int _{0} ^{\infty} d x \left[ x \frac{I _{\nu} ^{\prime} (x)}{I _{\nu} (x)} + x \frac{K _{\nu} ^{\prime} (x)}{K _{\nu} ( x)} + 2 -D \right] , \label{FLVT}
\end{align}
where $\nu = n - 1 + D/2$. Clearly, this expression is proportional to that obtained in the absence of Lorentz violation ($\lambda=0, \Lambda_T=1$), i.e.
\begin{align}
\frac{F _{t} (\Lambda _{T})}{A} \ = \ \frac{1}{\sqrt{ \Lambda} _{T}} \, \frac{F _{t} (\Lambda_{T} = 1)}{A} . \label{FTR1}
\end{align}
Therefore, if $\lambda < 0$ the Casimir stress (\ref{FTR1}) is enlarged as compared with that of the Lorentz-symmetric case whilst it is diminished when $\lambda > 0$. In particular, for the one-dimensional case $D=1$ the Eq. (\ref{FTR1}) reduces to
\begin{align}
\frac{F_ {t} ^{(D=1)}}{A} & \ = \ - \frac{1}{\sqrt{\Lambda _{T}}}\,\frac{\pi}{96\, R ^{2}}\ ,
\end{align}
c.f. (\ref{FR1a}). We emphasize that, unlike the radial spacelike case, the factorization in (\ref{FTR1}) holds for an arbitrary dimension $D$. The LV contributions appear in the global multiplicative factor ${ \Lambda} _{T}^{-1/2}$ only.
\section{Towards numerical Evaluation of the Casimir force}\label{Numerical Section}
The infinite series in Eqs. (\ref{FLVR}) and (\ref{FLVT}) are divergent. They do not even exist for some values of $D$ where the poles of the Gamma function take place. Partly, this behavior is a manifestation of the nonzero vacuum energy inherent to the CE. Therefore, in order to remove non-physical divergences a method of regularization is required. In this Section, we adapt to the problem at hand the regularization method used in Ref. \cite{Bender-Milton}.
\subsection{Radial spacelike case}
In this section we manipulate the form of the expression (\ref{FLVR}) so that each integral in the series exists. Afterwards, we can evaluate the resulting series numerically. To render the integrals finite we should remove the contact terms that arise from the local behavior near the boundaries. This can be done by replacing the constant $- 2(\alpha + D - 2)$ in Eq. (\ref{FLVR}) by $1$, leaving unchanged the value of the integral (\ref{FLVR}), and yet leads to well-defined (finite) integrals. For the formal proof see Refs. \cite{Bender-Milton,Milton's book,Milton 3} . Eventually, after integration by parts, the series (\ref{FLVR}) can be rewritten in the form
\begin{align}
\frac{F_{r}}{A} = \sqrt{ \Lambda_R} \, \sum _{n = 0} ^{\infty} \frac{(n - 1+ D/ 2) \Gamma (n+D-2)}{2 ^{D-1} R ^{D+1} \pi ^{\frac{D+1}{2}} n ! \, \Gamma \left( \frac{D-1}{2} \right)} \int _{0} ^{\infty} d x \,\textrm{ln}\left[ 2x I_{s_n}(x)K_{s_n}(x) \right] . \label{Rd2}
\end{align}
Now, at a noneven integer $D>2$ each term of the above series exists for any value of $n$. However, the above expression is still not useful since the series does not converge. In order to obtain a convergent reformulation, we analyze the asymptotic behavior of the integrals in Eq. (\ref{Rd2}) for large $n$. This will help us to better understand the source of divergences. From the uniform asymptotic expansions of the modified Bessel functions for large \mbox{$\mu$ \cite{Abramowitz}},
\begin{align}
I _{\mu} (\mu z) \sim \frac{e^{\mu \eta}}{\sqrt{2\pi \mu}(1+z^2)^{\frac{1}{4}}}\sum_{k=0} ^{\infty} \frac{U _{k} (p)}{ \mu ^{k}} , \quad\quad\quad
K _{\mu} (\mu z) \sim \bigg( \frac{\pi}{2\mu} \bigg) ^{\frac{1}{2}} \frac{e ^{- \mu \eta}}{(1+z^2)^{\frac{1}{4}}} \sum _{k=0} ^{\infty} (-1) ^{k} \frac{U _{k} (p)}{\mu ^{k}} ,
\label{expansiones}
\end{align}
where
\begin{align}
\eta = (1+z^2) ^{\frac{1}{2}} + \textrm{ln} \frac{z}{1+\sqrt{1+z^2}} \quad , \qquad \quad p = (1+z^2)^{-\frac{1}{2}}
\end{align}
and
\begin{align}
U _{0} (p) &= 1 , \notag \\ U _{1} (p) &= \frac{1}{24}(3 p - 5 p ^{3}) , \notag \\ U _{2} (p) &= \frac{1}{1152}(81 p ^{2} - 462 p ^{4} + 385 p ^{6}) , \notag \\ U _{3} (p) &= \frac{1}{414720}(30375 p ^{3} - 369603 p ^{5} + 765765 p ^{7} - 425425 p ^{9}),
\end{align}
we obtain the asymptotic expansion of integrals
\begin{align}
Q _{n} & \equiv - \int _{0} ^{\infty} dx \, \textrm{ln}[2 x I _{s _{n}}(x) K _{s _{n}}(x)] = - s _{n} \int _{0} ^{\infty} dy \, \textrm{ln}[2 s _{n} y I _{s _{n}}(s _{n} y) K _{s _{n}}( s _{n} y)] \notag \\ & \phantom{=} \sim \frac{\nu\pi}{2\sqrt{\Lambda_R}} + \frac{\pi \sqrt{\Lambda _{R}}}{128 \nu} -\frac{35 \pi (\sqrt{\Lambda _{R}}) ^{3}}{32768 \nu ^{3}} + \frac{565 \pi( \sqrt{\Lambda _{R}}) ^{5}}{1048576 \nu ^{5}} ,
\end{align}
for $n \rightarrow \infty$, where $s _{n} \sim \frac{\nu}{ \sqrt{\Lambda _{R}}}$ with $\nu=n-1+\frac{D}{2}$. The first term will give rise to divergences in Eq. (\ref{FLVR}), except for the special case $D=1$, where the series truncates. To solve this problem, let us introduce an analytic summation procedure based on the properties of the Riemann $\zeta$ function \cite{Bender-Milton}. The leading $n$-large behavior of the summand in Eq. (\ref{Rd2}) is given by
\begin{align}
- \frac{(n-1+\frac{D}{2})\Gamma(n+D-2)}{2^{D-1}\pi^{\frac{D+1}{2}}R^{D+1}n!\Gamma(\frac{D-1}{2})} Q _{n} \sim \frac{1}{2^{D} \pi ^{\frac{D+1}{2}} R ^{D+1}\Gamma(\frac{D-1}{2})} (c _{1} n ^{D-1} + c _{2} n ^{D-2} + \cdots + c _{k} n ^{D-k} + \cdots) , \quad (n \rightarrow \infty) , \label{Exp1}
\end{align}
where the coefficients $c _{k}$ depend on the dimension $D$ and the LV parameter $\Lambda _{R}$, i.e. $c _{k} = c _{k} (\Lambda _{R} , D)$. The first lowest coefficients are presented in Appendix \ref{AP1}. Since
\begin{align}
\sum _{n=1} ^{\infty} c _{k} n ^{D-k} = c _{k} \zeta(k-D) , \label{zeta1}
\end{align}
we can add to Eq. (\ref{Rd2}) $K$ terms of the form (\ref{zeta1}) and correspondingly subtract the right-hand side of Eq. (\ref{Exp1}) with the same number of terms. Thereby, the series (\ref{Rd2}) takes the form
\begin{align}
\frac{F _{r}}{A} = \sqrt{\Lambda _{R}} \bigg[ \frac{\Gamma(\frac{D}{2})}{4R^{D+1} \pi ^{1+\frac{D}{2}}} Q _{0} + \sum _{n=1} ^{\infty} \bigg( \frac{(n-1+\frac{D}{2})\Gamma(n+D-2)}{2 ^{D-1} \pi ^{\frac{D+1}{2}} R ^{D+1} n! \Gamma(\frac{D-1}{2})} Q _{n} + \frac{1}{2 ^{D} \pi ^{\frac{D-1}{2}} R ^{D+1}\Gamma(\frac{D-1}{2})} \sum _{k=1} ^{K} c _{k} n ^{D-k} \bigg) \nonumber \\
- \frac{1}{2 ^{D} \pi ^{\frac{D-1}{2}} R ^{D+1}} \sum _{k=1} ^{K} c _{k} \zeta(k-D)\bigg] , \label{RFLV}
\end{align}
and quickly converges since the $n$th term in the series tends to zero as $n ^{D-K-1}$, provided that $D<K$. Note that the Casimir stress is singular at all even positive integer values of $D$, where Eq. (\ref{RFLV}) has simple poles at $D = 2N$, with $N = 1, 2, 3,\ldots $ .
\subsection{Timelike case}
For the timelike case we simply present the final result for the Casimir force given by
\begin{align}
\frac{F_{t}}{A} = \frac{1}{\sqrt{\Lambda_T}} \bigg[ \frac{\Gamma(\frac{D}{2})}{4\,R^{D+1} \pi ^{1+\frac{D}{2}}} Q _{0} + \sum_{n=1} ^{\infty} \bigg( \frac{(n-1+\frac{D}{2})\Gamma(n+D-2)}{2^{D-1} \pi ^{\frac{D+1}{2}} R ^{D+1} n! \Gamma(\frac{D-1}{2})} Q _{n} + \frac{1}{2 ^{D} \pi ^{\frac{D-1}{2}}R^{D+1}\Gamma(\frac{D-1}{2})} \sum_{k=1} ^{K} b _{k} n ^{D-k} \bigg) \nonumber \\
- \frac{1}{2^D \pi^{\frac{D-1}{2}}R^{D+1}} \sum_{k=1}^K b_k \zeta(k-D)\bigg] , \label{Ftr}
\end{align}
where the coefficients $b _{k}$ are those given in the Appendix \ref{AP1} evaluated at $\Lambda _{R} = 1$, i.e. $b _{k} = c _{k} \vert _{\Lambda _{R} = 1}$.
\section{Numerical results}\label{NumResultsSec}
We have evaluated numerically the expressions (\ref{RFLV}) and (\ref{Ftr}) for the radial spacelike and timelike cases, respectively. In particular, for $D=3$ and $\Lambda _{R} = 1$ (i.e. without LV) the Casimir stress we obtain is
\begin{align}
F _{r} ^{(D=3)} = \frac{0.0028172}{R ^{2}} , \label{Fr3dn}
\end{align}
which, up to the numerical precision used in the present study, agrees with the value $0.0028168 / R ^{2}$ reported by Bender and Milton in Ref. \cite{Bender-Milton}. We observe that, unlike the attractive Casimir stress between parallel plates, in this case the force is repulsive, i.e. it tends to inflate the sphere.
To visualize the LV contribution in more detail, in Fig. \ref{ForceDR} we plot the Casimir pressure $F_{r} / A$ as a function of the dimension $D$ for different values of the LV parameter $\lambda$. There we observe that the force can be repulsive or attractive, depending on the sign and strength of the parameter $\lambda$. Remarkably, we observe that for any $D>2$ there exists a \emph{critical value} $\lambda_c\neq0$ at which the Casimir force (\ref{RFLV}) vanishes, i.e.
\begin{align}
F _{r} (\lambda = \lambda _{c} ) = 0 . \label{ZeroCond}
\end{align}
The force $F_r$ is positive when $\lambda < \lambda _{c}$, and flips its sign for $\lambda > \lambda _{c}$ thus tending to implode the sphere. Such a transition does not occur in the Lorentz-symmetric case \cite{Bender-Milton} as well as in the LV scalar Casimir stress for parallel plates \cite{Escobar-Medel-Martin, PLB}.
\begin{figure*}
\centering
\includegraphics[scale=0.45]{Fig1.pdf}
\caption{Plot of the Casimir pressure $F_{r}/A$ for a $D$-dimensional spherical shell, with $2<D<5$.}
\label{ForceDR}
\end{figure*}
The most relevant physical case corresponds to $D=3$. In Fig. \ref{ForceD3R} we plot the Casimir pressure for the specific cases considered in this work as a function of $R$ and different values of $\lambda$. In the timelike case, presented in the right panel of Fig. \ref{ForceD3R}, the Casimir pressure in the presence of Lorentz violation behaves exactly as the usual attractive Lorentz-symmetric Casimir pressure. As Eq. (\ref{Ftr}) shows, the only difference is that the strength of $F _{t} /A$ can be either larger or smaller than $F _{t} (\Lambda _{t} = 1) /A$, depending on the sign of $\lambda$: a positive $\lambda$ yields to a smaller pressure, while a negative $\lambda$ renders to a larger pressure. On the other hand, as shown in the left panel of Fig. \ref{ForceD3R}, the radial spacelike case exhibits a more interesting behavior. In this case, the solution of Eq. (\ref{ZeroCond}) produces the critical value
\begin{align}
\lambda _{c} \approx 0.0025 . \label{CritLambda}
\end{align}
As we can see in the plot, the Casimir pressure changes sign when $\lambda$ transits across the critical value $\lambda _{c}$. From a high-energy physics perspective, the critical value of Eq. (\ref{CritLambda}) would not be admissible, since it is expected to be much smaller than one. However, Lorentz-violating effective field theories also emerge in condensed matter systems, where the symmetry breaking parameters are not necessarily small. For instance, topological insulators, which are bulk insulators with Dirac fermions on the surface \cite{TIs}, have enabled the theoretical possibility of realizing axion electrodynamics \cite{Wilczek} in a condensed matter system. Also, a Lorentz-violating extension of quantum electrodynamics can be realized with a novel class of materials known as Weyl semi-metals, which are systems which host low-energy quasiparticles that are described by the Weyl equations \cite{Grushin}. Undoubtedly, these kind of low-energy systems, where high-energy phenomena take place, represent a promising arena in which our predictions could be tested. An outstanding example of this is the chiral magnetic effect, which is an electric current along an externally applied magnetic field due to chirality imbalance. This effect was first predicted to occur in quark-gluon plasma, but it was experimentally observed in the Dirac semimetal ZrTe$_{5}$ \cite{CME}.
Reversing the Casimir force have been a topic of interest since its inception. To revert the sign, one must usually search non symmetric situations or vacuum mediated proposals. The first Casimir repulsion proposal, known as Dzyaloshinskii repulsion \cite{Dzyaloshinskii}, involves the presence of a dielectric fluid filling the space between two dielectrics in a parallel configuration. Recently, it was proposed that switching between repulsive and attractive Casimir forces could be realized with two topological insulator plates \cite{Cortijo, MCU}, as well as between two Weyl semimetallic plates \cite{Wilson}. In the context of the Lorentz-violating scalar field theory described by the Lagrangian (\ref{Lagrangian1}), it was recently shown that the Casimir force retains its attractive character in the parallel plate configuration \cite{Escobar-Medel-Martin, PLB}. However, as our results support, the Lorentz-violating parameter $\lambda$ allows us to tune between the standard repulsive to attractive Casimir force acting on a $D$-dimensional sphere, with $D>2$, when $u ^{\mu}$ points along the radial direction.
\begin{figure*}
\centering
\includegraphics[scale=0.45]{Fig2.pdf}
\includegraphics[scale=0.43]{Fig4.pdf}
\caption{Plots of the Casimir pressure $F_{r}/A$ (left) and $F_{t}/A$ (right) for a massless three-dimensional scalar field as a function of the radius $R$.}
\label{ForceD3R}
\end{figure*}
\section{Conclusions} \label{ConcluSection}
We have analyzed the effects of Lorentz symmetry violation in the scalar Casimir self-stress on a $D$-dimensional spherical shell with $D>2$. By considering a flat background spacetime, we have chosen two possible scenarios of violation of Lorentz symmetry through a fixed timelike $u ^{\mu} = (1,0,...,0)$ and a fixed spacelike $u ^{\mu} = (0,1,0,...,0)$ vectors. For each case the Casimir stress was obtained by using Green's function techniques.
In the timelike case, for any $D$ we found that Lorentz violation manifests in the Casimir force through a global rescaling factor only, i.e. $F _{t} (\Lambda _{T}) = F _{t} (\Lambda _{T} = 1) / \sqrt{\Lambda _{T}}$, where $\Lambda _{T} = 1 + \lambda$. This is because the Lorentz breaking term in the equation of motion (\ref{EQLV}) can be absorbed by redefining the frequency as $\omega \to \sqrt{\Lambda _{T}} \, \omega$, thus going back to the standard Klein-Gordon equation.
The radial spacelike case is quite more interesting. The corresponding GF depends on the parameter $\Lambda _{R} = 1 - \lambda$ in a nontrivial fashion. The expression for the Casimir pressure is not proportional to the Lorentz-symmetric case. The main difference with the timelike case lies on the $\lambda$-dependence in the order of the Bessel functions appearing in Eq. (\ref{RFLV}), thus making this case counter intuitive. The special case $D=1$ admits an analytical solution and it was discussed in detail. For dimensions $2\leq D\leq5$, we numerically evaluated the Casimir stress. An interesting aspect is that there exists a critical value $\lambda _{c}$ for which the Casimir stress vanishes. Moreover, for $\lambda < \lambda _{c}$ the Casimir stress is positive (tends to expand the sphere) while for $\lambda > \lambda _{c}$ the force flips its sign (tending to implode the sphere).
It is worth mentioning that the Casimir effect for a sphere dates back to H. B. Casimir himself, who proposed that vacuum fluctuations might cause a conducting spherical shell to attract itself, in a way analogous to the case of two conducting plates. This conclusion was incorrect, as Boyer showed in 1968, since the Casimir stress in a perfectly conducting spherical shell is repulsive \cite{Boyer}. In this paper we showed that Lorentz violation, as described by the scalar field theory defined by the Lagrangian density (\ref{Lagrangian1}), allows us to tune the sign of the Casimir stress for a sphere in $D$-dimensions for $D>2$. In the three-dimensional case $D=3$, the critical value is $\lambda _{c} \approx 0.0025$. From a high-energy physics perspective, $\lambda _{c}$ is far from the allowed values for this parameter, since Lorentz symmetry violation is expected to be small. However, there are scenarios in which fundamental symmetries are naturally broken, e.g. condensed matter systems. Indeed, as shown in Ref. \cite{Grushin}, a condensed matter realization of a Lorentz-violating quantum electrodynamics (as described by the Standard-Model Extension) is the Weyl semimetal phase. This bridge between high-energy and condensed matter offers an opportunity to test phenomena initially predicted to occur at high-energies in solid state systems.
The present work can be extended in relevant ways. For example, the first task would be to incorporate the LV angular spacelike case, where in three-dimensions would correspond to add LV contributions to the $\hat{\theta}$- and $\hat{\phi}$-directions \footnote{Work in progress. These cases represent a different problem from the calculation and the physical point of view.}. Besides, since any realistic setup is necessarily immersed in a bath with a nonzero temperature, it would be interesting to determine the effects that thermal fluctuations would have in the Casimir stress. Studies in different geometries, as the cylindrical one, would also be of great value. We leave these problems for future works.
\acknowledgements
A. M.-R. acknowledges support from DGAPA-UNAM Project No. IA101320. C. A. E. is supported by a UNAM- DGAPA postdoctoral fellowship and Project PAPIIT No. IN111518. O. J. F. acknowledges support from DGAPA-UNAM Project No. IN103319. A.M.E-R. is supported in part by CONACyT grant 237351 (Mexico).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 103 |
\section{Introduction}
The question of whether or not a neutral molecule can attach an excess electron to form a bound anion is not simple to answer \cite{desfrancois96_114,compton01_b60,jordan03_352,simons08_1079}. Fermi and Teller, in their pioneering work \cite{fermi47_110}, demonstrated the existence of the minimal dipole moment required to bind an electron in an external point dipolar field. This result stimulated many theoretical and experimental investigations on multipole-bound anions using effective potential methods \cite{desfrancois95_205,desfrancois96_114,abdoul98_108,abdoul02_1321,garrett70_118,garrett71_106,garrett82_104,ard09_122,desfrancois04_199,desfrancois98_1336,abdoul98_108,abdoul02_1321,fossez13_552,fossez15_1028,fossez16_1775} and ab-initio approaches \cite{jordan77_232,gutsev98_1097,adamowicz89_346,smith99_355,clary99_356,kalcher00_560,skurski02_575,peterson02_123,sommerfeld04_1573,sommerfeld14_1187,gutsev99_1324,gutsev98_1317,gutowski99_1186}.
Because of the similarity of single-electron and rotational energy scales, there appears a strong, nonadiabatic coupling between the valence-electron and molecular rotational motions that impact the critical multipole moment required to form an anion \cite{herrick84_1248,clary88_345,clary89_303,brinkman93_208,ard09_122,garrett10_117}. Moreover, while the evidence for dipole-bound anions is solid, this is not the case for higher multipolarities \cite{simons08_1079,klahn98_752,fossez16_1775}. In our previous study on the resonant spectrum of quadrupole-bound anions \cite{fossez16_1775} we predicted narrow resonances above the detachment threshold. The energies and widths of those resonances appear to be rather insensitive to details of the potential and are almost identical for prolate and oblate charge distributions.
While the binding of multipole-bound anions is fragile, low-energy resonances in such systems are expected to be less sensitive to details of the short-range molecular potential as the spatial extension of the valence electron is huge.
This situation resembles universal behavior, independent of the details of the interaction, exhibited by other weakly-bound/unbound quantum systems, such as nuclear and hadronic halos, cold atomic gases near a Feshbach resonance, and helium dimers and trimers, see, e.g., Refs.~\cite{Stecher2009,Hadizadeh2011,Hadizadeh2012,Lazauskas2013,Kievsky2014,konig17_1986,Deltuva2016,Deltuva2017,Shalchi2017,Miller2018}. In all of those cases, simple arguments based on scale separation and effective field theory capture the essential physics \cite{braaten06_823,bertulani02_869,bedaque03_1085,hammer10_1093,hammer00_1683,hammer17_1959,konig17_1986}. Consequently, to investigate generic properties of multipole-bound anions, we consider a schematic model, which contains the following crucial physics ingredients: (i) a short-range multipole potential and (ii) nonadiabatic coupling between electronic and molecular motion.
In the present study, we investigate the generic near-threshold behavior of multipole-bound anions at the transition between the subcritical and supercritical regimes. The main interest of this study is to show the role of low-$\ell$ partial waves in shaping the properties of low-lying states. We assume that the potential representing the molecular core is given by a Gaussian radial form-factor with a multipolar angular distribution. The particular choice of the radial form-factor is not important as it represents an a priori unknown short-range attraction. One can view this particular realization as a regularized zero-range interaction. The continuum couplings are included using the Berggren expansion method as in Refs.~\cite{fossez13_552,fossez15_1028,fossez16_1775}.
To study the threshold behavior of the system we investigate the pattern of resonant poles as a function of four parameters: the strength and range of the Gaussian form factor, the multipolarity of the potential, and the molecular moment of inertia.
The paper is organized as follows. Section~\ref{s:mm} presents the model and method used. The results obtained in this study are discussed in Sec. \ref{s:rd}. Finally, Sec. \ref{s:conclusion} contains a summary and conclusions.
\section{Model and method} \label{s:mm}
\subsection{Hamiltonian}
In this work, we use the electron-plus-molecule Hamiltonian similar to that of Refs.~\cite{fossez13_552,fossez15_1028}. As shown in Fig.~\ref{fig:model} the valence electron is weakly coupled to the core. Without considering the spin-orbit interaction or the vibrational motion of the core, the Hamiltonian can be written as:
\begin{equation}
\hat{H}=\frac{\hat{j}^2}{2I} + \frac{\hat{p}_e^2}{2m_e} + V(\vec{r}).
\end{equation}
The first term is the rotational energy of the molecule with angular momentum $\hat{j}$ and moment of inertia $I$. The second term represents the kinetic energy of the electron of mass $m_e$ and linear momentum $\hat{p}_e$. The interaction between the rotor and the valence electron is modeled by the axially-deformed Gaussian potential of multipolarity $\lambda$:
\begin{equation}
V(\vec{r})=-V_0 \exp\left(-\frac{r^2}{2r_0^2}\right) P_{\lambda}(\cos\theta),
\end{equation}
where $\vec{r}$ is the electron's position vector in the molecular reference frame, $V_0$ is the potential strength, and $r_0$ is the potential range. The angular part of the potential is given by a Legendre polynomial of order $\lambda$, with $\theta$ being the angle between the direction of the valence electron $\hat{r}$ and the symmetry axis of the rotor.
\begin{figure}[t!]
\includegraphics[width=0.7\linewidth]{fig1_particle_plus_rotor_model.pdf}
\caption{A schematic illustration of the electron-plus-molecule model used in this work.}
\label{fig:model}
\end{figure}
\subsection{Coupled-channel equations}
The total angular momentum of the system $\hat{J}$ is given by the sum of the angular momentum of the rotor $\hat{j}$ and the valence electron $\hat{\ell}$.
Because the system is rotationally invariant in the laboratory reference frame, $\hat{J}$ commutes with the Hamiltonian $\hat{H}$ and the eigenvectors can be written as:
\begin{equation}
\Psi^J=\sum_c u^J_c(r) \Theta_c^J
\label{eq:wf}
\end{equation}
where $c$ labels all possible channels $(j,\ell)$ for a given $J$, $u^J_c(r)$ is the radial channel wave function, and $\Theta^J_c$ is the angular channel wave function. The eigenstates of Eq. \eqref{eq:wf} are also labeled by means for the parity quantum number $\pi$; hence, in the following we use the spectroscopic notation $J^\pi_n$, where $n=1$ marks the lowest $J^\pi$-state, $n=2$ - the next one, and so on.
Due to the symmetries of $V(\vec{r})$, the ground-state rotational band of the molecule has states with $j=0,2,4,\dots$ and $\pi=+$ for $\lambda$-even and
$j^\pi=0^+,1^-,2^+,3^-,\dots$ for $\lambda$-odd \cite{Herzberg1}.
The coupled-channel equations are obtained by inserting the wave function \eqref{eq:wf} into the Schr\"odinger equation:
\begin{equation}
\begin{aligned}
&\left[\frac{d^2}{dr^2} - \frac{j_c(j_c+1 )}{I} - \frac{\ell _c(\ell_c +1)}{r^2} +E_J\right] u^J_c (r) \\
&= \sum_{c^{\prime}} V^J_{cc^{\prime}} (r) u^J_{c^{\prime}} (r),
\end{aligned}
\label{eq:cc_eq}
\end{equation}
where $V^J_{cc^{\prime}} (r)$ is the channel-channel coupling potential. Throughout the paper, we will be using Rydberg units (energy expressed in Ry and distance in $a_0$).
In the coupled-channel approach, the motion of the electron is weakly coupled to the rotation of the molecule. The adiabatic, or strongly-coupled, limit corresponds to an infinite moment of inertia where the rotational band of the molecule collapses to the band-head energy.
\subsection{Berggren expansion method}
The coupled-channel equations \eqref{eq:cc_eq} can be solved by means of the direct integration method (DIM), but a good initial guess is required to ensure convergence \cite{fossez16_1775}; this can be difficult for weakly bound states and broad resonances. Also, higher-multipolarity potentials require a larger number of channels, which makes this method computationally demanding.
An alternative to the DIM is the Berggren expansion method (BEM), previously applied in the context of multipole-bound anions \cite{fossez15_1028,fossez16_1775} and nuclear halos \cite{hagen06_464,papadimitriou11_277,fossez16_1335}. The Berggren basis \cite{berggren68_32,berggren93_481} used in this work is defined in the complex momentum plane; it contains explicitly resonant states (poles of the one-body $S$-matrix) and scattering states defined along a contour $\mathcal{L}^+$ in the fourth quadrant of the momentum plane. The completeness relation for the Berggren ensemble can be written as:
\begin{equation}
\sum_{b,a,d} \ket{\tilde{u}_n} \bra{u_n} + \int_{\mathcal{L}^+} \ket{\tilde{u} (k)} \bra{u(k)} dk =1,
\label{eqBerggren}
\end{equation}
where the sum over discrete resonant states includes bound states $b$, antibound (or virtual) states $a$, and decaying poles $d$ lying between the positive real axis and the contour $\mathcal{L}^+$. The tilde symbol indicates time reversal. In the unlikely situation that bound states of energies higher than antibound states are present, they must be excluded from the sum in~Eq.~(\ref{eqBerggren}). The decaying poles in the fourth quadrant, which lie close to the real $k$-axis and have a real energy Re$(E)>0$ and a width $\Gamma$=$-$2Im$(E)>0$ can be interpreted as narrow resonances. The poles with Re$(E)<0$ and $\Gamma>0$, located below the $-45^\circ$ line in Fig.~\ref{figcomplexk} and close to the origin, can be associated with subthreshold resonances \cite{Kok1980,mukhamedzhanov10_210,Mukhamedzhanov2017,Sofianos1997}.
In practical applications, one often considers a contour $\mathcal{L}_1^+$ of Fig.~\ref{figcomplexk} that starts at the origin, extends into the fourth quadrant up to $k_{\text{peak}}$, comes back to the real axis at $k_{\text{mid}}$, and continues along the real axis up to the cutoff momentum $k_{\text{max}}$. To be able to explore $S$-matrix poles in other regions of the complex momentum plane, two other contours are used in this work. With the contour $\mathcal{L}_2^+$ we explore the region of subthreshold resonances. The contour $\mathcal{L}_3^+$ can be employed to reveal antibound states lying on the negative imaginary momentum axis and the capturing resonances lying in the third complex-$k$ quadrant.
\begin{figure}[tb]
\includegraphics[width=0.9\linewidth]{fig2_contour.pdf}
\caption{Berggren ensemble in the complex-$k$ plane. Bound, antibound, decaying, and capturing resonant states are marked. The distribution of poles is symmetric with respect to the imaginary $k$-axis because of time reversal symmetry. Three different scattering contours $\mathcal{L}_1^+$, $\mathcal{L}_2^+$, and $\mathcal{L}_3^+$ reveal $S$-matrix poles in different sectors in complex momentum/energy plane. The $-45^\circ$ line
separating decaying resonances from subthreshold resonances is marked.}
\label{figcomplexk}
\end{figure}
For each channel, the basis is generated using the diagonal part of the potential in the channel basis $V_{cc}$ \cite{fossez15_1028}. Bound states and decaying resonances entering the Berggren basis for a given partial wave are obtained by a direct integration of the Schr\"odinger equation for the diagonal term of the potential, while the selected scattering states along the contour $\mathcal{L}^+$ are discretized in the momentum space using a Gauss-Legendre quadrature as in Refs.~\cite{fossez15_1028,fossez16_1775}. The non-resonant continuum is limited by the momentum cutoff $k_{\text{max}}$ that has to be sufficiently large to ensure the completeness of the Berggren basis. While the bound states are normalized in the standard way, decaying resonances are normalized using the exterior complex scaling method \cite{gyarmati71_38,simon79_436}. The scattering states are normalized to Dirac-delta function. This representation provides a natural way to include continuum couplings for each desired partial wave.
The spectrum of the system is obtained by diagonalizing the complex-symmetric Hamiltonian matrix. It consists of resonant eigenstates representing bound states and narrow resonances, and non-resonant scattering solutions. Differentiating resonant states from the non-resonant scattering background requires special treatment. Since resonant states do not depend on a detailed choice of the contour $\mathcal{L}^+$, by moving the contour slightly a new spectrum can be obtained, where non-resonant states move according to the contour change and resonant states stay invariant \cite{fossez16_1775}. In this way, resonant states can be located. As a further test, these resonant states are used in the DIM as an initial guess, and it is checked that the BEM results are reproduced.
\section{Results} \label{s:rd}
\subsection{Threshold trajectories for multipolar Gaussian potentials in the adiabatic limit \label{sub:threshold_line}}
\begin{figure}[htb]
\includegraphics[width=0.9\linewidth]{fig3_threshold_line_multipolar_all.pdf}
\caption{Threshold trajectories $({V}_{0}, r_0)_{c}^{\pm}$ for multipolar Gaussian potentials with $\lambda=1-4$ in the adiabatic limit.}
\label{fig:threshold_line}
\end{figure}
Multipole-bound anions can be characterized by their critical multipole moments $Q_{\lambda,c}^\pm$, which mark the limit between the subcritical and supercritical regimes.
We note that for odd-multipole potentials $Q_{\lambda,c}^-=-Q_{\lambda,c}^+$, but there is no such relation for even-multipole potentials. For instance, there are two critical values of the quadrupole moment for a quadrupole-bound anion ($\lambda=2$): $Q_{2,c}^+$ (prolate) and $Q_{2,c}^-$ (oblate), and $Q_{2,c}^- \ne -Q_{2,c}^+$.
As the usual ${ -1/{r}^{\lambda+1} }$ radial dependence of multipolar potentials is replaced in our work by the Gaussian form factor, the detachment threshold is obtained at the critical trajectories of $({V}_{0}, r_0)_{c}^{\pm}$. Figure~\ref{fig:threshold_line} shows such trajectories obtained in the adiabatic limit for the $J^{\pi}=0^+_1$ ground states of anions with multipolarities $\lambda=1-4$.
The complex-momentum contours used in the Berggren basis are defined by the points $k=(0,0), k_{\text{peak}}=(0.5,-0.1)$, $k_{\text{mid}}=1.0$, and $k_{\text{max}}=14.0$ (in units of ${ {a}_{0}^{-1} }$), with each segment being discretized by 40 Gauss-Legendre points. To ensure convergence, we took $\ell_{\text{max}}=4$ for $\lambda=1,2,3$ and $\ell_{\text{max}}=8$ for $\lambda=4,5$.
As one would expect, the absolute value of the critical potential strength ${ | {V}_{0,c} | }$ required to bind an excess electron is decreasing with the range $r_0$ and for a fixed range ${ | {V}_{0,c} | }$ increases with multipolarity. Also, as noted in previous studies \cite{neirotti97_1466,ferron04_130,pupyshev04_161,fossez16_1775}, for even multipolarities, the value of ${ | {V}_{0,c} | }$ for negative-$V_0$ potentials (``prolate") is larger than that for positive-$V_0$ potentials (``oblate").
It is interesting to note that at the threshold, the wave functions are dominated by the $\ell=0$ component. Dividing the intrinsic wave function into the inner region $(r < R)$ and outer region $(r > R)$ contributions, where $R$ is the distance at which the core potential becomes practically unimportant, one can show \cite{misu97_1181,riisager92_615,yoshida05_1895} that the probability of finding the electron in the outer region approaches one at the detachment threshold, if the $\ell=0$ component is present in the intrinsic wave function. This has been practically demonstrated in our previous work on quadrupole-bound anions \cite{fossez16_1775} in the context of the scaling properties of root-mean-square radii.
\subsection{Resonances of the near-critical quadrupolar Gaussian potential \label{sub:resonances}}
In order to study the role of low-$\ell$ partial waves in multipole-bound anions at the interface between the subcritical and supercritical regimes, one has to recognize the impact of $\ell = 0$ partial waves on resonant states near threshold \cite{yoshida05_1895}. In our coupled-channel formalism, resonant states appear through the mixing of different channels. To study general features of near-threshold resonances, we consider three states of the quadrupolar potential in the adiabatic approximation. Namely, we investigate: (i) the ${J}^{\pi} = {0}_{1}^{+}$ ground state dominated by the $\ell = 0$ partial wave; (ii) an excited ${ {J}^{\pi} = {0}_{d}^{+} }$ state dominated by the $\ell=2$ channel; and (iii) the lowest ${ {J}^{\pi} = {1}_{1}^{-} }$ state, which is primarily $\ell = 1$. The quadrupolar case discussed here is characteristic of other multipolar potentials.
\subsubsection{Resonant states dominated by the $\ell=0$ channel}\label{swave}
The ground state (g.s.) of the quadrupolar potential is computed with the BEM, using the extended contour $\mathcal{L}_3$ of Fig.~\ref{figcomplexk} defined by the points: $k=(0,0)$, $(-0.1,-0.4)$, $(0.1,-0.4)$, $(2,0)$, and $(14,0)$ (all in $a_0^{-1} $), each segment being discretized with 40 Gauss-Legendre points. By considering the contour that extends into the third quadrant of the complex momentum plane, antibound states can be revealed, see Fig.~\ref{figcomplexk} and Refs.~\cite{betan04_37,michel06_16,michel09_2}.
\begin{figure}[htb]
\includegraphics[width=0.9\linewidth]{fig4_quadrup_0+_virtualstate_c1_E_K_channel_ratio.pdf}
\caption{The lowest $0^+$ resonant state of the quadrupolar Gaussian potential with
$ r_0 = a_0 $ as a function of $V_0$. Top: real energy and imaginary momentum. Bottom: the channel decomposition of the real part of the norm. The critical strength ${V}_{0,c}$ is marked by arrow.
}
\label{fig:antibound}
\end{figure}
Figure~\ref{fig:antibound}(a) shows the energy and momentum of the ${0}_{1}^{+}$ state for different values of the potential strength $V_0$. For large values of $V_0$, the g.s. is bound (Re($E$)$<$0) and has a positive imaginary momentum. As the potential strength decreases, the energy of the ground state moves up and approaches the $E=0$ threshold at ${V}_{0,c} = 8.7$\,Ry. For $V_0<{V}_{0,c}$ the lowest ${0}^{+}$ state becomes antibound (Re($E)<0$, Im$(k)<0$). As illustrated in Fig.~\ref{fig:antibound}(b), the contributions ${ {N}_{\ell} }$ to the complex norm of the wave function from different $\ell$-channels ($\ell = 0, 2$, 4) vary smoothly when crossing the threshold. The norm is largely dominated by the $\ell = 0$ component. At the critical strength, the $\ell>0$ contributions to the norm vanish, cf. discussion in Sec.~\ref{sub:threshold_line}. The presence of near-threshold antibound states impacts the structure of the low-energy continuum and can manifest their existence through peaks in the scattering cross section at low-energy \cite{rohr75_283,rohr76_284,rohr78_1426,heiss11_776,mukhamedzhanov10_210}.
\subsubsection{Resonant states dominated by a $\ell \neq 0$ channel}
We now consider the evolution of an excited state of the quadrupolar potential with $ r_0 = 4 \, a_0 $. At $V_0$ =1.1\,Ry the lowest $0^+$ state is bound and the second ${J}^{\pi} = {0}_2^+$ state is a decaying resonance, see Fig.~\ref{fig:2nd_resonant}. Figure \ref{fig:0_2_norm}(a) shows the channel decomposition for this state. It is seen that its configuration has the predominant $\ell=2$ component.
\begin{figure}[htb]
\includegraphics[width=0.9\linewidth]{fig5_quadrup_0+_2_reso_c1_E_K_channel_ratio.pdf}
\caption{Trajectory of the $0^+$ resonant state in the complex-$k$ plane of the quadrupolar potential with
$r_0 = 4\,a_0 $ as the
potential strength $V_0$ increases in the direction indicated by an arrow. At the lowest value $V_0$ =1.1\,Ry, the $0^+$ ground state is bound and the state of interest is an excited ${0}_2^+$ state associated with a decaying resonance. At $V_0=1.8$\,Ry the pole crosses the $-45^\circ$ line and becomes a subthreshold resonance $0_d^+$. At $V_0=2.857$\,Ry the decaying pole reaches the imaginary-$k$ axis and coalesces with the capturing pole with Im$(k)<0$ forming an exceptional point. The antibound states at $V_0=1.8\,$Ry and
$V_0=2.7$\,Ry are marked.}
\label{fig:2nd_resonant}
\end{figure}
As the potential gets deeper, the pole crosses the $-45^\circ$ line at $V_0 \approx 1.8$\,Ry and becomes a subthreshold resonance labeled as $0_d^+$. At $V_0=2.7$\,Ry a rapid transition to a configuration dominated by the $\ell=4$ partial wave takes place, which is indicative of a level crossing in the complex-$k$ plane. At $V_0=2.857$\,Ry the decaying pole arrives at the imaginary-$k$ axis and coalesces with the symmetric capturing pole forming an exceptional point \cite{heiss12_1392,Muller2008_1973,Okolowicz2009_1974}. At still larger values of $V_0$, the exceptional point splits up into two antibound states moving up and down along the imaginary $k$-axis as shown in Fig.~\ref{fig:2nd_resonant}. A similar situation was discussed in Refs.~\cite{domcke81_289,Garmon2015_1975} in the context of electron-molecule scattering and optical lattice arrays, respectively.
\begin{figure}[htb]
\includegraphics[width=0.9\linewidth]{fig6_norm_configuration_exchanging.pdf}
\caption{Real norms of the channel wave functions for the decaying pole $0_d^+$ shown in Fig.~\ref{fig:2nd_resonant} and the antibound states $0_b^+$ and $0_c^+$ of Fig.~\ref{fig:antibound_transition}.}
\label{fig:0_2_norm}
\end{figure}
In the range of $V_0$ corresponding to the trajectory $0^+_2 \rightarrow 0^+_d$ shown in Fig.~\ref{fig:2nd_resonant}, there appear antibound states in the threshold region. Their trajectories along the imaginary $k$-axis are shown in Fig.~\ref{fig:antibound_transition} and their channel decompositions are given in Fig.~\ref{fig:0_2_norm}(b) and (c). As $V_0$ increases, the antibound states $0_a^+$, $0_b^+$, and $0_c^+$ emerge as bound physical states of the system labeled as $0_1^+$, $0_2^+$, and $0_3^+$, respectively. The lowest antibound state $0^+_a$ has a dominant $\ell=0$ configuration, similar to that of Fig.~\ref{fig:antibound}. At low values of $V_0$, the wave function of the antibound state $0^+_b$ is predominantly $\ell=2$. As seen in Fig.~\ref{fig:2nd_resonant}, this state appears close to the decaying pole $0^+_d$ at $V_0 \approx 1.8$\,Ry and the crossing between these two poles in the complex-$k$ plane is seen in their wave function decompositions. Following the crossing, the state $0^+_b$ acquires a large $\ell=0$ component. The antibound state $0^+_c$ begins as an $\ell=4$ configuration. At $V_0\approx 2.7$\,Ry, this state interacts with $0^+_d$ and its configuration changes to $\ell=2$. One can thus see that the presence of antibound states results in the particular shape of the $0^+_d$-pole trajectory in the complex-$k$ plane.
\begin{figure}[htb]
\includegraphics[width=0.9\linewidth]{fig7_with_line_antibound_state_transitions.pdf}
\caption{Trajectories of antibound and bound $0^+$ states along the imaginary $k$-axis as a function of $V_0$ for the quadrupolar potential with $r_0 = 4\,a_0$. With increasing potential strength, the antibound states $0_a^+$, $0_b^+$, and $0_c^+$ become bound states of the system $0_1^+$, $0_2^+$, and $0_3^+$, respectively. The open circle marks the exceptional point of Fig.~\ref{fig:2nd_resonant}, which is the source of two antibound states. The particular values of $V_0$ discussed around Fig.~\ref{fig:2nd_resonant} are marked.}
\label{fig:antibound_transition}
\end{figure}
The dependence of the $0^+_d$-pole trajectory on the potential range is illustrated in Fig.~\ref{fig:0+_dif_c}.
\begin{figure}[htb]
\includegraphics[width=0.9\linewidth]{fig8_traj_quadrupole_dif_c_2nd_pole.pdf}
\caption{Trajectory of the $0_d^+$ resonant state in the complex-$k$ plane for different values $r_0$ of quadrupolar potential as indicated by numbers (in units of $a_0$). The ranges of $V_0$ (in Ry) are: (25.6-29.0) for $r_0=a_0$;
(9.7-14.5) for $r_0=1.5\,a_0$;
(4.8-10) for $r_0=2\,a_0$;
(1.7-4.79) for $r_0=3\,a_0$; and
(1.1-2.85) for $r_0=4\,a_0$.}
\label{fig:0+_dif_c}
\end{figure}
For potentials with longer ranges, pole trajectories appear closer to the origin. In all the cases shown, a transition from decaying to subthreshold resonances takes place. These poles have large widths, and are expected to impact the structure of the low-energy scattering continuum.
\subsubsection{Resonant states without a $\ell = 0$ component}
Here we discuss the lowest ${ {J}^{\pi} = {1}_{1}^{-} }$ state, which is primarily $\ell = 1$ with a small admixture of the $\ell = 3$ channel. This case closely follows the discussion of Ref.~\cite{domcke81_289} for $p$-wave scattering from short-range potentials.
\begin{figure}[htb]
\includegraphics[width=0.8\linewidth]{fig9_quadrup_1-_reso_c1_E_K_channel_ratio.pdf}
\caption{Top: trajectory of the lowest $1_1^-$ resonant state of the quadrupolar potential with $r_0=a_0$ as a function of $V_0$ in the range of (9-12.7)\,Ry.
The potential strength $V_0$ increases along the direction indicated by an arrow. The positions of the bound and antibound states at $V_0=12.34$\,Ry and 12.4\,Ry are marked.
Bottom: real norms of channel functions for this state.
}
\label{fig:1-_resonant}
\end{figure}
The corresponding trajectory of this state in the complex momentum plane is shown in Fig.~\ref{fig:1-_resonant}(a). At larger values of $V_0$, the $1_1^-$ state is bound. As $V_0$ decreases, this state crosses the detachment threshold and becomes a narrow decaying resonance. The trajectory of the capturing resonance, symmetric with respect to the Im$(k)$ axis, is not shown. As discussed in Ref.~\cite{domcke81_289}, the exceptional point appears at the origin at $V_{0,c}$. Close to the threshold, the bound state and the antibound state are located symmetrically to the origin. For the $p$-wave dominated state, the transition from the subcritical to the supercritical regime is smooth, i.e., the wave function amplitudes hardly change with $V_0$, see Fig.~\ref{fig:1-_resonant}(b). This is because the contributions from antibound and bound state poles cancel each other out. In this case, the structure of the low-energy continuum is not expected to be affected by the presence of threshold poles.
The situation presented in Fig.~\ref{fig:1-_resonant} is rather generic for $p$-wave dominated resonant poles. Increasing the potential range moves the pole trajectory closer to the real-$k$ axis. Consequently, states containing no $s$-wave component are likely to appear as isolated narrow resonances.
For odd-multipolarity potentials, a ($j=J, \ell=0$) component of a $J^\pi$ state becomes large as the detachment threshold is approached, see Sec.~\ref{swave}. On the other hand, for even-multipolarity potentials, odd-$J$ states cannot have an $s$-wave component, as the core's angular momentum $j$ must be even, and narrow near-threshold resonances can appear.
\subsection{Rotational motion}\label{sub:rotational}
To describe multipole-bound anions, one has to take into account the nonadiabatic coupling between the rotational motion of the molecule and the single-particle motion of the electron. Whether a multipole-bound anion can exhibit rotational bands depends on the molecule's multipolarity. For instance, it was shown in Ref.~\cite{fossez15_1028} that rotational bands of dipolar anions do not extend above the detachment threshold while a similar study for quadrupole-bound anions \cite{fossez16_1775} demonstrated that the rotational motion of the anion is hardly affected by the continuum effects. The reason for this difference might be due to the existence of two coupling regimes in the dipolar case: a strong coupling regime below the threshold (valence electron follows the rotational motion of the core) and a weak coupling regime in the continuum region (valence electron is almost entirely decoupled from the molecular rotation).
\begin{figure}[htb]
\includegraphics[width=0.8\linewidth]{fig10_rotational_band_lambda_1_c1.pdf}
\caption{The rotational band built upon the
${ {J}^{\pi} = {0_1^+} }$ state of a dipole-bound anion.
The parameters $V_0= 5.33$\,Ry, $r_0=a_0$, and $I=10^{3} \, m_e a_0^2$
have been chosen to place the band-head energy slightly below the zero-energy threshold, where rotational motion of the rotor can excite the system into the continuum.
The energy is plotted as a function of $J(J+1)$.
}
\label{fig:lambda_1_rota_band}
\end{figure}
Figure~\ref{fig:lambda_1_rota_band} illustrates the case of a rotational band built upon the subthreshold ${ {J}^{\pi} = 0_1^+ }$ state of the Gaussian dipolar potential. It is seen that the rotational band is not affected when the zero-energy threshold is crossed below $J=4$. This result indicates that the presence of the two coupling regimes predicted to exist in realistic calculations for dipole-bound anions~\cite{fossez15_1028} must be due to difficulties in imposing proper boundary conditions at infinity for the dipolar potential ($\sim r^{-2}$) when the rotational motion of the molecule is considered nonadiabatically \cite{fossez13_552}. Since in the present work the radial part of the dipolar pseudopotential is replaced by a Gaussian, the outgoing boundary condition can be readily imposed.
\begin{figure}[htb]
\includegraphics[width=0.8\linewidth]{fig11_rotational_band_lambda_2_c1.pdf}
\caption{Similar to Fig.~\ref{fig:lambda_1_rota_band} but for rotational bands built upon the ${ {J}^{\pi} = {0_1}^{+} }$ and ${1_1}^{-}$ bandheads of a quadrupolar Gaussian potential with $V_0=12.38$\,Ry, $r_0=a_0$, and for $I=50\, m_e a_0^2$ and $I=100\, m_e a_0^2$.}
\label{fig:lambda_2_rota_band}
\end{figure}
A similar result is obtained for the quadrupolar case shown in Fig.~\ref{fig:lambda_2_rota_band} for two rotational bands built upon the ${ {J}^{\pi} = 0_1^+ }$ and $1_1^-$ bandheads. The existence of rotational bands extending above the detachment threshold is consistent with the findings of Ref.~\cite{fossez16_1775} employing the realistic quadrupolar pseudopotential. The results for higher-multipolarity potentials follow the pattern obtained for the dipolar and quadrupolar cases; hence, they are not shown here.
We now investigate the impact of the molecular rotation on the anion's energy spectrum.
By definition, changing the moment of inertia of the rotor is expected to have a larger effect on states dominated by channels with large $j$, but in practice such channels are unlikely to dominate at low energies.
As an illustrative example, we study the $3_1^-$ state of the quadrupolar ($\lambda=2$) Gaussian potential. Figure~\ref{fig:3-_contour}(a,b) shows, respectively, the energy and decay width of the $3_1^-$ resonance as a function of the potential strength and the inverse moment of inertia.
\begin{figure}[htb]
\includegraphics[width=1.0\linewidth]{fig12_3-_E_G_threshold_I_V.pdf}
\caption{Energy (a) and decay width (b), both in Ry, of the $3^-_1$ resonance of
the quadrupolar Gaussian potential with $r_0=a_0$ as a function of the inverse of the moment of inertia and the potential strength. The detachment threshold ($E=0$) is indicated.
The dominant $(j,\ell)$ channel is marked in panel (b).
When the rotational energy of the molecule $E_{\rm rot}^{j=4}$ lies below/above the energy of the $3^-_1$ resonance, the (4,1) decay channel is open/closed.
The line $E_{\rm rot}^{j=4}=E(3^-_1)$ (thick solid) separating these two regimes is marked, so is the line $E_{\rm rot}^{j=2}=E(3^-_1)$ (thick dotted) which corresponds to the threshold energy for the opening of the (2,1) channel. The norms of the two dominant channels (2,1) (solid line) and (4,1) (dotted line) are shown as a function of $V_0$ for $1/I=0.04 \,m_e^{-1}a_0^{-2}$ (c) and 0.02$\,m_e^{-1}a_0^{-2}$ (d).}
\label{fig:3-_contour}
\end{figure}
At large values of $V_0$ when the $3_1^-$ resonance lies close to the threshold, its wave function is primarily described in terms of two
channels with $(j,\ell) = (2,1)$ and $(4,1)$ with the dominant (2,1) amplitude, see Fig.~\ref{fig:3-_contour}(c,d). At a finite value of $I$, as the energy of the resonance increases, a transition takes place to a state dominated by the (4,1) component that is associated with a reduction of the decay width. This transition can be explained in terms of channel coupling. At very low values of $1/I$ the resonance's energy $E(3^-_1)$ lies above the rotational $4^+$ state of the molecule. As the moment of inertia decreases, the $4^+$ member of the ground-state rotational band of the molecule moves up in energy, and at some value of $I$ it becomes degenerate with the energy of the $E(3^-_1)$ resonance, i.e., $ E_{\rm rot}^{j=4}=E(3^-_1)$.
At still higher values of $1/I$, the (4,1) channel is closed to the anion's decay. As seen in Fig.~\ref{fig:3-_contour}(b), the irregular behavior seen in the width of the resonance can be attributed to the (4,1) channel closing effect \cite{fossez16_1335}.
A second irregularity in Fig.~\ref{fig:3-_contour}(c,d), seen at large potential strengths, corresponds to $E_{\rm rot}^{j=2}=E(3^-_1)$.
As the resonance approaches the threshold, its tiny decay width can be associated with the (0,3) channel. Due to its higher centrifugal barrier, (0,3) channel contributes around $1\%$ to the total norm in the threshold region.
\subsection{Unbound threshold solutions in the supercritical region}
\label{sub:subspucritical}
\begin{figure}[tb]
\includegraphics[width=0.9\linewidth]{fig13_contour_quadrupolar_0+.pdf}
\caption{Unbound threshold 0$^+$ states of the quadrupolar Gaussian potential obtained by using scattering ${\cal L}$-contours with different $k_{\rm peak}$ (in units of $a_0^{-1}$) in the complex momentum plane, see Fig.~\ref{figcomplexk}. The potential has $V_0=8$\,Ry and $r_0=a_0$.}
\label{fig:contour_2_0+}
\end{figure}
In our previous study on quadrupole-bound anions \cite{fossez16_1775}, based on a realistic pseudopotential, it was shown that there appear series of narrow resonances at energies close to the rotor energies, exhibiting fairly regular patterns. Similar sequences of threshold states, predicted by the present model, are shown in Fig.~\ref{fig:contour_2_0+}, which displays unbound 0$^+$ states of the quadrupolar Gaussian potential computed
with different scattering contours in the complex momentum plane obtained by varying
$k_{\rm peak}$, see Fig.~\ref{figcomplexk}. It is seen that the calculated states exhibit appreciable contour dependence.
Similar results have been obtained for other $J^\pi$ states and Gaussian potentials with higher-multipolarity potentials. Since the general pattern of near-threshold solutions obtained in different calculations seems to be fairly generic, and primarily depends on the shape of the contour used, they should be interpreted in terms of non-resonant scattering continuum states rather than resonance poles.
\section{Conclusions}\label{s:conclusion}
In this work, we studied properties of near-threshold states of multipole-bound anions using the Berggren expansion method within the coupled-channel formalism.
We considered a Hamiltonian of a nonadiabatic electron-plus-molecule model with the particle-core interaction being represented by a multipolar Gaussian potential. Such a four-parameter model, rooted in scale-separation arguments of halo effective field theory, is expected to describe general trends of near-threshold resonant poles for multipolarities $\lambda \ge 2$.
By calculating the threshold lines for anions of different multipolarity, we predicted that within this model, higher-$\lambda$ anions can exist as marginally-bound open systems.
The role of the low-$\ell$ channels in shaping the transition between subcritical and supercritical regimes has been explored. We demonstrate the presence of a complex interplay between bound states, antibound states, subthreshold resonances, and decaying resonances as the strength of the Gaussian potential is varied. In some cases, we predict the presence of exceptional points. The fact that antibound states and subthreshold resonances can be present in multipolar anions is of interest as they can affect scattering cross sections at low energy.
For Gaussian potentials, the outgoing boundary condition can be readily imposed. Consequently, the
rotational band of the anion is not affected when the zero-energy threshold is reached. This indicates that the presence of two coupling regimes of rotation predicted to exist in realistic calculations for dipole-bound anions~\cite{fossez15_1028} must be due to specific asymptotic behavior of the dipolar pseudo-potential in the presence of molecular rotation.
The non-adiabatic coupling due to the collective rotation of the molecular core
can give rise to a transition into the supercritical region. We also predict interesting channel-coupling effects resulting in variation of an anion's decay width due to rotation.
In summary, by looking systematically at the pattern of resonant poles of multipole-bound anions near the electron detachment threshold we uncover a rich structure of the low-energy continuum. These simple systems are indeed splendid laboratories of generic phenomena found in marginally-bound molecules and atomic nuclei.
\begin{acknowledgments}
Discussions with Simin Wang are gratefully acknowledged as well as useful comments from Erik Olsen. The original version of the particle-rotor code used in this work was written by Nicolas Michel.
This work was supported by the U.S.\ Department of Energy, Office of Science, Office of Nuclear Physics under award number DE-SC0013365 and by the National Science Foundation under award number PHY-1403906.
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,723 |
\section{Introduction and statement of the main results}
Let $T\geq 2M$ be two positive integers and consider the complex Grassmannian $\mathbb{G}r(M,{\mathbb{C}}^{T})$, i.e. the space of $M$--dimensional complex vector subspaces of $\mathbb C^T$. Finite collections of points (also called {\em codes} or {\em packings}) in $\mathbb{G}r(M,{\mathbb{C}}^{T})$ with different desired separation properties have been investigated by several authors (in Section \ref{sec:history} we describe some relevant references). The most frequent criterium for ``well--separated'' codes is the maximization of the minimal mutual squared chordal distance, which is the sum the squared sines of the principal angles of two subspaces. However, following \cite{Hochwald00,Varanasi02,CuevasTCOM} (see also Section \ref{sec:aplicacion} below), a more relevant measure for its application to information theory is given by the {\em chordal product energy}, related to the product of the squared sines of the principal angles, which justifies its name.
Given a code $[{\bf X}_1],\ldots,[{\bf X}_k]\in\mathbb{G}r(M,{\mathbb{C}}^{T})$, its chordal product energy with parameter $N$ is
\begin{equation}\label{eq:unionbound2}
\mathcal E({\bf X}_1,\ldots,{\bf X}_K)= \sum_{i\neq j}\operatorname{det} ( {\bf I}_M -{\bf X}_i^H{\bf X}_j {\bf X}_j^H {\bf X}_i) ^{-N},
\end{equation}
where ${\bf I}_M$ is the identity matrix and we have chosen representatives ${\bf X}_i$ of each point $[{\bf X}_i]$ satisfying ${\bf X}_i^H{\bf X}_i={\bf I}_M$. Note that the energy is well defined in the sense that it does not change if other representatives with that property are chosen.
Recall that the SVD
of ${\bf X}_i^H{\bf X}_j$, e.g. ${\bf U}{\bf D}{\bf V}^H$, is given in terms of the cosines of the principal angles between the subspaces $[{\bf X}_i]$ and $[{\bf X}_j]$, $\cos \theta_1, \ldots, \cos\theta_M$, cf. \cite{HanTIT06}, so
\begin{equation}
\operatorname{det} \left( {\bf I}_M - {\bf X}_i^H{\bf X}_j {\bf X}_j^H{\bf X}_i \right) =\operatorname{det} \left( {\bf I}_M - {\bf D}^2 \right)= \prod_{i=1}^M \sin^2 \theta_i,
\end{equation}
while the squared chordal distance between the two subspaces $[{\bf X}_i]$ and $[{\bf X}_j]$ is given by $\sum_{i=1}^M \sin^2 \theta_i$.
The sum in \eqref{eq:unionbound2} is a pairwise interaction energy in the spirit of the well--studied Riesz or logarithmic energies of importance in Potential Theory (see \cite{SaffBook} for a complete monograph dedicated to energy minimization in the sphere and other spaces). We refer to the function (again, choosing representatives ${\bf A}$ and ${\bf B}$ such that ${\bf A}^H{\bf A}={\bf B}^H{\bf B}={\bf I}_M$)
\begin{equation}
[{\bf A}],[{\bf B}]\in\mathbb{G}r(M,{\mathbb{C}}^{T})\mapsto\operatorname{det} ( {\bf I}_M -{\bf A}^H{\bf B} {\bf B}^H {\bf A}) =\operatorname{det} ( {\bf I}_M -{\bf B}^H{\bf A} {\bf A}^H {\bf B}) ,
\label{eq:coherenceDet}
\end{equation}
as the {\em chordal product determinant} or, simply, the chordal product, and note that is {\em not} a metric in $\mathbb{G}r(M,{\mathbb{C}}^{T})$, for it may happen that $[{\bf A}]\neq[{\bf B}]$ and yet $\operatorname{det} ( {\bf I}_M -{\bf A}^H{\bf B} {\bf B}^H {\bf A})=0$, if the intersection of $[{\bf A}]$ and $[{\bf B}]$ is nontrivial.
In this paper we perform the first theoretical study of the chordal product energy, for numerical results, see \cite{CuevasTCOM} and references therein. We start describing the context where the problem arises, following \cite{Hochwald00}.
\subsection{The importance of Grassmannian codes in information theory}\label{sec:aplicacion}
Consider a {\em transmitter}, i.e. some device that is able to send a signal, which is indeed a collection of numbers ordered in a complex $T\times M$ matrix ${\bf X}$. Physically, this corresponds to the setting where the transmitter has $M$ antennas and there is a total amount of $T$ time slots where the communication channel is assumed to be constant (i.e. the contour conditions of the communication are considered constant during the time that these $TM$ numbers are sent). The {\em receiver} is another device, that we consider equipped with $N$ antennas, and the signal it receives is
\begin{equation*}
\mathbf{Y} = \mathbf{X} \mathbf{H} + \sqrt{\frac{M}{T \rho}} \mathbf{W},
\end{equation*}
where $\mathbf{H}$ is an unknown $M\times N$ matrix (termed {\em the channel}), $\mathbf{W}$ describes the noise and $\rho$, called the signal-to-noise-ratio (SNR), measures the magnitude of the signal against the noise.
\subsubsection{The zero--noise case}
Since $\mathbf{H}$ is unknown (it is common to assume that it has random complex Gaussian entries), even in the event that $\mathbf W=0$ the receiver cannot recover the whole matrix ${\bf X}$:
\begin{itemize}
\item If two matrices ${\bf X}_1$ and ${\bf X}_2$ have the same column span, then one can easily find an full--rank matrix $\mathbf H$ such that $\mathbf{X}_1 \mathbf{H}=\mathbf{X}_2 \mathbf{H}$, hence the receiver just cannot distinguish which of these two matrices was the original signal.
\item On the other hand, if two matrices ${\bf X}_1$ and ${\bf X}_2$ have the property that the intersection of the column span of ${\bf X}_1$ and ${\bf X}_2$ is trivial, the the receiver can easily distinguish if a given matrix ${\bf Y}$ has been constructed by ${\bf X}_1\mathbf{H}$ or by ${\bf X}_2\mathbf{H}$: if the column span of ${\bf Y}$ intersected with the column span of ${\bf X}_1$ (resp. ${\bf X}_2$) is nontrivial, then ${\bf X}_1$ (resp. ${\bf X}_2$) was sent.
\end{itemize}
Summarizing, if a previously agreed code of possible signals $[{\bf X}_1],\ldots,[{\bf X}_K]\in\mathbb{G}r(M,{\mathbb{C}}^{T})$ is fixed with the property that the column spans of ${\bf X}_i$ and ${\bf X}_j$ have trivial intersection for $i\neq j$, the receiver will be able to recover, at least in the zero--noise scenario, {\em the element of the Grassmannian represented by the sent signal, but not the concrete representative of that element.} Hence, collections of points in $\mathbb{G}r(M,{\mathbb{C}}^{T})$ are searched with that property.
\subsubsection{The general case}
In the more realistic context of the presence of non--zero noise, the analysis is quite more involved since there is always a non--zero probability of error in the detection procedure. The pioneer work \cite{Hochwald00} showed that, in order to recover the element ${\bf X}_i$ of $\mathbb{G}r(M,{\mathbb{C}}^{T})$ just by knowing ${\bf Y}$, the optimal method is to use the so called maximum--likelihood decoder:
\[
i=\operatorname{argmax}_{j=1,\ldots,K}\operatorname{tr}{({\bf Y}^H{\bf X}_j{\bf X}_j^H{\bf Y})}=
\operatorname{argmax}_{j=1,\ldots,K}\operatorname{tr}{({\bf X}_j^H{\bf Y}\Y^H{\bf X}_j)},
\]
where $\cdot^H$ holds for Hermitian conjugate. Then, \cite{Varanasi01} showed that if only $2$ codewords are permitted, i.e. if $K=2$, and assuming that the entries of $\mathbf{H}$ and $\mathbf{W}$ are complex Gaussian $\mathcal{N}(0,1)$ numbers, then the probability $P_e({\bf X}_1, {\bf X}_2,\rho)$ of erroneously decoding ${\bf X}_1$ if ${\bf X}_2$ was sent can be given by a (quite complicated) formula involving the residues of a certain rational function. Luckily, the asymptotic expansion of this {\em Pairwise Error Probability} (PEP) in the case $\rho\to\infty$, called the high-SNR asymptotic analysis, admits a much more concise expression, see \cite{Varanasi02,CuevasTCOM}:
\begin{equation}
P_e({\bf X}_1, {\bf X}_2,\rho) \approx C \rho^{-NM} \operatorname{det} ( {\bf I}_M -{\bf X}_1^H{\bf X}_2 {\bf X}_2^H {\bf X}_1) ^{-N},\quad \rho\to\infty,
\label{eq:PEPbis}
\end{equation}
where $C= \frac12\left( \frac{4M}{T}\right)^{NM} \frac{(2NM-1)!!}{(2NM)!!)}$, it is assumed that any two distinct points have trivial intersection as linear subspaces, and the representatives ${\bf X}_i$ of each $[{\bf X}_i]$ are such that ${\bf X}_i^H{\bf X}_i={\bf I}_M$. If we have $K$ elements $[{\bf X}_1],\ldots,[{\bf X}_K]$ in the code of possible signals and we assume that we send one of them at random, all with equal probability $1/K$, then the total probability of erroneously decoding a signal is bounded above by
\begin{equation}\label{eq:unionbound}
\frac1K\sum_{i\neq j}P_e({\bf X}_i, {\bf X}_j,\rho)
\approx
\frac{C}{K} \rho^{-NM} \sum_{i\neq j}\operatorname{det} ( {\bf I}_M -{\bf X}_i^H{\bf X}_j {\bf X}_j^H {\bf X}_i) ^{-N}.
\end{equation}
The determinant in \eqref{eq:PEPbis} is the chordal product \eqref{eq:coherenceDet} and the sum in the right--hand side in \eqref{eq:unionbound} is the energy \eqref{eq:unionbound2}.
\subsubsection{Criteria for the design of Grassmannian codes}
It follows from the previous discussion that reasonable criteria for the design of a code $[{\bf X}_1],\ldots,[{\bf X}_K]$ would be to maximize the pairwise chordal product \eqref{eq:coherenceDet}, or to minimize the chordal product energy \eqref{eq:unionbound2}. In \cite{CuevasTCOM} these approaches are considered, numerically showing that the obtained codes are very well suited for their use in non--coherent communications, with a slight advantage in the use of the chordal product energy. Yet, little or no theory exists about the behavior of the optimal pairwise chordal product or energy. The main purpose of this paper is to put the basis for the study of this question.
\subsection{Main results of the paper}
We will start our study by computing the moments of the chordal product when $[{\bf B}]$ is fixed and $[{\bf A}]$ is chosen at random uniformly in $\mathbb{G}r(M,{\mathbb{C}}^{T})$, w.r.t. the unique, standard rotation--invariant probability measure. This yields a complete statistical characterization of the chordal product as a product of beta--distributed random variables:
\begin{thm}\label{th:expected}
Assume that $T\geq 2M$. Let $p\in(2M-T-1,\infty)$ (notice that $p$ may be negative and/or noninteger). Let $[{\bf B}] \in\mathbb{G}r(M,{\mathbb{C}}^{T})$ be any fixed element and let $[{\bf A}] \in\mathbb{G}r(M,{\mathbb{C}}^{T})$ be uniformly distributed on the Grassmannian. Then, the $p$--th moment of $\operatorname{det}({\bf I}_M-{\bf B}^H {\bf A}\A^H {\bf B})$ is:
\begin{equation}\label{eq:detmoment}
{\rm E}_{{\bf A}}[\operatorname{det}({\bf I}_M-{\bf B}^H {\bf A}\A^H {\bf B})^p] =
\prod_{m=1}^M\frac{\Gamma(T-m+1)\Gamma(T+p-m-M+1)}{\Gamma(T-m-M+1)\Gamma(T+p-m+1)},
\end{equation}
where $\Gamma(\cdot)$ is Euler's Gamma function. Moreover, $\operatorname{det}({\bf I}_M-{\bf B}^H {\bf A}\A^H {\bf B})$ is distributed as the product of $M$ independent beta random variables, $z_m$, with parameters $\alpha_m = T-M+1-m$ and $\beta_m = M$, $m=1,\ldots,M$, i.e.
\begin{equation}
\operatorname{det}({\bf I}_M-{\bf B}^H {\bf A}\A^H {\bf B}) \sim \prod_{m=1}^M z_m, \quad z_m \sim {\rm Beta} (T-M+1-m,M).
\end{equation}
\end{thm}
An immediate consequence is that, at least for moderate values of $N$, we can upper bound the energy \eqref{eq:unionbound2} and hence the probability of error \eqref{eq:unionbound} of random codes $[{\bf X}_1],\ldots,[{\bf X}_K]$ when they are all independently and uniformly distributed:
\begin{cor}\label{cor:optimalrandom}
Assume that $N\leq T-2M$. For i.i.d. chosen $[{\bf X}_1],\ldots,[{\bf X}_K]$, the expected value of the chordal product energy \eqref{eq:unionbound2} is
\[
K(K-1)\prod_{m=1}^M\frac{(T-m)!(T-N-m-M)!}{(T-m-M)!(T-N-m)!},
\]
In particular, there exists a code such that the union bound \eqref{eq:unionbound} is at most:
\[
C(K-1)\rho^{-NM}\prod_{m=1}^M\frac{(T-m)!(T-N-m-M)!}{(T-m-M)!(T-N-m)!},
\]
where $C= \frac12\left( \frac{4M}{T}\right)^{NM} \frac{(2NM-1)!!}{(2NM)!!)}$.
\end{cor}
In Section \ref{sec:pdfcdf} we use Theorem \ref{th:expected} to compute exactly the probability density function and the cumulative density function of the random variable $\operatorname{det}({\bf I}_M-{\bf B}^H{\bf A}\A^H{\bf B})$ in the same hypotheses of the theorem. The expressions we get are exact and can be obtained in closed form for any fixed value of $M$. A reduced version of that result for $M=2$ is now shown:
\begin{cor}\label{cor:M2} Fix any $[{\bf B}]\in \mathbb{G}r(2,\mathbb C^T)$ with $T\geq4$. The probability that a randomly chosen $[{\bf A}]\in\mathbb{G}r(2,\mathbb C^T)$ satisfies $\operatorname{det}({\bf I}_2-{\bf B}^H{\bf A}\A^H{\bf B})\leq\delta\in(0,1]$ is exactly:
\[
F_2(\delta,T)=\frac12(T-1)(T-2)^2(T-3)\delta^{T-3}\left(\frac{1}{T-3}-\frac{2\delta}{(T-2)^2} -\frac{\delta^2}{T-1}+\frac{2\delta\log\delta}{T-2} \right),
\]
and similar formulas can be computed for the probability density functions $F_M(\delta,T)$ for higher values of $M$ (the case $M=1$ which yields $F_1(\delta,T)=\delta^{T-1}$ is quite trivial but it also follows from our approach).
\end{cor}
See the complete result in Corollary
\ref{cor:pdfdet}. As an illustrative example, Figs. \ref{Fig:PDFs} and \ref{Fig:CDFs} depict, respectively, the computed pdf and cdf of the chordal product for different values of $T$ and $M$.
\begin{figure}
\centering
\includegraphics[width=.60\textwidth]{images/PDFs.pdf}
\caption{Probability density functions of $\operatorname{det}({\bf I}_M-{\bf B}^H {\bf A}\A^H {\bf B})$, when $[{\bf B}]\in\mathbb{G}r(M,{\mathbb{C}}^{T})$ is fixed and $[{\bf A}]\in\mathbb{G}r(M,{\mathbb{C}}^{T})$ is uniformly distributed on the Grassmannian.}
\label{Fig:PDFs}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=.60\textwidth]{images/CDFs.pdf}
\caption{Cumulative distribution function of $\operatorname{det}({\bf I}_M-{\bf B}^H {\bf A}\A^H {\bf B})$, when $[{\bf B}]\in\mathbb{G}r(M,{\mathbb{C}}^{T})$ is fixed and $[{\bf A}]\in\mathbb{G}r(M,{\mathbb{C}}^{T})$ is uniformly distributed on the Grassmannian.}
\label{Fig:CDFs}
\end{figure}
Using the statistical characterization above, we have derived a lower bound
on the number of elements in any code {in the Grassmannian} with a given minimum value of chordal product $\delta$. Following \cite{Barg_TIT02}, we call this result a Gilbert-Varshamov bound since its proof mimics the argument of that classical result.
\begin{cor}[Gilbert--Varshamov lower bound]\label{cor:lowerbounddet}Assume that $T\geq2M$.
For any fixed $K\geq2$, there exists a code $[{\bf X}_1],\ldots,[{\bf X}_K]\in\mathbb{G}r(M,{\mathbb{C}}^{T})$ such that $\operatorname{det}({\bf I}_M-{\bf X}_i^H{\bf X}_j{\bf X}_j^H{\bf X}_i)\geq\delta$ where $\delta$ is the unique solution of the equation:
$$
F_M(\delta;T)= \frac{1}{K};\text{ that is }\delta=F_M^{-1}\left(\frac{1}{K};T\right)
$$
Equivalently, given $\delta\in(0,1)$, there exists a code consisting of $K\geq\frac{1}{F_M(\delta;T)}$ elements and satisfying $\operatorname{det}({\bf I}_M-{\bf X}_i^H{\bf X}_j{\bf X}_j^H{\bf X}_i)\geq\delta$ for $i\neq j$.
\end{cor}
\begin{exmpl}
In the case $M=1$ we have that
\[
\operatorname{det}({\bf I}_M- {\bf X}_i^H{\bf X}_j {\bf X}_j^H{\bf X}_i) = \sin^2 \theta,
\]
where $\theta\in[0,\pi/2]$ is the principal angle between the one-dimensional subspaces $[{\bf X}_i]$ and $[{\bf X}_j]$ in $\mathbb{G}(1,\mathbb{C}^T)$. That is to say, the chordal product coincides with the squared chordal distance. For uniformly distributed subspaces, the squared sine of the pairwise principal angle has cdf $F_1(\delta,T) = \delta^{T-1}$. The Gilbert-Varshamov bound shows that, for $\delta\in(0,1)$, there exist codes with cardinality $K$ and minimum chordal product $\delta = \sin^2 \theta$ such that
\begin{equation}
K > \delta^{-(T-1)} = \left( \sin \theta \right)^{-2(T-1)}.
\end{equation}
\end{exmpl}
\begin{exmpl}
Let us now take $T=10, M=2$. Assume that we want to allocate $K=2^{9}$ points in $\mathbb{G}r(M,{\mathbb{C}}^{T})$. Then, Corollary \ref{cor:lowerbounddet} says that there exists a code $[{\bf X}_1],\ldots,[{\bf X}_K]$ such that for all these points the chordal product is at least $\delta$, the unique solution of:
$$
\frac12(T-1)(T-2)^2(T-3)\delta^{T-3}\left(\frac{1}{T-3}-\frac{2\delta}{(T-2)^2} -\frac{\delta^2}{T-1}+\frac{2\delta\log\delta}{T-2} \right)=\frac1{K},
$$
that is
$$
\log_2\left(2016\delta^7\left(\frac17-\frac{2\delta}{64}-\frac{\delta^2}{9}+ \frac{\delta\log\delta}{4}\right)\right)=-9,
$$
which yields $\delta\approx0.2129$. The numerical algorithm in \cite{CuevasTCOM} produces in this case $[{\bf X}_1],\ldots,[{\bf X}_{2^9}]$ with minimum determinantal value $0.3958>0.2129$.
\end{exmpl}
\subsection{Historical discussion}\label{sec:history}
There exist several results on packings on Grassmannian spaces but they are rather centered in finding codes such that the mutual chordal distance between different elements $[{\bf X}_i],[{\bf X}_j]$ is close to maximal. For example, in \cite{rankin} we find bounds for the mutual distance of any code with a fixed number of elements (this is known as the Rankin bound). Gilbert--Varshamov bounds have also been obtained for that chordal distance by resorting to calculations of the volume of a metric ball of radius $\delta$ in $\mathbb{G}r(M,{\mathbb{C}}^{T})$, see \cite{Barg_TIT02} and \cite{Dai_TIT08}. The case that $\delta$ is sufficiently small was analyzed in \cite{Dai_TIT08}, \cite{Henkel2005}, and the real case has also been studied, see \cite{Conway96} and references therein. But to our knowledge our results are the first theoretical bounds on codes focusing on the explicit use of the chordal product, which is the key figure of merit in non--coherent communications.
In the case $M=1$ the Grassmannian becomes the projective space, the chordal product equals the squared chordal distance, and the literature is much more prolific, going back to \cite{Shannon_59} (although Shannon studied the case of the geodesic, not chordal, distance), \cite{Barg_TIT02} for the chordal and the geodesic distance and more recently \cite{jasper2019game} where a more complete set of references can be found. The optimal value of \eqref{eq:unionbound2} in that case has been studied in \cite{BE2018} and \cite{Anderson} as a case of Riesz energy, showing that the minimum value is equal to the average computed in Corollary \ref{cor:optimalrandom}, minus a term of the form
\[
O\left(K^{1+\frac{N}{T-1}}\right)=o(K^2),\quad\text{ for }N\leq T-2.
\]
\section{Proof of Theorem \ref{th:expected}}\label{sec:statcharact}
First assume that $p$ is an integer in the range of the hypotheses. Let ${\mathrm E}(p,T)$ be the expected value in the theorem (we omit the dependence on $M$ in the notation). By unitary invariance, we can assume that ${\bf B}=\binom{{\bf I}_M} {\bf 0}$. If we write the expected value using Proposition \ref{prop:integrales} and we pass to polar coordinates we get
\begin{align*}
{\rm E}(p,T)=&C(T)\int_{\tilde{{\bf A}}\in\mathbb{C}^{(T-M)\times M}}\frac{\operatorname{det}\left({\bf I}_M-({\bf I}_M+\tilde {\bf A}^H\tilde {\bf A})^{-1}\right)^p}{\operatorname{det}({\bf I}_M+\tilde {\bf A}^H\tilde {\bf A})^{T}}\,d\tilde{{\bf A}}\\
&=C(T)\int_{\tilde{{\bf A}}\in\mathbb{C}^{(T-M)\times M}}\frac{\operatorname{det}\left( {\bf A}^H\tilde {\bf A}\right)^p}{\operatorname{det}({\bf I}_M+\tilde {\bf A}^H\tilde {\bf A})^{T+p}}\,d\tilde{{\bf A}}\\
=&C(T)\int_0^\infty\rho^{2M(T-M)+2p M-1}\int_{\underset{\|\tilde{{\bf A}}\|_F=1}{\tilde {\bf A}\in\mathbb{C}^{(T-M)\times M}}}\frac{\operatorname{det}(\tilde {\bf A}^H\tilde {\bf A})^p }{\operatorname{det}({\bf I}_M+\rho^2\tilde {\bf A}^H\tilde {\bf A})^{T+p}}\,d\tilde{{\bf A}}\,d\rho,
\end{align*}
where we omit the dependence on $M$ in the constant:
$$
C(T)=\frac{1}{Vol(\mathbb{G}r(M,{\mathbb{C}}^{T}))}\stackrel{Lemma~\ref{lem:volumenG}}{=}\frac{(T-M)!\cdots(T-1)!}{\pi^{M(T-M)}1!\cdots(M-1)!}.
$$
Since the integrand of the inner integral depends only on the singular values of $\tilde {\bf A}$, we can take it to the set $\mathbb S_M^+$ consisting of ordered tuples of positive numbers $\sigma_1>\ldots>\sigma_M$ with the property that $\sigma_1^2+\cdots+\sigma_M^2=1$, see for example \cite[Th. 3.3]{IMAJNA}, that yields
\begin{multline*}
{\rm E}(p,T)=D(T)\times\\\int_0^\infty\rho^{2M(T-M)+2p M-1} \int_{\mathbb S_M^+}\frac{(\sigma_1\cdots\sigma_M)^{2p+2T-4M+1}\prod_{j\neq k}(\sigma_k^2-\sigma_j^2)^2 }{\prod_{m=1}^M(1+\rho^2\sigma_m^2)^{T+p}}\,d\sigma_1\cdots d\sigma_M\,d\rho,
\end{multline*}
where
\begin{align*}
D(T) =& \frac{C(T)Vol({\mathcal U}_{T-M})Vol({\mathcal U}_M)}{Vol({\mathcal U}_{T-2M})2^{M(T-M)}\pi^M}.
\end{align*}
It follows immediately that ${\rm E}(p,T)/D(T)= {\rm E}(0,T+p)/D(T+p)=1/D(T+p)$, that is,
\begin{align*}
{\rm E}(p,T)=& \frac{D(T)}{D(T+p)} \\
=& \frac{C(T)Vol({\mathcal U}_{T-M}) Vol({\mathcal U}_M)}{2^{M(T-M)}Vol({\mathcal U}_{T-2M})}\frac{2^{M(T+p-M)}Vol({\mathcal U}_{T+p-2M})}{C(T+p) Vol({\mathcal U}_{T+p-M}) Vol({\mathcal U}_M)}
\\
=& \frac{(2\pi)^{Mp}(T-M)!\cdots(T-1)!Vol({\mathcal U}_{T-M}) Vol({\mathcal U}_{T+p-2M})}{Vol({\mathcal U}_{T-2M})(T+p-M)!\cdots(T+p-1)! Vol({\mathcal U}_{T+p-M})}.
\end{align*}
The volume of the unitary group is known (see \cite[p. 28]{IMAJNA}):
$$
Vol({\mathcal U}_k)=\frac{(2\pi)^{k(k+1)/2}}{1!\cdots(k-1)!}.
$$
The theorem (for integer $p$ in the range) follows by substituting the known values in the constants above.
On the other hand, it is known that the $p$th moment of a beta distributed random variable with parameters $\alpha>0$ and $\beta >0$ denoted as $x \sim {\rm Beta}(\alpha,\beta)$ is \cite{Srivastavabook}
\begin{equation}
\label{eq:betamoments}
{\rm E}[x^p] = \frac{\Gamma(\alpha+\beta)\Gamma(\alpha+p)}{\Gamma(\alpha+\beta+p)\Gamma(\alpha)},
\end{equation}
so the $m$th product term in \eqref{eq:detmoment} corresponds to the $p$th moment of a beta distributed random variable with parameters $\alpha_m = T-M+1-m$ and $\beta_m = M$, thus proving that the distribution of $\operatorname{det}({\bf I}_M-{\bf B}^H {\bf A}\A^H {\bf B})$ is equivalent to the distribution of the product of $M$ independent beta random variables (this is an instance of the Hausdorff moments problem, hence the distribution is uniquely determined by its moments). Notice that \eqref{eq:betamoments} is valid for $p+ \alpha >0$, and since $m$ can get up to $M$ this entails to $p>2M-T-1$. Now that we have characterized $\operatorname{det}({\bf I}_M-{\bf B}^H {\bf A}\A^H {\bf B})$ as a product of beta distributed random variables, we can write down the formula for its moments for noninteger $p>2M-T-1$, finishing the proof of the theorem.
\section{Probability density function of the chordal product}\label{sec:pdfcdf}
Can we effectively recover the pdf of the random variable $x= \operatorname{det}({\bf I}_M-{\bf B}^H {\bf A}\A^H {\bf B})$ from its moments? If the density function is $f(x)$ and the moments are $\mathcal M_n$ then we have the classical formula:
\begin{equation}\label{eq:pdffrommoments}
f(x)=\int_{-\infty}^\infty e^{2i\pi xs}\sum_{n=0}^\infty\frac{(-2i\pi s)^n}{n!}\mathcal M_n\,ds.
\end{equation}
Following \cite[Th. 7]{SpringerThomson} a closed-form expression for the pdf can actually be written down in terms of certain special functions called Meijer $G$--functions. However, this expression is quite involved and requires extra work in practice for the derivation of bounds. In the following, we show that we can obtain simpler closed-form formulas for small values of $M =1,2,3$. They represent the most practical use cases in noncoherent communications. Moreover, we also provide a general recursive procedure to obtain the pdfs for higher values
\begin{cor}\label{cor:pdfdet}
Let $T\geq2M$. The probability density function (pdf) of $\operatorname{det}({\bf I}_M-{\bf B}^H {\bf A}\A^H {\bf B})$, when $[{\bf B}]\in\mathbb{G}r(M,{\mathbb{C}}^{T})$ is fixed and $[{\bf A}]\in\mathbb{G}r(M,{\mathbb{C}}^{T})$ is uniformly distributed on the Grassmannian, for $M = 1,2,3$ is:
\begin{align*}
M=1 \, \to \, f_{1}(x;T)=&(T-1)x^{T-2},\\
M=2 \, \to \, f_{2}(x;T)=&\frac12(T-1)(T-2)^2(T-3)x^{T-4}\left(1-x^2+2x\log x\right),\\
M=3 \, \to \, f_{3}(x;T)=&\frac1{288}(T-1)(T-2)^2(T-3)^3(T-4)^2(T-5)x^{T-6}\times\\
&\left(1+80x-162x^2+80x^3+x^4+24x\log x-24x^3\log x-36x^2\log^2x\right)
\end{align*}
The cumulative distribution function (cdf) $F_M(x,T)=\int_0^xf_M(s,T)\,ds$ for these three cases is respectively:
\begin{align*}
M=1\to&F_{1}(x;T)=x^{T-1}\\
M=2\to&F_{2}(x;T)=\frac12(T-1)(T-2)^2(T-3)x^{T-3}\left(\frac{1}{T-3}-\frac{2x}{(T-2)^2} -\frac{x^2}{T-1}+\frac{2x\log x}{T-2} \right)\\
M=3\to&F_{3}(x;T)=\frac1{288}(T-1)(T-2)^2(T-3)^3(T-4)^2(T-5)x^{T-5}\times Q,
\end{align*}
with
\begin{multline*}
Q=\frac{1}{T-5}
+\frac{80x}{T-4}
-\frac{24x}{(T-4)^2}
-\frac{162x^2}{T-3}
-\frac{72x^2}{(T-3)^3}
+\frac{80x^3}{T-2}\\
+\frac{24x^3}{(T-2)^2}
+\frac{x^4}{T-1}
+\frac{24x\log x}{T-4}
+\frac{72x^2\log x}{(T-3)^2}
-\frac{24x^3\log x}{T-2}
-\frac{36x^2\log^2x}{T-3}.
\end{multline*}
For arbitrary higher values of $M$ the pdf has the form:
\begin{align*}
& f_M(x;T) = (T-M)^M\prod_{m=1}^{M-1}(T-m)^m(T-M-m)^{M-m}\cdot\\
&\left[
\sum_{m=1}^M\sum_{l=1}^m\frac{A_{ml}(-1)^{l-1}}{(l-1)!}x^{T-m-1}\log^{l-1}x +
\sum_{m=1}^{M-1}\sum_{l=1}^{M-m}\frac{B_{ml}(-1)^{l-1}}{(l-1)!}x^{T-m-M-1}\log^{l-1}x
\right]
\end{align*}
where the $M^2$ coefficients $A_{ml}, B_{ml}$ can be obtained (e.g. with the aid of symbolic computation software) by solving the linear system of $M^2$ equations resulting from equating coefficients on both sides for the polynomial identity:
\begin{align*}
&\sum_{m=1}^M\sum_{l=1}^m A_{ml}(x-m)^{m-l}\prod_{i\neq m}^M (x-i)^i\prod_{i=1}^{M-1}(x-M-i)^{M-i} + \\
& +\sum_{m=1}^{M-1}\sum_{l=1}^{M-m} B_{ml}(x-M-m)^{M-m-l}\prod_{i=1}^M (x-i)^i\prod_{i\neq m}^{M-1}(x-M-i)^{M-i} = 1.
\end{align*}
\end{cor}
\begin{proof}
For $M=1$ and integer $p\geq0$ note that
$$
\int_0^1x^p \underbrace{(T-1)x^{T-2}}_{f_{1}(x;T)}\,dx= \frac{T-1}{T+p-1},
$$
and hence the claimed pdf satisfies Theorem \ref{th:expected} and must be the searched distribution. With the help of some integral formulas for the $\log$ function it is easy to check that
$$
\int_0^1x^pf_{2}(x;T)\,dx=\frac{(T-1)(T-2)^2(T-3)}{(T+p-1)(T+p-2)^2(T+p-3)},
$$
which again satisfies Theorem \ref{th:expected} and we are done. A more lengthy but trivial computation gives the case $M=3$.
These formulas and the general case for higher values of $M$ can be derived from the following procedure. The moments of the chordal product determinant from Theorem \ref{th:expected} are
$$
\mathcal{M}_p(T,M) = \prod_{m=1}^M\frac{(T-m)!(T+p-m-M)!}{(T-m-M)!(T+p-m)!}.
$$
By expanding the factorials and collecting terms in the product, this can be rewritten as
$$
\mathcal{M}_p(T,M) = \prod_{m=1}^M\left(\frac{T-m}{T+p-m}\right)^m\;\cdot\;\prod_{m=1}^{M-1}\left(\frac{T-M-m}{T+p-M-m}\right)^{M-m}
$$
so the denominator $D$ is the value at $x=T+p$ of the polynomial
$$
D(x) = \prod_{m=1}^M(x-m)^m\prod_{m=1}^{M-1}(x-M-m)^{M-m}.
$$
Notice that this is a product of all-different real root factors $(x-\alpha)$ with varying multiplicities, so its inverse has a partial fraction decompositio
$$
\frac{1}{D} = \sum_{m=1}^M\sum_{l=1}^m\frac{A_{ml}}{(x-m)^l} + \sum_{m=1}^{M-1}\sum_{l=1}^{M-m}\frac{B_{ml}}{(x-M-m)^l},
$$
for some coefficients $A_{ml},\, B_{ml}\in\mathbb{R}$. Following one of the usual procedures to solve for these coefficients, for general $x$, multiplying by $D(x)$ on both sides yields the polynomial equation of order $M^2-1$:
\begin{align*}
&\sum_{m=1}^M\sum_{l=1}^m A_{ml}(x-m)^{m-l}\prod_{i\neq m}^M (x-i)^i\prod_{i=1}^{M-1}(x-M-i)^{M-i} + \\
& +\sum_{m=1}^{M-1}\sum_{l=1}^{M-m} B_{ml}(x-M-m)^{M-m-l}\prod_{i=1}^M (x-i)^i\prod_{i\neq m}^{M-1}(x-M-i)^{M-i} = 1.
\end{align*}
Expanding and gathering terms by powers of $x$, one can equate the coefficient of $x^0$ to $1$ and the coefficients of $x^n$, for $n=1,\dots, M^2-1$, to $0$ to obtain a linear system of $M^2$ equations in the $M^2$ coefficients $A_{ml},\, B_{ml}$, and solve for them.
Now, notice that by the Laplace transform properties for $a,b$ nonnegative integers
$$
\int_0^1 x^a\log^b x\,dx = (-1)^b\mathcal{L}[t^b](a+1) = (-1)^b\frac{b!}{(a+1)^{b+1}},
$$
and thus
$$
\frac{1}{(T+p-m)^l}=\frac{(-1)^{l-1}}{(l-1)!}\int_0^1 x^p x^{T-m-1}\log^{l-1}x\,dx.
$$
Hence, by expressing every term of the partial fraction decomposition in this integral form, the function $f_M(x;T)$ can be identified inside the moment function written as an integral:
\begin{align*}
\mathcal{M}_p(T,M) = & (T-M)^M\prod_{m=1}^{M-1}(T-m)^m(T-M-m)^{M-m}\cdot\\
& \cdot\int_0^1 x^p\left[
\sum_{m=1}^M\sum_{l=1}^m\frac{A_{ml}(-1)^{l-1}}{(l-1)!}x^{T-m-1}\log^{l-1}x + \right.\\
&\left. \sum_{m=1}^{M-1}\sum_{l=1}^{M-m}\frac{B_{ml}(-1)^{l-1}}{(l-1)!}x^{T-m-M-1}\log^{l-1}x
\right] dx,
\end{align*}
where the factors in front of the integral all come from the numerator over $D$ in $\mathcal{M}_h(T,M)$. This finishes the proof of the corollary.
\end{proof}
\section{Proof of Corollary \ref{cor:lowerbounddet}} \label{sec:GilbertVarshamov}
Let $\delta$ be the unique solution of the equation $F_M(\delta;T)=K^{-1}$. Let $G$ be the maximum number of points in $\mathbb{G}r(M,{\mathbb{C}}^{T})$ that can be allocated with the claimed property. We must prove that $G\geq K$. Indeed, assume that $G<K$ an let ${\bf X}_1,\ldots,{\bf X}_G$ be a code with $\operatorname{det}({\bf I}_M-{\bf X}_i^H{\bf X}_j{\bf X}_j^H{\bf X}_i)\geq\delta$ for all $1\leq i,j\leq G$. We note that
\begin{multline*}
\frac{1}{Vol(\mathbb{G}r(M,{\mathbb{C}}^{T}))}Vol\left(\cup_{i=1}^G\{[{\bf A}]\in\mathbb{G}r(M,{\mathbb{C}}^{T}):\operatorname{det}({\bf I}_M-{\bf X}_i^H{\bf A}\A^H{\bf X}_i)\leq\delta\}\right)\leq\\\frac{1}{Vol(\mathbb{G}r(M,{\mathbb{C}}^{T}))}\sum_{i=1}^G Vol\left(\{[{\bf A}]\in\mathbb{G}r(M,{\mathbb{C}}^{T}):\operatorname{det}({\bf I}_M-{\bf X}_i^H{\bf A}\A^H{\bf X}_i)\leq\delta\}\right)=\\
GF_M(\delta;T)=\frac{G}{K}<1,
\end{multline*}
and we thus deduce that there exists $[{\bf A}]\in\mathbb{G}r(M,{\mathbb{C}}^{T})$ such that
$$
\operatorname{det}({\bf I}_M-{\bf X}_i^H{\bf A}\A^H{\bf X}_i)>\delta\quad \forall\;1\leq i\leq G.
$$
But then the code ${\bf X}_1,\ldots,{\bf X}_G,{\bf X}_{G+1}$ with ${\bf X}_{G+1}={\bf A}$ also satisfies the claimed property and has $G+1$ points, which contradicts the definition of $G$.
\section[Alternative parameterization of $\mathbb{G}r(M,{\mathbb{C}}^{T})$ ]{Alternative parameterization of the Grassmannian and the density function of $\tilde {\bf A}$ in $\binom{{\bf I}_M}{\tilde {\bf A}}\in\mathbb{G}r(M,{\mathbb{C}}^{T})$}\label{sec:parametrization}
We recall the volume of the Grassmannian for completeness.
\begin{lem}\label{lem:volumenG}
The volume of the Grassmannian $\mathbb{G}r(M,{\mathbb{C}}^{T})$ is:
$$
Vol(\mathbb{G}r(M,{\mathbb{C}}^{T}))=\frac{\pi^{M(T-M)}1!\cdot 2!\cdots(M-1)!}{(T-M)!\cdot (T-M+1)!\cdots(T-1)!}
$$
\end{lem}
\begin{proof}
This is a classical fact: since the Grassmannian is formally defined as a quotient of the Stiefel manifold $\mathbb{S}t(M,{\mathbb{C}}^{T})$ (i.e. the set of $T\times M$ complex matrices ${\bf X}$ such that ${\bf X}^H{\bf X}={\bf I}_M$) by the unitary group $\mathcal U_M$, the volume of $\mathbb{G}r(M,{\mathbb{C}}^{T})$ is the quotient of the volumes of the Stiefel and unitary matrices which is well--known, see for example \cite{Hua} (note that there exist several normalizations for the Riemannian structure of the classical groups, leading to different volume formulas. We use the standard that considers $\mathbb{S}t(M,{\mathbb{C}}^{T})$ and $\mathcal U_M$ as submanifolds of their ambient affine spaces, with the inherited structure).
\end{proof}
Recall that a $p\times n$ complex matrix ${\bf X}$ is distributed as a complex matrix-variate $t$ distribution with $\nu$ degrees of freedom when its density is given by
\begin{equation}
p({\bf X}) = C^{-1} \operatorname{det}({\bf I}_p + {\bf X}^H{\bf X})^{-(\nu +p+n-1)}.
\end{equation}
\begin{prop}\label{prop:integrales}
Let $T\geq 2M$. If $[{\bf A}]$ is uniformly distributed in $\mathbb{G}r(M,{\mathbb{C}}^{T})$ and we write $[{\bf A}]=\begin{bmatrix}{{\bf I}_M}\\{\tilde {\bf A}}\end{bmatrix}$ (note that there exists a unique representative of that form), then $\tilde {\bf A}$ has density
\begin{equation}
\frac{1}{Vol(\mathbb{G}r(M,{\mathbb{C}}^{T}))\operatorname{det}({\bf I}_M+\tilde {\bf A}^H\tilde {\bf A})^{T}}.\label{eq:matrix_t_dist}
\end{equation}
Hence, $\tilde{\bf A}\in\mathbb C^{(T-M)\times M}$ follows a matrix--variate $t$ distribution with $\nu=1$ degrees of freedom.
In other words, for any measurable non--negative or integrable function $f:\mathbb{G}r(M,{\mathbb{C}}^{T})\to\mathbb{C}$,
\begin{align*}
\int_{[{\bf A}]\in\mathbb{G}r(M,{\mathbb{C}}^{T})}f([{\bf A}])\,d[{\bf A}]=&\int_{\tilde{{\bf A}}\in\mathbb{C}^{(T-M)\times M}}\frac{f\binom{{\bf I}_M}{\tilde {\bf A}}}{\operatorname{det}({\bf I}_M+\tilde {\bf A}^H\tilde {\bf A})^{T}}\,d\tilde{{\bf A}}
\\
=&\int_{\tilde{{\bf A}}\in\mathbb{C}^{(T-M)\times M}}\frac{f\binom{({\bf I}_M+\tilde{\bf A}^H\tilde{\bf A})^{-1/2}}{\tilde {\bf A}({\bf I}_M+\tilde{\bf A}^H\tilde{\bf A})^{-1/2}}}{\operatorname{det}({\bf I}_M+\tilde {\bf A}^H\tilde {\bf A})^{T}}\,d\tilde{{\bf A}}.
\end{align*}
\end{prop}
\begin{proof}
This result has been proved in \cite[Prop. 1 and Cor. 1]{GrassLattice}, by showing that both sides of the equality are equal to
\[
\int_{{\bf W}\in \mathbb C^{(T-M)\times M}, \|{\bf W}\|_{op}<1}f\left(\begin{bmatrix}\sqrt{{\bf I}_M-{\bf W}^H{\bf W}}\\{\bf W}\end{bmatrix}\right)\,d{\bf W},
\]
with $\|\cdot\|_{op}$ the operator norm. Note that the two integrals on ${\bf C}^{(T-M)\times M}$ are equal since $f$ is a function defined in the Grassmannian and hence its value is independent of the choice of representatives. Moreover, the advantage of the last expression in the proposition is that
$$
{\bf X}=\binom{({\bf I}_M+\tilde{\bf A}^H\tilde{\bf A})^{-1/2}}{\tilde {\bf A}({\bf I}_M+\tilde{\bf A}^H\tilde{\bf A})^{-1/2}}
$$
is a Stiefel matrix, i. e. it satisfies ${\bf X}^H{\bf X}={\bf I}_M$.
\end{proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,954 |
\subsection*{Acknowledgement}
We appreciate for the insightful discussion by S. Nakajima. This work was partially supported by grants from the Japanese Ministry of Education, Culture, Sports, Science and Technology to K.T. (17H04948, 25106002), J.H. (16H00881), A.S. (15H04116), M.K. (17H04694, 16H06538), M.S. (16H00736, 16H02866), and I.T. (17H00758, 16H06538), and JST PRESTO to A.S. (Grant Number JPMJPR15N7), M.K. (Grant Number JPMJPR15N2), M.S. (Grant Number JPMJPR16N6), JST CREST to I.T. (Grant Number JPMJCR1302, JPMJCR1502), RIKEN Center for Advanced Intelligence Project to J.H and I.T., and by JST support program for starting up innovation-hub on materials research by information integration initiative to A.S., M.K., K.S., A.K., and I.T.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 487 |
Międzyleś – dawna gromada, czyli najmniejsza jednostka podziału terytorialnego Polskiej Rzeczypospolitej Ludowej w latach 1954–1972.
Gromady, z gromadzkimi radami narodowymi (GRN) jako organami władzy najniższego stopnia na wsi, funkcjonowały od reformy reorganizującej administrację wiejską przeprowadzonej jesienią 1954 do momentu ich zniesienia z dniem 1 stycznia 1973, tym samym wypierając organizację gminną w latach 1954–1972.
Gromadę Międzyleś z siedzibą GRN w Międzylesiu utworzono – jako jedną z 8759 gromad na obszarze Polski – w powiecie wołomińskim w woj. warszawskim, na mocy uchwały nr VI/10/23/54 WRN w Warszawie z dnia 5 października 1954. W skład jednostki weszły obszary dotychczasowych gromad Dąbrowica, Grabów, Franciszków, Józefin, Jaźwie, Międzyleś, Międzypole, Pawłów, Rudniki, Szczepanek, Szlędaki i Wólka Dąbrowicka ze zniesionej gminy Międzyleś w tymże powiecie. Dla gromady ustalono 27 członków gromadzkiej rady narodowej.
31 grudnia 1961 do gromady Międzyleś włączono wsie Beredy, Borucza, Gołębiowizna, Kąty Wielgie i Paluchy ze zniesionej gromady Kąty-Miąski w tymże powiecie.
1 stycznia 1969 do gromady Międzyleś włączono przysiółek Stasinów z gromady Ostrówek w tymże powiecie.
Gromada przetrwała do końca 1972 roku, czyli do kolejnej reformy gminnej.
Przypisy
Miezzxdzyleszzx (powiat wolxominxski) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 131 |
Q: Clicking outside of an Angular Bootstrap Datepicker in IE throws 'contains' error When I click outside of an open datepicker box in IE, I get the following error:
'object doesn't support property or method 'contains' Bootstrap datepicker'
The datepicker does not close. I've come across many fixes which involve modifying the Bootstrap source, but I would prefer to not go this route. (IE doesn't have a contain method)
I was able to fix the issue by calling this function on the top parent div:
<div class="clearfix" ng-click="formClicked($event)">
<div class="form-group required">
<label for="shipTo">Ship-To #</label>
<select id="shipTo" class="form-control input-sm" ng-model="model.orderInfo.accountId"></select>
</div>
<div class="col-md-6">
<div class="form-group required">
<label for="shipDate">Ship Date</label>
<div class="input-group calendar-box">
<input id="shipDate" ng-model="model.orderInfo.shipDate" min-date="model.shipDateMin" max-date="model.shipDateMax" class="form-control input-sm" ng-click="dateOpen($event, 'shipDateOpened')" type="text" datepicker-popup="{{model.datePickFormat}}" is-open="model.shipDateOpened" ng-change="setCancelDate()" ng-readonly="true" required>
<div class="input-group-addon cursor-pointer calendar-icon" ng-click="dateOpen($event, 'shipDateOpened')"></div>
</div>
</div>
</div>
<div class="col-md-6">
<div class="form-group required">
<label for="cancelDate">Cancel Date</label>
<div class="input-group calendar-box">
<input id="cancelDate" ng-model="model.orderInfo.cancelDate" min-date="model.cancelDateMin" max-date="model.cancelDateMax" class="form-control input-sm" ng-click="dateOpen($event, 'cancelDateOpened')" type="text" datepicker-popup="{{model.datePickFormat}}" is-open="model.cancelDateOpened" ng-change="checkCancelDate()" ng-readonly="true" required>
<div class="input-group-addon cursor-pointer calendar-icon" ng-click="dateOpen($event, 'cancelDateOpened')"></div>
</div>
</div>
</div>
</div>
And the function looks like this:
$scope.formClicked = function($event){
$log.debug('form clicked');
$event.preventDefault();
$event.stopPropagation();
$scope.model.shipDateOpened = false;
$scope.model.cancelDateOpened = false;
};
The problem is, now on my mobile view this formClicked($event) function is fired when I try to tap on the select dropdown and it will not open. Is there a better solution to this problem, or is there a way I can conditionally render an ng-click when I am in my desktop view?
A: I believe it is this piece of code that causes the problem :
var documentClickBind = function(event) {
var popup = $popup[0];
var dpContainsTarget = element[0].contains(event.target);
// The popup node may not be an element node
// In some browsers (IE) only element nodes have the 'contains' function
var popupContainsTarget = popup.contains !== undefined && popup.contains(event.target);
if (scope.isOpen && !(dpContainsTarget || popupContainsTarget)) {
scope.$apply(function() {
scope.isOpen = false;
});
}
};
As the comment says, and as MDN states - neither IE nor Edge have a Node.contains() and apparently MS have no plans for implementing it (the problem is raised several times, this for example is simply just closed). So a polyfill would be the way to deal with this problem for good :
if (!Node.prototype.contains) {
Node.prototype.contains = function contains(node) {
if (!(0 in arguments)) {
throw new TypeError('1 argument is required');
}
do {
if (this === node) {
return true;
}
} while (node = node && node.parentNode);
return false;
}
}
A slightly modified version of this, originally based on this suggestion.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,154 |
Q: How to target the specific JSF component with unique id? In my case I want to refresh the component with ID baseTab. It is itself contained if form main and tabView tabs. The absolute reference to the component is main:tabs:baseTab. and this is what I need to refer to for example in update attribute:
<p:commandButton update="main:tabs:baseTab"/>
The problem with such full ID is that it is long and can be easily change when I change something in component hierarchy. The ID baseTab is itself unique so I should be, at least theoretically, able to refer this component direcly. But how I can do that?
What is the syntax to refer the component via unique ID? I've tried the following:
*
*:baseTab
*baseTab
*main:baseTab
*:tabs:baseTab
And none of that was working, each was causing the page error that such component does not exist....
A: I can see the value in not making this reference depend on the component tree structure.
It should be possible to leverage the component binding for this because update can take a ValueExpression.
Define a map in request scope using (for example) a faces-config.xml:
<managed-bean>
<managed-bean-name>bind</managed-bean-name>
<managed-bean-class>java.util.HashMap</managed-bean-class>
<managed-bean-scope>request</managed-bean-scope>
</managed-bean>
Bind the target to the map:
<h:foo binding="#{bind.someIdForBinding}" />
Reference this control's client identifier in your button:
<p:commandButton update=":#{bind.someIdForBinding.clientId}"/>
The target component will be put into the binding map when the tree is created/restored.
Note that this code is untested.
A: Don't think it is possible unless you components are in the same parent.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,873 |
Uganda holidays
Trail of hope for Uganda's lost Pygmy tribe
A new initiative is taking Uganda's Batwa Pygmies back into the land they lost to conservation, and giving tourists a rare glimpse of their lives
Back on track in Uganda ... the Batwa community pose for a farewell photo. Photograph: Jessica Deane
Simon Collis
We've barely entered the national park when our guide stops suddenly and holds up a hand to call for silence. His eyes gaze deep into the forest and he crouches. We do the same. Our hearts beat fast. Has he spotted something already? A golden monkey? A forest elephant?
Without warning, he yells, claps his hands and chants an ancient prayer to Biheko, god of the forest, just as his ancestors have always done. Stephen is no ordinary guide, and this, the Batwa Cultural Trail, is no ordinary walk in the woods. The woods in question are the dense forest of Mgahinga Gorilla national park, a 33 sq km area among the volcanic Virunga mountains in the far southwestern corner of Uganda. Though small, the park is home to a vast diversity of plant, animal and bird life, including the famed mountain gorilla, a mainstay of Uganda's tourist industry. Yet these rare beasts are not the only ones to lay claim to the Virungas.
Stephen is one of the Batwa, the "Pygmy people" indigenous to these mountain slopes. Evicted from their homes when the forest was gazetted as a national park in 1991, they are now a displaced ethnic group threatened with extinction. And though interaction between tourists and Batwa is hardly a new thing, control over excursions has always rested solely with outsiders. The Batwa, mainly excluded from Ugandan society, sing when they're told to, dance when they're told to and must be grateful for what they're given. Now, for the first time, the Batwa are taking control. This trail is the first time that the Batwa have had a direct stake in the tourism they're engaged in, bringing income direct to their communities.
These nomadic hunter-gatherers are widely acknowledged to have been the first human residents of forest areas stretching across much of what is now Uganda, Rwanda, Burundi and the Democratic Republic of Congo. As other ethnic groups arrived, cutting the forest to provide land for crops and livestock, Batwa populations became fragmented – but at least there was enough woodland for their environmentally sustainable way of life to endure.
For Ugandan Batwa, everything changed in 1991 with the creation of formal conservation areas that outlawed all human activity in the Virungas and in nearby Bwindi. Suddenly forced to live outside of the forest, unable to return to hunt small animals, collect wild honey or gather fruits, the Batwa found their traditional skills and vast knowledge of the forest ill-suited to life outside it.
They never sought to own the land they lived on, and so when they were evicted they were deemed unworthy of compensation. Now the majority are landless squatters, some of the poorest inhabitants of one of the world's poorest countries, watching as tourists buy $500 permits to visit gorillas in the forest that was once theirs.
The Batwa Cultural Trail is a new initiative launched by the Uganda Wildlife Authority and the United Organisation for Batwa Development in Uganda. It is a fairly gentle five- to six-hour nature walk through the lower slopes of the Virungas and means that for the first time the Batwa have a stake in the conservation and management of the national park, even though they still live outside it. Communities such as Stephen's receive payments that allow them to buy food, clothes, soap and other daily necessities; for tourists, this is an increasingly rare opportunity to see the forest as it has been viewed for millennia, not just as an animal habitat but as a human one too, a vastly complex combination of larder, medicine cabinet, home and temple.
Our day begins in Kisoro, the battered and dusty town at the base of the Virungas, about eight hours by bus from Uganda's capital, Kampala. Black volcanic rocks litter the roadsides, broken only by a central patch of greenery where goats, sheep and cattle spend the day grazing. Despite the presence of the Kindly Service Station and the Peace and Loving Bar, Kisoro would be considered pleasant but easily forgettable were it not for the 4,127m volcanic peak of Muhabura that glowers overhead, its cone invariably shrouded in a mass of swirling cloud.
The imposing slopes of Mount Muhabara. Photograph: Jessica Deane
It's towards Muhabura that we depart. At Muhabura base camp we meet our interpreter, Benjamin, and our Batwa guides, Stephen, George Wilson and Safari. They aren't as short as we expect but they all exude a wiry toughness, and from the moment of Stephen's prayer to Biheko it's clear that the Batwa's relationship with the forest is as deep as the woods themselves.
We're introduced to the fruits that form a pre-hunting breakfast, and to others that the Batwa give to their children as toys or even dolls. We're shown leaves that when ground into a paste can ward off evil spirits, roots that can cure infestation, plants that lower blood pressure and the black crust of an ants' nest that alleviates fungal skin disease. Nothing here is as it seems; where we see bright yellow fruits, the Batwa see the ingredients of natural soap.
Of everything that we're shown, it's the long, sinuous creepers that hang from the trees that best exemplify the Batwa's connection with the forest. Stephen tells us that, when cut into strings and dried, the creepers could be woven into bags. "They were so strong they would last all the way to Congo or Rwanda," he tells us proudly. When the Batwa wore animal hides, these strings would form their belts. The name of the creepers, umuse, translates as "a cousin whom you pray with and respect". The forest isn't just the Batwa's home – it's their family.
Interpreter Benjamin, right, and guide Stephen with the venerated umuse creepers Photograph: Jessica Deane
Looking at Mgahinga through the Batwa's eyes is a source of wonder but also of sadness. Stephen picks at the umuse with his fingers, then looks us in the eye. "Once we could exchange these strings for food or money. Now we've lost them, and our lives have changed completely."
When we stop for lunch in a clearing, we ask Stephen about his life since he left the forest. Like most of his people, his family squats on poor-quality land belonging to non-Batwa. Its value, two million Ugandan shillings (around £580), is way beyond anything Stephen will ever be able to afford. Instead, his community's human waste will gradually raise the land's fertility, and after about a year he'll be moved on.
"It isn't like when we were in the forest," he says. "There we could sleep in a hollow, or a shelter made from branches. Now we need real shelters, and when we move we have to build them again, and again, and again. Because we move from place to place, our children can't go to school."
With this initiative so new, it's difficult to know how much the trail will benefit Batwa communities. Stephen knows that the income will help his people survive from one day to the next but he's unsure about the long-term future. "We don't have a way to support ourselves," he says. "Our children aren't studying, they have no food to eat, they get sick and die. We're dependent on others to lend us land. If that ended, the Batwa would be finished forever."
The children of Safari, 38, and Stephen, 40, can barely remember the lives their people once led, yet losing touch with the forest isn't Stephen's biggest fear. "Of course we're sad. We're worried. We used to live long lives because of what the forest gave us. One woman lived to 120 because she was here. Now, we three are some of the oldest in our communities.
"The hope is to be given land of our own. That would be peace. Even if our children had no access to the forest, they could have a future."
As the trail continues, we're more aware than ever that the skills we're witnessing – trapping animals with snares made from branches and vines, finding and collecting wild honey, tracking buffalo and bushbucks (antelopes) – are consigned to history. For our guides, this was real, everyday life, learned from their fathers in the isolation of the forest. Our children, like the Batwa's, are unlikely ever to witness it first-hand.
We arrive at Garama cave, a 200m-long lava tube beneath Mount Gahinga. This chamber was once a royal residence, the sacred heart of the forest, a meeting place, food store, court of law. We enter and Stephen tells us that, in the past, non-Batwa would not have been allowed access. Crouching beneath the low, damp ceiling, we make our way to the depths of the cavern. Benjamin extinguishes his lantern and we stand for a moment in the pure darkness, listening to drops of water echoing in the blackness, the sudden fluttering of bats. Then we hear a low hum, which becomes a voice, and another, and another, music piercing the darkness, 20, 30 men and women together, filling Garama with song as in centuries past.
The lantern is lit again, and the chambers in front and to our right are filled with Batwa from local communities. First they sing to us a lament, remembering what they had and what they have lost, the forest they love and the lives they now lead. Then they sing to us a welcome song, because we are visitors to those who have received so few, new friends to those who have been forgotten.
This was no ordinary walk in the woods.
Ethical holidays
Ethical and green living | {
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Home / Artists / Nancy Bowen
Nancy Bowen
Region: Brooklyn, NY
MacDowell fellowships: 2010, 2017
More: nancybowenstudio.com
Nancy Bowen is a mixed media artist known for her eclectic mixtures of imagery and materials in both two and three dimensions. Her work offers a poetic commentary on our quickly changing material culture. Like an artistic archeologist in this age of globalization and post-industrialization, she salvages (often disappearing) ornament and craft traditions, and incorporates them into sculpture and collages.
Bowen has had over a dozen solo exhibitions throughout the United States and Europe including the Kentler International Drawing Space, Lesley Heller Gallery, and Annina Nosei Gallery in NYC, Galerie Farideh Cadot in Paris, the Betsy Rosenfield gallery in Chicago, and the James Gallery in Houston. At MacDowell, she worked on a series of collages for her solo exhibition "For Each Ecstatic Instant" at the Kentler International Drawing Space in September 2018.
In 2017 she won an Anonymous was a Woman Award. She has also won award from the National Endowment for the Arts, the New York Foundation for the Arts, MacDowell, Yaddo, The Jentel Foundation, the Brown Foundation Fellowship at Dora Maar House, and the European Ceramic Work Center among others. She received a B.F.A. from the School of the Art Institute of Chicago and an M.F.A. from Hunter College (CUNY). She is currently an associate professor of Sculpture at Purchase College, S.U.N.Y. She maintains a studio in the Brooklyn Navy Yard.
Eastman (formerly Shop)
Nancy Bowen worked in the Eastman (formerly Shop) studio.
Thanks to the generous support of MacDowell Fellow and board member Louise Eastman, a century-old farm building has been reinvented as a modern, energy efficient live and workspace for visual artists. Originally built to provide storage when the residency program was expanding, this small barn was simply converted for studio use in the mid-1950s with the… | {
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{"url":"https:\/\/tomate.wordpress.com\/2016\/06\/02\/tightening-the-uncertainty-principle-for-the-currents\/","text":"# Tightening the uncertainty principle for the\u00a0currents\n\nWe have a new paper out in the arXiv! It\u2019s a rather technical paper on how to\u00a0improve on certain \u00a0thermodynamic uncertainty principles recently proved for stochastic dynamics.\n\nThe uncertainty principle of Heisenberg is one of the coolest concepts in all of physics, so it is not surprising that fields of research close and far to quantum mechanics have searched for similar principles, at times successfully. It has recently been the case for Stochastic Thermodynamics, the theory that formulates in a proper manner thermodynamics, extending it to nonequilibrium and fluctuating systems. The thermodynamic uncertainty principle roughly states that\n\n\u201cThe more precise the measurement of a thermodynamic current, the greater the total mean dissipation\u201d.\n\nThe more significant of our contributions to this\u00a0subject is that we managed to replace the \u201ctotal mean dissipation\u201d by the \u201cleast possible mean dissipation in a system that can sustain that particular current\u201d (given a whole bunch of other assumptions, of course\u2026). A sort of minimum entropy production principle. Furthermore, we argued\u00a0that uncertainty relation for the so-called Fano factor the dissipation rate $\\sigma$\n\n$\\frac{\\mathrm{var}\\;\\sigma}{\\mathrm{mean}\\;\\sigma} \\geq 2$\n\nis the more significant of these uncertainty relations.\n\nBut that\u2019s not what I want to talk about here. I\u2019d rather\u00a0collect a few random thoughts on quantum vs. stochastic uncertainties.\n\nUncertainty relations in quantum physics relate the variances of two canonically conjugate observables, such as position and momentum. The thermodynamic uncertainty displayed above relates the variance of the\u00a0dissipation rate and its mean. This is indeed quite different: as usually the case, the temptation to reconnect quantum and stochastic phenomena hits against a square too much: probabilities are squared* wave functions, variances are squared* means etc. (*of course, not literally!).\n\nHowever, in quantum physics a number of very different uncertainty relations have been derived. In particular I like\u00a0the energy-time uncertainty relation.\u00a0Strictly speaking, since time is not an operator but a parameter, there is no time-energy uncertainty in the same sense that there is position-momentum uncertainty. A quote attributed to Landau says that \u201cTo violate the time-energy uncertainty relation all I have to do is measure the energy very precisely and then look at my watch.\u201d However, one can indeed make sense of such a relation in the following form:\n\n\u201cThe uncertainty in time is expressed as the average time taken, starting in state \u03c8, for the expectation of some arbitrary operator A to change by its standard deviation.\u201d\n\nThe quote is taken from the always excellent diary\u00a0of John Baez. Now, that makes things much closer in spirit to the thermodynamic uncertainty principle, in that we are not talking about the variance of time, but of the actual time it takes to do something. And moreover, it confronts the variance and the expectation value of the same observable. This starts to echo to thermodynamic uncertainty principle.\u00a0In particular, in this recent paper\n\nE. Roldan\u00a0et al., Decision making in the arrow of time, Phys. Rev. Lett. 115, 250602 (2015)\n\nit is stated that\n\n\u201cThe steady state entropy production rate of a stochastic process is inversely proportional to the minimal time needed to decide on the direction of the arrow of time\u201d.\n\nI find that things might actually converge somehow\u2026","date":"2018-01-18 00:18:48","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\": 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\": 0.7950387597084045, \"perplexity\": 598.5273097711549}, \"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\/1516084887024.1\/warc\/CC-MAIN-20180117232418-20180118012418-00134.warc.gz\"}"} | null | null |
❮ # 441 Jean-Michel Besnier # 443 Charles Johnson ❯
# 442 Dennis Washington $5.76B
Random fact: Worked as a mechanic and newspaper delivery man as a teenager.
Washington is the owner of Washington Companies, an industrial conglomerate with interests in copper mining, waste remediation, shipyards and railroads. The Missoula, Montana-based company sold about 900 miles of track in 2022 for about $2 billion. He also controls a major stake in Seaspan, which has more than 100 vessels.
Last change -$1.41M ( -0.0%)
YTD change +$204M ( +3.7%)
Industry Industrial
Dennis Washington's net worth of $5.76B can buy ...
The majority of Washington's fortune is derived from his sole ownership of the Washington Companies, a Missoula, Montana-based conglomerate that has businesses split among eight main subsidiaries. The value of each subsidiary is based on information disclosed by Washington Companies and the metrics of publicly traded peer companies.
Washington's Montana Resources unit is valued by multiplying US Geological Survey average mineral prices by the company's production numbers as provided in a January 2021 article in the Montana Standard. Southern Railway of British Columbia is valued based on rail company BNSF's February 2022 agreement to claim the rail properties of what was once the Washington Companies' most valuable asset, Montana Rail Link, for approximately $2 billion. His closely held marine assets, owned through an association of Canadian companies called Seaspan, are valued by using fleet data provided by Seaspan and ship-broker sales reports.
The valuation of Washington's Vancouver Drydock, one of the Seaspan firms, is based on Canadian government projections of revenue issued in conjunction with a federal ship contract. It's then calculated at 1x sales.
Washington separately owns around one-fifth of the publicly traded asset management firm Atlas Corporation, incorporated in the Marshall Islands. One of the two companies wholly owned by Atlas is Seaspan Corporation, a manager and charterer of more than 100 container ships.
The value of winglet manufacturer Aviation Partners is based on 2020 revenue of $422 million, derived from disclosures on the total number of aircraft sporting its winglets and production costs.
In April 2022, Washington's net worth was recalculated to account for the sale of his Montana Rail Link assets for below their estimated prior value, resulting in a decline of $2.5 billion.
Larry Simkins, Washington's chief financial officer, said the billionaire declined to comment on the calculation of his net worth.
Family: Married, No children
Dennis Washington was born in Spokane, Washington, during the Great Depression. He moved around the Pacific Northwest with his family to places where work could be found.
At age 8, while his father was working as a dock hand, Washington contracted polio. An aunt smuggled him out of the quarantined town where they were living to a hospital in Seattle, where he was one of the early recipients of heat therapy. Returning home a year later, he learned his parents were divorcing. Over the following years, he lived with relatives in three states before settling in Missoula, Montana, with his grandmother, whom he's described as the greatest influence on him as a young man.
Washington was self-sufficient by age 14, selling newspapers, working as a mechanic and shining shoes, according to a biography provided by Washington Companies. By age 26, he was an executive at a construction company. He decided in 1964 to go into business for himself, backed by a loan from a Caterpillar dealer. Early work included cutting a parking lot at the summit of Glacier National Park and building roads for the U.S. Forest Service. He was the largest contractor in the state five years later.
He purchased the dormant Continental Mine in Butte, Montana in 1985 and restarted copper production. He diversified into other businesses, including heavy equipment sales -- he is a Komatsu dealer -- and railroads such as Montana Rail Link, which operates around 900 miles of track mostly in the state of Montana.
Washington's business centered around construction until 2007, when the business was sold for about $2.6 billion. He also owns several closely held entities that use the Seaspan name and operate shipyards, ferries, tugboats and barges, primarily in the Vancouver, British Columbia, region.
With his wife, Phyllis, Washington splits his time between Montana; Palm Springs, California; and his yacht, the 332-foot Attessa IV. Their philanthropic efforts include providing life-changing experiences for disadvantaged children. Two sons, Kevin and Kyle, are executives at Seaspan.
1934 Dennis R. Washington is born in Spokane, Washington.
1942 Contracts polio at age 8 while living in Bremerton, Washington.
1964 Forms construction company to repair roads for U.S. Forest Service.
1976 Buys heavy equipment dealer Modern Machinery.
1987 Montana Rail Link formed with lease of southern Montana train route.
1992 Purchase of Canadian tugboat company marks foray into marine transport.
2005 Seaspan sells shares in an initial public offering.
2011 Closely held Seaspan wins Canadian contract to build non-combat vessels. | {
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{"url":"https:\/\/www.qb365.in\/materials\/stateboard\/11th-standard-physics-kinetic-theory-of-gases-model-question-paper-2767.html","text":"#### Kinetic Theory of Gases Model Question Paper\n\n11th Standard\n\nReg.No. :\n\u2022\n\u2022\n\u2022\n\u2022\n\u2022\n\u2022\n\nPhysics\n\nTime : 02:00:00 Hrs\nTotal Marks : 50\n5 x 1 = 5\n1. A sample of ideal gas is at equilibrium. Which of the following quantity is zero?\n\n(a)\n\nrms speed\n\n(b)\n\naverage speed\n\n(c)\n\naverage velocity\n\n(d)\n\nmost probable speed\n\n2. Two identically sized rooms A and B are connected by an open door. If the room A is air conditioned such that its temperature is 4\u00b0 lesser than room B, which room has more air in it?\n\n(a)\n\nRoom A\n\n(b)\n\nRoom B\n\n(c)\n\nBoth room has same air\n\n(d)\n\nCannot be determined\n\n3. The ratio\u00a0$\\gamma =\\frac { { C }_{ p } }{ { C }_{ V } }$\u00a0for a gas mixture consisting of 8 g of helium and 16 g of oxygen is\n\n(a)\n\n23\/15\n\n(b)\n\n15\/23\n\n(c)\n\n27\/11\n\n(d)\n\n17\/27\n\n4. If sP and sV denote the specific heats of nitrogen gas per unit mass at constant pressure and constant volume respectively, then\n\n(a)\n\nsP - sV = 28R\n\n(b)\n\nsP - sV = R\/28\n\n(c)\n\nsP - sV = R\/14\n\n(d)\n\nsP - sV = R\n\n5. The perfect gas equation can be written as\n\n(a)\n\nPV = $\\mu$RT\n\n(b)\n\nPV = $\\mu$R\n\n(c)\n\nPV = RT\n\n(d)\n\nP\u00a0= $\\mu$RTV\n\n6. 10 x 2 = 20\n7. What is the microscopic origin of pressure?\n\n8. Why moon has no atmosphere?\n\n9. Define the term degrees of freedom.\n\n10. Deduce Charles\u2019 law based on kinetic theory.\n\n11. List the factors affecting the mean free path.\n\n12. A fresh air is composed of nitrogen N2(78%) and oxygen O2(21%). Find the rms speed of N2 and O2 at 20\u00b0C.\n\n13. Calculate the temperature at which the rms velocity of a gas triples its value at S.T.P.\n\n14. Calculate the mean free path of air molecules at STP. The diameter of N2 and O2 is about 3 \u00d7 10-10 m\n\n15. Estimate the total number of air molecules in a room of capacity 25 m3 at a temperature of 27\u00b0C.\n\n16. What type of motion is associated with the\u00a0molecule of a gas?\n\n17. 5 x 3 = 15\n18. Write down the postulates of kinetic theory of gases.\n\n19. Explain in detail the kinetic interpretation of temperature.\n\n20. Explain in detail the Maxwell Boltzmann distribution function.\n\n21. Describe the Brownian motion.\n\n22. State and explain Boyle's law.\n\n23. 2 x 5 = 10\n24. Calculate the rms speed, average speed and the most probable speed of 1 mole of hydrogen molecules at 300 K. Neglect the mass of electron.\n\n25. An oxygen molecule is travelling in air at 300 K and 1 atm, and the diameter of oxygen molecule is 1.2 \u00d7 10\u221210m. Calculate the mean free path of oxygen molecule.","date":"2020-01-18 23:25:54","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\": 0.584904670715332, \"perplexity\": 1270.5790569207015}, \"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-2020-05\/segments\/1579250593994.14\/warc\/CC-MAIN-20200118221909-20200119005909-00405.warc.gz\"}"} | null | null |
Утіда Кадзуо (, 18 квітня 1962, Сідзуока) — японський футбольний тренер. З 2017 року тренує команду китайського «Сучжоу Дуну».
Тренерська кар'єра
2004 року входив до тренерського штабу юнацької збірної Японії.
Самостійну тренерську кар'єру розпочав 2010 року у клубі «Ванфоре Кофу», а наступний рік тренував збірну Гуаму.
2012 року повернувся на батьківщину, увійшовши до тренерського штабу команди «Сімідзу С-Палс». Частину 2015 року був головним тренером другої команди цього клубу.
З 2016 року працює в Киитаї — спочатку очолював тренерський штаб «Іньчуань Хеланшань», а за рік став головним тренером «Сучжоу Дуну».
Посилання
transfermarkt.com
J.League
Японські футбольні тренери
Тренери збірної Гуаму з футболу
Тренери ФК «Ванфоре Кофу»
Тренери ФК «Сімідзу С-Палс»
Уродженці Сідзуоки | {
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\section{\uppercase{Introduction}}
\label{sec:introduction}
\noindent Augmented \& Virtual Reality (AR-VR) systems and applications have seen massive development and have been studied extensively over the last few decades \cite{azuma1997survey,billinghurst2015survey,moustakas2015}.
Virtual reality (VR) is an artificial 3D environment generated with software. Users are immersed in this 3D world, and they tend to accept it as a real environment. On the other hand, Augmented Reality (AR) is a technology that blends digital content into our real world. Thus, AR combines real and virtual imagery, is interactive in real-time, and registers the virtual imagery with the real world. AR \& VR systems require specialized hardware and most of the time they are quite expensive.
In contrast with Virtual Reality, where the user is completely immersed in a virtual environment, AR allows the user to interact with the AR digital world and manipulate the virtual content via special input devices. Three-dimensional visualization would be ideally accompanied by 3D interaction, therefore 3D input devices are highly desirable \cite{reitmayr2005iorb}. To reduce the complexity, the input device we choose, is a simple, colored ball at the end of a stick with three degrees of freedom (DOF). We will refer to it as the \textsc{ar-pointer}.
Building an AR system we have to decide on how to implement its three basic functions. \textit{Display}, where we have to combine images from the real and virtual world, \textit{Tracking}, where we have to find the position of the user's viewpoint in the real world and register its view in the 3D virtual world and a \textit{User Interface} (Input-Interaction), where a computer responds in real-time to the user input and generates interactive graphics in the digital world.
With the advances in mobile device technology, handheld computing devices are becoming powerful enough to support the functionalities of an AR System. Google's ARcore platform is such an example. Considering that we want to build a low-cost AR system with a 3D tangible input interface, we choose a mobile phone, running Unity and Vuforia, to implement the AR Video-based \textit{Display} and the \textit{Traking} modules while the tangible \textit{User Interface} is implemented in a "custom-made" device running Open Source software.
The contributions of this paper are threefold.
First, we describe the development of a DIY (Do It Yourself) low-cost AR-VR working prototype system with a 3D tangible user interface that can be used as a test-bed to examine a variety of problems related to 3D interaction in VR or AR environments. Second, the usage of the real 3D position of the tangible input device obtained via an adaptive color and distance camera registration algorithm, offers a powerful and flexible environment for interactivity with the digital world of AR-VR. Third, we present cMinMax, a new algorithm that can estimate the corners of a convex polygon. This algorithm is suitable for the fast registration of markers in augmented reality systems and in applications where real-time feature detector is necessary. cMinMax is faster, approximately by a factor of 10, and more robust compared to the widely used Harris Corner Detection algorithm.
\section {\uppercase{Related Work}} \label{sec:Related}
\noindent During the last two decades AR research and development have seen rapid growth and as more advanced hardware and software becomes available, many AR systems with quite different interfaces are moving out of the laboratories to consumer products.
Concerning the interactivity of an AR system, it appears that users prefer for 3D object manipulation to use the so-called Tangible User Interface (TUI) \cite{billinghurst2008tangible,ishii2008tangible},
Thus for the Interactivity interface, we follow the Tangible Augmented Reality approach, a concept initially proposed by T. Ishii \cite{ishii1997tangible}.
This approach offers a very intuitive way to interact with the digital content and it is very powerful since physical objects have familiar properties and physical constraints, therefore they are easier to use as input devices \cite{ishii2008tangible,shaer2010tangible}.
Besançon et al. compared the mouse-keyboard, tactile, and tangible input for AR systems with 3D manipulation \cite{besanccon2017mouse}. They found that the three input modalities achieve the same accuracy, however, tangible input devices are more preferable and faster.
To use physical objects as input devices for interaction requires accurate tracking of the objects, and for this purpose, many Tangible AR applications use computer vision-based tracking software.
An AR system with 3D tangible interactivity and optical tracking is described in \cite{martens2004experiencing}, where 3D input devices tagged with infrared-reflecting markers are optically tracked by well-calibrated infrared stereo cameras. Following this approach, we attempted to build an AR system with a 3D tangible input interface using a multicamera smartphone. Unfortunately, it did not succeed because neither Android or iOS SDKs were offering adequate support for multiple cameras nor the smartphone we used was allowing full access to their multiple cameras images to estimate the distance of the input device using binocular stereo vision. In addition, the fact that the cameras were too close and not identical it was one more problem.
With the recent technical advances and commercialization of depth cameras (e.g. Microsoft Kinect) more accurate tracking of moving physical objects became available for VR-AR applications. Such an approach is described in \cite{hernandez2012detecting} where the 3D position of a moving object is estimated utilizing the images of an RGB camera and a depth sensor. Taking into consideration that depth cameras start to appear in mobile devices, we decided to follow a similar approach and instead of using stereo vision to estimate the distance of the 3D input device we use one RGB and one depth camera.
A different approach is used in \cite{teng2017augmented}, where a user with a mobile device and two AR "markers" can perform a 3D modeling task using a tangible interface. Markers are realized as image targets. The first image target is applied to create the virtual modeling environment and the second image target is used to create a virtual pen. Using Vufuria's platform they estimate its position in the 3D world and interact accordingly. However, this input device, a stick with an image target attached to its end, is difficult to use, it is not accurate and it is not a real 3D input device since the system knows its 3D position in the virtual world but not in the real one.
\section{\uppercase{System Architecture}} \label{sec:System}
\noindent
Our system architecture implements the three modules of an AR-system, \textit{Tracking}, \textit{Display} and \textit{User Interface},
\begin{figure}[ht]
\centering
\includegraphics[width=1.0\columnwidth]{images/architecture.png}
\caption{System Architecture.}
\label{fig:archit}
\end{figure}
in two separate subsystems that communicate via WiFi (\autoref{fig:archit}).
The \textbf{\textit{first subsystem}} implements \textit{Tracking} and \textit{Display}, and it contains an Android mobile phone (Xiaomi Red-mi Note 6 pro)
\begin{figure}[ht]
\centering
\includegraphics[width=1.0\columnwidth]{images/hardware.png}
\caption{Hardware.}
\label{fig:hard1}
\end{figure}
(\autoref{fig:hard1}) running a 3D Unity Engine with Vuforia where by utilizing an "image target" the front camera projects the augmented environment on our screen.
The \textbf{\textit{second subsystem}} implements the 3D tangible AR \textit{User's interface} (TUI) and it consists of a Raspberry Pi 4 with an RGB Raspberry Camera and a depth camera (Structure Sensor) housed in a homemade 3D-printed case. The Structure Sensor projects an infrared pattern which is reflected from the different objects in the scene. Its IR camera captures these reflections and computes the distance to every object in the scene while at the same time the Raspberry camera captures the RGB image.
All the processing power in the second subsystem is done in the Raspberry Pi 4. It uses python as well as the OpenCV library for image processing, Matlab was also used as a tool for testing main algorithms before their final implementation.
\input{paperImplementation.tex}
\section{Applications and Experiments} \label{sec:App}
\noindent Using the tangible user interface three AR application were developed. The 3D position of the \textsc{ar-pointer} is used to interact with the virtual world where it appears as a red ball.
\subsection{ The Tic-Tac-Toe game application}
\noindent A Tic-Tac-Toe game was implemented on top of our system to demonstrate its interactivity. In \autoref{fig:TicTac}, where a screenshot of the application is shown, the option of selecting and deselecting an object (such as X or O) into the Virtual world is highlighted. It can be seen that our system offers the simple but important commands of that every AR application does require.
\subsection{ The Jenga game application}
\noindent Additionally, a Jenga game was implemented to demonstrate the precision and stability of our system. This application can be seen in (\autoref{fig:Jenga}. Since this game demands such features in real life, its implementation in our system, showcases the system's practicality and functionality.
\begin{figure}[h]
\centering
\begin{minipage}{0.53\columnwidth}
\includegraphics[width=\linewidth]{images/Screenshot_4.jpg}
\caption{Tic-Tac-Toe Game.}
\label{fig:TicTac}
\end{minipage}
\begin{minipage}{0.43\columnwidth}
\includegraphics[width=\linewidth]{images/Jenqa_4.jpg}
\caption{Jenga Game.}
\label{fig:Jenga}
\end{minipage}
\end{figure}
\subsection{ The 3D Contour Map application}
\noindent The last game that was designed is the creation of 3D height maps (\autoref{fig:MapCountour}) using our tangible interface. In this application we are able to create mountains and valleys, building our terrain. Also in this particular application, we have the ability to create and then process 3D terrains from real height maps after an image process running at raspberry creating the 3d mesh. This process is based on the previous work of \cite{panagiotopoulos2017generation}. In this particular application, we are showing the advantages of having the 3D coordinates giving us the ability to set much more complicates commands such as setting the height.
\begin{figure}[h]
\centering
\includegraphics[width=.95\columnwidth]{images/MapAR.jpg}
\caption{A screenshot from the Contour Map App.}
\label{fig:MapCountour}
\end{figure}
A video clip with these applications is available at \url{ https://www.youtube.com/watch?v=OyU4GOLoXnA} .
\section{\uppercase{Discussion \& Conclusion}}
\label{sec:conclusion}
\noindent
This work was a proof of concept that a marker-based low-cost AR with 3D TUI running in real-time (25-30 fps) is feasible to implement and use it as a testbed for identifying various problems and investigate possible solutions. If we add one or more input devices with a different color and/or different shape, then the current implementation is scalable to co-located collaborative AR, supporting two or more users. The knowledge in real-time of the 3D position of the input device offers a powerful and flexible environment for interactivity with the digital world of AR.
Some advantages of the system are Fast 3D Registration Process, Fast Corner Detection Algorithm, Depth Adaptive Camera Calibration, Data Fusion from RGB and Depth Camera, Simple and Fast Image segmentation, Real-Time Operation, Versatility,
Open Source Implementation and Hardware Compatibility.
\subsection {Future Directions}
\noindent Today, there is a lot of effort towards developing high-quality AR systems with tangible user interfaces. Microsoft-Hololense \footnote{\href{https://www.microsoft.com/en-us/hololens}{https://www.microsoft.com/en-us/hololens}} and Holo-Stylus \footnote{\href{https://www.holo-stylus.com/}{https://https://www.holo-stylus.com}} are just two of them. However, all of them are build on specialized hardware and proprietary software and there are expensive. On the other side, smartphones are continuously evolving, adding more computer power, more sensors, and high-quality display. Multi cameras and depth sensors are some of their recent additions. Therefore, we expect that it will be possible to implement all the functionalities of an AR system just in a \textit{smartphone}. In this case, computing power will be in demand. We will need to develop new fast and efficient algorithms. One way to achieve this is to make them task-specific. cMinMax is such an example, where we can find the corners of a marker (convex quadrangle) almost ten times faster than the commonly used Harris Corner Detection algorithm. The fusion of data obtained from different \textit{mobile} sensors (multiple RGB cameras, Depth Camera, Ultrasound sensor, Three‑axis gyroscope, Accelerometer, Proximity sensor, e.t.c) to locate in real-time 3D objects in 3D space and register them to the virtual world is another challenging task. A simple example is presented in \autoref{subsub:MapRealVirtual}, where we combine data from an RGB and a Depth camera in order to find the 3D coordinates of a small ball (approximated with a point) in space.
\subsection {Conclusions}
\noindent This paper has presented the implementation of an inexpensive single-user realization of a system with a 3D tangible user interface build with off the selves components. This system is easy to implement, it runs in real-time and it is suitable to use as an experimental AR testbed where we can try new concepts and methods. We did optimize its performance either by moving computational complexity out of the main loop of operation or by using task-specific fast procedures. cMinMax, a new algorithm for finding the corners of a markers mask, is such an example, where we have sacrifice generality in order to gain speed.
\section*{\uppercase{Acknowledgements}}
\noindent We would like to thank the members of the Visualization and Virtual Reality Group of the Department of Electrical and Computer Engineering of the University of Patras as well as the members the Multimedia Research Lab of the Xanthi's Division of the "Athena" Research and Innovation Center, for their comments and advice during the preparation of this work.
\bibliographystyle{apalike}
{\small
\section{\uppercase{Implementation}} \label{sec:Implement}
\noindent An illustrative example of the setup of our system in the real world is shown in \autoref{fig:setup}.
\begin{figure}[h]
\includegraphics[width=1.0\columnwidth]{images/space.png}
\caption{The AR system.}
\label{fig:setup}
\end{figure}
As it was described in \autoref{sec:System} our system is composed of two different subsystems.
The first subsystem, the mobile phone, is responsible for the visualization of the Augmented Reality environment. The virtual objects are overlaid on a predefined target image printed on an A4 paper. Thus the mobile phone is responsible for graphics rendering, tracking, marker calibration, and registration as well as for merging virtual images with views of the real world.
The second subsystem is attached to a "desk lamb arm", and faces the target image. This system is responsible to align the images from the two cameras, locate the 3D coordinates of a predefined physical object(yellow ball), namely the \textsc{ar-pointer}, transform its XYZ coordinates to Unity coordinates and send them to mobile via WiFi. The physical pointer, which has a unique color, is localized via an adaptive color and distance camera registration algorithm.
The physical \textsc{ar-pointer} has its virtual counterpart in the AR world, a virtual red ball, which represents the real 3D input device. Thus, by moving the real \textsc{ar-pointer} in the real 3D world, we move the virtual \textsc{ar-pointer} in the virtual world, interacting with other virtual objects of the application. This is a tangible interface, which gives to the user the perception of a real object interacting with the virtual world. The subsystems use the marker as the common fixed frame and communicate through wi-fi. \autoref{fig:archit} shows the building blocks of our subsystems and \autoref{fig:flow1} displays the flowchart of the processes running in the second subsystem.
\begin{figure}[ht]
\centering
\includegraphics[width=.99\columnwidth]{images/flow_rasp.png}
\caption{Flow Chart of Processes Running in Raspberry.}
\label{fig:flow1}
\end{figure}
\subsection{Camera Registration} \label{subsec:Camera}
\noindent Since the two different cameras (RGB, Depth) are connected to the Raspberry and the position of each other is different in space, the images taken by the two cameras are slightly misaligned.
To correct this, we find a homographic transformation that compensates differences in the geometric location of the two cameras. Using a plug-in of matlab called registration-Estimator we select SIFT algorithm to find the matching points and to return affine transformation.
\subsection{Initialization} \label{subsec:Init}
\noindent At initialization, masks that filter the background, the color bounds of the physical \textsc{ar-pointer} and the real to virtual world coordinates mappings are calculated.
\subsubsection{Find Mask} \label{subsub:FindMask}
\noindent During initialization, the image target, and the \textsc{ar-pointer}) need to be separated from their background. To this end, two binary masks are created to filter out the background with the following method:
\begin{enumerate}
\item Capture background image.
\item Place object (Marker or \textsc{ar-pointer}) and capture the second image.
\item Subtract images in absolute value.
\item Apply adaptive threshold (Otsu) \& blur filter.
\item Edge detection ( Canny algorithm) \& fill contours.
\item Create a binary image (mask) by selecting the contour with the largest area.
\end{enumerate}
In \autoref{fig:createMaskPointer} we see the inputs to create the mask (steps 1 and 2) and the obtained mask (step 6)
\begin{figure}[h]
\includegraphics[width=.32\columnwidth]{images/0_backGround_img.png}
\hfill
\includegraphics[width=.32\columnwidth]{images/1_tracker_img.png}\hfill
\includegraphics[width=.32\columnwidth]{images/7_imgContour.png}
\caption{The background without and with the object and the mask. }
\label{fig:createMaskPointer}
\end{figure}
\subsubsection{Find Color Bounds}
\label{subsec:ColorBounds}
\noindent Variations in the room illumination can make the same object appear with different RGB values. To address this, the following method was developed that detects the color bounds of the \textsc{ar-pointer} under the light conditions of the room. The decided to use the HSV representation since the colors are separable in the HUE axis whereas in RGB all the three axes are needed.
\begin{enumerate}
\item Find mask of \textsc{ar-pointer} (see \ref{subsub:FindMask}).
\item Isolate pointer from image.
\item Convert RGB image to HSV.
\item Calculate histogram \& find max value of Hue HSV.
\item Create new bounds $\pm 15$ at HSV.
\end{enumerate}
In \autoref{fig:histogram2} the histogram of the example in \autoref{fig:WithRegister} is shown. It can be derived (step 4) that the \textsc{ar-pointer} is near Hue=20 \footnote{8bit pixel value, to obtain real HUE values multiply by 2, since in OpenCV max HUE is 180}.
\begin{figure}[ht]
\centering
\includegraphics[width=.85\columnwidth]{images/HistogramYellowTarget.png}
\caption{HSV Histogram.}
\label{fig:histogram2}
\end{figure}
By applying the derived bounds $(5 \leq \text{HUE} \leq 35)$ on the RGB color spectrum only the yellow spectrum is kept (\autoref{fig:colorSpectrurm2}).
\begin{figure}[ht]
\centering
\includegraphics[width=.40\columnwidth]{images/map-saturation.png}
\hfill
\includegraphics[width=.40\columnwidth]{images/HistogramResult.png}
\caption{Derived bounds isolate the yellow spectrum.}
\label{fig:colorSpectrurm2}
\end{figure}
Identifying the color bounds makes our system robust to light variations and enables the use of multiple differently colored \textsc{ar-pointer} objects.
\subsubsection{AR Registration} \label{subsec:ARreg}
To allow the interaction with the digital world via the motion of the \textsc{ar-pointer} we need to map its real-world 3D coordinates $(x_r, y_r, z_r)$ to the digital world coordinates $(x_v, y_v, z_v)$. To calculate this mapping the common object of reference is the Image Target as shown in \autoref{fig:setup}.
Since the image frames are almost vertically aligned we can approximate the relation between the z coordinates (distance) with a scalar factor $\rho _ z \approx \rho _z (x,y)$ which is proportional to the size of the image (see also \ref{subsub:MapRealVirtual}). This can be derived to obtain $z_v= \rho_z z_r $. To map the $(x_r ,y_r )$ coordinates we need to find a projective transformation matrix ($T_{RV}$) to account mainly for translation and rotation offset. To calculate $T_{RV}$, we need at least four points. To this end, we used the four corners of the image target mask, and map them to the Unity four corners of the marker (\autoref{fig:RealVirtual}). Unity has constant coordinates where each corner of the marker is located at $(\pm 0.5, \pm 0.75)$, So first we will find the corners of the marker running the appropriate software in Raspberry and then we will find the transformation matrix that moves those points to the Unity virtual space.
\begin{figure}[ht]
\includegraphics[width=.45\columnwidth]{images/MaskedPoints.png}
\hfill
\includegraphics[width=.45\columnwidth]{images/0_RGB_image.png}
\caption{Corner Detection.}
\label{fig:matchWorlds}
\end{figure}
To find the corners of the marker we first create the mask of the marker as described in \ref{subsub:FindMask} and then find its corners. Initially, we used the Harris Corner Detection Algorithm from OpenCV \cite{opencv03}, but later on, we developed another simpler and faster algorithm, the cMinMax (see \autoref{subsec:cMinMax}). After we found the four corners (see \autoref{fig:matchWorlds}) we add a fifth point, the center of gravity of the marker for better results. We calculate the center of gravity as the average of X \& Y coordinates for all the pixels in the mask.
Now, we use the two sets of points (Real World points, Unity points) and with the help of OpenCV, we get the projective transformation matrix.
This process needs to be done only once at the start of the program and not in the main loop gaining in computational power. The result of matching the real world with the virtual one is that we can now project the virtual \textsc{ar-pointer} at the same position where the real \textsc{ar-pointer} is on the smartphone screen, making those 2 objects (real-yellow,virtual-red) to coincide.
\subsection{cMinMax: A Fast Algorithm to Detect the Corners in a Quadrangle}
\label{subsec:cMinMax}
\noindent
A common problem in image registration (see section \ref{subsec:ARreg}) is to find the corners of an image. One of the most popular algorithms to address this problem is the Harris Corner Detection \cite{harris1,opencv03}. However, most of the time the image is the photo of a parallelogram, which is a convex quadrangle. To address this specific problem we have developed a specific algorithm, referred to as \textit{cMinMax}, to detect the four corners in a fast and reliable way. The algorithm utilizes the fact that if we find the x-coordinates of the pixels that belong to the mask, then their maximum, $x_{max}$, is a corner's coordinate. Similarly for $x_{min}$, $y_{min}$ and $y_{max}$. The proposed algorithm is approximately 10 times faster and more robust than the Harris Corner Detection Algorithm, but its applicability is limited only to convex polygons.
The basic steps of the algorithm are:
\begin{enumerate}
\item \textbf{Preprocessing:} Generate a bi-level version of the image with the mask.
\item Project the image on the vertical and horizontal axis and find the $( x_{min} , x_{max} , y_{min} , y_{max} )$. These are coordinates of four corners of the convex polygon.
\item If $N$ is the expected maximum number of angles, then for $k=1,..,int(N/2)-1$, rotate the image by $\Delta \theta = k * pi /N $ and repeat the previous step. Identify the four additional corners, rotate the image backward by $- \Delta \theta$ and find their position in the original image.
\item In the end, we have found $ 2N $ points which is greater than the number of expected polygon corners. Hence, there are more than one pixels around each corner. The centroids of these bunches are the estimated corners of the convex polygon.
\item If the number of detected corners is less than N, repeat the previous three steps by rotating the image with $\Delta \theta = (k * pi /N)- pi/2N $
\end{enumerate}
\begin{figure}[h!]
\centering
\includegraphics[width=.90\columnwidth]{images/hexagon_rotate.jpg}
\caption{Detected corners in a hexagon for M=3.}
\label{fig:hexagon_rotate}
\end{figure}
In \autoref{fig:hexagon_rotate} we apply the algorithm in a hexagon and we find all the corners with three rotations.
\noindent
\subsection{\textsc{ar-pointer} detection (Main Loop)} \label{subsec:ARpointer}
In the main loop the 3D position of the physical \textsc{ar-pointer} is continuously estimated and transmitted to the mobile phone.
\subsubsection{RGB Color Segmentation} \label{subsub:RGBSegm}
\noindent The X \& Y coordinates will be acquired from the RGB image. The \textsc{ar-pointer} that we used is a "3D-printed" yellow ball. Our segmentation is based on color, thus we can use any real object with the limitation to have a different color from the background. Since we have our HSV color bounds (see section \ref{subsec:ColorBounds}) the detection of the object used as the \textsc{ar-pointer} is straightforward.
\begin{enumerate}
\item Convert RGB input image to type HSV.
\item Keep pixel values only within preset HSV color bounds (section \ref{subsec:ColorBounds}).
\item Edge detection ( Canny algorithm).
\item Find and save the outlined rectangle of contour with maximum area.
\end{enumerate}
Filtering the RGB image with the color bounds results to the first image of \autoref{fig:isolation2}. Then, we create a rectangle that contains the \textsc{ar-pointer} and use its center as the coordinates of it (see the second image of \autoref{fig:isolation2}).
\begin{figure}[h]
\centering
\includegraphics[width=.45\columnwidth]{images/trackerFoundFiltred.png}
\hfill
\includegraphics[width=.45\columnwidth]{images/trackerFoundRGB.png}
\caption{Color detection.}
\label{fig:isolation2}
\end{figure}
\subsubsection{Depth Estimation} \label{subsub:DepthFind}
The 3D coordinates of the \textsc{ar-pointer} will be acquired from the depth image. Knowing where the \textsc{ar-pointer} is located in the RGB image from the Color Segmentation, and since the images are aligned, we crop a small rectangle from the depth image that contains the area of the \textsc{ar-pointer} (\autoref{fig:WithRegister}).
\begin{figure}[h]
\includegraphics[width=.45\columnwidth]{images/RGB_image_withMatrix.png}
\hfill
\includegraphics[width=.45\columnwidth]{images/Depth_image_withMatrix.png}
\caption{\textsc{ar-pointer} 3D detection}
\label{fig:WithRegister}
\end{figure}
This rectangle contains all the depth information we need and since it is a small part of the image it reduces also the computational cost. In this rectangle there are 3 different depth information (see \autoref{image:trackerDepth2}):
\begin{enumerate}
\item Depth information of the \textsc{ar-pointer} (pixel values: 1000-8000).
\item Depth information of background (pixel values: 7000-8000).
\item Depth information for the area which is created by the \textsc{ar-pointer} that blocks the IR emission creating a shadow of the object (pixel value: 0 ).
\end{enumerate}
\begin{figure}[h]
\centering
\includegraphics[width=.75\columnwidth]{images/1_TrackerDepth_image.png}
\caption{Pre-process of the depth image.}
\label{image:trackerDepth2}
\end{figure}
\noindent Given the fact that the background always corresponds to the maximum value, we do the following on.
\begin{enumerate}
\item We calculate the average of non-zero elements of rectangle image, (\autoref{image:trackerDepth2} first image).
\item Set to zero all pixels with values $> 10 \%$ of average, (\autoref{image:trackerDepth2} second image).
\item Recalculate average of non-zero elements and use it as the final depth value for the \textsc{ar-pointer}.
\end{enumerate}
With this approach, we get stable values for small changes of \textsc{ar-pointer}, fast results and precision below 1 cm.
\subsubsection{Map \textsc{ar-pointer} Coordinates to Virtual Word} \label{subsub:MapRealVirtual}
\noindent At this point we know the distance of the \textsc{ar-pointer} from the Image Target plane (see \autoref{subsub:DepthFind}), as well its position on the RGB image (see \autoref{subsub:RGBSegm}). Since the \textsc{ar-pointer} is not in the Image Target plane, the position of the \textsc{ar-pointer} on the RGB image is the B point and not the A (see \autoref{fig:RealVirtual} ) as it should be. We know
$h$ and $H$, therefore the correction vector $\overrightarrow{(AB)}$ is given from the relation $ \overrightarrow{(AB)}= \overrightarrow{(OA)}*h/H$.
\begin{figure}[h]
\includegraphics[width=.95\columnwidth]{images/RealVirtualMap.png}
\caption{Depth Correction.}
\label{fig:RealVirtual}
\end{figure}
Therefore the coordinates of the \textsc{ar-pointer} in the real word are
\begin{align*}
x_{rA}&=x_{rB} *\frac{ (OB)-(AB)}{(OB)} \\
y_{rA}&=y_{rB} * \frac{(OB)-(AB)}{(OB)}, \hspace{0.5cm}
z_{r}=h
\end{align*}
\noindent and the coordinates in the virtual world are
\begin{center}
$
\begin{bmatrix}
x_{vA}\\
y_{vA}\\
1
\end{bmatrix}
= T_{RV}
\begin{bmatrix}
x_{rA}\\
y_{rA}\\
1
\end{bmatrix}
$
, $z_{v} = \rho_z * z_{r}$
\end{center}
\noindent where $T_{RV}$ and $\rho_{z}$ were define in \autoref{subsec:ARreg}.
\subsection{AR Engine} \label{subsec:ArEng}
\noindent The final AR engine is a marker-based AR-system with AR video display.
It runs exclusively in the mobile phone, \autoref{fig:archit}, and is based on the 3D Unity platform. It executes the following steps.
\begin{enumerate}
\item Capture images with the mobile's built in camera.
\item Detects the image target(marker) in the real world.
\item Displays the virtual environment on top of the image target and the virtual \textsc{ar-pointer} in the mobile screen.
\end{enumerate}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,332 |
Beer Makes History
Ye Olde Tavern Tours
This podcast series explores the American Revolution in Boston. Each episode covers historic events and is accompanied by a craft beer pairing. Huzzah!
Top 10 Beer Makes History Episodes
The Battle of Bunker Hill and the Evacuation of Boston
Shortly after war broke out in the countryside in 1775, the Battle of Bunker Hill raged. The colonists organized an army and forced British soldiers to evacuate Boston in March 1776. The colonies then approved the Declaration of Independence, capping off 10 ten years of resistance. Key Player: Benjamin Church
"Boston in the American Revolution"
Lexington, Concord, and the Start of the Revolutionary War
General Gage tried a third failed powder raid in April 1775. British troops marched towards Concord and encountered local militia in Lexington. This standoff led to the "shot heard round the world" and the Battles of Lexington and Concord. Key Player: Thomas Gage
Brooke's book: "Boston in the American Revolution"
Punishment, Powder Raids, and the First Continental Congress
Parliament passed several laws in 1774 called the Coercive Acts as punishment for the Boston Tea Party. The laws inspired action in the countryside and a meeting in Philadelphia. General Gage began a new strategy to shut down colonial resistance by seizing colonial powder stores. Key Player: Paul Revere
Boston's Tea Party
Things quieted down in Boston from 1770 to 1773 until Parliament passed another tax—the Tea Act. Bostonians targeted the Loyalists who were charged with enforcing it. When that didn't get results, rebels revolted in a unique way: dumping the tea into the harbor. Key Player: Richard Clarke
Trouble was mounting between redcoats and rebels in Boston in 1770. Within 10 days, a British customs official killed a young boy, a huge fight between Bostonians and soldiers broke out, and redcoats shot and killed 5 people, in what would become known as the Boston Massacre. Key Player: Joseph Warren
Military Occupation: Standoffs, Trolling, and an Execution
Because of the increasingly violent mobs in Boston, Parliament sent 2,000 British troops to occupy and control the town. Townspeople struggled to live alongside the troops, their customs, and their debauchery. The tension ultimately resulted in a standoff. Key Player: Samuel Adams
The Townshend Duties and Defending Liberty
The cycle of Parliamentary taxation and resistance in Boston continued through 1768. One prominent merchant tries to get around the latest British tax by smuggling alcohol, but gets caught. Boston rebels riot in a spectacular fashion and Parliament responds with a heavy hand. Key Player: John Hancock
Boston's Violent Mobs Take on the Stamp Act
Without much resistance against the Sugar Act, Parliament passed the Stamp Act in 1765. This time, Boston reacted violently, with two riots breaking out against top crown officials and their homes. Key Player: Thomas Hutchinson
Craft Beer Cellar
The Sugar Act: Rum, Smuggling, and Resistance
To decrease the British Empire's debt, Parliament passed the Sugar Act in 1764. This law, and its tax on molasses, will disproportionately affect Massachusetts because it was home to the most prolific rum producers in the North American colonies. Key Players: Mercy Otis Warren and James Otis
Boston in 1763: Taverns, Brawls, and Crippling Debt
In 1763, Boston and its residents loved to visit taverns, drink alcohol, fight in the streets, and debate politics. But the French and Indian War had just ended and an indebted British Empire was looking to tax their North American colonies, making life much more difficult. Key Player: Ebenezer Mackintosh | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 576 |
The Emergency Department Operations Study Group is a multi-institutional research consortium committed to patient care processes in emergency medicine. Our mission is to facilitate the development of clinical operations research means, methods and knowledge in response to the health industry's growing demand for system performance monitoring for improved care quality. Our members include a diversity of emergency departments including academic, community practice, large high volume and small rural critical access centers.
We strive to better understand how process and practice variation, across institutions, impacts patient outcomes. Our approach involves a focus on the concept of Evidence-Based Clinical Practice, where clinical practice and process changes are driven by the scientific evaluation of process metrics and patient outcomes data. We do this by collecting and disseminating comparative data across emergency departments, participating in the development of operations study design and research methods, advocating for standard data reporting, and serving as a data sharing network across emergency departments and performance measuring organizations.
Our focus is on high volume and resource intensive emergency department presentations with particular interest in the care processes of acute coronary syndrome and congestive heart failure patients. We promote and facilitate research projects in line with our mission.
Our published work continues to grow. Catch up on recent papers and past work. | {
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\section{The ALICE Collaboration}
\label{app:collab}
\input{authorlist1.tex}
\end{document}
\section{Introduction}
Hadron production measurements in proton-proton collisions at
the Large Hadron Collider (LHC)~\cite{Evans:2008zzb}
energies open a new, previously unexplored domain in particle phy\-sics,
which allows validation of the predictive power of Quantum
Chromo Dynamics (QCD) \cite{Gross:1973ju}. A quantitative description of hard
processes is provided by perturbative QCD (pQCD) supplemented with
parton distribution functions (PDF) $f(x)$ and fragmentation functions
(FF) $D(z)$, where $x$ is the fraction of the proton longitudinal
momentum carried by a parton and $z$ is the ratio of the observed hadron
momentum to the final-state parton momentum.
Due to the higher collision energy at the LHC, the PDF and FF can
be probed at lower values of $x$ and $z$, respectively, than in
previous experiments. Such measurements can provide further
constraints on these functions, which are crucial for pQCD predictions for LHC energies.
Furthermore, while pion production at the Relativistic Heavy Ion Collider
(RHIC) \cite{Reardon:1988hg} is considered to be dominated by gluon fragmentation
only for $\pT < 5-8$~GeV/$c$ \cite{Vogt,Adare:2010cy}, at
LHC energies it should remain dominant for $\pT < 100$~GeV/$c$
\cite{Sassot:2010bh,Chiappetta:1992uh}. Theoretical estimates
\cite{Sassot:2010bh} suggest that the fraction of pions
originating from gluon fragmentation remains above 75~\% in the $\pT$
range up to 30~GeV/$c$. Here, the measurement of the
$\pi^0$ production cross section at LHC energies provides constraints
on the gluon to pion fragmentation \cite{deFlorian:2007aj} in a new energy
regime. In addition, the strange quark content of the $\eta$ meson makes the
comparison to pQCD relevant for possible differences of
fragmentation functions with and without strange quarks
\cite{Aidala:2010bn}. Furthermore, the precise measurement of $\pi^0$ and
$\eta$ meson spectra over a large $\pT$ range is a
prerequisite for understanding the decay photon (electron) background for
a direct photon (charm and beauty) measurement. Finally, a significant
fraction of hadrons at low $\pT$ is produced in pp ~collisions via
soft parton interactions, which cannot be well described within the
framework of pQCD. In this kinematic region commonly used event generators like
PYTHIA~\cite{Sjostrand:2006za} or PHOJET~\cite{Engel:1995sb} have to resort to
phenomenological models tuned to available experimental
data delivered by lower-energy colliders like Sp$\bar{\mbox{p}}$S,
RHIC, and Tevatron~\cite{Aaltonen:2009ne}, to adequately describe hadron production.
The large increase in center-of-mass energy at the
LHC provides the possibility for a stringent test of the
extrapolations based on these models.
This paper presents the first measurement of neutral pion and $\eta$
meson production in proton-proton collisions at center-of-mass
energies of $\sqrt{s}=0.9$~TeV and 7~TeV in a wide $\pT$ range with
the ALICE detector~\cite{Aamodt:2008zz}. The paper is organized as
follows: description of the subdetectors used for these measurements,
followed by the details about the data sample, as well as about event
selection and photon identification, is given in
section~\ref{sec:Detector}. Section~\ref{sec:Reconstruction} describes
the algorithms of neutral meson extraction, methods of production
spectra measurement, and shows the systematic uncertainty
estimation. Results and their comparison with pQCD calculations are
given in section~\ref{sec:Results}.
\section{Detector description and event selection}
\label{sec:Detector}
Neutral pions and $\eta$ mesons are measured in ALICE via the
two-photon decay channel. The photons are detected with two methods
in two independent subsystems, with the Photon Spectrometer (PHOS)
\cite{PHOS_TDR} and with the photon conversion method (PCM) in the
central tracking system employing the Inner Tracking System (ITS)
\cite{Aamodt:2010ys} and the Time Projection Chamber (TPC)
\cite{Alme:2010ke}. The latter reconstructs and identifies photons
converted to $\rm{e}^+\rm{e}^-$ pairs in the material of the inner
detectors. The simultaneous measurements with both methods with
completely different systematic uncertainties and with momentum
resolutions having opposite dependence on momentum provide a
consistency check of the final result.
The PHOS detector consists at present of three modules installed at a
distance of 4.60~m from the interaction point. PHOS covers the
acceptance of $260^\circ<\varphi<320^\circ$ in azimuthal angle and
$|\eta|<0.13$ in pseudorapidity. Each module has 3584 detection
channels in a matrix of $64\times56$ cells. Each detection channel
consists of a lead tungstate, $\mbox{PbWO}_4$, crystal of $2.2\times
2.2$~cm$^2$ cross section and 18~cm length, coupled to an avalanche
photo diode and a low-noise charge-sensitive preamplifier. PHOS
operates at a temperature of $-25~^\circ$C at which the light yield of
the $\mbox{PbWO}_4$ crystal is increased by about a factor 3 compared
to room temperature. PHOS was calibrated in-situ by equalizing mean
deposited energies in each channel using events with pp collisions.
The Inner Tracking System (ITS) \cite{Aamodt:2008zz} consists of six layers
equipped with Silicon Pixel Detectors (SPD) positioned at a radial
distance of 3.9 cm and 7.6 cm, Silicon Drift Detectors (SDD) at 15.0
cm and 23.9 cm, and Silicon Strip Detectors (SSD) at 38.0 cm and 43.0 cm. The two
innermost layers cover a pseudorapidity range of $|\eta|<2$ and
$|\eta| < 1.4$, respectively.
The Time Projection Chamber (TPC)
\cite{Alme:2010ke} is a large (85~m$^3$) cylindrical drift detector filled
with a Ne/CO$_2$/N$_2$ (85.7/9.5/4.8\%) gas mixture. It is the main
tracking system of the Central Barrel. For the maximum track length
of 159 clusters it covers a pseudorapidity range of $|\eta|<0.9$ over
the full azimuthal angle. In addition, it provides particle
identification via the measurement of the specific ionisation energy loss
(d$E$/d$x$) with a resolution of 5.5\% \cite{Alme:2010ke}. The ITS and the
TPC are aligned with respect to each other to the level of few hundred
$\mu$m using cosmic-ray and proton-proton collision data \cite{Aamodt:2010ys}.
The event selection was performed with the VZERO
detector \cite{VZERO} in addition to the SPD. The VZERO is a forward scintillator hodoscope with two
segmented counters located at $3.3$~m and $-0.9$~m from the
interaction point. They cover the pseudorapidity ranges $2.8<\eta<5.1$
and $-3.7<\eta<-1.7$, respectively.
The proton-proton collision data used in this analysis were collected
by the ALICE experiment in 2010 with the minimum bias trigger MB$_{\rm
OR}$~\cite{Aamodt:2010pp}. This trigger required the crossing of two
filled bunches and a signal in at least one of the two SPD pixel
layers or in one of the VZERO counters. An offline selection based on
time and amplitude signals of the VZERO detectors and the SPD was
applied to reject beam-induced and noise background
\cite{Aamodt:2010pp}. \repl{Pileup collision events were identified imposing
a criterion based on multiple primary vertices reconstructed with the SPD detector,
and removed from the further analysis.}
The cross sections for the MB$_{\rm OR}$
trigger have been calculated from other measured cross sections at the
same energies with appropriate scaling factors. At $\sqrt{s}=7$~TeV
the cross section for the coincidence between signals in the two VZERO
detectors, $\sigma_{\rm{MB_{\rm AND}}}$, was measured in a
Van-der-Meer scan~\cite{alice_sigma7TeV}, and the relative factor
$\sigma_{\rm MB_{\rm AND}}$/$\sigma_{\rm MB_{\rm OR}}=0.87$\orig{$2$}\repl{$3$}
\orig{$\pm 0.003$}\repl{with negligible error} as obtained from data was used.
At $\sqrt{s}=0.9$~TeV the cross
section $\sigma_{\rm MB_{\rm OR}}$ has been calculated from the
inelastic cross section measured in $\rm{p}\bar{\rm{p}}$ collisions at
$\sqrt{s}=0.9$~TeV~\cite{Alner:1986iy} and relative factor
$\sigma_{\rm MB_{OR}}/\sigma_{\rm inel}=0.91$\orig{$6 \pm 0.013$}\repl{$^{+0.03}_{-0.01}$}
estimated from Monte Carlo simulations~\orig{[21]}\repl{\cite{alice_sigma7TeV}}.
Table~\ref{table-cs} shows the values of the cross section obtained at
both energies as well as the integrated luminosity of the total data
samples used. In the photon conversion analysis, only events with a
reconstructed vertex ($\sim$90\% of the total) are inspected, and
those events with a longitudinal distance (i.e. along the beam
direction) between the position of the primary vertex and the
geometrical center of the apparatus larger than 10~cm are discarded.
The analysis using PHOS as well as Monte Carlo simulations show that
the number of $\pi^0$s in events without a reconstructed vertex is
below 1\% of the total number of $\pi^0$s.
\begin{table}[b]
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
\s (TeV) & $\sigma_{\rm MB_{\rm OR}}$ (mb) & $\sigma_{pp}^{\rm INEL}$ (mb) \\ \hline
$0.9$ & \orig{$46.1 \pm 1.1$}\repl{$47.8^{+2.4}_{-1.9}$}(syst) & \orig{$50.3 \pm 0.4(\rm stat) \pm 1.1$}\repl{$52.5 \pm 2$}(syst)\\ \hline
$7$ & $62.$\orig{$5$}\repl{$2$}$ \pm 2.2$(syst) & $73.2$ \orig{$\pm 1.1^{\rm model}$}\repl{$^{+2.0}_{-4.6}$} $\pm 2.6^{\rm lumi}$ \\ \hline
\end{tabular}\\[2pt]
\begin{tabular}{|c|c|c|c|}
\hline
\s (TeV) &\multicolumn{3}{|c|}{ \rule{54pt}{0pt} $\cal{L}$ $({\rm nb}^{-1})$ \rule{54pt}{0pt} } \\
\cline{2-4}
& \rule{8pt}{0pt} PCM \rule{7pt}{0pt} & \rule{3pt}{0pt} PHOS $\pi^0$ \rule{2pt}{0pt} & PHOS $\eta$ \\ \hline
0.9 & 0.14 & 0.14 & \\ \hline
7 & 5.6 & 4.0 & 5.7 \\ \hline
\end{tabular}
\end{center}
\caption{Cross sections of the reactions and integrated luminosities of the measured data samples for the two beam energies (top),
and luminosities used in the different analyses for the 7 TeV data (bottom).}
\label{table-cs}
\end{table}%
To maximize the pion reconstruction efficiency in PHOS, only
relatively loose cuts on the clusters (group of crystals with deposited energy and common edges) were used:
the cluster energy was required to be above the minimum ionizing
energy $E_{\rm cluster}>0.3$~GeV and the minimum number of crystals in a
cluster was three to reduce the contribution of non-photon clusters.
Candidate track pairs for photon conversions were reconstructed using a secondary
vertex (V0) finding algorithm \cite{Alessandro:2006yt}.
In order to select photons among all secondary vertices (mainly
$\gamma$, K$^0_S$, $\Lambda$ and $\bar{\Lambda}$), electron selection
and pion rejection cuts were applied. The main particle identification
(PID) selection used the specific energy loss in the TPC (d$E$/d$x$).
The measured d$E$/d$x$\ of electrons was required to lie in the interval
$[-4\sigma_{{\rm d}E/{\rm d}x},+5\sigma_{{\rm d}E/{\rm d}x}]$ around the expected value. In
addition, pion contamination was further reduced by a cut of
$2\sigma$ above the nominal pion d$E$/d$x$\ in the momentum range
of $0.25$~GeV/$c$ to $3.5$~GeV/$c$ and a cut of 0.5$\sigma$
at higher momenta.
For the $\gamma$ reconstruction constraints on the
reconstructed photon mass and on
the opening angle between the reconstructed photon momentum vector and
the vector joining the collision vertex and the conversion point were
applied. These constraints were implemented as a cut on the $\chi^2(\gamma)$ defined using a
reconstruction package for fitting decay particles \cite{alikf}.
The photon measurement contains information on the direction which allows
to reduce the contamination from secondary neutral pions.
With the conversion method a precise $\gamma$-ray tomograph of the
ALICE experiment has been obtained \cite{ALICE_X0}. The integrated
material budget for $r < 180$~cm and $|\eta|<$0.9 is
11.4\orig{$^{+0.39}_{-0.71}$}\repl{$\pm0.5$}\% $X_0$ as extracted from detailed comparisons
between the measured thickness and its implementation in Monte Carlo
simulations based on the GEANT 3.21 package\orig{.}\repl{ using the same simulation runs
for the material studies as for the $\pi^0$ measurement.}
Photon pairs with an opening angle larger than 5~mrad were selected for the meson analysis.
\section{Neutral meson reconstruction}
\label{sec:Reconstruction}
Neutral pions and $\eta$ mesons are reconstructed as excess yields,
visible as peaks at their respective rest mass, above the
combinatorial background in the two-photon invariant mass
spectrum. Invariant mass spectra demonstrating the $\pi^0$ and
$\eta$ mesons peak in some selected \pT\ slices are
shown in Fig.\ref{fig:InvMassSpec} by the histogram.
\begin{figure}[htb]
\hfil
\includegraphics[width=0.45\textwidth]{figures/PCM_InvMass.pdf}
\hfil
\includegraphics[width=0.45\textwidth]{figures/PHOS_InvMass.pdf}
\hfil
\caption{Invariant mass spectra in selected \pT\ slices in PCM
(left) and PHOS (right) in the $\pi^0$ and $\eta$ meson mass
regions. The histogram and the bullets show the data before and
after background subtraction, respectively. The curve is a
fit to the invariant mass spectrum after background subtraction.}
\label{fig:InvMassSpec}
\end{figure}
The background is determined by mixing photon pairs from different
events and is normalized to the same event background at the right
side of the meson peaks. A residual correlated background is further
subtracted using a linear or second order polynomial fit. The
invariant mass spectrum after background subtraction, depicted by
bullets in Fig.\ref{fig:InvMassSpec}, was fitted to obtain the
$\pi^0$ and $\eta$ peak parameters (a curve). The number
of reconstructed $\pi^0$s ($\eta$s) is obtained in each $\pT$ bin by
integrating the background subtracted peak within 3 standard
deviations around the mean value of the $\pi^0$ ($\eta$) peak position
in the case of PHOS. In the PCM measurement the integration windows
were chosen to be asymmetric ($m_{\pi^0}$-0.035~GeV/$c^2$,
$m_{\pi^0}$+0.010~GeV/$c^2$) and ($m_{\eta}$-0.047~GeV/$c^2$,
$m_{\eta}$+0.023~GeV/$c^2$) to take into account the left side tail of
the meson peaks due to bremsstrahlung. For the same reason in the case
of PCM the full width at half maximum (FWHM) instead of the Gaussian
width of the peak was used. We vary the normalization and integration
windows to estimate the related systematic uncertainties. The peak
position and width from the two analyses compared to Monte Carlo
simulations are shown in Fig.~\ref{fig:InvMass} as a function of
$\pT$.
\begin{figure}[htb]
\centering
\includegraphics[width=0.48\textwidth]{figures/data_Pi0_CombinedMassAndWidthALLMeasurementsDP_7TeVLogX_Paper.pdf}
\caption{Reconstructed $\pi^0$ peak width (a) and position (b)
as a function
of $\pT$ in pp ~collisions at $\sqrt{s}=7$~TeV
in PHOS and in the photon
conversion method (PCM) compared to Monte Carlo simulations.
The horizontal line in (b) indicates the nominal $\pi^0$
mass.}
\label{fig:InvMass}
\end{figure}
The reconstruction efficiency $\epsilon $ and acceptance $A$ are
calculated
in Monte Carlo simulations tuned to reproduce the detector
response. In the PHOS case, the tuning included a
4.5\% energy non-linearity observed in real data at $E<1$~GeV and not
reproduced by the GEANT simulations and an additional 6\%
channel-by-channel decalibration.
In the PCM case, an additional smearing in
each momentum component given by $\sigma=\sqrt{\sigma_0^2+\sigma_1^2\cdot p^2}$
with $\sigma_0=0.011$~GeV/$c$ and $\sigma_1=0.007$ was necessary to reproduce the measured
width of the $\pi^0$ peak.
PYTHIA~\cite{Sjostrand:2006za} and PHOJET~\cite{Engel:1995sb} event
generators and single particle simulations were used as input.
The small photon conversion
probability of about 8.5\%, compensated by the large TPC acceptance,
translates into $\epsilon\cdot A$ of about
$2\times10^{-3}$ at $\pT > 1$~GeV/$c$
and decreases at lower $\pT$ due to the decrease of the efficiency of soft
electron reconstruction and conversion probability. In the PHOS case,
the acceptance $A$ is zero for $\pT<0.4$~GeV/$c$,
$\epsilon\cdot A$ increases with $\pT$ and
saturates at about $2.0\times 10^{-2}$ at $\pT>15$~GeV/$c$. At high
$\pT>25$~GeV/$c$ the efficiency decreases due to cluster merging.
The invariant differential cross section of $\pi^0$ and $\eta$ meson
production were calculated as
\begin{equation}
E \frac{{\rm d}^3 \sigma}{{\rm d}p^3} =
\frac{1}{2\pi} \frac{\sigma_{\rm MB_{\rm OR}}}{N_{\rm events}} \frac{1}{\pT}
\frac{1}{\epsilon \,A\,Br}\frac{N^{\pi^0 (\eta)}}{\Delta y \Delta \pT}\,,
\end{equation}
where $\sigma_{\rm {MB_{\rm OR}}}$ is the interaction cross section
for the MB$_{\rm OR}$ trigger for pp ~collisions at $\sqrt{s}=0.9$~TeV
or $\sqrt{s}=7$~TeV, $N_{\rm events}$ is the number of MB$_{\rm OR}$
events, $\pT$ is the transverse momentum within the bin to which
the cross section has been assigned after the correction for the finite bin width $\Delta \pT$ (see below),
$Br$ is the branching ratio of the $\pi^0$ ($\eta$) meson to the two
$\gamma$ decay channel and $N^{\pi^0(\eta)}$ is the number of
reconstructed $\pi^0$ ($\eta$) mesons in a given $\Delta y$ and
$\Delta \pT$ bin. Finally, the invariant cross sections were corrected
for the finite $\pT$ bin width following the prescription in
\cite{Lafferty:1994cj}, keeping the $y$ values equal to the bin averages and
calculating the $\pT$ position at which the differential cross section coincides with
the bin average. The Tsallis fit (see below) was used for the correction.
Secondary $\pi^0$'s from weak decays or hadronic interactions in the detector
material are subtracted using Monte Carlo simulations. The contribution
from K$^0_S$ decays is scaled using the measured K$^0_S$ spectrum at
$\sqrt{s}=0.9$~TeV \cite{Aamodt:2011zza} or the charged kaon spectra
at $\sqrt{s}=7$~TeV \cite{Floris:2011ru}.
The measured $\pi^0$ and $\eta$ meson spectra at the center-of-mass
energy of $\sqrt{s}=7$~TeV cover a $\pT$ range from 0.3 to 25~GeV/$c$ and
from 0.4 to 15~GeV/$c$, respectively; the $\pi^0$ spectra at
$\sqrt{s}=0.9$~TeV cover a $\pT$ range from 0.4 to 7~GeV/$c$.
\begin{table}[ht]
\centering
\begin{tabular}{|c|c|c|c|c|}
\hline
& \multicolumn{2}{|c|}{PHOS} \\
\cline{2-3}
& $\pT=1.1$~GeV/$c$ & $\pT=7.5$~GeV/$c$ \\
\hline
Yield extr. & $\pm$2.1 & $\pm$2.5 \\
Non-linearity & $\pm$9.0 & $\pm$1.5 \\
Conversion & $\pm$3.5 & $\pm$3.5 \\
Absolute energy scale & $\pm$0.7 & $\pm$1.0 \\
Acceptance & $\pm$1.0 & $\pm$1.0 \\
\repl{Calibration and alignment} & \repl{$\pm$7.0} & \repl{$\pm$3.0} \\
\repl{Pileup} & \repl{$\pm$0.8} & \repl{$\pm$0.8} \\
\hline
Total & $\pm$\orig{10.0}\repl{12.5}\% & $\pm$\orig{4.8}\repl{6.0}\% \\
\hline
\hline
& \multicolumn{2}{|c|}{PCM} \\
\cline{2-3}
& $\pT$=1.1~GeV/$c$ & $\pT$=7.5~GeV/$c$ \\
\hline
Material Budget & \orig{$-8.8~,~+4.8$}\repl{$\pm 9.0$} & \orig{$-8.8, +4.8$}\repl{$\pm 9.0$} \\
Yield extraction & \orig{$-1.2~,~+0.2$}\repl{$\pm 0.6$} & \orig{$-9.3, +9.5$}\repl{$\pm 4.9$} \\
PID & \orig{$-0.15,~+0.1$}\repl{$\pm 0.1$} & \orig{$-8.9, +1.7$}\repl{$\pm 5.4$} \\
$\chi^2 (\gamma)$ & \orig{$-0.6~,~+0.1$}\repl{$\pm 0.3$} & \orig{$-7.7, +4.7$}\repl{$\pm 6.2$} \\
Reconstruction $\epsilon$ & \orig{$-0.4~,~+3.8$}\repl{$\pm 1.9$} & \orig{$-9.8, +7.8$}\repl{$\pm 4.9$} \\
\hline
Total & \orig{$-8.9\%,~+6.1$}\repl{$\pm 9.2$}\% & \orig{$-20\%,~+14$}\repl{$\pm 14.0$}\%\\
\hline
\end{tabular}
\caption{Summary of the relative systematic errors for the PHOS and the PCM analyses. }
\label{tab:SysErrs}
\end{table}
A summary of the systematic uncertainties is shown in Table\ \ref{tab:SysErrs} for
two different $\pT$ values.
In PHOS, the significant source of systematic errors both at low and
high $\pT$ is the raw yield extraction. It was estimated by varying
the fitting range and the assumption about the shape of the background around the peak.
The uncertainty related to the non-linearity of PHOS which dominates
at low $\pT$ was estimated by introducing different non-linearities into
the MC simulations under the condition that the simulated $\pT$-dependence
of the $\pi^0$ peak position and peak width is still consistent with data.
\repl{The uncertainties on the calibration and alignment were estimated in Monte Carlo simulations
by varying the calibration parameters and the relative module positions within the expected tolerances.
The uncertainty related to the pileup event rejection was evaluated in data by estimating the fraction of
unidentified pileup events by extrapolating the distance and the number of contributing tracks of found
pileup vertices to zero.}
The uncertainty of the conversion probability was estimated comparing
measurements without magnetic field to the standard measurements with
magnetic field.
In the measurements with converted photons, the main sources of systematic errors are the
knowledge of the material budget (dominant at low $\pT$), raw yield extraction, PID,
the photon $\chi^2$ cut and reconstruction efficiency. The contribution from the raw yield
extraction was estimated by changing the normalization range, the integration window,
and the combinatorial background evaluation. The PID, photon $\chi^2(\gamma)$ cut and
reconstruction efficiency was estimated by evaluating stability of the results
after changing the cut values.
\section{Results and comparison with pQCD}
\label{sec:Results}
The combined spectrum is calculated as a weighted average using
statistical and systematic errors of the individual analyses
\cite{Nakamura:2010zzi}. The combined production cross sections are
shown in Fig.~\ref{fig:CrossSection} a).
\begin{figure}[htp]
\centering
\includegraphics[width=0.55\textwidth]{figures/InvXSectionNLOVar2_Paper.pdf}
\caption{a) Differential invariant cross section of $\pi^0$
production in pp ~collisions at $\sqrt{s}=7$~TeV (circles) and
0.9~TeV (squares) and of $\eta$ meson production at
$\sqrt{s}=7$~TeV (stars). The lines and the boxes represent the
statistical and systematic error of the combined measurement respectively.
The uncertainty on the pp ~cross section is not included.
NLO pQCD calculations using
the CTEQ6M5 PDF and the DSS (AESS for $\eta$ mesons) FF for three scales $\mu=0.5\pT$,
$1\pT$ and $2\pT$ are shown.
Dotted lines in panels b) and
c) correspond to the ratios using the BKK FF. Ratio of the NLO
calculations to the data parametrisations are shown in panels b),
c) and d). The full boxes represent the uncertainty on the pp~
cross sections.}
\label{fig:CrossSection}
\end{figure}
The combined spectra \repl{including statistical and systematic errors} are fitted with the Tsallis function \cite{Tsallis:1987eu}
\begin{equation}
\displaystyle
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
E \frac{{\rm d}^3 \sigma}{{\rm d}p^3} =
\displaystyle
\frac{\sigma_{pp}^{\rm INEL}}{2\pi}A
\frac{c \cdot (n-1)(n-2)}{nC\left[ nC+m(n-2)\right]}
\displaystyle\left(1+\frac{\mT-m}{nC}\right)^{-n},
\label{eq:Tsallis}
\end{equation}
where the fit parameters are $A$, $C$ and $n$, $\sigma_{\rm pp}$ is
the proton-proton inelastic cross section, $m$ is the meson rest mass
and $\mT=\sqrt{m^2+\pT^2}$ is the transverse mass. The fit parameters
are shown in Table~\ref{lab:TsallisParam}\orig{, where the uncertainties are
the quadratic sum of the statistical and systematical
uncertainties}. The property of the Tsallis function (\ref{eq:Tsallis})
is such that the parameter $A$ is equal to the integral of this
function over $\pT$ from 0 to infinity, $A={\rm d}N/{\rm d}y$, and
thus can be used as an estimation of the total yield at $y=0$\orig{.}
\repl{per inelastic pp collision.} The additional uncertainty on the parameter
$A$ due to the spectra normalization of \orig{$1.4$}\repl{$^{+3.2}_{-1.1}$}\% and
\orig{$2.8$}\repl{$^{+7.0}_{-3.5}$}\% at $\sqrt{s}=900$~GeV,
and $7$~TeV respectively\orig{, which is not shown in the table}\repl{, is not included}.
The found parameters of the Tsallis function for $\pi^0$ production spectrum in
pp collisions at $\sqrt{s}=900$~GeV are in agreement with those for
the $\pi^+ + \pi^-$ spectra measured by the ALICE collaboration at the
same energy \cite{Aamodt:2011zj}.
\begin{table}[t]
\centering
\begin{tabular}{|c|c|c|c|c|} \hline
Meson & \s & $A$ & $C$ & $n$ \\
& TeV & & (MeV/$c^2$) & \\\hline
$\pi^0$& 0.9 & $1.5 \pm 0.3$ & $132 \pm 15$ & $7.8 \pm 0.5$ \\\hline
$\pi^0$& 7 & \orig{$2.45 \pm 0.07$}\repl{$2.40 \pm 0.15$} & \orig{$140$}\repl{$139$}$ \pm 4$ & \orig{$6.90$}\repl{$6.88$}$ \pm 0.07$ \\\hline
$\eta$ & 7 & \orig{$0.22$}\repl{$0.21$}$ \pm 0.03$ & $229 \pm 21$ & \orig{$6.9$}\repl{$7.0$}$ \pm 0.5$ \\\hline
\end{tabular}
\caption{Fit parameters of the Tsallis parametrisation
(\ref{eq:Tsallis}) to the combined invariant production yields of
$\pi^0$ and $\eta$ mesons for inelastic events. \orig{The errors are
statistical and systematic added in quadrature.}
The uncertainty on the parameter $A$ due to the spectra normalization
of \orig{$1.4$}\repl{$^{+3.2}_{-1.1}$}\% and
\orig{$2.8$}\repl{$^{+7.0}_{-3.5}$}\% at $\sqrt{s}=900$~GeV,
and $7$~TeV respectively, is not included.}
\label{lab:TsallisParam}
\end{table}
The ratio of the data points of the two methods to the combined fit,
shown in Fig.~\ref{fig:YieldNormalized}, illustrates the consistency
between the two measurements.
\begin{figure}[htp]
\centering
\includegraphics[width=0.48\textwidth]{figures/InvXSection_OnlyRatioPi07TeV_Paper.pdf}
\caption{Ratio of the two independent $\pi^0$ meson measurements
to the fit of the combined normalized invariant production cross
section of $\pi^0$ mesons in pp ~collisions at
$\sqrt{s}=7$~TeV.}
\label{fig:YieldNormalized}
\end{figure}
We compare our results with Next-to-Leading Order~(NLO) pQCD calculations
using the PDF CTEQ6M5 and DSS $\pi^0$ \cite{deFlorian:2007aj}, BKK
$\pi^0$ \cite{Binnewies:1994ju} and AESSS $\eta$ \cite{Aidala:2010bn}
NLO fragmentation functions, see Fig.~\ref{fig:CrossSection} a). The
data and NLO predictions are compared via a ratio with the fit to the
measured cross section. This is shown in the bottom panels (b), (c)
and (d) in Fig.~\ref{fig:CrossSection}. In the NLO calculations the
factorization, renormalization and fragmentation scales are chosen to
have the same value given by $\mu$. The uncertainty in the inelastic
pp\ cross section is represented by the full boxes at unity. At
$\sqrt{s}=0.9$~TeV the NLO calculations at $\mu=1\,\pT$ describe the
measured $\pi^0$ data well, while at $\sqrt{s}=7$~TeV the higher scale
($\mu = 2\, \pT$) and a different set of fragmentation functions are
required for a description of the data. However, the latter parameter
set does not provide a good description of the low energy data. In any
case, the NLO pQCD calculations show a harder slope compared to the
measured results. Using the INCNLO program~\cite{Aurenche:1999nz}, we
tested different parton distribution functions (CTEQ5M, CTEQ6M, MRS99)
and different fragmentation functions (BKK, KKP, DSS) and found a
similar result: pQCD predicts harder slopes, and variation of PDFs and
FFs does not change the shape, but results mainly in the variation of
the absolute cross section. A similar trend is observed for the
$\eta$ meson (a higher scale $\mu = 2 \pT$ is required), although the
discrepancy is less significant due to the larger error bars and
smaller $\pT$ reach.
The ratio $\eta/\pi^0$ is shown in Fig.~\ref{fig:EtaToPi0}. It has the
advantage that systematic uncertainties in the measurement partially
cancel. This is also the case for the NLO pQCD calculation, where in particular the
influence of the PDF is reduced in the ratio. Here,
predictions that failed to reproduce the measured $\pi^0$ and $\eta$ cross
section are able to reproduce the $\eta/\pi^0$ ratio.
\begin{figure}[hbtp]
\centering
\includegraphics[width=0.48\textwidth]{figures/data_Pi0EtaRatioTheory_Paper.pdf}
\caption{$\eta/\pi^0$ ratio measured in pp ~collisions at
$\sqrt{s}=7$~TeV compared to NLO pQCD predictions.}
\label{fig:EtaToPi0}
\end{figure}
\section{Conclusion}
In summary, the invariant differential cross sections for inclusive
$\pi^0$ production in pp\ collisions at $\sqrt{s}=7$~TeV and 0.9~TeV and for $\eta$
meson production at 7~TeV have been measured in a wide $\pT$ range
taking advantage of two independent methods available in the ALICE
experiment at the LHC. NLO pQCD calculations cannot provide a
consistent description of measured data at both beam
energies. State-of-the-art calculations describe the data at 0.9 TeV
and 0.2 TeV \cite{Adare:2007dg},
however this is not the case at
7~TeV, where the calculations overestimate the cross sections and exhibit
a different slope compared to the data. Thus, this measurement provides an
important input for the tuning of pQCD calculations and represents
crucial reference data for the measurement of the nuclear modification
factor $R_{\rm AA}$ of the $\pi^0$ production in heavy-ion collisions at
the LHC.
Furthermore, the NLO predictions for the $\eta$ mesons using the
newest fragmentation functions require a value $\mu=2\pT$ in order to get
closer to the experimental results.
\section{Acknowledgments}
\input{acknowledgements_Nov2011}\\
This job was supported partially by the grant RFBR~10-02-91052.
We would like to thank W.\ Vogelsang for providing the NLO pQCD
calculations used in this paper.
\section*{Affiliation notes}
\renewcommand\theenumi{\roman{enumi}}
\begin{Authlist}
\item \Adef{0}Deceased
\item \Adef{Dipartimento di Fisica dell'Universita, Udine, Italy}Also at: Dipartimento di Fisica dell'Universita, Udine, Italy
\item \Adef{M.V.Lomonosov Moscow State University, D.V.Skobeltsyn Institute of Nuclear Physics, Moscow, Russia}Also at: M.V.Lomonosov Moscow State University, D.V.Skobeltsyn Institute of Nuclear Physics, Moscow, Russia
\item \Adef{Institute of Nuclear Sciences, Belgrade, Serbia}Also at: "Vin\v{c}a" Institute of Nuclear Sciences, Belgrade, Serbia
\end{Authlist}
\section*{Collaboration Institutes}
\renewcommand\theenumi{\arabic{enumi}~}
\begin{Authlist}
\item \Idef{org1279}Benem\'{e}rita Universidad Aut\'{o}noma de Puebla, Puebla, Mexico
\item \Idef{org1220}Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine
\item \Idef{org1262}Budker Institute for Nuclear Physics, Novosibirsk, Russia
\item \Idef{org1292}California Polytechnic State University, San Luis Obispo, California, United States
\item \Idef{org14939}Centre de Calcul de l'IN2P3, Villeurbanne, France
\item \Idef{org1197}Centro de Aplicaciones Tecnol\'{o}gicas y Desarrollo Nuclear (CEADEN), Havana, Cuba
\item \Idef{org1242}Centro de Investigaciones Energ\'{e}ticas Medioambientales y Tecnol\'{o}gicas (CIEMAT), Madrid, Spain
\item \Idef{org1244}Centro de Investigaci\'{o}n y de Estudios Avanzados (CINVESTAV), Mexico City and M\'{e}rida, Mexico
\item \Idef{org1335}Centro Fermi -- Centro Studi e Ricerche e Museo Storico della Fisica ``Enrico Fermi'', Rome, Italy
\item \Idef{org17347}Chicago State University, Chicago, United States
\item \Idef{org1118}China Institute of Atomic Energy, Beijing, China
\item \Idef{org1288}Commissariat \`{a} l'Energie Atomique, IRFU, Saclay, France
\item \Idef{org1294}Departamento de F\'{\i}sica de Part\'{\i}culas and IGFAE, Universidad de Santiago de Compostela, Santiago de Compostela, Spain
\item \Idef{org1106}Department of Physics Aligarh Muslim University, Aligarh, India
\item \Idef{org1121}Department of Physics and Technology, University of Bergen, Bergen, Norway
\item \Idef{org1162}Department of Physics, Ohio State University, Columbus, Ohio, United States
\item \Idef{org1300}Department of Physics, Sejong University, Seoul, South Korea
\item \Idef{org1268}Department of Physics, University of Oslo, Oslo, Norway
\item \Idef{org1132}Dipartimento di Fisica dell'Universit\`{a} and Sezione INFN, Bologna, Italy
\item \Idef{org1315}Dipartimento di Fisica dell'Universit\`{a} and Sezione INFN, Trieste, Italy
\item \Idef{org1145}Dipartimento di Fisica dell'Universit\`{a} and Sezione INFN, Cagliari, Italy
\item \Idef{org1270}Dipartimento di Fisica dell'Universit\`{a} and Sezione INFN, Padova, Italy
\item \Idef{org1285}Dipartimento di Fisica dell'Universit\`{a} `La Sapienza' and Sezione INFN, Rome, Italy
\item \Idef{org1154}Dipartimento di Fisica e Astronomia dell'Universit\`{a} and Sezione INFN, Catania, Italy
\item \Idef{org1290}Dipartimento di Fisica `E.R.~Caianiello' dell'Universit\`{a} and Gruppo Collegato INFN, Salerno, Italy
\item \Idef{org1312}Dipartimento di Fisica Sperimentale dell'Universit\`{a} and Sezione INFN, Turin, Italy
\item \Idef{org1103}Dipartimento di Scienze e Tecnologie Avanzate dell'Universit\`{a} del Piemonte Orientale and Gruppo Collegato INFN, Alessandria, Italy
\item \Idef{org1114}Dipartimento Interateneo di Fisica `M.~Merlin' and Sezione INFN, Bari, Italy
\item \Idef{org1237}Division of Experimental High Energy Physics, University of Lund, Lund, Sweden
\item \Idef{org1192}European Organization for Nuclear Research (CERN), Geneva, Switzerland
\item \Idef{org1227}Fachhochschule K\"{o}ln, K\"{o}ln, Germany
\item \Idef{org1122}Faculty of Engineering, Bergen University College, Bergen, Norway
\item \Idef{org1136}Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava, Slovakia
\item \Idef{org1274}Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague, Czech Republic
\item \Idef{org1229}Faculty of Science, P.J.~\v{S}af\'{a}rik University, Ko\v{s}ice, Slovakia
\item \Idef{org1184}Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-Universit\"{a}t Frankfurt, Frankfurt, Germany
\item \Idef{org1215}Gangneung-Wonju National University, Gangneung, South Korea
\item \Idef{org1212}Helsinki Institute of Physics (HIP) and University of Jyv\"{a}skyl\"{a}, Jyv\"{a}skyl\"{a}, Finland
\item \Idef{org1203}Hiroshima University, Hiroshima, Japan
\item \Idef{org1329}Hua-Zhong Normal University, Wuhan, China
\item \Idef{org1254}Indian Institute of Technology, Mumbai, India
\item \Idef{org1266}Institut de Physique Nucl\'{e}aire d'Orsay (IPNO), Universit\'{e} Paris-Sud, CNRS-IN2P3, Orsay, France
\item \Idef{org1277}Institute for High Energy Physics, Protvino, Russia
\item \Idef{org1249}Institute for Nuclear Research, Academy of Sciences, Moscow, Russia
\item \Idef{org1320}Nikhef, National Institute for Subatomic Physics and Institute for Subatomic Physics of Utrecht University, Utrecht, Netherlands
\item \Idef{org1250}Institute for Theoretical and Experimental Physics, Moscow, Russia
\item \Idef{org1230}Institute of Experimental Physics, Slovak Academy of Sciences, Ko\v{s}ice, Slovakia
\item \Idef{org1127}Institute of Physics, Bhubaneswar, India
\item \Idef{org1275}Institute of Physics, Academy of Sciences of the Czech Republic, Prague, Czech Republic
\item \Idef{org1139}Institute of Space Sciences (ISS), Bucharest, Romania
\item \Idef{org27399}Institut f\"{u}r Informatik, Johann Wolfgang Goethe-Universit\"{a}t Frankfurt, Frankfurt, Germany
\item \Idef{org1185}Institut f\"{u}r Kernphysik, Johann Wolfgang Goethe-Universit\"{a}t Frankfurt, Frankfurt, Germany
\item \Idef{org1177}Institut f\"{u}r Kernphysik, Technische Universit\"{a}t Darmstadt, Darmstadt, Germany
\item \Idef{org1256}Institut f\"{u}r Kernphysik, Westf\"{a}lische Wilhelms-Universit\"{a}t M\"{u}nster, M\"{u}nster, Germany
\item \Idef{org1246}Instituto de Ciencias Nucleares, Universidad Nacional Aut\'{o}noma de M\'{e}xico, Mexico City, Mexico
\item \Idef{org1247}Instituto de F\'{\i}sica, Universidad Nacional Aut\'{o}noma de M\'{e}xico, Mexico City, Mexico
\item \Idef{org23333}Institut of Theoretical Physics, University of Wroclaw
\item \Idef{org1308}Institut Pluridisciplinaire Hubert Curien (IPHC), Universit\'{e} de Strasbourg, CNRS-IN2P3, Strasbourg, France
\item \Idef{org1182}Joint Institute for Nuclear Research (JINR), Dubna, Russia
\item \Idef{org1143}KFKI Research Institute for Particle and Nuclear Physics, Hungarian Academy of Sciences, Budapest, Hungary
\item \Idef{org18995}Kharkiv Institute of Physics and Technology (KIPT), National Academy of Sciences of Ukraine (NASU), Kharkov, Ukraine
\item \Idef{org1199}Kirchhoff-Institut f\"{u}r Physik, Ruprecht-Karls-Universit\"{a}t Heidelberg, Heidelberg, Germany
\item \Idef{org20954}Korea Institute of Science and Technology Information
\item \Idef{org1160}Laboratoire de Physique Corpusculaire (LPC), Clermont Universit\'{e}, Universit\'{e} Blaise Pascal, CNRS--IN2P3, Clermont-Ferrand, France
\item \Idef{org1194}Laboratoire de Physique Subatomique et de Cosmologie (LPSC), Universit\'{e} Joseph Fourier, CNRS-IN2P3, Institut Polytechnique de Grenoble, Grenoble, France
\item \Idef{org1187}Laboratori Nazionali di Frascati, INFN, Frascati, Italy
\item \Idef{org1232}Laboratori Nazionali di Legnaro, INFN, Legnaro, Italy
\item \Idef{org1125}Lawrence Berkeley National Laboratory, Berkeley, California, United States
\item \Idef{org1234}Lawrence Livermore National Laboratory, Livermore, California, United States
\item \Idef{org1251}Moscow Engineering Physics Institute, Moscow, Russia
\item \Idef{org1140}National Institute for Physics and Nuclear Engineering, Bucharest, Romania
\item \Idef{org1165}Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark
\item \Idef{org1109}Nikhef, National Institute for Subatomic Physics, Amsterdam, Netherlands
\item \Idef{org1283}Nuclear Physics Institute, Academy of Sciences of the Czech Republic, \v{R}e\v{z} u Prahy, Czech Republic
\item \Idef{org1264}Oak Ridge National Laboratory, Oak Ridge, Tennessee, United States
\item \Idef{org1189}Petersburg Nuclear Physics Institute, Gatchina, Russia
\item \Idef{org1170}Physics Department, Creighton University, Omaha, Nebraska, United States
\item \Idef{org1157}Physics Department, Panjab University, Chandigarh, India
\item \Idef{org1112}Physics Department, University of Athens, Athens, Greece
\item \Idef{org1152}Physics Department, University of Cape Town, iThemba LABS, Cape Town, South Africa
\item \Idef{org1209}Physics Department, University of Jammu, Jammu, India
\item \Idef{org1207}Physics Department, University of Rajasthan, Jaipur, India
\item \Idef{org1200}Physikalisches Institut, Ruprecht-Karls-Universit\"{a}t Heidelberg, Heidelberg, Germany
\item \Idef{org1325}Purdue University, West Lafayette, Indiana, United States
\item \Idef{org1281}Pusan National University, Pusan, South Korea
\item \Idef{org1176}Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum f\"ur Schwerionenforschung, Darmstadt, Germany
\item \Idef{org1334}Rudjer Bo\v{s}kovi\'{c} Institute, Zagreb, Croatia
\item \Idef{org1298}Russian Federal Nuclear Center (VNIIEF), Sarov, Russia
\item \Idef{org1252}Russian Research Centre Kurchatov Institute, Moscow, Russia
\item \Idef{org1224}Saha Institute of Nuclear Physics, Kolkata, India
\item \Idef{org1130}School of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom
\item \Idef{org1338}Secci\'{o}n F\'{\i}sica, Departamento de Ciencias, Pontificia Universidad Cat\'{o}lica del Per\'{u}, Lima, Peru
\item \Idef{org1146}Sezione INFN, Cagliari, Italy
\item \Idef{org1115}Sezione INFN, Bari, Italy
\item \Idef{org1313}Sezione INFN, Turin, Italy
\item \Idef{org1133}Sezione INFN, Bologna, Italy
\item \Idef{org1155}Sezione INFN, Catania, Italy
\item \Idef{org1316}Sezione INFN, Trieste, Italy
\item \Idef{org1286}Sezione INFN, Rome, Italy
\item \Idef{org1271}Sezione INFN, Padova, Italy
\item \Idef{org1322}Soltan Institute for Nuclear Studies, Warsaw, Poland
\item \Idef{org1258}SUBATECH, Ecole des Mines de Nantes, Universit\'{e} de Nantes, CNRS-IN2P3, Nantes, France
\item \Idef{org1304}Technical University of Split FESB, Split, Croatia
\item \Idef{org1168}The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Cracow, Poland
\item \Idef{org17361}The University of Texas at Austin, Physics Department, Austin, TX, United States
\item \Idef{org1173}Universidad Aut\'{o}noma de Sinaloa, Culiac\'{a}n, Mexico
\item \Idef{org1296}Universidade de S\~{a}o Paulo (USP), S\~{a}o Paulo, Brazil
\item \Idef{org1149}Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil
\item \Idef{org1239}Universit\'{e} de Lyon, Universit\'{e} Lyon 1, CNRS/IN2P3, IPN-Lyon, Villeurbanne, France
\item \Idef{org1205}University of Houston, Houston, Texas, United States
\item \Idef{org20371}University of Technology and Austrian Academy of Sciences, Vienna, Austria
\item \Idef{org1222}University of Tennessee, Knoxville, Tennessee, United States
\item \Idef{org1310}University of Tokyo, Tokyo, Japan
\item \Idef{org1318}University of Tsukuba, Tsukuba, Japan
\item \Idef{org21360}Eberhard Karls Universit\"{a}t T\"{u}bingen, T\"{u}bingen, Germany
\item \Idef{org1225}Variable Energy Cyclotron Centre, Kolkata, India
\item \Idef{org1306}V.~Fock Institute for Physics, St. Petersburg State University, St. Petersburg, Russia
\item \Idef{org1323}Warsaw University of Technology, Warsaw, Poland
\item \Idef{org1179}Wayne State University, Detroit, Michigan, United States
\item \Idef{org1260}Yale University, New Haven, Connecticut, United States
\item \Idef{org1332}Yerevan Physics Institute, Yerevan, Armenia
\item \Idef{org15649}Yildiz Technical University, Istanbul, Turkey
\item \Idef{org1301}Yonsei University, Seoul, South Korea
\item \Idef{org1327}Zentrum f\"{u}r Technologietransfer und Telekommunikation (ZTT), Fachhochschule Worms, Worms, Germany
\end{Authlist}
\endgroup
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,719 |
You Can Count on Governments to Conceal the Truth about Islamic Crimes
The FBI Stops, Then Muffs, Jihadist Hostage-Taking at a Texas Synagogue
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National Security & Defense
Police patrol outside the main railway station in Cologne. (Roberto Pfeil/AFP/Getty)
At this point, it's a sad joke — a form of gallows humor shared in times of trouble. When there's a shooting, and the hours tick by without any identification of the suspect, one can presume that it's a jihadist. When there's a riot in Europe, and the perpetrators are described as "youths," one can presume it's Muslim men. When an Islamist goes on a shooting spree or stabbing spree or beheads a coworker, authorities will latch onto any explanation but the obvious.
But now even gallows humor is inappropriate. Western denial of Islamic crimes is so common, so systematic, that we can no longer have any confidence that we understand the true dimensions of the jihadist threat. Consider the following:
In Germany, police actively "tried to obfuscate" what happened on New Year's Eve, when thousands of Muslim men systematically sexually assaulted hundreds of German women — an act that my colleague Andrew McCarthy has aptly termed a "rape jihad."
The New York Times reported yesterday that Swedish authorities now stand accused of covering up a wave of sexual assaults at a concert last summer. A Swedish newspaper wrote today that national media refused at the time to report factual accounts from the concert assaults, claiming they were nothing but far-right "propaganda."
RELATED: Mass Sex Abuse in Cologne: Part of a Disturbing Trend among Middle East and North African Immigrants
Yet for sheer scale, nothing touches the infamous scandal in Rotherham, England, where gangs of Pakistani men brutalized an estimated 1,400 women and girls. Over at least 16 years, from 1997 to 2013, girls and women were trafficked, tortured, and raped while authorities turned a blind eye to the abuse. According to the Rotherham Borough Council's belated report on the mass abuse, authorities were concerned that they might give "oxygen" to racism claims. Up to 160 British police officers now face investigation for systematically ignoring abuse complaints.
#share#Here at home, willful blindness seems to be a deliberate strategy. The Obama administration for years called Nidal Hasan's deadly terror attack at Fort Hood "workplace violence," and in November, the FBI said that it may never release a report into the motivations of the Muslim man who attacked two Chattanooga recruiting stations, killing five. FBI director James Comey said that the Bureau didn't want to "smear people." Finally, in December — five months after the event — Comey unequivocally declared the Chattanooga shooting a "terror attack."
RELATED: When Worlds Collide: Unassimilable Muslim Migrants Crash Europe's Fantasy Islam
But for sheer brazenness, it's hard to top Philadelphia mayor Jim Kenney. Immediately after police apprehended Edward Archer for attempting to assassinate a Philadelphia police officer, Archer started telling anyone who would listen that he did it "in the name of Islam." But don't tell Kenney. He broke land-speed records to get in front of the cameras and declare that the attack "has nothing to do with being a Muslim or following the Islamic faith."
The consequences of the lies, cover-ups, and evasions are serious. American and European elites hector their respective publics over accepting increasing numbers of migrants. They belittle the public's concerns over the terrorist threat and instead praise Islam to the heavens for its tolerance — often with full knowledge of systematic criminal acts. Even now — after Cologne and after Rotterham — those of us outside the halls of power can't have any confidence that we know the truth about the impact of mass Muslim immigration on Europe, or that we know the true dimensions of the domestic terror threat in the United States.
RELATED: Mass Muslim Immigration Will Bring Islam's Problems Here
When, for instance, a Muslim man beheaded his coworker in Oklahoma and attempted to behead another, was his attack "merely" the enraged action of a disgruntled worker? Or was it an act of "lone wolf" jihadist? After all, on social media, he'd made the popular hand sign of ISIS fighters, posted pictures of jihadists (including ISIS fighters and Osama bin Laden), and appeared to call for Jihad.
What about the recent stabbing spree at the University of California, Merced? Authorities say it was over a study-group dispute, but the young Muslim attacker reportedly carried an image of an ISIS flag and a handwritten manifesto that "included instructions to behead a student."
#related#How many Americans are aware of these incidents? How many Americans are aware that New York City police are now looking at a Muslim terror suspect as a "person of interest" in the stabbing of a nine-year-old boy? To learn that, you might have to read a British newspaper that reported on a student named Fareed Mumuni who allegedly stabbed the child in a "botched audition to join ISIS." His long-term goal was reportedly the bombing of Times Square.
The truth disrupts the elite's preferred multicultural narrative, which places all faiths and cultures on equal footing — except for our despised Western civilization. The truth must therefore be suppressed. But the lies are starting to backfire. Victims can't be ignored indefinitely, and it's hard to hide mass-scale public assaults. Since 9/11, Western governments and mainstream media have relentlessly pounded their people with deception and wishful thinking about Islam, jihad, and the Middle East. The lies are now being exposed by the light of day. Will enough people care?
Also from David French
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— David French is a staff writer at National Review and a veteran of Operation Iraqi Freedom.
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,518 |
Animal Planet – Discovery UK – Britian Wildest Places
Animal Planet – Discovery UK – Britian Wildest Places2019-10-152020-11-02https://justynjones.dns-systems.net/smallworld2020/wp-content/uploads/2016/02/SmallWorldLogo.pngSmall World TVhttps://www.smallworldtv.co.uk/wp-content/uploads/2016/02/Small-World-Tv-Dolphin.jpg200px200px
TX: January 2008
Dolphins Episode
In this series made by Telscope of Swansea, Small World Productions made an episode featuring the dolphins and other cetaceans under threat in UK waters.
The film raises concerns about the plight of the British dolphins expressed by the Whale and Dolphin Conservation Society and features footage of very unusual behaviour, including rare clips of dolphins leaping strangely, playing with jellyfish and chasing a porpoise.
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Border of Life & DeathBroadcast Films | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,202 |
Nomenclature: Current taxonomic status: accepted. Taxonomic rank: variety. Synonyms: Uncinula ljubarskii var. aduncoides (R. Y. Zheng & G. Q. Chen) R. Y. Zheng & G. Q. Chen; Erysiphaceae Tul. & C. Tul.; Erysiphales.
Type Information: Basionym: Uncinula aduncoides R. Y. Zheng & G. Q. Chen. Type: Uncinula aduncoides R. Y. Zheng & G. Q. Chen.
Taxonomic Literature: Taxonomic notes: This variety differs from var. ljubarskii by ascomata characters, see below. Braun U., Beih. Nova Hedwigia 89: 1-700 [479-481] (1987).
Ecology: Biotroph; phytopathogenic; growing on leaves, epiphyllous (mostly) or amphigenous. Host or Phorophyte Taxonomy: Acer trifidum Hook. & Arn.; Acer, Aceraceae.
Reproduction Strategy: With sexual (and possible asexual) stages. Ascocarps: Cleistothecioid, orbicular, forming independently from the host thallus or mycelium, scattered or gregarious, .085-.13-(.14) mm in diam.. Margin: External filaments present (the width of the appendages is usually somewhat increasing upwards and especially the apex is mostly enlarged); circinate or sub-helicoid, (1)-1.5-2-(2) µm long, 4.5-8-(11) µm in diameter, hyaline or pigmented (at the very base), numerous, (10)-20-40-(45) per mm², growing between the lower and upper hald of the ascocarp, flexuose (curved, subundulate, rarely straight and stiff), stiff and straight, or sub-geniculate (bent), not ramified, aseptate or septate (with a single septum at the base).
Asci: 5-16 asci per ascocarp, not stipitate or indistinctly stipitate, 40-65 µm long, 25-45 µm wide; dehiscence unitunicate.
Ascospores: c. 4 or c. 8 per ascus, spores (4)-5-6-(7) per ascus, ellipsoid or ovoid, 16-25 µm long, 10-15 µm wide; septa absent.
Conidium Formation: Conidiogenous cells single. Conidia: 22-27 µm long, 12-16-(18) µm wide. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,088 |
Etta Driscoll Pisano is an American breast imaging researcher. She is a professor in residence of radiology at the Beth Israel Deaconess Medical Center and chief research dean at the American College of Radiology. In 2008, she was elected a member of the National Academy of Medicine.
Early life and education
Pisano was born in New York City but raised in the suburbs of Philadelphia. She grew up the oldest of seven children and after her mother died when she was a teenager, decided to pursue a career in medicine. Her father, a radiologist, frequently took her around his hospital and introduced her to the women doctors working there. Pisano received her Bachelor of Arts degree in Philosophy from Dartmouth College before enrolling at the Duke University School of Medicine. When discussing with her guidance counsellor about medical school options, she was informed that she was "wasting her time" due to her gender. She ignored the advice and applied to both medical and law schools, being accepted into every one including Harvard Law. She chose to attend Duke and earned membership into the Alpha Omega Alpha Society, where she also met and married her husband Jan Kylstra. Pisano completed her medical residency in radiology at the Beth Israel Deaconess Medical Center (BIDMC) and served as their Chief of Breast Imaging and Instructor in Radiology for one year.
Career
Pisano left Beth Israel to become an assistant professor in radiology at the UNC School of Medicine in 1989 and served as their Chief of Breast Imaging until 2005. In her last year as chief, Pisano was the principal investigator of the Digital Mammographic Imaging Screening Trial, a study on the importance of digital mammography for young women. She led cancer studies across the United States, Canada, and Germany which found that digital mammography is more accurate in women under the age of 50. Upon stepping down as chief, Pisano was appointed the vice dean for academic affairs and granted the title of Kenan Professor of Radiology and Biomedical Engineering. She was later named the inaugural director of UNC's Center for Research Excellence in Breast Cancer Imaging in 2007. The following year, she was named the principal investigator of a five-year Clinical and Translational Science Award grant after Paul Watkins stepped down. As a result of her research, Pisano was elected a member of the National Academy of Medicine that same year. Pisano ended her tenure at UNC in 2010 by accepting a deanship and vice president position at the Medical University of South Carolina (MUSC). She became the first female to lead the MUSC College of Medicine and one of the few women who are deans of medical schools in the country.
Pisano worked at MUSC from 2010 until 2014, when she stepped down to focus on breast cancer imaging research. At the time, she was considered one of the top 10 experts in women's imaging and one of the 20 most influential people in radiology. The following year, Pisano was appointed the vice-chair of Research in the Department of Radiology at her alma mater, BIDMC. In this role, she led the first randomized trial to compare two types of digital mammography for breast cancer screening. In 2017, she was named the American College of Radiology's (ACR) chief science officer of their Center for Research and Innovation. Pisano stayed in this role for one year before becoming the ACR's first female chief research officer.
Personal life
Pisano and her husband have four children together.
References
External links
Living people
American radiologists
Women radiologists
Members of the National Academy of Medicine
Dartmouth College alumni
Duke University School of Medicine alumni
Harvard University alumni
Harvard Medical School faculty
University of North Carolina School of Medicine faculty
Medical University of South Carolina faculty
Physicians from New York City
Physicians from Pennsylvania
20th-century American physicians
20th-century American women scientists
21st-century American physicians
21st-century American women scientists
Year of birth missing (living people)
American women academics | {
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Have you been dreaming of that PC for a while now and still can't find the savings? Or have you been waiting months to repair your favourite device but the unexpected thing called life keeps getting in the way??
Don't wait any longer as we offer Skye Interest Free Finance at all of our Tech House locations!
Currently we are running a promotion offering 12 months interest free. This promotion will run until the end of July 2019 covering you for all your EOFY needs.
You will find all of the nitty gritty below.
If you want to apply before coming in store, or are in store applying please use one of the links below depending on which store you are visiting. | {
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{"url":"https:\/\/experts.mcmaster.ca\/display\/publication1965794","text":"# Distributed model validation with Epsilon Academic Article\n\n\u2022\n\u2022 Overview\n\u2022\n\u2022 Research\n\u2022\n\u2022 Identity\n\u2022\n\u2022 View All\n\u2022\n\n### abstract\n\n\u2022 AbstractScalable performance is a major challenge with current model management tools. As the size and complexity of models and model management programs increases and the cost of computing falls, one solution for improving performance of model management programs is to perform computations on multiple computers. In this paper, we demonstrate a low-overhead data-parallel approach for distributed model validation in the context of an OCL-like language. Our approach minimises communication costs by exploiting the deterministic structure of programs and can take advantage of multiple cores on each (heterogeneous) machine with highly configurable computational granularity. Our performance evaluation shows that the implementation is extremely low overhead, achieving a speed up of 24.5$$\\times$$ \u00d7 with 26 computers over the sequential case, and 122$$\\times$$ \u00d7 when utilising all six cores on each computer.\n\n### publication date\n\n\u2022 March 25, 2021","date":"2021-09-21 00:08:16","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\": 1, \"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\": 0.23339195549488068, \"perplexity\": 2748.7599499351504}, \"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-2021-39\/segments\/1631780057119.85\/warc\/CC-MAIN-20210920221430-20210921011430-00428.warc.gz\"}"} | null | null |
golang-groups
=============
This is the implementation of a web page showing all the Go meetup groups around the world.
It uses the Meetup API and runs on App Engine.
<img src="screenshot.png" width="400" height="400">
I originally created this as a demo for a talk I gave at GoSV in San Mateo.
There's some [slides](http://go-talks.appspot.com/github.com/campoy/golang-groups/talk/talk.slide)
### Disclaimer
This is not an official Google product (experimental or otherwise), it is just
code that happens to be owned by Google.
| {
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Who has the time to shift by way of reams of product opinions, or spend a half a day driving round malls speaking with salespeople? Every time a shot is fired, these lenses work with their respective mirrors and sensors to capture 10+ images, that are later fused collectively. Some cameras don't require any cables in any respect, as they will transmit the pictures to a PC wirelessly. With camera plus the stabilize function eliminates movement blur so the pictures are clearer and have more correct coloration.
The picture quality from a degree and shoot digital camera is mostly ok for common makes use of, equivalent to auction photos, and even four X 6 prints. This year the S9+ will come with more memory and storage compared to the smaller S9, and also will have a dual-camera system much like that fitted to Samsung's larger Word eight , which includes a telephoto camera.
This way, you might take extra photos and revel in it because you don't need to pay for an extra movie. The good thing about this method is that you simply're not restricted to the tremendous-vast-open aperture in vivid gentle, which would reduce depth of field and due to this fact restrict how a lot of the image is in sharp focus. This time round, Samsung is differentiating its two smartphones internally and externally.
Will probably be easy for the Samsung Galaxy S9 to stand out on the mobile conference in Spain this week as a result of different main smartphone manufacturers such as Huawei and LG are usually not anticipated to launch new flagship telephones, in accordance with Thomas Husson, vp at research agency Forrester. The outside safety cameras come in many fashions.
These cameras used two an identical lenses, arranged one on high of the other within the manner of an over-and-below shotgun. If the camera decides that too many photos have been taken at your location, it retracts the shutter and blocks the viewfinder. Virtually 10 years after by the winter of 2005 disposable cameras turned stapled to the patron movie camera market and the flash – equipped disposables had been normally used. | {
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Q: Scala None instance not == None I've got an intermittent problem in Scala code where I'm working with values from an immutable map with String keys. Here's the basic code, including debug logging I added:
val compStruct = subsq.comps get (ident)
compStruct match {
...
case None =>
logger.info(s"Found None, of type ${compStruct.getClass.getName}, at position $position (ident $ident)")
...
case x =>
logger.info(s"Illegal structure of type ${x.getClass.getName} at position $position (ident $ident) - x == None is ${x == None}, x.getClass == None.getClass is ${x.getClass == None.getClass}, x.getClass.getName == None.getClass.getName (${None.getClass.getName}) is ${x.getClass.getName == None.getClass.getName}")
...
}
the problem is that case x is sometimes taken when the value is actually a None, as shown by the (sanitized) debug output:
INFO ...: Found None, of type scala.None$, at position 3000 (ident XX)
INFO ...: Illegal structure of type scala.None$ at position 3200 (ident XX) - x == None is false, x.getClass == None.getClass is true, x.getClass.getName == None.getClass.getName (scala.None$) is true
(The first line is what I'd expect to happen and indeed does happen normally; the rest are the error case)
So if my logging is to be believed (and I haven't messed up my expression somehow) I've got a case where the map is returning x, where x is an instance of class scala.None$ (the same class scala.None$ as seen by the compiled code) but doesn't match the case None and x == None is false.
A classloading issue would be the obvious cause, but x.class == None.class seems to preclude that.
Added: As I suggested in the comments, I can reproduce the instances of None not matching with the following code:
object Test {
def main(args: Array[String]): Unit = {
val none1 = None
val clas = this.getClass.getClassLoader.loadClass("scala.None$")
val constr = clas.getDeclaredConstructors()(0)
constr.setAccessible(true)
val none2 = constr.newInstance()
println(s"none1 == none2 is ${none1 == none2}")
println(s"none1 == None is ${none1 == None}")
println(s"none2 == None is ${none2 == None}")
}
}
Which gives:
none1 == none2 is false
none1 == None is false
none2 == None is true
I don't think that has anything to do with what's happening in the application, though.
Added: I modified the actual None$ classfile to both print a message when the constructor executes and throw an exception if the None$.MODULE$ value is non-null when the constructor is called, and even moved the store to the static MODULE$ value to the static constructor block (the original code had this store in the constructor, which I think is technically a violation of JVM rules since the object is not considered initialized until after the constructor returns).
This does block the reflection call to the constructor (above code sample) that duplicated the symptoms of the problem, but doesn't change anything in the actual application. The value of None$.MODULE$ changes from one execution of the code to the next, even though the class remains the same (same System.identityHashCode), yet the constructor is only called once.
A: Regarding your test:
None is an object in Scala. When you define
val none1 = None, you assign to none1 this object, which is supposed to be a singleton.
By using reflection, you are bypassing the private constructor and creating a new instance of None class. The == operator will only return true if the two pointers are pointing to the same object.
You can verify the memory address of these objects by using System.identityHashCode(none1) and compare it.
Also, if you try to run your match against none1 in your object Test, you will run into a match error, as the second instance of None does not match either None or x.
I was able to reproduce your error. By running this code:
val a = Map("a" -> "b", "b" -> None)
a.get("b") match {
case None => print("None")
case x => print("X")
} // Prints X
a.get("c") match {
case None => print("None")
case x => print("X")
} // Prints None
I know that this does not explains why it prints X, but at least you know when...
Because your HashMap has None values, it is a HashMap[String,java.io.Serializable] rather than HashMap[String,String].
And the call to get will return a java.io.Serializable rather than a String.
To solve your problem and get it to match when it is None, you can do:
case x if(x.isInstanceOf[None$]) =>
A: Take note that the way a None is processed in your context of code is via an Option[A] where by not specifying the case for your second condition, it means that you are allowing None to be a part of the second case.
What you should do is process the map.get into this way if you are trying to achieve on getting the case which is not a None
val compStruct = subsqs.comp.get(ident)
compStruct match {
case None => ...
case x: Some(_) => ...
}
| {
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} | 4,147 |
{"url":"https:\/\/phys.libretexts.org\/Under_Construction\/Purgatory\/3%3A_Applying_Particle_Models_to_Matter\/3.3_Intro_Particle_Model_of_Matter","text":"$$\\require{cancel}$$\n\n# 3.3 Intro Particle Model of Matter\n\n\nIn this introductory model of the particle nature of matter we focus primarily on the force that acts between two atoms or molecules. We make extensive use of the relation of force to the potential energy describing the interaction between two atomic sized particles. Initially we assume the particles are electrically neutral, but we will see how to take this into account a little later in the chapter. The next two models will use the basic ideas established here to help us develop a much deeper understanding of both bond energy and thermal energy.\n\nThe particle model of matter that we introduce is the familiar picture of matter as composed of atoms and molecules. Our particle model for ordinary matter is simple and universal. It is not restricted to a particular kind of matter, but encompasses all ordinary matter. That is what makes this model so useful. Of course, being very general, it can\u2019t predict many of the details that depend on the \u201cparticulars\u201d, but it can predict many of the universal properties.\n\n## Construct Definitions\n\n\u2022 Particle: This label applies to microscopic constituents of matter, typically an atom or molecule, but it could also refer, for example, to the constituents of the nucleus, if that were the focus of interest.\n\u2022 Attractive and Repulsive Forces: Atomic sized particles exert forces on each other in the same way that large-scale objects do. These forces can be attractive or repulsive, which one typically depends on their separation.\n\u2022 Interaction between two particles: The basis for making sense out of how particles interact is to focus on the interaction of only two particles at a time. There are several properties that keep reoccurring in our description of this interaction.\\\n\u2022 Center-to-center separation: We consistently refer to the distance between particles as being the center-to-center separation, rather than the distance between their surfaces. Usually we will use the symbol r to indicate this separation distance.\n\u2022 Equilibrium position or equilibrium separation: When we are focusing on just two particles, we will find that there is often a \u201cspecial\u201d separation, often referred to as the equilibrium separation. The reason this separation is special is that at this equilibrium separation the interparticle force is zero. What does this say about the slope of the PE at this point? It has similarities to the spring-mass system in this respect. A mass hanging on a spring hangs at a particular \u201cseparation\u201d from the point at which the spring is supported. This is a favored position. If the mass finds itself closer to the point of support, the \u201cspring force\u201d pushes it away, back toward the equilibrium position. Conversely, if it finds itself too far form the support, the spring force pulls it back toward the equilibrium position. The exact same thing happens with two atomic sized particles. We label the equilibrium separation for two particles with the symbol ro.\n\u2022 Pair-wise Potential Energy: We will consistently refer to the potential energy between two atomic size particles as the pair-wise potential energy, PEpair-wise. This potential energy has a fairly generic shape to it that you need to become familiar with. In general terms, it becomes very repulsive if the two particles begin to get too close to each other. The potential has a minimum and becomes \u201chorizontal\u201d\u2013slope is zero\u2013at the two particle\u2019s equilibrium separation. As the particles begin to separate, the potential at first \u201clooks like\u201d a spring-mass potential, but then begins to flatten out and becomes perfectly flat (horizontal, so zero force acting between the two particles here) once the separation is a few times that of the equilibrium separation, ro. The parameter that describes how \u201cdeep\u201d the potential is, that is, the difference in energy between where the potential is flat at large separations and at its lowest value where the equilibrium separation occurs, is often called the \u201cwell depth\u201d and designated with the lowercase Greek letter e. The well depth, e, is the magnitude of this energy difference, so is always a positive quantity.\n\u2022 Single Particle Potential Energy: In a solid or liquid, each particle has multiple pair-wise interactions, because it has lots of neighbors to interact with. It will sometimes be useful to focus on just one particle at a time, and to \u201cadd up\u201d all the interactions it has with its neighbors to obtain a potential energy function that describes the forces acting on just this one particle from all of its neighbors. We call this the single-particle potential energy to make it clear that it is not PEpair-wise.\n\n### Graphical Representation of the Pair-Wise Potential Energy\n\nThe PEpair-wise PE curve shown has the typical shape of almost all atomic-size pair-wise potentials. It has a simple mathematical form, so is useful for that reason alone. This particular shape describes rather well the interaction between the atoms of the noble elements (He, Ne, etc) in all their phases. Note that the equilibrium separation occurs at a slightly larger separation distance than one particle diameter, designated by the lower-case Greek sigma, s. If these particles acted like billiard balls, there would be a little space between them, even when they were as close as they \u201cwanted\u201d to get to each other.\n\nIt is customary, much to many beginners\u2019 consternation, to define the zero of PEpair\u2011wise to be the value the PE has when the particles are separated by a great difference. Also notice how steep the curve gets as the particles begin to get closer than the equilibrium separation. Remember, a steep PE curve means a strong force, which is repulsive in this case. We will frequently return to this graph.\n\n# Meaning of the Model Relationships\n\n1) All \u201cnormal\u201d matter is comprised of tiny particles (atoms and molecules) that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another.\n\nThis is a slightly paraphrased quote from Nobel Laureate Richard Feynman in which he stated that if all scientific information were to be lost, these would be the most valuable ideas to pass on to future generations.\n\n2) The part about the particles attracting and repelling each other is most easily visualized in terms of the slope of the pair-wise potential energy acting between the two particles. Be sure you review the previous chapter if you need to so that the relationship between force and PE ( |F| = |d(PE)\/dr| ) is absolutely clear to you. There is no point in going any further, if you are still stumbling over this relationship.\n\n3) Make sure you can use the relationship in (2) to explain to your fellow chemistry students (or your chemistry TA) the three bulleted features of the pair-wise potential using relationship (2) and the general shape of the pair-wise potential.\n\n4) When there are many particles, the phase (s, l, g) of those particles depends on their total energy. At sufficiently high total energy, the particles are unbound and in the gas phase. At sufficiently low energy the particles are in the liquid or solid phase and are bound. The average particle-particle separation in the bound state is approximately equal to the separation corresponding to the minimum of the pair-wise potential energy. In the unbound state it is much greater than the separation corresponding to the minimum PE.\n\nOne common mistake that many students make is to attempt to ascribe macroscopic properties (like solid, liquid, gas) to the interaction of only a small number of particles using PEpair-wise. The macrostate of matter, whether it is in a solid, liquid or gas phase, for example, is due to the simultaneous interactions of something like 1025 pair-wise interactions if we have a mole of the substance. These ideas are not easy, so be patient. Initially, try to imagine a solid at very low temperatures. Each particle \u201cwants\u201d to be at the right distance with respect to all of its neighbors. If there is a way for the system to \u201cget rid\u201d of its energy (by giving it to some colder system, for example), it will continue to settle down and reduce its thermal energy. Eventually, all the random motion comes to a stop (if we can keep cooling the sample) and the particles find their \u201cmagic\u201d places, each near the \u201cbottom\u201d of the PEpair-wise with each of each neighbors.\n\nNow, imagine we start adding energy to the sample. All the particles begin acting like little spring-masses, oscillating back and forth around their equilibrium positions. Eventually they move sufficiently far, so that some \u201cjump\u201d out of where they are \u201csupposed to be.\u201d Particles at or near the surface might even leave the sample if their vibrations get vigorous enough. Picturing what happens when a substance melts, i.e., turns from a solid to a liquid, is difficult, even for the experts. Don\u2019t worry about picturing that transition. But you can imagine continuing to add energy until all the particles, even 1023 particles, have sufficient energy to separate far apart from each other, causing them to be in the gas phase. Recall what value all ~1025 pair-wise potentials will have, if all particles are separated by many particle diameters. So, what is the bottom line here at this point in our making sense of all this? Without getting into a lot of detail, it should make sense to you that at some sufficiently low temperature, everything will be a solid and at some sufficiently high temperature, everything should be a gas. That is plenty for right now.\n\n5) The interactions of one particle in a liquid or solid with all of its neighbors add together to form one three-dimensional potential energy for a particular particle with a minimum that defines the equilibrium position of that particle (where the net force due to all of the pair-wise interactions is zero). We refer to this potential as the single-particle potential energy to emphasize its distinction from the pair-wise potential energy.\n\n6) Each particle in a solid or liquid oscillates in three dimensions about its equilibrium position as determined by its single-particle potential.\n\nOK, so here is where we are attempting to make our mental picture a little clearer regarding what is happening to a single particle (which could be an atom or a tightly bound-together molecule) when it finds itself somewhere in the middle of similar particles in a solid or liquid. It really is acting like it is attached to a bunch of springs with all of its neighbors (and nearby neighbors). But here is the \u201creally neat\u201d thing. No matter how complicated the actual chemical bonds are, and no matter how many there are, or in what directions they point, they all add up to exactly what would happen if you had only three (that\u2019s right, only three) little springs of exactly the right strength, one going out in each of the three x, y, and z directions of the three-dimensional space we seem to occupy in this universe (at least on our scale and on the scale of atoms and molecules). So the picture you want to get into your head is something like that shown below, remembering that the spring constant of the springs can be different in the three directions.\n\nWe will come back to this picture shortly when we make more sense of thermal energy.\n\nBut, what about the bond energy? Well, it really does depend on the real bonds, the real chemical bonds. However, we can develop reasonable estimates in terms of the well depths of the pair-wise interactions for the bond energy that work for practically all pure substances. Carrying out the analysis to make sense of bond energy and to make sense of thermal energy is what the next two models are about.\n\n3.3 Intro Particle Model of Matter is shared under a not declared license and was authored, remixed, and\/or curated by Dina Zhabinskaya.","date":"2022-07-07 11:39:52","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\": 1, \"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\": 0.6825353503227234, \"perplexity\": 350.68698906092203}, \"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-2022-27\/segments\/1656104690785.95\/warc\/CC-MAIN-20220707093848-20220707123848-00644.warc.gz\"}"} | null | null |
About TRT
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Melissa Guat
Tanjong Katong Secondary School
I first joined Mr Wong's English tuition when I was in Secondary 1, back in 2015, and my improvement in English has been evident ever since.
Due to his progressive teaching methodologies and not to mention, his engaging tuition classes, I could always gain an edge over most of my classmates in terms of the command of the language. This "edge" that I have gained has proven to be effective when I came in 2nd in the whole school.
Mr Wong is extremely patient in his teaching and he never fails to adopt an inclusive culture as he frequently asks for our opinions on ways to better cater to our needs.
Besides his aptitude for teaching, an added bonus is the notes he painstakingly prepares for us every week - which include Linguistics for comprehension, essay writing and Oral. With due credit to his efficacious teaching, I attained an A1 for my English O Level paper. I truly believe that if a student comes to Mr Wong as early as in Sec. 1, like myself, he/she stands a good chance to excel in his/her English! Now, I am in Victoria Junior College and have been awarded the MOE Humanities Scholarship.
Mr Wong is not your ordinary tutor. He has gone beyond his duties as a tutor to ensure that his students not only excel in their studies, but also in their future education and career. He meets up with students outside tuition hours to catch up with them and find out if he can help his students in any other areas. Mr Wong really puts in the effort to understand his students and their capabilities, so that every student can excel. For me, I was no exception. Mr Wong has met me a couple of times to ensure that I was doing well at home and at school. Even after my O level, he still set aside time despite his packed schedule to discuss with me about my next step, what courses should I take in Polytechnic. As I was interested in the Architecture course, Mr Wong went all the way to link me up with one of his friends, who is an architect in a well-established architecture firm, for internship and I am very appreciative for the support he has given me. I have never regretted joining The Rationale Thinking and I believe many more students can benefit too.
I first joined Mr Wong's English Tuition in late March 2015, the same year in which I was taking my 'O' level, I was struggling with the monthly Secondary School English common tests, frequently obtaining only a C5 or a C6 grade. The first day I stepped into his tuition class was like an enlightenment as he helped me understand what I had been doing wrong the entire time, His meticulous focus on global issues, critical thinking and current affairs, topics that neither my teachers nor previous tutors ever taught, was the key to me excelling whilst writing Argumentative or Discursive essays.
Mr Wong's English tuition places emphasis on attention grabbing vocabulary and adjectives assisted me during my final examinations where I tackled a descriptive essay, a risky decision on my part but when I received my results I knew it was worth it. The skills and knowledge he provided me with were not only useful for English, but also gave me an advantage for subjects like History, Social studies and Geography.
Mr Wong has helped me tremendously and I improved from a C5 to an A1 in the short span of 7 months. If it were not for him, I doubt I would have ever come close.
Watch Melissa's video testimonial here:
Winston Ong
St. Hilda's Secondary School
View More Testimonial
Ryan Wielson Lim
Currently in Temasek Junior College,
Formerly from Bedok Green Secondary
The short journey I had with Mr Wong of 3 months was a really phenomenal one. He is an inspiring teacher who imparts his knowledge to us as students, and personally, lit a fire in me. In Mr Wong's English tuition class, he provides a comprehensive coverage of real-life events and his emphasis on vocabulary has enabled me to drastically improve my linguistic skills.
Mr Wong's tuition classes are never mundane as he encourages students to enunciate our views and questions. His notes are all prepared scrupulously by himself and never fails to include occasional tests to engrain our newfound knowledge of vocabulary, enabling us to utilise it in the examination hall.
Mr Wong's concept of moulding his students into global thinkers, with unique insights of global happenings has allowed me to grasp an A1 in my 'O' Level Examinations after getting a C5 in my SA1. I cherish the humbling opportunity I had to work with one of the most passionate and caring teachers, and I cannot be more thankful to have come under his tutelage.
When I joined Mr Wong's English tuition I was receiving an E8 for my SA1 English. I was very worried and not doing well for English would affect my chances of going to a JC.
Prior to attending Mr Wong's English tuition, my grammar and writing skills were very weak and I could only write argumentative or expository essays. Sometimes I was able to do well while other times I failed. Mr Wong reviewed my papers and immediately came up with a strategy to do well for the final exam and that was to focus on writing Hybrid essays, which is a combination of "Descriptive" and "Reflective" components. I had never done Hybrid essays before and didn't take English Literature at Sec 4. Initially I wasn't sure if I was able to write descriptive essays but he convinced me to try it out and with his model essays and in-house notes on the skills I attempted Hybrid essay during my O Level and got an A2 grade!
Mr Wong's English tuition curriculum provides his students with a lot of notes on comprehension linguistic skills. These notes definitely helped me to understand more on how to identify the type of comprehension questions and allowed me to be able to answer the question correctly.
Most of us know that when our foundation is weak, it actually takes a long time and effort to correct the problems but I must say that Mr Wong is very effective in correcting the problems and his teaching style is very dynamic. Within 5 months, I actually jumped from E8 to A2!!!
Watch Yuxin's video testimonial here:
Kaelyn Lim
Geylang Methodist
Secondary School
Melissa joined TRT since Sec. 1 and is now a Humanities Scholar at VJC
Joshua Teo
Bedok Green Secondary School
Thaddaeus Kwok
Bedok South Secondary School
When I first came to Mr Wong's English Tuition in middle of Sec 3, my English grade was C5. Back then, my command of English was really poor. Furthermore, I didn't know how to write argumentative essays as I didn't know any real world examples and was very bad at explaining my points.
Mr Wong immediately advised me on how to improve. For example, reading news articles that he gave out during class, and listening to the BBC world service when I commute has helped me to know more current affairs. My writing style also improved a lot under Mr Wong. Mr Wong teaches us to use critical thinking and takes us step by step through the thinking process. His methods of teaching English is very logical and has helped me improve a lot.
Mr Wong does not just care about academics, he is also very supportive. He is always concerned for the welfare of his students. Mr Wong also never stops pushing me to aim high. Knowing that I wanted to major in Computer Engineering, he always encourages me to aim high and work towards a PSC scholarship and study at MIT.
Mr Wong is a very inspiring teacher and motivates me to do well in my studies. Before I met him, I was a playful student and didn't care about my studies. But he encouraged me to work hard, so I studied hard for my O Level and am now a student at Raffles Institution. Thanks Mr Wong!
Winston improved from a C5 to an A1 within 3 months
Not Just a tutor, but an Advisor and Mentor
Yuxin's transformation under Mr Wong's mentorship
- From a playful student to studying at Raffles Institution
In a short span of 7 months, Joshua Jumped from a C5 to an A1 for his English 'O' Level Examinations and also came in Top 20 in his cohort
Within 5 months, Kaelyn jumped from a B4 to an A1 and came in top of her
cohort. Now, she is pursuing a Diploma in Psychology at Ngee Ann Polytechnic
Ryan took advice from Mr Wong and attained an A in his 'O' Level
Watch Kaelyn's video testimonial here:
I joined Mr Wong's English tuition in June 2017 after getting a B4 grade for my mid-year exam. I can still vividly remember how pleasantly surprised I was when I attended the first lesson. Unlike other English tuition classes I previously attended, there is a distinct difference between him and other tutors.
He made sure that all of us understood what he taught and when we are still not sure, will patiently answer any of our queries. He makes sure that we are aware of the relevant news articles we can use to build the content of our essays and equips us with A1 quality essays written personally by him.
As for oral, we practise reading aloud a short portion of a passage, this helps us to build our confidence and gradually improve our pronunciation, not forgetting the spoken interaction segment! Additionally, he often shares his personal experiences in different industries that are relevant to deepen our understanding of various topics.
Apart from the knowledge he imparts in his tuition class, he goes out of his way to advise me where I should go after I received my O levels, even though I was no longer attending tuition classes with him. Mr Wong is by far the best tutor I know as he is a very caring, inspiring, diligent and meticulous teacher who only has his students best interest at heart and I feel privileged to be taught by him.
Wei Yuxin
Currently in Raffles Institution,
Formerly from Temasek Secondary School
Watch Ryan's video testimonial here:
Copyright The Rationale Thinking Learning Centre. All rights reserved.
No content of this website can be used without prior permission from its author
The Rationale Thinking
121 Bishan Street 12, #01-89singapore, singapore570121SG
Phone: 9168 8775 Website: therationalethinking.com | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,392 |
Based on the best-selling Stephen King novel, this "absolutely spellbinding horror movie" (Roger Ebert) has become a pervasive, pop-culture touchstone for anyone who's ever wanted to get even. Sissy Spacek and Piper Laurie deliver Oscar®-nominated* performances and John Travolta and Amy Irving are terrific in this ultimate revenge fantasy that has become one of the all-time great horror classics, and is now, finally, offered as a definitive, two-disc Collector's Edition Blu-ray!
Released a few days after Halloween in 1976, it's considered one of the great American horror films. A box-office hit with two Oscar nominations for acting, it contains one of the genre's most memorable images and an ending that has been ripped off dozens of times.
It also gave a career lift to a young novelist named Stephen King.
Yet revisit "Carrie" 40 years later, as you can do on a new Blu-ray version from Shout! Factory, and you might think, "This ... is not that scary." A high school drama, yes. A resonant look at the emotional damage caused by bullying, yes. But a horror movie? Not so much. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,181 |
With nobody vulnerable, his partner passed and RHO opened a weak 2 . I'd be fine with a 3 overcall (showing an intermediate hand like this and a good 6+ card spade suit). However, at the table, 2 was chosen. LHO raised to 4 and partner's 4 bought the contract.
The defense played ace, king and another diamond, ruffed and overruffed. Declarer drew trumps, finding the 2 bidder with all 3 of them. Now what?
West correctly played low on the second club ("splitting" his honors could have proved disastrous). Declarer could try dummy's Q (resulting in down 2), but he guessed to play low, but still down 1 (East took the K and the defense still had to get a heart trick).
So, how could declarer have made it? The bidding and play marked East with 3=6=2=2 shape (he opened a weak two bid in hearts and showed up with 3 spades and 2 diamonds). Knowing he had only 2 clubs, the right play is clear. Go to dummy and lead a club to the 8! This "intrafinesse" wins the contract. West wins this trick, but later the A drops the king and a marked finesse in clubs allows declarer to throw his heart on the fourth club for +420.
Note that the defense could have prevailed with a heart shift at trick 2. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,150 |
Coronation Global Capital Plus, Global Managed and Global Equity Select Funds - July 2020
Neil Padoa
Neil is a portfolio manager and Head of Global Developed Markets. He joined Coronation in May 2012 and has 13 years' investment experience.
OUR FUND RANGE with a developed market bias includes Global Equity Select and two multi-asset funds – the long-term, growth-oriented Global Managed Fund and the more conservative Global Capital Plus Fund.
As you would expect during a period when markets recovered strongly, the funds performed in line with their risk budgets over the quarter in review, with Global Equity Select, Global Managed and Global Capital Plus producing US dollar returns of 18.4%, 12.3% and 6.5%, respectively.
At quarter-end, Global Managed was positioned in 62% growth assets and 38% more stable, diversifying assets. The growth-asset allocation consists of 53% effective equity exposure and smaller positions in listed property, convertible bonds and high-yield corporate bonds. The more stable part of the portfolio consists of Treasury bills, hedged equity, inflation-protected securities, commodities and investment-grade corporate bonds.
The more conservative Global Capital Plus owned 40% in growth assets, including 22% effective equity exposure and 60% more stable and diversifying assets, including 20% in investment-grade corporate bonds and 13% in Treasury bills.
STREAMING RETURNS
Spotify, which more than doubled in value over the quarter, was the largest single contributor to returns in Global Equity Select and Global Managed. Streaming now accounts for the majority of music industry revenue. While there are now over 300 million paying music streamers globally, music remains extremely under-monetised in our view, given the 3.5 billion smartphones in the world today. Headline subscription prices have not changed much in years and average revenue per user has in fact declined due to family and student discount plans.
Music spending per capita has halved in real terms since 1999. As the largest audio platform outside of China with 130 million paying subscribers and an additional 163 million ad-supported users (compared to Apple Music, which has 60 million to 70 million subscribers), and with a better product and ongoing innovation, Spotify is well placed for long-term growth.
We are also bullish on Spotify's podcast strategy. Terrestrial radio remains a large advertising revenue pool globally and Spotify is trying to disrupt this, acting decisively and investing in leading podcast creation tools, studios and exclusive content from top podcasters such as Joe Rogan.
Since 2015, Spotify has grown its revenue by 37% per annum and we expect strong growth to continue. In the words of co-founder and CEO Daniel Ek, "everything linear dies". As the leading player and innovator in the fast-growing audio streaming market that is led by an exceptional management team, we believe Spotify is well positioned to capitalise on this trend.
Other contributors to returns include long-held positions in Alphabet, Charter Communications, Naspers, UnitedHealth and Bayer.
SMOKE WITHOUT FIRE
Philip Morris International (PMI) was the largest detractor from performance, although with a decline of -2.6% the effect was only marginally negative. PMI is a global tobacco company and the leader in potentially reduced-risk, next-generation products through its IQOS heated tobacco franchise. IQOS is already contributing c.20% to company revenues. PMI has invested significantly in the IQOS franchise over a sustained period and has first-mover advantage in the heated tobacco category. IQOS has been a phenomenal success in our view, ranging from truly extraordinary results in Japan to solid, steady progress across many European markets.
To date, c.11 million smokers have completely quit smoking combustible cigarettes and moved to IQOS. At the time of writing, the US Food and Drug Administration has just authorised IQOS to be sold in the US with a reduced risk exposure claim.
As IQOS grows, it is accretive to PMI's revenues and profits, and there is still a long runway of growth for IQOS globally. Despite the resilience of tobacco as a consumer category, PMI has not been immune to Covid-19 lockdowns. PMI has been negatively impacted by lost duty-free sales, lockdowns and temporarily slower IQOS user conversion.
We expect that over the medium term these lost sales should be recovered and that IQOS should fairly quickly resume its growth trajectory. PMI remains a top 10 holding.
THE WHEAT FROM THE CHAFF
Last quarter, we felt there were attractive opportunities for those investors with a long time horizon and the ability to filter companies whose prices had been dislocated with little impact to their sustainable earnings power. After a sharp rally, these opportunities are now harder to find.
In addition, the need to reassess the prospects of many businesses continues as investors assess fundamental virus-induced behavioural changes versus short-term noise. Fundamental changes, however, play to the strengths of fundamental investors, and we continue to find a select number of stocks with attractive long-term prospects that are reasonably priced, while appropriately managing exposures across a range of asset classes.
Thank you for your continued support and interest in the funds. +
Coronation Global Capital Plus: Highest annual return 17.1% Jul 2010 - Jun 2011;Lowest annual return (7.4%) Sep 2014 - Aug 2015
Coronation Global Managed: Highest annual return 23.4% Jan 2019 - Dec 2019; Lowest annual return (14.4%) Mar 2015 - Feb 2016
Coronation Global Equity Select: Highest annual return 37.1% Jan 2019 – Dec 2019; Lowest annual return (20.4%) Jan 2018 – Dec 2018 | {
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} | 1,660 |
A XXV-a ediție a Jocurilor Olimpice este un eveniment multi-sportiv internațional major ce va avea loc în perioada 6-22 februarie 2026 în orașele italiene Milano și Cortina d'Ampezzo.
Legături externe
https://www.milanocortina2026.org/
Milan Cortina 2026 (CIO)
2026
Jocurile Olimpice de iarnă
Milano
Cortina d'Ampezzo | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,097 |
Q: rxJava opposite for throttleFirst operator (not throttling but collecting) I need following operator which when element comes, starts the timer (creates window for elements) and collects them into List or Observable/Flowable. When time specified for the timer ends and no element comes, the operator doesn't send empty event. When next element comes, new timer is created and it starts to collect elements.
Rx java have Buffer and Window operator but this operators have disadvantages:
*
*Buffer http://reactivex.io/documentation/operators/buffer.html
with signature
buffer(long timespan, TimeUnit unit) almost fits but it produces events with empty list when no elements in timespan comes.
*Window http://reactivex.io/documentation/operators/window.html
with signature
window(long timespan, TimeUnit unit) almost fits but it produces empty Observables/Flowables when no elements in timespan comes.
It is possible to filter this empty elements but I would like to avoid polluting scheduler with empty events (based on the timer) with List/Observable-s/Flowable-s.
I spend some time and found very similar formally, but functionally fulfilling the opposite role
throttleFirst(long windowDuration, TimeUnit unit)
https://raw.githubusercontent.com/wiki/ReactiveX/RxJava/images/rx-operators/throttleFirst.png
But unfortunately it throttles, but not collect items.
A: I don't think one could save on object creation so if you want to avoid empty lists, filter them out.
As for starting a new periodic timer when a new item arrives late, I can't think of any combination of existing operators that could do it without possibly losing items.
I created the following "contraption" that could do it without item loss:
public static final class BufferWithTimeout<T> {
Scheduler.Worker trampoline = Schedulers.trampoline().createWorker();
final long timeout;
final TimeUnit unit;
final Scheduler.Worker worker;
final SerialDisposable timer = new SerialDisposable();
final PublishSubject<List<T>> output = PublishSubject.create();
List<T> current;
long bufferIndex;
BufferWithTimeout(long timeout, TimeUnit unit, Scheduler scheduler) {
this.worker = scheduler.createWorker();
this.timeout = timeout;
this.unit = unit;
}
void onValue(T value) {
trampoline.schedule(() -> {
if (timer.isDisposed()) {
return;
}
if (current == null) {
current = new ArrayList<>();
long bi = ++bufferIndex;
timer.set(worker.schedulePeriodically(() -> {
onTime(bi);
}, timeout, timeout, unit));
}
current.add(value);
});
}
void onTime(long index) {
trampoline.schedule(() -> {
if (index == bufferIndex && current != null) {
if (current.isEmpty()) {
current = null;
bufferIndex++;
timer.set(null);
} else {
output.onNext(current);
current = new ArrayList<>();
}
}
});
}
void onTerminate(Throwable error) {
timer.dispose();
worker.dispose();
trampoline.schedule(() -> {
if (current != null && !current.isEmpty()) {
output.onNext(current);
current = null;
}
if (error != null) {
output.onError(error);
} else {
output.onComplete();
}
});
}
void dispose() {
timer.dispose();
worker.dispose();
trampoline.schedule(() -> {
current = null;
});
}
public static <T> ObservableTransformer<T, List<T>> create(
long timeout, TimeUnit unit, Scheduler scheduler) {
return o ->
Observable.defer(() -> {
BufferWithTimeout<T> state = new BufferWithTimeout<>(
timeout, unit, scheduler);
return o
.doOnNext(v -> state.onValue(v))
.doOnError(e -> state.onTerminate(e))
.doOnComplete(() -> state.onTerminate(null))
.ignoreElements()
.<List<T>>toObservable()
.mergeWith(state.output.doOnDispose(state::dispose));
});
}
}
You could try it via:
// generate events over time
Observable.fromArray(1, 2, 3, 5, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31)
.flatMap(v -> Observable.timer(v * 100, TimeUnit.MILLISECONDS).map(w -> v))
// apply operator
.compose(BufferWithTimeout.create(
700, TimeUnit.MILLISECONDS, Schedulers.computation()
))
// wait for it all
.blockingSubscribe(System.out::println);
Note though that this creates way more objects per source element, there is way around it but it would get way more complicated.
A: What would you like to happen instead of the onComplete? Could you combine a buffer() or window() with a switchIfEmpty()?
//No emissions or on complete
source.window(...).switchIfEmpty(Observable.never());
or
//Empty list emission
source.window(...).switchIfEmpty(Observable.just(Collections.emptyList()));
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,264 |
FROM php:7.0-alpine
MAINTAINER Rob Loach <robloach@gmail.com>
# Packages
RUN apk --update add \
autoconf \
build-base \
curl \
git \
subversion \
freetype-dev \
libjpeg-turbo-dev \
libmcrypt-dev \
libpng-dev \
libbz2 \
libstdc++ \
libxslt-dev \
openldap-dev \
make \
unzip \
wget && \
docker-php-ext-install mcrypt zip bz2 mbstring pcntl xsl && \
docker-php-ext-configure gd --with-freetype-dir=/usr/include/ --with-jpeg-dir=/usr/include/ && \
docker-php-ext-install gd && \
docker-php-ext-configure ldap --with-libdir=lib/ && \
docker-php-ext-install ldap && \
apk del build-base && \
rm -rf /var/cache/apk/*
# PEAR tmp fix
RUN echo "@testing http://dl-4.alpinelinux.org/alpine/edge/testing/" >> /etc/apk/repositories && \
apk add --update php7-pear@testing && \
rm -rf /var/cache/apk/*
# Memory Limit
RUN echo "memory_limit=-1" > $PHP_INI_DIR/conf.d/memory-limit.ini
# Time Zone
RUN echo "date.timezone=${PHP_TIMEZONE:-UTC}" > $PHP_INI_DIR/conf.d/date_timezone.ini
# Register the COMPOSER_HOME environment variable
ENV COMPOSER_HOME /composer
# Add global binary directory to PATH and make sure to re-export it
ENV PATH /composer/vendor/bin:$PATH
# Allow Composer to be run as root
ENV COMPOSER_ALLOW_SUPERUSER 1
# Setup the Composer installer
RUN curl -o /tmp/composer-setup.php https://getcomposer.org/installer \
&& curl -o /tmp/composer-setup.sig https://composer.github.io/installer.sig \
&& php -r "if (hash('SHA384', file_get_contents('/tmp/composer-setup.php')) !== trim(file_get_contents('/tmp/composer-setup.sig'))) { unlink('/tmp/composer-setup.php'); echo 'Invalid installer' . PHP_EOL; exit(1); }"
# Set up the volumes and working directory
VOLUME ["/app"]
WORKDIR /app
# Set up the command arguments
CMD ["-"]
ENTRYPOINT ["composer", "--ansi"]
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,935 |
{"url":"https:\/\/stacks.math.columbia.edu\/tag\/0FRB","text":"Lemma 24.10.2. In the situation above, let $\\mathcal{M}$ be a right graded $\\mathcal{A}_ U$-module and let $\\mathcal{N}$ be a left graded $\\mathcal{A}$-module. Then\n\n$j_!\\mathcal{M} \\otimes _\\mathcal {A} \\mathcal{N} = j_!(\\mathcal{M} \\otimes _{\\mathcal{A}_ U} \\mathcal{N}|_ U)$\n\nas graded $\\mathcal{O}$-modules functorially in $\\mathcal{M}$ and $\\mathcal{N}$.\n\nProof. Recall that the degree $n$ component of $j_!\\mathcal{M} \\otimes _\\mathcal {A} \\mathcal{N}$ is the cokernel of the canonical map\n\n$\\bigoplus \\nolimits _{r + s + t = n} j_!\\mathcal{M}^ r \\otimes _\\mathcal {O} \\mathcal{A}^ s \\otimes _\\mathcal {O} \\mathcal{N}^ t \\longrightarrow \\bigoplus \\nolimits _{p + q = n} j_!\\mathcal{M}^ p \\otimes _\\mathcal {O} \\mathcal{N}^ q$\n\nSee Section 24.6. By Modules on Sites, Lemma 18.27.9 this is the same thing as the cokernel of\n\n$\\bigoplus \\nolimits _{r + s + t = n} j_!(\\mathcal{M}^ r \\otimes _{\\mathcal{O}_ U} \\mathcal{A}^ s|_ U \\otimes _{\\mathcal{O}_ U} \\mathcal{N}^ t|_ U) \\longrightarrow \\bigoplus \\nolimits _{p + q = n} j_!(\\mathcal{M}^ p \\otimes _{\\mathcal{O}_ U} \\mathcal{N}^ q|_ U)$\n\nand we win. An alternative proof would be to redo the Yoneda argument given in the proof of the lemma cited above. $\\square$\n\nIn your comment you can use Markdown and LaTeX style mathematics (enclose it like $\\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).","date":"2023-03-29 13:31:45","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\": 2, \"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\": 2, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9849157929420471, \"perplexity\": 271.474731444144}, \"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-2023-14\/segments\/1679296948976.45\/warc\/CC-MAIN-20230329120545-20230329150545-00301.warc.gz\"}"} | null | null |
Q: Kubectl appears to be discarding standard output I'm trying to copy the contents of a large (~350 files, ~40MB total) directory from a Kubernetes pod to my local machine. I'm using the technique described here.
Sometimes it succeeds, but very frequently the standard output piped to the tar xf command on my host appears to get truncated. When that happens, I see errors like:
<some file in the archive being transmitted over the pipe>: Truncated tar archive
The files in the source directory don't change. The file in the error message is usually different (ie: it appears to be truncated in a different place).
For reference (copied from the document lined to above), this is the analog to what I'm trying to do (I'm using a different pod name and directory names):
kubectl exec -n my-namespace my-pod -- tar cf - /tmp/foo | tar xf - -C /tmp/bar
After running it, I expect the contents of my local /tmp/bar to be the same as those in the pod.
However, more often than not, it fails. My current theory (I have a very limited understanding of how kubectl works, so this is all speculation) is that when kubectl determines that the tar command has completed, it terminates -- regardless of whether or not there are remaining bytes in transit (over the network) containing the contents of standard output.
I've tried various combinations of:
*
*stdbuf
*Changing tar's blocking factor
*Making the command take longer to run (by adding && sleep <x>)
I'm not going to list all combinations I've tried, but this is an example that uses everything:
kubectl exec -n my-namespace my-pod -- stdbuf -o 0 tar -b 1 -c -f - -C /tmp/foo . && sleep 2 | tar xf - -C /tmp/bar
There are combinations of that command that I can make work pretty reliably. For example, forgetting about stdbuf and -b 1 and just sleeping for 100 seconds, ie:
kubectl exec -n my-namespace my-pod -- tar -c -f - -C /tmp/foo . && sleep 100 | tar xf - -C /tmp/bar
But even more experimentation led me to believe that the block size of tar (512 bytes, I believe?) was still too large (the arguments of -b are a count of blocks, not the size of those blocks). This is the command I'm using for now:
kubectl exec -n my-namespace my-pod -- bash -c 'dd if=<(tar cf - -C /tmp/foo .) bs=16 && sleep 10' | tar xf - -C /tmp/bar
And yes, I HAD to make bs that small and sleep "that big" to make it work. But this at least gives me two variables I can mess with. I did find that if I set bs=1, I didn't have to sleep... but it took a LONG time to move all the data (one byte at a time).
So, I guess my questions are:
*
*Is my theory that kubectl truncates standard output after it determines the command given to exec has finished correct?
*Is there a better solution to this problem?
A: Maybe you haven't been specific enough for kubectl regarding what the full command that it must contend with really is. There might be ambiguity as to who should be responsible for the pipe process. The "--" probably doesn't direct kubectl to include that as part of the command. That is probably being intercepted by the shell.
Have you tried wrapping all of it in double-quotes ?
CMD="tar cf - /tmp/foo | tar xf - -C /tmp/bar"
kubectl exec -n my-namespace my-pod -- "${CMD}"
That way it would include the scope of saving at the target as part of the process to monitor for completion.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,446 |
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margin-right:-10px;
margin-left:-10px;
}
div[class*=col-]{
padding-right:10px;
padding-left:10px;
}
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.bg-success{background:#5cb85c;color:#fff;}
.bg-info{background:#5bc0de;color:#fff;}
.bg-warning{background:#f0ad4e;color:#fff;}
.bg-danger{background:#d9534f;color:#fff;}
.bg-white{background:#fff;color:#222;}
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.bg-cyan{background-color:#00bcd4!important;color:#fff!important;}
.bg-teal{background-color:#009688!important;color:#fff!important}
.bg-green{background-color:#259b24!importantcolor:#fff!important;}
.bg-lightgreen{background-color:#8bc34a!important;color:#fff!important}
.bg-lime{background-color:#cddc39!important;color:#fff!important;}
.bg-yellow{background-color:#ffeb3b!important;color:#fff!important;}
.bg-amber{background-color:#ffc107!important;color:#fff!important;}
.bg-orange{background-color:#ff9800!important;color:#fff!important;}
.bg-brown{background-color:#795548!important;color:#fff!important;}
.bg-grey{background-color:#9e9e9e!important;color:#fff!important;}
.bg-cream{background-color:#f9ffe2!important;}
.text-white{color:#fff;}
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} | 1,339 |
\section{Introduction}
\label{Introduction}
Benefiting from the flexible mobility and deployment, unmanned aerial vehicles (UAVs) have been employed in numerous applications in recent years, such as remote surveillance, photography, traffic control, and cargo transportation, etc \cite{1}. By the reasonable design of its trajectory, UAV tends to obtain more dominant line-of-sight (LoS) channels and thus greatly improves the communication performance \cite{2}. However, these advantages are confronted with many challenges. Due to the broadcast nature of wireless transmission and the strong LoS links, UAVs are more vulnerable to attacks from jamming and eavesdropping \cite{3}.
Due to its capability of smartly configuring the wireless propagation environment, intelligent reflecting surface (IRS) has emerged as a promising technology for future wireless networks \cite{5,huang1}. Specifically, IRS is a uniform planar array consisting a large number of passive reflecting units, which can reflect the wireless signal with adjustable phase shift and/or amplitude. The main advantage of IRS-assisted wireless communications lies in that it can considerably increase the channel capacity with low power consumption and high deployment flexibility. Benefiting from above superiority, IRS has been applied to cognitive radio \cite{cog2}, device-to-device (D2D) networks \cite{8}, secure wireless communication \cite{11} and so on. A comprehensive tutorial on IRS-aided wireless communications can be found in \cite{12}.
Thanks to the appealing advantages, IRS has been exploited as a promising choice for improving the performance of UAV communication for combating security threats. In \cite{17}, the average secrecy rate was maximized by jointly optimizing the UAV trajectory, the transmit beamforming and the phase shift of IRS. In \cite{ICC} and \cite{WCL}, the UAV equipped with one IRS serves as a passive relay is proposed to maximize the secrecy rate of the system.
By exploiting the IRS, higher average rate of the UAV communication has been achieved to mitigate the jamming signal from the malicious jammer \cite{21}. However, all of these works are based on the perfect knowledge of the eavesdropper/jammer's location, which is generally unavailable in practical systems and thus brings new challenges to the IRS passive beamforming, deployment, and UAV trajectory. To the best of our knowledge, this still remains an open problem and needs further study.
\begin{figure}
\centering
\includegraphics[width=8.0cm]{Model.eps}
\caption{IRS-assisted UAV communication in the presence of a jammer.} \label{Model}
\vspace{-5mm}
\end{figure}
In this paper, we study a UAV communication system aided by IRS, and the communication process incurs malicious jamming from a jammer with imperfect location information, as shown in Fig. \ref{Model}. We consider a joint robust design to maximum the achievable average rate in the uplink. However, the formulated optimization problem is difficult to tackle due to its non-convexity and coupled variables. To deal with the problem, we propose an alternating algorithm by using the successive convex approximation (SCA), semidefinite relaxation (SDR), and S-procedure methods to require the sub-optimal solution. For the case that IRS is deployed near the GN, numerical results show that our proposed robust algorithm can greatly improve the worst system performance compared to the benchmark algorithms.
However, for the case that IRS is deployed near the jammer, numerical results show that the proposed algorithm is effective only with perfect jammer's location informantion.
\textit{Notations}: ${\text{diag}}\left\{ \cdot \right\}$, ${\text{tr}}\left( \cdot \right)$ returns the diagonal matrix whose diagonals are the elements of input vector. $\Re \left( \cdot \right)$ and $\angle \left( \cdot \right)$ represent the real part and the phase of the input complex value. By denoting ${\mathbf{\Theta}}\underline \succ 0$, it is used to represent ${\mathbf{\Theta }}$ as positive semidefinite.
\section{System Model and Problem Formulation}
\label{sec:System Model}
In this paper, a UAV communication system is considered as shown in Fig.\ref{Model}, where an IRS is deployed to assist in the information transmission from the GN to a UAV in the presence of a jammer with imperfect location information. All communication nodes are placed in the three dimensional (3D) Cartesian coordinates. The location of the jammer, GN and IRS are expressed as ${\bf{q_m}} = [{x_m},{y_m},{z_m}]$, ${\bf{q_g}} = [{x_g},{y_g},0]$ and ${\bf{q_r}} = [{x_r},{y_r},{z_r}]$, respectively. Due to the uncertain location region in 3D space, we regard the jammer's location as a hemisphere, which roughly simulates the practical situation. In the result, the location of the jammer can be denoted by
\begin{equation}\label{eq0}
\left\{ {\begin{array}{*{20}{c}}
{{x_m} = {\bar x_{m}} + \Delta {x_m},} \\
{{y_m} = {\bar y_{m}} + \Delta {y_m},} \\
{{z_m} = {\bar z_{m}} + \Delta {z_m},}
\end{array}} \right.
\end{equation}
where ${\bar q_{m}} = [{\bar x_{m}},{\bar y_{m}},{\bar z_{m}}]$
denotes the center of hemisphere and $(\Delta {x_m}, \Delta {y_m}, \Delta {z_m}) \in {\varepsilon _m}$ denotes the possible estimated errors, which is confined to the condition
\begin{equation}\label{eq0}
{\varepsilon _m} \triangleq \{ \Delta x_m^2 + \Delta y_m^2 + \Delta z_m^2 \leqslant D_m^2,\Delta z \geqslant 0\},
\end{equation}
where ${D_m}$ is the distance deviated from the center of the hemisphere.
The UAV is assumed to fly at a fixed height $H_u$. The flying time of the UAV is $T$, which is divided into $N$ time slots, i.e., $\Delta t = T/N$, where ${\Delta t}$ is the length of a time slot. The UAV has the fixed starting point and destination, which are denoted by ${{\mathbf{q}}_{0}}$ and ${{\mathbf{q}}_{N}}$. The trajectory of the UAV can be expressed as ${\bf q}[n] = {[x[n],y[n],z[n]]^T},n \in {\cal{N}} = \{ 1,2,...,N\}$, ${\bf{Q}} \buildrel \Delta \over = \{ {\bf q}[n],\forall n\}$, which meets the mobility constraints as
\begin{equation}\label{eq0}
{\bf{q}}\left[ 0 \right] = {{\bf{q}}_{0}},{\bf{q}}\left[ N \right] = {{\bf{q}}_{N}},\\
\end{equation}
\begin{equation}\label{eq0}
\left\| {{\mathbf{q}}[n] - {\mathbf{q}}[n - 1]} \right\| \leqslant {D_{\max }},n = 1,...,N,\\
\end{equation}
where ${{D_{\max }}}$ is the maximum flying length in each time slot. $p[n]$ is the transmit power of the GN in time slot $n$ and ${\mathbf{P}} = \left\{ {p\left[ n \right],\forall n} \right\}$, thus the power constraints are expressed as
\begin{equation}\label{eq0}
\frac{1}{N}\sum\limits_{n = 1}^N {p[n]} \leqslant \bar p,\\~~
p[n] \leqslant {p_{{\text{max}}}},\forall n,
\end{equation}
where $\bar p$ and ${p_{max}}$ are the average transmit power and the maximum transmit power of the GN, respectively.
We assume that both the GN and the UAV are equipped with single omni-directional antenna, and the IRS composes of a uniform planar array (UPA) with $L=L_x \times L_z$ elements, where $L_x$ and $L_z$ denote the number of elements along the x-axis and z-
axis, respectively. Then we denote the diagonal phase-shift matrix for the IRS as ${\mathbf{\Gamma }} \triangleq \left\{ {{\mathbf{\Gamma }}[n] = {\text{diag}}\left( {{e^{j{\theta _1}[n]}},{e^{j{\theta _2}[n]}},...,{e^{j{\theta _L}[n]}}} \right),\forall n} \right\}$, where ${\theta _i}[n] \in [0,2\pi ),i \in \{ 1,...,L\}$ is the phase shift of the $i$-th reflecting element in time slot $n$.
Due to high flying altitude of UAV and the flexible deployment of IRS, we assume that all channels are LoS channels in the considered system. Specifically, the GN-UAV channel in time slot $n$ is expressed as
\begin{equation}\small
{{\text{h}}_{gu}}[n] = \sqrt {\rho d_{gu}^{ - 2}\left[ n \right]} {e^{{{ - j2\pi {d_{gu}}\left[ n \right]} \mathord{\left/
{\vphantom {{ - j2\pi {d_{gu}}\left[ n \right]} \lambda }} \right.
\kern-\nulldelimiterspace} \lambda }}},
\end{equation}
where ${d_{gu}}\left[ n \right] = \left\| {{\bf{q}}\left[ n \right] - {{\bf{q}}_G}} \right\|$ is the distance between the GN and the UAV. $\lambda$ is the carrier wavelength and $\rho$ is the path loss at the reference distance ${D_0} = 1{\rm{m}}$. The same channel model is adopted for the channel from the jammer to the UAV, i.e., ${h_{mu}[n]}$.
Further, the reflecting channel such as GN-IRS-UAV channel is composed of two parts, namely, the GN-IRS channel and the IRS-UAV channel. Specifically, the IRS-UAV channel denoted by ${{\bf{h}}_{ru}}\left[ n \right] \in {\mathbb{C}}^{L \times 1}$, can be given by
\begin{equation}\small
{{\mathbf{h}}_{ru}}[n] = \sqrt {\rho d_{ru}^{ - 2}\left[ n \right]} {{\mathbf{\tilde h}}_{ru}}\left[ n \right],
\end{equation}
where ${{\mathbf{\tilde h}}_{ru}}\left[ n \right]$ denotes the phase of the channel, which is expressed as
\begin{equation}\small
{{\mathbf{\tilde h}}_{ru}}\left[ n \right]={{{e}}^{{{ - j2\pi {d_{ru}}\left[ n \right]} \mathord{\left/
{\vphantom {{ - j2\pi {d_{ru}}\left[ n \right]} \lambda }} \right.
\kern-\nulldelimiterspace} \lambda }}}{u_x}\left[ n \right] \otimes {u_z}\left[ n \right],
\end{equation}
where
${u_x^T}\left[ n \right] \!=\! \left[ {1\!,\!{e^{ - j{{{2\pi d{\phi _{ru,x}}\left[ n \right]} \mathord{\left/{\vphantom {{2\pi d{\phi _{ru,x}}\left[ n \right]} \lambda }} \right.
\kern-\nulldelimiterspace} \lambda }}}},\!...\!,{e^{ - j\left( {{L_x} - 1} \right){{{2\pi d{\phi _{ru,x}}\left[ n \right]} \mathord{\left/{\vphantom {{2\pi d{\phi _{ru,x}}\left[ n \right]} \lambda }} \right.
\kern-\nulldelimiterspace} \lambda }}}}} \right],
$ $
{u_z^T}\left[ n \right] \!=\! \left[ {1,{e^{ - j{{{2\pi d{\phi _{ru,z}}\left[ n \right]} \mathord{\left/{\vphantom {{2\pi d{\phi _{ru,z}}\left[ n \right]} \lambda }} \right.
\kern-\nulldelimiterspace} \lambda }}}},...,{e^{ - j\left( {{L_z} - 1} \right){{{2\pi d{\phi _{ru,z}}\left[ n \right]} \mathord{\left/{\vphantom {{2\pi d{\phi _{ru,z}}\left[ n \right]} \lambda }} \right.
\kern-\nulldelimiterspace} \lambda }}}}} \right] \!,\!
$ $
{{\phi _{ru,x}}}\left[ n \right] = \sin \vartheta _{ru}^{\left( v \right)}\left[ n \right]\cos \vartheta _{ru}^{\left( h \right)}\left[ n \right] = {{\left( {x\left[ n \right] - {x_r}} \right)} \mathord{\left/
{\vphantom {{\left( {x\left[ n \right] - {x_r}} \right)} {{d_{ru}}\left[ n \right]}}} \right.
\kern-\nulldelimiterspace} {{d_{ru}}\left[ n \right]}},
$ $
{\phi _{ru,z}}\left[ n \right] = \sin \vartheta _{ru}^{\left( v \right)}\left[ n \right]\sin \vartheta _{ru}^{\left( h \right)}\left[ n \right] = {{\left( {{H_u} - {z_r}} \right)} \mathord{\left/
{\vphantom {{\left( {{H_u} - {z_r}} \right)} {{d_{ru}}\left[ n \right]}}} \right.
\kern-\nulldelimiterspace} {{d_{ru}}\left[ n \right]}},
$
$\vartheta _{ru}^{\left( v \right)}$ and $\vartheta _{ru}^{\left( h \right)}$ represent the vertical and horizontal angle of departure (AoD) at the IRS, respectively.
${d_{ru}}\left[ n \right] = \left\| {{\mathbf{q}}\left[ n \right] - {{\mathbf{q}}_r}} \right\|$ is the distance between the IRS and the UAV. $d$ is the antenna distance.
The same model is adopted for GN-IRS channel and jammer-IRS channel. Then the channel gain is expressed as
\begin{equation}\label{5}
{g_0}\left[ n \right]{\text{ = }}{\left| {{h_{gu}}\left[ n \right] + {\mathbf{h}}_{gr}^H\left[ n \right]{\mathbf{\Gamma }}\left[ n \right]{{\mathbf{h}}_{ru}}\left[ n \right]} \right|^2},
\end{equation}
\begin{equation}\label{6}
{g_m}\left[ n \right]{\text{ = }}{\left| {{h_{mu}}\left[ n \right] + {\mathbf{h}}_{mr}^H\left[ n \right]{\mathbf{\Gamma }}\left[ n \right]{{\mathbf{h}}_{ru}}\left[ n \right]} \right|^2}.
\end{equation}
Then, the achievable average rate is given by
\begin{equation}\label{R0}\small
R = \frac{1}{N}\sum\limits_{n \in \cal{N}} {{{\log }_2}\left( {1 + \frac{{p[n]{g_0}\left[ n \right]}}{{{p_m}{g_m}\left[ n \right] + {\sigma ^2}}}} \right)} ,
\end{equation}
where ${p_m}$ denote the transmit power of the jammer, ${\sigma ^2}$ denotes the power of additive white Gaussian noise (AWGN) at the receiver. Thus, the problem is formulated as
\begin{equation*}\small
\begin{split}{\left( {{\rm{P0}}} \right)}
:{\rm{ }}&\mathop {\max}\limits_{\bf{P},\bf{Q},{\mathbf{\Gamma }} } {R}\\
{\rm{ }}{\rm s.t}.& ~ {\theta _i}[n] \in [0,2\pi ),i \in \{ 1,...,L\} ,\forall n,\\
&~\left( {\rm{1}} \right)- \left( {\rm{6}} \right).
\end{split}
\end{equation*}
It is challenging to solve (P0) optimally due to the non-convex objective function and coupled optimization variables. However, it can be effectively solved by dividing the problem into three sub-problems with consideration for the worst case. Thus, we exploit an alternating optimization (AO) to ensure that the sub-problems are convex in each iteration, while fixing the other two in each iteration until convergence is achieved.
\section{The Proposed Alternating Algorithm}
\label{The Proposed Alternating Algorithm}
\subsection{Sub-Problem 1: Optimizing {\bf P} for Given {\bf Q} and $\bf \Gamma$ }
For given the UAV trajectory ${\bf{Q}}$ and the phase shift ${\mathbf{\Gamma }} $, we consider the worst situation to deal with the jammer's uncertain location. When the jammer source is located closest to the UAV, we can obtain the lower bound of the average rate. $(\rm{P0})$ can be rewritten as
\begin{equation*}
\begin{split}{\left( {{\rm{P1}}} \right)}
:{\rm{ }}&\mathop {\max }\limits_{\bf{P}} \frac{1}{N}\sum\limits_{n \in {\cal{N}}} {{{\log }_2}\left( {1 + \frac{{p[n]{{g_0}\left[ n \right]}}}{{p_m}{{{{\tilde g}_m}}}\left[ n \right] + {\sigma ^2}}} \right)}\\
&{\rm{s}}{\rm{.t}}{\rm{.}}~\left( {\rm{5}} \right), \left( {\rm{6}} \right).
\end{split}
\end{equation*}
This is a standard convex optimization problem that can be efficiently solved by some existing algorithms, such as CVX's interior point method.
\subsection{Sub-Problem 2: Optimizing $\bf \Gamma$ for Given {\bf Q} and {\bf P}}
For given trajectory ${\bf{Q}}$ and transmit power ${\bf{P}}$, $(\rm{P0})$ can be transformed as
\begin{equation*}
\begin{gathered}
\left( {{\rm{P}}2} \right):\mathop {\max }\limits_{\bf{\Gamma} } \frac{1}{N}\sum\limits_{n \in \cal{N}} {{{\log }_2}\left( {1 + \frac{{p[n]{g_0}\left[ n \right]}}{{{p_m}{g_m}\left[ n \right] + {\sigma ^2}}}} \right)} \hfill \\
~~~~~~~~{\text{s}}{\text{.t}}{\text{.}}~{\theta _i}[n] \in [0,2\pi ),i \in \{ 1,...,L\} ,\forall n, \hfill \\
\end{gathered}
\end{equation*}
it is hard to obtain the optimal solution due to the unit modulus constraints and the imperfect CSI caused by jammer's uncertain location. Thus, we consider a practical scheme to estimate the ${{\mathbf{h}}_{mr}}$ and ${{\mathbf{h}}_{mu}}$, which belong to a given range, i.e.
\begin{equation*}\small
\begin{gathered}
{{\mathbf{\Psi }}_1} = \left\{ {{{\mathbf{h}}_{mr}}\left| {\left| {{\beta _{mr}}} \right| \in \left[ {\beta _{mr}^{\min },\beta _{mr}^{\max }} \right]} \right.,} \right. \hfill \\
\left. {\phi _{mr,i} \in \left[ {{{\left( {\phi _{mr,i}} \right)}^{\min }},{{\left( {\phi _{mr,i}^{\left( i \right)}} \right)}^{\max }}} \right],i \in \left\{ {x,z} \right\}} \right\}, \hfill \\
\end{gathered}
\end{equation*}
\begin{equation*}\small
{{\mathbf{\Psi }}_2} = \left\{ {{{\mathbf{h}}_{mu}}\left| {\left| {{\beta _{mu}}} \right| \in \left[ {\beta _{mu}^{\min },\beta _{mu}^{\max }} \right]} \right.} \right\},
\end{equation*}
where ${\beta _{mu}}$, ${\beta _{mr}}$ denote the amplitude of path loss, superscripts min and max respectively denote the lower and upper bound. We denote ${\mathbf {\hat v}}^H[n] = [{e^{j{\theta _1}[n]}},{e^{j{\theta _2}[n]}},...,{e^{j{\theta _L}[n]}}]$, ${\mathbf v}^{H}\left[ n \right] = {e^{j\tau }} \left[{\mathbf {\hat v}^H}\left[ n\right],1 \right]$, where $\tau$ is an arbitrary phase rotation. As such, (\ref{5}) and (\ref{6}) can be rewritten as
\begin{equation*}\small
{g_0}\left[ n \right] = {\left| {{{\mathbf{v}}^H}\left[ n \right]{\text{diag}}\left\{ {{{\mathbf{h}}_0}\left[ n \right]} \right\}{{\mathbf{h}}_g}\left[ n \right]} \right|^2},
\end{equation*}
\begin{equation*}\small
{g_m}\left[ n \right] = {\left| {{{\mathbf{v}}^H}\left[ n \right]{\text{diag}}\left\{ {{{\mathbf{h}}_0}\left[ n \right]} \right\}{{\mathbf{h}}_m}\left[ n \right]} \right|^2},
\end{equation*}
where ${{\mathbf{h}}_i}\left[ n \right] = {\left[ {{\mathbf{h}}_{ir}^H\left[ n \right],{h_{iu}}\left[ n \right]} \right]^H},i \in \left\{ {m,g} \right\} $, ${{\mathbf{h}}_0}\left[ n \right] = {\left[ {{\mathbf{h}}_{ru}^H\left[ n \right],1} \right]^H} $. (P0) can be equivalently written as
\begin{equation*}
\begin{gathered}
\left( {{\rm{P}}2.1} \right):\mathop {\max }\limits_{\mathbf{v}} \mathop {\min }\limits_{{{\mathbf{w}}_m}} \frac{{{{\mathbf{v}}^H}\left[ n \right]{{\mathbf{w}}_g}\left[ n \right]{\mathbf{w}}_g^H\left[ n \right]{\mathbf{v}}\left[ n \right]}}{{{{\mathbf{v}}^H}\left[ n \right]{{\mathbf{w}}_m}\left[ n \right]{\mathbf{w}}_m^H\left[ n \right]{\mathbf{v}}\left[ n \right] + {\sigma ^2}}}\\
~~~~~~~~{\text{s}}{\text{.t}}{\text{.}}~{\theta _i}[n] \in [0,2\pi ),i \in \{ 1,...,L\} ,\forall n, \hfill \\
\end{gathered}
\end{equation*}
where ${{\mathbf{w}}_m}\left[ n \right] = \sqrt {{p_m}} {\text{diag}}\left\{ {{{\mathbf{h}}_0}\left[ n \right]} \right\}{{\mathbf{h}}_m}\left[ n \right]$, ${{\mathbf{w}}_g}\left[ n \right] = \sqrt {p\left[ n \right]} {\text{diag}}\left\{ {{{\mathbf{h}}_0}\left[ n \right]} \right\}{{\mathbf{h}}_g}\left[ n \right]$.
To overcome the difficulty of the imperfect CSI, we consider the method which constructs a convex hull based on weighted sum of $T$ discrete samples i.e. \cite{24}
\begin{equation*}\small
\Omega {\text{ = }}\left\{ {\sum\limits_{t = 1}^T {{\alpha _t}{\mathbf{h}}_t^H{{\mathbf{h}}_t}} \left| {\sum\limits_{t = 1}^T {{\alpha _t} = 1,} } \right.{\alpha _t} \geqslant 0} \right\},
\end{equation*}
where ${{\mathbf{h}}_t} = {\left( {{\text{diag}}\left\{ {{{\mathbf{h}}_0}} \right\}{{\mathbf{h}}_m}} \right)_t}$ and ${\alpha _t}$ denotes the weighted coefficient of the $t$-th discrete sample. (P2.1) is written as
\begin{equation*}
\begin{gathered}
\left( {{\rm{P}}2.2} \right):\mathop {\max }\limits_{\mathbf{v}} \mathop {\min }\limits_\Omega \frac{{{{\mathbf{v}}^H}\left[ n \right]{{\mathbf{w}}_g}\left[ n \right]{\mathbf{w}}_g^H\left[ n \right]{\mathbf{v}}\left[ n \right]}}{{\sum\limits_{t = 1}^T {{\alpha _t}\left[ n \right]} {{\mathbf{v}}^H}\left[ n \right]{{\mathbf{h}}_t}\left[ n \right]{\mathbf{h}}_t^H\left[ n \right]{\mathbf{v}}\left[ n \right] + {\sigma ^2}}}\\
~~~~~~~~{\text{s}}{\text{.t}}{\text{.}}~{\theta _i}[n] \in [0,2\pi ),i \in \{ 1,...,L\} ,\forall n. \hfill \\
\end{gathered}
\end{equation*}
(P2.2) is still an intractable problem. However, it's observed that the worst case of the objection can be efficiently solved when ${\mathbf{v}}$ is fixed. When ${\mathbf{v}}$ is fixed, (P2.2) can be rewritten as
\begin{equation*}\small
\mathop {\min }\limits_\Omega \frac{{{{\mathbf{v}}^H}\left[ n \right]{{\mathbf{w}}_g}\left[ n \right]{\mathbf{w}}_g^H\left[ n \right]{\mathbf{v}}\left[ n \right]}}{{\sum\limits_{t = 1}^T {{\alpha _t}} {{\mathbf{v}}^H}\left[ n \right]{{\mathbf{h}}_t}\left[ n \right]{\mathbf{h}}_t^H\left[ n \right]{\mathbf{v}}\left[ n \right] + {\sigma ^2}}}.
\end{equation*}
Let ${{\mathbf{\bar w}}_t}\left[ n \right] = {{\mathbf{v}}^H}\left[ n \right]{{\mathbf{h}}_t}\left[ n \right]$, the optimal weighted coefficient is obtained by maximizing $\sum\limits_{t = 1}^T {{\alpha _t}} {\mathbf{\bar w}}_t^H\left[ n \right]{{\mathbf{\bar w}}_t}\left[ n \right]$. By using the Cauchy-Schwarz's inequality, we have
\begin{equation}\label{11}\small
\begin{gathered}
{\left( {\sum\limits_{t = 1}^T {{\alpha _t}\left[ n \right]} {\mathbf{\bar w}}_t^H\left[ n \right]{{{\mathbf{\bar w}}}_t}\left[ n \right]} \right)^2} \!\leqslant\! \left( {\sum\limits_{t = 1}^T {\alpha _t^2\left[ n \right]} } \right)\sum\limits_{t = 1}^T {{{\left( {{\mathbf{\bar w}}_t^H\left[ n \right]{{{\mathbf{\bar w}}}_t}\left[ n \right]} \right)}^2}}. \hfill \\
\end{gathered}
\end{equation}
The inequality (\ref{11}) holds equal only when ${{{\alpha _1}\left[ n \right]} \mathord{\left/
{\vphantom {{{\alpha _1}\left[ n \right]} {{\mathbf{\bar w}}_1^H\left[ n \right]{{{\mathbf{\bar w}}}_1}\left[ n \right]}}} \right.
\kern-\nulldelimiterspace} {{\mathbf{\bar w}}_1^H\left[ n \right]{{{\mathbf{\bar w}}}_1}\left[ n \right]}} = {{{\alpha _2}\left[ n \right]} \mathord{\left/
{\vphantom {{{\alpha _2}\left[ n \right]} {{\mathbf{\bar w}}_2^H\left[ n \right]{{{\mathbf{\bar w}}}_2}\left[ n \right]}}} \right.
\kern-\nulldelimiterspace} {{\mathbf{\bar w}}_2^H\left[ n \right]{{{\mathbf{\bar w}}}_2}\left[ n \right]}} = ... = {{{\alpha _T}\left[ n \right]} \mathord{\left/
{\vphantom {{{\alpha _T}\left[ n \right]} {{\mathbf{\bar w}}_T^H\left[ n \right]{{{\mathbf{\bar w}}}_T}\left[ n \right]}}} \right.
\kern-\nulldelimiterspace} {{\mathbf{\bar w}}_T^H\left[ n \right]{{{\mathbf{\bar w}}}_T}\left[ n \right]}}$ is satisfied. Since we have $\sum\limits_{t = 1}^T {{\alpha _t} = 1}$, ${\alpha _t}$ can be obtained as
\begin{equation}\label{12}\small
{\alpha _t}\left[ n \right] = {\mathbf{\bar w}}_t^H\left[ n \right]{{\mathbf{\bar w}}_t}\left[ n \right]{\left( {\sum\limits_{t = 1}^T {{\mathbf{\bar w}}_t^H\left[ n \right]{{{\mathbf{\bar w}}}_t}\left[ n \right]} } \right)^{ - 1}}.
\end{equation}
Based on the given ${\alpha _t}$, (P2.2) can be transformed as
\begin{equation*}\small
\begin{aligned}
\left( {{\rm{P}}2.3} \right):&\mathop {\max }\limits_{\mathbf{v}} \frac{{{{\mathbf{v}}^H}\left[ n \right]{{\mathbf{w}}_g}\left[ n \right]{\mathbf{w}}_g^H\left[ n \right]{\mathbf{v}}\left[ n \right]}}{{\sum\limits_{t = 1}^T {{\alpha _t}\left[ n \right]} {{\mathbf{v}}^H}\left[ n \right]{{\mathbf{h}}_t}\left[ n \right]{\mathbf{h}}_t^H\left[ n \right]{\mathbf{v}}\left[ n \right] + {\sigma ^2}}} \hfill \\
&{\text{s}}{\text{.t}}{\text{.}}~{\theta _i}[n] \in [0,2\pi ),i \in \{ 1,...,L\} ,\forall n. \hfill \\
\end{aligned}
\end{equation*}
Let ${\mathbf{V}}\left[ n \right] = {\mathbf{v}}\left[ n \right]{{\mathbf{v}}^H}\left[ n \right]$, ${\mathbf{A}}\left[ n \right] = {{\mathbf{w}}_g}\left[ n \right]{\mathbf{w}}_g^H\left[ n \right]$ and ${\mathbf{B}}\left[ n \right] = \sum\limits_{t = 1}^T {{\alpha _t}\left[ n \right]} {{\mathbf{\bar w}}_t}\left[ n \right]{\mathbf{\bar w}}_t^H\left[ n \right]$, then (P2.3) can be reformulated as
\begin{equation*}
\begin{aligned}
\left( {{\rm{P}}2.4} \right):\mathop {\max }\limits_{\mathbf{V}}& \frac{{{\text{tr}}\left( {{\mathbf{A}}\left[ n \right]{\mathbf{V}}\left[ n \right]} \right)}}{{{\text{tr}}\left( {{\mathbf{B}}\left[ n \right]{\mathbf{V}}\left[ n \right]} \right) + {\sigma ^2}}} \hfill \\
{\text{s}}{\text{.t}}{\text{.}}~&{{\mathbf{V}}_{l,l}}\left[ n \right] = 1,l = 1,2...,L + 1,\forall n, \hfill \\
&{{\mathbf{V}}}\left[ n \right]\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \succ } 0,{\text{rank}}\left( {{\mathbf{V}}\left[ n \right]} \right) = 1,\forall n,
\end{aligned}
\end{equation*}
where ${{\mathbf{V}}_{l,l}}\left[ n \right]$ denotes the $(l,l)$-th element of ${\mathbf{V}}\left[ n \right]$. However, note that ${\text{rank}}\left( {{\mathbf{V}}\left[ n \right]} \right) = 1$ is non-convex, we consider applying SDR. By defining $k\left[ n \right]{\text{ = }}{1 \mathord{\left/
{\vphantom {1 {\left( {{\text{tr}}\left( {{{\mathbf{0}}_2}{\mathbf{V}}\left[ n \right]} \right) + \sigma _M^2} \right)}}} \right.
\kern-\nulldelimiterspace} {\left( {{\text{tr}}\left( {{{\mathbf{0}}_2}{\mathbf{V}}\left[ n \right]} \right) + \sigma _M^2} \right)}}$ and ${\mathbf{\tilde V}}\left[ n \right] = k\left[ n \right]{\mathbf{V}}\left[ n \right]$, we can rewrite (P2.4) as a convex semidefinite programming (SDP) problem as follow
\begin{equation*}
\begin{aligned}
\left( {{\rm{P}}2.5} \right):\mathop {\max }\limits_{{\mathbf{\tilde V}},{\mathbf{k}}}& \quad {\text{tr}}\left( {{\mathbf{A}}\left[ n \right]{\mathbf{\tilde V}}\left[ n \right]} \right) \hfill \\
{\text{s}}{\text{.t}}{\text{.}}~&{{\mathbf{V}}_{l,l}}\left[ n \right] = k\left[ n \right],l = 1,2...,L + 1,\forall n, \hfill \\
&{{\mathbf{\tilde V}}_{l,l}}\left[ n \right]\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \succ } 0,k \left[ n \right] \geqslant 0,\forall n, \hfill \\
&{\text{tr}}\left( {{\mathbf{B}}\left[ n \right]{\mathbf{\tilde V}}\left[ n \right]} \right) + k\left[ n \right]{\sigma ^2} = 1,\forall n,
\end{aligned}
\end{equation*}
(P2.5) can be efficiently solved by using convex optimization toolbox such as CVX. However, the constraint ${\text{rank}}\left( {{\mathbf{V}}\left[ n \right]} \right) = 1$ may be not always guaranteed. Specifically, if the obtained ${{\mathbf{V}}\left[ n \right]}$ is of rank-1, we can obtain the ${{\mathbf{v}}\left[ n \right]}$ by applying
eigenvalue decomposition, thus the obtained ${{\mathbf{v}}\left[ n \right]}$ is the optimal solution to (P2.3). Otherwise, Gaussian randomization is
needed for recovering ${{\mathbf{v}}\left[ n \right]}$ approximately. The details can be found in \cite{5} and thus are omitted here. Therefore, the coefficients are obtained as
\begin{equation*}
{\hat v_i}\left[ n \right] = {e^{j\angle \left( {\frac{{{v_i}\left[ n \right]}}{{{v_{N + 1}}\left[ n \right]}}} \right)}},n = 1,...,L.
\end{equation*}
The proposed algorithm to solve the (P2) is summarized as {\bf{Algorithm 1}}, where ${R_{irs}}$ is the objective function of (P2).
\begin{algorithm}[t]\small
\renewcommand{\algorithmicrequire}{\textbf{Input:}}
\renewcommand{\algorithmicensure}{\textbf{Output:}}
\caption{Algorithm for (P2) in each time slot}\label{Algorithm1}
\begin{algorithmic}[1]
\STATE\textbf{Input:}
${\mathbf{P}}$, ${\mathbf{Q}}$, $T$, ${\mu _1}$.
\STATE\textbf{Onput:} ${{\mathbf{v}}\left[ n \right]}$.
\STATE Initialize ${{\mathbf{v}}\left[ n \right]}=I_{L+1}, i=0$, $R_{irs}^{\left( 0 \right)}=0$.
\STATE Let ${\varepsilon _{m,t}} = \left[ {\vartriangle {x_{m,t}},\vartriangle {y_{m,t}},\vartriangle {z_{m,t}}} \right]$, ${\beta _{mr}} = \beta _{mr}^{\min }$, and ${\beta _{mu}} = \beta _{mu}^{\min }$, compute ${\phi _{x,t}}\left[ n \right]$, ${\phi _{z,t}}\left[ n \right]$ and obtain ${{\mathbf{\tilde h}}_{mr,t}}$ and ${\tilde h_{mu,t}}$. Then construct ${{{\mathbf{h}}_t}}\left[ n \right]$, where $t = 1,2,...T$.
\STATE\textbf{~repeat:}
\STATE~ i:=i+1.
\STATE~ Compute $\alpha _t^{\left( i \right)}$ based on (\ref{12}).
\STATE~ Compute ${{\mathbf{v}}\left[ n \right]}$ for given $\alpha _t^{\left( i \right)}\left[ n \right]$ by solving (P2.5).
\STATE{\textbf{~until} $\left| {R_{irs}^{\left( i \right)}\left[ n \right] - R_{irs}^{\left( {i - 1} \right)}} \left[ n \right]\right| \leqslant {\mu _1}$. }
\end{algorithmic}
\end{algorithm}
\subsection{Sub-Problem 3: Optimizing {\bf Q} for Given {\bf P} and ${\bf \Gamma}$}
For given ${\bf{P}}$ and $\bf{\Gamma} $, (P0) can be rewritten as
\begin{equation*}
\begin{split}{\left( {{\rm{P3}}} \right)}
:{\rm{ }}&\mathop {\max}\limits_{\bf{Q} } {R}\\
{\rm{ }}{\rm s.t}.&~\left( {\rm{1}} \right)-\left( {\rm{4}} \right),
\end{split}
\end{equation*}
(P3) is difficult to solve due to the uncertain location of the jammer and its non-convex objective function. To overcome the imperfect location of the jammer, we consider introducing the slack variables ${\mathbf{D}}{\text{ = }}\left\{ {d[n],\forall n} \right\}$, which are used to approximately denote the square of distance from the jammer to the UAV. For the jammer-IRS channel, we just consider the worst case that the jammer is farthest to the IRS. As such, we can conveniently to transform the problem into a convex form, which is given by
\begin{equation}\label{15}
d[n] \leqslant {\left\| {{\mathbf{q}}[n] - {{\mathbf{q}}_m}} \right\|^2},d[n] \geqslant 0,\forall n,\\
\end{equation}
where ${q_m}$ contains infinite number of variables due to the uncertain location information of the jammer. By resorting to $\mathcal{S} - procedure$, the constraints of imperfect location information can be rewritten as
\begin{equation*}\small
\Delta x_m^2 + \Delta y_m^2 + \Delta z_m^2 - D_m^2 \leqslant 0,
\end{equation*}
\begin{equation*}
\begin{aligned}
- {\left( {{{\bar x}_m} + \Delta x_m - x[n]} \right)^2} - {\left( {{\bar y}_m} + \Delta y_m - y[n] \right)^2} - \\~~{\left( {{{\bar z}_m} + \Delta z_m - z[n]} \right)^2} + d[n] \leqslant 0,
\end{aligned}
\end{equation*}
and there exists $\left( {\Delta {{\hat x}_m},\Delta {{\hat y}_m},\Delta {{\hat z}_m}} \right) = \left( {0,0,0} \right)$ satisfying constraint. Therefore, according to \textbf{ \emph{Lemma}} \emph{1}, we introduce ${\mathbf{\delta }} \triangleq \left\{ {\delta \left[ n \right] \geqslant 0,\forall n} \right\}$, the equality constraint is given by
\begin{equation}\label{16}
\begin{gathered}
\Theta (x[n],y[n],d[n],\delta [n])= \hfill \\
\left[ {\begin{array}{*{20}{c}}
{\delta \left[ n \right] + 1}&0&0&{{{\bar x}_m} - x[n]} \\
0&{\delta \left[ n \right] + 1}&0&{{{\bar y}_m} - y[n]} \\
0&0&{\delta \left[ n \right] + 1}&{{{\bar z}_m} - H_u} \\
{{{\bar x}_m} - x[n]}&{{{\bar y}_m} - y[n]}&{{{\bar z}_m} - H_u}&{E[n]}
\end{array}} \right], \hfill \\
\end{gathered}
\end{equation}
where
$E\left[ n \right] = - D_m^2\delta [n] + {x^2}[n] - 2{\bar x_m}x[n] + \bar x_m^2 + \\{y^2}[n] - 2{\bar y_m}y[n] + \bar y_m^2 + {H_u^2}[n] - 2{\bar z_m}H_u[n] + \bar z_m^2 - d[n],\forall n$. To make the problem solvable, we use the first-order Taylor expansion of the convex function to obtain the lower bound. At the given feasible points ${{\mathbf{x}}_0} = \left\{ {{x_0}\left[ n \right],\forall n} \right\}$ and ${{\mathbf{y}}_0} = \left\{ {{y_0}\left[ n \right],\forall n} \right\}$, $E[n]$ can be transformed as
\begin{equation*}
\begin{gathered}
\tilde E\left[ n \right]= \!-\! D_m^2\delta \left[ n \right] - x_0^2\left[ n \right] \!+\! 2{x_0}\left[ n \right]x\left[ n \right] \!-\! 2{{\bar x}_m}x[n]
+ \bar x_m^2 - y_0^2[n] \\ + 2{y_0}\left[ n \right]y\left[ n \right] - 2{{\bar y}_m}y[n] + \bar y_m^2+H_u^2-2{{\bar z}_m}z[n] + \bar z_m^2 - d[n]. \\
\end{gathered}
\end{equation*}
Then we substitute the $\tilde E\left[ n \right]$ into (\ref{16}) and obtain the convex formation
\begin{equation}
\tilde \Theta (x[n],y[n],d[n],\delta [n])\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \succ } 0,
\end{equation}
which is a semidefinite programming problem that can be solved optimally by CVX. Next, to deal with the non-convex objective function of (P3), we observe that ${{{\mathbf{\tilde h}}}_{ru}}\left[ n \right]$, ${{{\mathbf{\tilde h}}}_{gu}}\left[ n \right]$ and ${{{\mathbf{\tilde h}}}_{mu}}\left[ n \right]$ are relative to the trajectory variables. However,it is observed that those are more complex to handle due to its non-linear formation. To overcome such problem, we use ($j-1$)th iteration to obtain the approximate ${{{\mathbf{\tilde h}}}_{ru}}\left[ n \right]$,${{{\mathbf{\tilde h}}}_{gu}}\left[ n \right]$ and ${{{\mathbf{\tilde h}}}_{mu}}\left[ n \right]$ \cite{20}. Based on the consideration above, we consider rewriting the objective function. By denoting
\begin{equation*}
\begin{gathered}
{{\bf{H}}_g}\left[ n \right] = \left[ {\sqrt \rho \tilde h_{gu}^{\left(i-1\right)}\left[ n \right],\sqrt \rho {\mathbf{h}}_{gr}^H\left[ n \right]{\mathbf{\Gamma }}\left[ n \right]{{{\mathbf{\tilde h}}}_{ru}^{\left(i-1\right)}}\left[ n \right]} \right] \hfill \\
{{\bf{H}}_m}\left[ n \right] = \left[ {\sqrt \rho \tilde h_{mu}^{\left(i-1\right)}\left[ n \right],\sqrt \rho {\mathbf{ h}}_{mr}^H\left[ n \right]{\mathbf{\Gamma }}\left[ n \right]{{{\mathbf{\tilde h}}}_{ru}^{\left(i-1\right)}}\left[ n \right]} \right], \hfill \\
\end{gathered}
\end{equation*}
where
${\tilde h_{gu}}[n] = {e^{ - j\frac{{2\pi {d_{gu}}\left[ n \right]}}{\lambda }}}$, (\ref{5}), (\ref{6}) can be transformed as
\begin{equation*}
{g_0}\left[ n \right] = {\mathbf{d}}_g^T[n]{\mathbf{H}}_g^H[n]{{\mathbf{H}}_g}[n]{{\mathbf{d}}_g}[n],
\end{equation*}
\begin{equation*}
{g_m}\left[ n \right] = {\mathbf{d}}_m^T[n]{\mathbf{H}}_m^H[n]{{\mathbf{H}}_m}[n]{{\mathbf{d}}_m}[n],
\end{equation*}
where ${{\mathbf{d}}_g}[n] = {\left[ {{ {{d_{gu}^{-1}}[n]} } ,{{{d_{ru}^{ - 1}}[n]} } } \right]^T}$, ${{\mathbf{d}}_m}[n] = {\text{ }}{\left[ {\sqrt {{ {d^{ - 1}[n]} }} , {{{d_{ru}^{ - 1}}[n]}} } \right]^T}$.
Moreover, the other problem is the non-convex objective function which is composed of the coupled variables. To tackle it, we consider introducing the slack variables and leveraging the SCA method. By introducing the slack variables ${\mathbf{S}} = \{ S[n],\forall n\}$ and ${\mathbf{G}} = \{ G[n],\forall n\}$, an optimization problem equivalent to (P3) is obtained as follows
\begin{equation*}\small
\begin{gathered}
\left( {{\rm{P}}3.1} \right):\mathop {\max }\limits_{{\mathbf{Q}},{\mathbf{S}},{\mathbf{G}}} \frac{1}{N}\sum\limits_{n \in {\text{N}}} {{{\log }_2}\left( {1 + \frac{{{S^{ - 1}}[n]}}{{G[n]}}} \right)} \hfill \\
~~~~~~~{\text{s.t.}}~{\rm C}1:p[n]{\mathbf{d}}_g^T[n]{\mathbf{H}}_g^H[n]{{\mathbf{H}}_g}[n]{{\mathbf{d}}_G}[n] \geqslant {S^{ - 1}}[n],\forall n, \hfill \\
~~~~~~~\quad ~~{\rm C}2:{p_m}{\mathbf{d}}_m^T[n]{\mathbf{H}}_m^H[n]{{\mathbf{H}}_m}[n]{{\mathbf{d}}_m}[n] + {\sigma ^2} \leqslant G[n],\forall n. \hfill \\
~~~~~~~~~~~\left( 3 \right)-\left( 4 \right), \left( 18 \right). \hfill \\
\end{gathered}
\end{equation*}
(P3) and (P3.1) share the same optimal solution when the constraints hold with equalities \cite{3}. And the objective function in (P3) has a lower bound at $\left( {{S_0}\left[ n \right],{G_0}\left[ n \right]} \right)$ by
\begin{equation*}\small
\begin{gathered}
\tilde R\left( {S[n],G[n]} \right) = {\text{l}}o{g_2}\left( {1 + {1 \mathord{\left/
{\vphantom {1 {\left( {{S_0}[n]{G_0}[n]} \right)}}} \right.
\kern-\nulldelimiterspace} {\left( {{S_0}[n]{G_0}[n]} \right)}}} \right) \hfill \\ + {\zeta _1}\left[ n \right]\left( {S\left[ n \right] - {S_0}\left[ n \right]} \right) + {\zeta _2}\left[ n \right]\left( {G\left[ n \right] - {G_0}\left[ n \right]} \right) \hfill, \\
\end{gathered}
\end{equation*}
where ${\zeta _1}[n] = - {\log _2}\left( {\frac{e}{{{S_0}[n] + {{({S_0}[n])}^2}{G_0}[n]}}} \right)$, ${\zeta _2}[n] = - {\log _2}\left( {\frac{e}{{{G_0}[n] + {{({G_0}[n])}^2}{S_0}[n]}}} \right)$. Thus (P3.1) is lower bounded by
\begin{equation*}
\begin{gathered}
\left( {{\rm{P}}3.2} \right):\mathop {\max }\limits_{{\mathbf{Q}},{\mathbf{S}},{\mathbf{G}}} \frac{1}{N}\sum\limits_{n \in {\cal{N}}} {{\tilde{R}\left( {S[n],G[n]} \right)}} \hfill \\
~~~~{\text{s}}{\text{.t}}{\text{.}}
~{\rm C}1, {\rm C}2, \left( 3 \right)- \left( 4 \right), \left( 18 \right). \hfill \\
\end{gathered}
\end{equation*}
However, the constraints ${\rm C}1$ and ${\rm C}2$ are non-convex because of the coupled variables. To deal with the non-convexity, we introduce the slack variables ${{\bf{\xi }}_1} = \left\{ {{\xi _1}\left[ n \right],\forall n} \right\}$, ${{\mathbf{\xi }}_2} = \left\{ {{\xi _2}\left[ n \right],\forall n} \right\}$, ${{\mathbf{\xi }}_3} = \left\{ {{\xi _3}\left[ n \right],\forall n} \right\}$, ${{\mathbf{\xi }}_4} = \left\{ {{\xi _4}\left[ n \right],\forall n} \right\}$. The problem is transformed into
\begin{equation*}
\begin{split}{\left( {{\rm{P3.3}}} \right)}
&:\mathop {\max }\limits_{~~~~\bf{Q},S,G, \hfill\atop
~ {\xi _1}, {\xi _2}, {\xi _3}, {\xi _4}\hfill} \frac{1}{N}\sum\limits_{n \in {\cal{N}}} {{\tilde{R}\left( {S[n],G[n]} \right)}} \\
&{\rm{s}}{\rm{.t}}{\rm{.}}~{\rm C}1.1:p[n]\tilde {\bf{d}}_{g}^T[n]{\bf{H}}_{g}^H[n]{{\bf{H}}_{g}}[n]{{\tilde {\bf{d}}}_{g}}[n] \ge {S^{ - 1}}[n],\\
&~~~~~{\rm C}2.1:{p_m}\tilde {\bf{d}}_{m}^T[n]{\bf{H}}_{g}^H[n]{{\bf{H}}_{g}}[n]{{\tilde {\bf{d}}}_{m}}[n] + {\sigma ^2}\le G[n],\\
&~~~~~ {d_{gu}^{ -1 }}[n] \ge {\xi _1}[n],{d_{ru}^{ -1 }}[n] \ge {\xi _2}[n],\forall n,\\
&~~~~~\sqrt {{d}^{ -1 }[n]} \le {\xi _3}[n],{d_{ru}^{ -1 }}[n] \le {\xi _4}[n],\forall n,\\
&~~~~~\left( 3 \right)-\left( 4 \right),\left( 18 \right),
\end{split}
\end{equation*}
where ${{\mathbf{\tilde d}}_g} = {\left[ {{\xi _1}\left[ n \right],{\xi _2}\left[ n \right]} \right]^T}$, ${{\mathbf{\tilde d}}_m} = {\left[ {{\xi _3}\left[ n \right],{\xi _4}\left[ n \right]} \right]^T}$.
To conveniently tackle the non-convex variables, we unfold as
\begin{equation}\label{18}
\begin{gathered}
{F_1}\left[ n \right] - {{\xi _1^{ - 2}\left[ n \right]}} \leqslant 0,{F_2}\left[ n \right] - {{\xi _2^{ - 2}\left[ n \right]} } \leqslant 0,\forall n, \hfill \\
{\xi _3^{ - 2}\left[ n \right]} - d\left[ n \right] \leqslant 0,{\xi _4^{ - 2}\left[ n \right]} - {F_2}\left[ n \right] \leqslant 0,\forall n, \hfill \\
\end{gathered}
\end{equation}
where ${F_1}\left[ n \right] = {\left( {x\left[ n \right] - {x_g}} \right)^2} + {\left( {y\left[ n \right] - {y_g}} \right)^2} + {H_u^2}$, ${F_2}\left[ n \right] = {\left( {x\left[ n \right] - {x_r}} \right)^2} + {\left( {y\left[ n \right] - {y_r}} \right)^2} + {\left( {H_u - {z_r}} \right)^2}$. However, there still exists several non-convex feasible regions in constraints (\ref{18}). The first-order Taylor expansions of ${ {\xi _1^{ - 2}\left[ n \right]}}$, ${{\xi _2^{ - 2}\left[ n \right]} }$, ${\mathbf{d}}_g^T[n]{\mathbf{H}}_g^H[n]{{\mathbf{H}}_g}[n]{{\mathbf{d}}_g}[n]$ at the feasible points ${{\mathbf{\xi }}_{{\mathbf{1}},{\mathbf{0}}}} = \left\{ {{\xi _{1,0}}\left[ n \right],\forall n} \right\}$, ${{\mathbf{\xi }}_{{\mathbf{2}},{\mathbf{0}}}} = \left\{ {{\xi _{2,0}}\left[ n \right],\forall n} \right\}$ and ${{\mathbf{\tilde d}}_{g,0}} = \left\{ {{{\tilde d}_{g,0}}[n],\forall n} \right\}$, which is given by
\begin{equation*}\small
\begin{split}
&\xi _1^{ - 2}\left[ n \right] \geqslant \xi _{1,0}^{ - 2}\left[ n \right] - 2\xi _{1,0}^{ - 2 - 1}\left[ n \right]\left( {{\xi _1}\left[ n \right] - {\xi _{1,0}}\left[ n \right]} \right)={ {{{\tilde \xi }_1}\left[ n \right]} },\hfill\\
&\xi _2^{ - 2}\left[ n \right] \geqslant \xi _{2,0}^{ - 2}\left[ n \right] - 2\xi _{2,0}^{ - 2 - 1}\left[ n \right]\left( {{\xi _2}\left[ n \right] - {\xi _{2,0}}\left[ n \right]} \right)={ {{{\tilde \xi }_2}\left[ n \right]} },\hfill\\
\end{split}
\end{equation*}
\begin{equation*}\small
\begin{split}
&{\mathbf{\tilde d}}_g^T[n]{\mathbf{H}}_g^H[n]{{\mathbf{H}}_g}[n]{{{\mathbf{\tilde d}}}_g}[n] \geqslant - {\mathbf{\tilde d}}_{g,0}^T[n]{\mathbf{H}}_g^H[n]{{\mathbf{H}}_g}[n]{{{\mathbf{\tilde d}}}_{g,0}}[n] \hfill\\
&+ 2\Re \left[ {{\mathbf{\tilde d}}_{g,0}^T[n]{\mathbf{H}}_g^H[n]{{\mathbf{H}}_{\text{g}}}[n]{{{\mathbf{\tilde d}}}_g}[n]} \right], \hfill \\
\end{split}
\end{equation*}
and ${F_2}\left[ n \right]$ can be transformed as
\begin{equation*}
\begin{gathered}
{{\tilde F}_2}\left[ n \right] = - x_0^2\left[ n \right] + 2{x_0}\left[ n \right]x\left[ n \right] + x_r^2 - y_0^2\left[ n \right] + \\ 2{y_0}\left[ n \right]y\left[ n \right]-y_r^2 \!+\! H_u^2-2{x_r}x\left[ n \right] - 2{y_r}y\left[ n \right] - 2{z_r}H_u. \hfill \\
\end{gathered}
\end{equation*}
Thus ${\rm C}1.1$ can be written as
\begin{equation*}\small
\begin{gathered}
{\rm C}1.2:p[n](2\Re \left[ {{\mathbf{\tilde d}}_{g,0}^T[n]{\mathbf{H}}_g^H[n]{{\mathbf{H}}_{\text{g}}}[n]{{{\mathbf{\tilde d}}}_g}[n]} \right]-\\
~~~~~~~~~~~~~~~~~~~~{{\tilde{\bf{d}}}_{g,0}}^T[n]{\bf{H}}_{g}^H[n]{{\bf{H}}_{g}}[n]{{\tilde {\bf{d}}}_{g,0}}[n]) \ge {S^{ - 1}}[n].
\end{gathered}
\end{equation*}
Then we can substitute all concave points to solvable ones finally. Accordingly, (\rm{P3.3}) can be formulated as
\begin{equation*}\label{20}
\begin{split}
\left( {{\rm{P}}3.4} \right):&\mathop {\max }\limits_{~~~~\bf{Q},S,G, \hfill\atop
~ {\xi _1}, {\xi _2}, {\xi _3}, {\xi _4}\hfill} \frac{1}{N}\sum\limits_{n \in {\cal{N}}} {{\tilde{R}}\left( {S[n],G[n]} \right)} \hfill\\
{\rm{s}}{\rm{.t}}{\rm{.}}~
&{F_1}\left[ n \right] -{ {{{\tilde \xi }_1}\left[ n \right]} } \leqslant 0,{F_2}\left[ n \right] - { {{{\tilde \xi }_2}\left[ n \right]} } \leqslant 0,\forall n,\\
&{{\xi _3}^{ - 2}\left[ n \right]} - d\left[ n \right] \leqslant 0,{{\xi _4}^{ - 2}\left[ n \right]} + {{\tilde F}_2}\left[ n \right] \leqslant 0,\forall n,\\
&{\rm C}1.2, {\rm C}2.1, \left( 3 \right)-\left( 4 \right), \left( 18 \right).\\
\end{split}
\end{equation*}
(\rm{P3.4}) is a standard convex optimization problem, and we can use CVX solver to solve it.
\begin{algorithm}[t]
\renewcommand{\algorithmicrequire}{\textbf{Input:}}
\renewcommand{\algorithmicensure}{\textbf{Output:}}
\caption{An alternating algorithm for solving {(\rm{P0}})}\label{Algorithm1}
\begin{algorithmic} [1]
\STATE\textbf{Input:}${\mu _2}$, ${i_{\max }}$
\STATE\textbf{Ontput:}$R$.
\STATE{Initialization:}
~Set~$i = 0$, as iteration index, ${\mu _2}$ as the threshold and original points ${\Upsilon _0} = \left\{ {{{\mathbf{Q}}^{\left( 0 \right)}},{{\mathbf{P}}^{\left( 0 \right)}},{{\mathbf{\Gamma }}^{\left( 0 \right)}}} \right\}$, thus obtaining the ${R^{\left( 0 \right)}}$ by using (\ref{R0}) and generating a series of initial points
${\mathbf{S}}_0^{\left( 0 \right)},{\mathbf{T}}_0^{\left( 0 \right)},{\mathbf{\xi }}_{10}^{\left( 0 \right)},{\mathbf{\xi }}_{20}^{\left( 0 \right)},{\mathbf{h}}_{ru}^{\left( 0 \right)},{\mathbf{ d}}_{g,0}^{\left( 0 \right)}, {\mathbf{ d}}_{m,0}^{\left( 0 \right)}$.
\STATE\textbf{~repeat:}
\STATE~ With given ${{\mathbf{Q}}^{\left( i \right)}}$, ${{\mathbf{\Gamma }}^{\left( i \right)}}$, update ${{\mathbf{P}}^{\left( i \right)}}$ to ${{\mathbf{P}}^{\left( i+1 \right)}}$ by solving sub-problem (\rm{P1}).
\STATE~ With given ${{\mathbf{Q}}^{\left( i \right)}}$, ${{\mathbf{P}}^{\left( {i + 1} \right)}}$ , update ${{\mathbf{\Gamma }}^{\left( i \right)}}$ to ${{\mathbf{\Gamma }}^{\left( i+1 \right)}}$ by solving sub-problem (\rm{P2.4}).
\STATE~ With given ${{\mathbf{\Gamma }}^{\left( {i + 1} \right)}}$, ${{\mathbf{P}}^{\left( {i + 1} \right)}}$ , update ${{\mathbf{Q}}^{\left( i \right)}}$ to ${{\mathbf{Q}}^{\left( i+1 \right)}}$ by solving sub-problem (\rm{P3.4}).
\STATE~ With given ${{\mathbf{Q}}^{\left( i+1\right)}}$, ${{\mathbf{P}}^{\left( i+1 \right)}}$, ${{\mathbf{\Gamma }}^{\left( i+1 \right)}}$, compute ${R^{\left( {j + 1} \right)}}$ and ${\mathbf{S}}_0^{\left({i+1}\right)}$, ${\mathbf{T}}_0^{\left( {i + 1} \right)}$, ${\mathbf{\xi }}_{10}^{\left( {i + 1} \right)}$, ${\mathbf{\xi }}_{20}^{\left( {i + 1} \right)}$, ${\mathbf{h}}_{ru}^{\left( {i + 1} \right)}$, ${\mathbf{d}}_{g,0}^{\left( {i + 1} \right)}$, ${\mathbf{d}}_{m,0}^{\left( {i + 1} \right)}$.
\STATE~ {Update $ i \leftarrow i + 1\ $}.
\STATE\textbf{~until}: $\left| {{R^{\left( {j + 1} \right)}} - {R^{\left( j \right)}}} \right| \leqslant {\mu _2}$ or $j = {j_{\max }}$.
\end{algorithmic}
\end{algorithm}
\subsection{Overall Algorithm}
The overall algorithm for solving (P0) is summarized in Algorithm 2. The ${\mu _2}$ is used to control the accuracy of convergence and ${j_{\max }}$ is the maximum number of iterations. In fact, solving sub-problem 2 and sub-problem 3 dominates the complexity of Algorithm 2.
Specifically, the computational complexities of Algorithm 2 is approximately ${\cal O}(N{(L + 1)^{3.5}}{I_1}{I_2} + 8{N^{3.5}}{I_2})$, where $I_1$ is the number of iterations required for solving (P2.4) and $I_2$ is for (P0), respectively.
\section{Numerical Results}
\label{Numerical Results}
In this part, we give the simulation results to verify the effectiveness of our proposed algorithm. Specifically, {\bf{``Proposed"}} refers to the scheme we proposed for jointly optimizing UAV trajectory, GN's power allocation and IRS passive beamforming. To study the impact of the deployment of IRS, we consider two setups. In particular, one is denoted by ``{\bf IRS-M}" corresponding to that the IRS is deployed at (251, 50, 5), i.e., nearby the jammer; while the other is denoted by ``{\bf IRS-G}" corresponding to that the IRS is deployed at (201, 100, 5), i.e., nearby the GN. Moreover, we consider benchmark algorithm that {\bf{``w/o IRS"}} refers to the case without the IRS.
The parameters are set following the suggestions and settings in \cite{3}, and are shown as follows: ${{\rm{{\bf{q}}}}_{0}} = \left( {0,0,100} \right)$, ${{\rm{{\bf{q}}}}_{N}} = \left( {400,200,100} \right)$, ${{\rm{{\bf{q}}}}_m} = \left( {250,50,0} \right)$, ${{\rm{{\bf{q}}}}_g} = \left( {200,100,0} \right)$, $H_{u} = 100$ m, ${V_{\max }} = 60$, ${\bar p} = 30$ dBm, ${p_{max}} = 31.76$ dBm, ${p_{m}} = 30$ dBm, $\rho = {10^{ - 3}}$, ${\sigma ^2} = - 169$ dBm/Hz, ${\Delta t} = 0.5$, ${\mu _1} {\rm{ = }}{\mu _2}{\rm{ = }}{10^{ - 3}}$, $T=10^3$.
\begin{figure}
\centering
\subfigure[$D_m=0$] {\includegraphics[width=.35\textwidth]{fig2atra.eps}}
\subfigure[$D_m=20$] {\includegraphics[width=.35\textwidth]{fig2btra.eps}}
\caption{UAV trajectories for different schemes.}
\label{fig2tra}
\vspace*{-5mm}
\end{figure}
Fig. \ref{fig2tra} shows the UAV's trajectories for different schemes when the number of IRS elements is $L=100$ and jammer's power is $p_m=30$ dBm. Fig. \ref{fig2tra}(a) shows the optimized UAV's trajectories when $D_m=0$. It is obviously shown that the trajectories in our proposed algorithm can significantly decrease the flying path length of the UAV compared to the case without IRS. Specifically, when deploying the IRS near the GN, the UAV keeps away from the jammer a bit first and finally hovers directly above the GN. While for the case when IRS is deployed near the jammer, it is observed that the UAV almost flies along the straight line from the start point to the end point. The reason is that the jamming signal power is drastically weakened by passive beamforming, which is roughly equivalent to the case that UAV flies under the low jamming level. Note that for the case ``w/o IRS", the UAV must keep away from the jammer farther and hovers for a while at a relative stable position so as to achieve an optimal system performance.
Fig. \ref{fig2tra}(b) shows the UAV's trajectories when $D_m=20$. It is observed that the trajectory for ``IRS-M" becomes similar to that for ``w/o IRS". The reason is that when deploying the IRS nearby the jammer, the passive beamforming is quite sensitive to the location of the jammer, which is however incurs a large uncertainty and thus renders the passive beamforming focusing on mitigating the jamming signal ineffective. In contrast, it can be observed that deploying the IRS nearby the GN is still helpful. This is because in this case the IRS mainly contributes to enhance the GN's signal received at the UAV but not reduce the jamming signal, which is thus not impacted by the uncertainty of the jammer's location..
\begin{figure}
\centerline
{\includegraphic
[width=0.80\columnwidth]
{fig3pm.eps}}
\caption{\label{fig3pm}Achievable average rate for different schemes versus $p_m$.}
\vspace{-5mm}
\end{figure}
Fig. \ref{fig3pm} shows the achievable average rate versus $p_m$ under the known and unknown jammer's location when $L=100$, respectively. For the case $D_m=0$, it can be observed that deploying the IRS nearby the GN and nearby the jammer both can increase the average uplink rate, for enhancing the information signal and reducing the jamming signal, respectively. Moreover, it is observed that ``IRS-G" outperforms ``IRS-M" first and then becomes less effective than the later. The reason is that when the jamming is under the low level, the SNR at the UAV is dominated by the noise, thus deploying the IRS nearby the GN for improving the information signal is more useful to increase the rate. However, as $p_m$ increases up to sufficiently large, the noise becomes ignorable and thus deploying the IRS nearby the jammer for jamming reduction becomes more effective. For the case $D_m=20$, it is observed that the achievable uplink rates for the three schemes all decreased and that for ``IRS-M" even becomes lower than that for ``w/o IRS". The reason is that the passive beamforming for IRS needs accurate location information to align the reflecting channels to the direct channel. Due to jammer's uncertain location, the jamming signal via IRS cann't be destructively combined with that from the direct channel for jamming reduction. Compared to the case ``IRS-M", ``IRS-G" is insensitive to the jammer's location and can still enhance the information-carrying signals.
\begin{figure}
\centerline
{\includegraphic
[width=0.80\columnwidth]
{fig4L.eps}}
\caption{\label{fig4L}Achievable average rate for different schemes versus $L$.}
\vspace{-5mm}
\end{figure}
In Fig. \ref{fig4L}, the achievable average rate for different schemes versus the number of IRS elements are illustrated given $p_m=15$ dBm. It is observed when $D_m=0$, the achievable average rate for ``IRS-M" increases at the beginning and then tends to stability while ``IRS-G" keeps a steady growth rate. The reason is that as $L$ increases up to sufficiently large, the jamming signal in ``IRS-M" case is well reduced and thus the reception at the UAV is no more jamming-dominant, leading the lower performance gain by increasing the number of IRS elements. While for ``IRS-G", the reception at the UAV can substantially benefit from increasing $L$ because the IRS in this case mainly focuses on enhancing the information signal from GN. When $D_m=20$, it is observed that the average rate for ``IRS-G" is much higher than that for the other two and still benefits from increasing the number of IRS elements. However, the rate for ``IRS-M" is lower than that for ``w/o IRS" and even decreases as increasing $L$. This is because for ``IRS-G", the passive beamforming gain is insensitive the the location of the jammer and a larger number of IRS elements is always favorable to improve the rate; while for ``IRS-M", the passive beamforming can no more reduce the jamming signal effectively due to the uncertain location of the jammer, and even helps enhance the jamming singal at the UAV statistically.
\section{Conclusions}\label{Conclusions}
In this paper, we study the UAV uplink transmission assisted by IRS in the presence of a jammer with imperfect location information. By considering the GN's power allocation, IRS passive beamforming, and UAV trajectory, an alternating optimization based algorithm is proposed to solve the problem by exploiting the SDR, SDA and S-procedure methods. Simulation results were shown to verify the performance of our proposed algorithm, by considering two setups of the IRS deployment for further exploiting the effect of IRS. Specifically, it is observed that by deploying the IRS nearby the GN, the proposed joint design can always improve the uplink transmission regardless of the knowledge of jammer's location; however, by deploying the IRS nearby the jammer, the proposed design is effective only when the jammer's location is perfectly known.
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,491 |
Most people need to work with a lawyer only a handful of times during their life. When they do, it means they are addressing something important. It may be resolving a civil dispute, tackling a municipal court matter, navigating divorce, completing a major real estate transaction or protecting hard-earned assets by creating a comprehensive estate plan.
At Weiss & Weiss, LLC, our attorneys provide experienced and accomplished legal representation across a broad spectrum of practice areas. We emphasize clear communication and always aim to reach effective solutions in the most efficient manner possible.
At Weiss & Weiss, LLC, we emphasize personal attention. We don't follow cookie-cutter formulas or follow any legal templates. We will take the time to learn the unique facts of your situation, then help you identify and prioritize objectives. You will work with the same lawyer from start to finish ensuring familiarity with your case at every stage.
Practice areas such as estate planning and real estate are loaded with legalese and jargon. We avoid this language, opting instead to explain things in clear terms that our clients can understand. Clarity increases client engagement, and an engaged client is more likely to achieve the most important goals.
Our lawyers work with clients throughout New Jersey from our offices in Manville and Hoboken. We welcome the opportunity to review the facts of your case and recommend the best steps to take next. Call 908-300-3461 or use our online contact form to schedule an appointment.
The attorneys at Weiss & Weiss, LLC, handle a diverse assortment of legal matters for an equally broad range of clients. Our lawyers advocate for clients in everything from DUI and criminal defense cases in New Jersey Municipal Courts to civil disputes, real esate issues, estate planning and divorce. | {
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} | 8,595 |
Night Time Is The Right Time
May 4, 2019 Andrew Martone Uncategorized Leave a comment
Another Aretha cover that isn't exactly well known, but packs one hell of a punch. Positioned near the end of Side A of 1968's Aretha Now, "Night Time Is The Right Time" is yet another example of Aretha transforming and elevating a hit into something all her own.
What separates Aretha's version from the rest is that it relies so heavily on the piano. Sure, early versions like that of Roosevelt Sykes have the piano, but they don't have the rhythm of later versions like Ray's. Both Ray and Nappy' Brown's arrangements. Once again, Aretha's brilliant piano playing is the focal point of her arrangement.
Aretha also does the opposite of her usual, and significantly scales back the background parts for much of the song, specifically the "night and day", instead utilizing her brass section to fill in the blanks. Of course, being one of the only female artists to cover it (especially that early on in the song's lineage), Aretha's version augments the male-female dynamic made so famous by Ray Charles' version.
Ray relied on the grit and tenacity of Margie Hendrix from his Raylettes to momentarily steal the show and elevate (other word) the moment. But nobody was stealing the show from Aretha. She was the show and the elevated moment. While Margie Hendrix's "baby"'s have some serious growl and intensity, they plateau melodically. Aretha's show-out has serious power, that demonstrates her range.
When Aretha unleashes her own "I gotta call you 'baby!'" a minute and a half in, it's game over. While The Sweet Inspirations reinforce her, "Night Time Is The Right Time" becomes not just a statement, but a climactic proclamation. Yet, in the midst of going off, Aretha yields her voice to bang out a fantastically bluesy piano solo, something that's always welcomed for Aretha. As soon as she's done tearing up her solo, she proclaims "baby won't you please me!" From there, the free for all of wails and screams resumes as Aretha's conquest of "Night Time Is The Right Time" concludes, victorious.
Listen to Aretha unleash voice and piano on "Night Time Is The Right Time":
1968Aretha NowAtlantic RecordsRay CharlesThe Sweet Inspirations
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Lavrente Indico Diaz (nascut el 30 de desembre de 1958) és un cineasta filipí i antic crític de cinema.. Conegut sovint com un dels membres clau del moviment del cinema lent, després d'haver fet diverses de les pel·lícules narratives més llargues de la història, Diaz és un dels cineastes filipins contemporanis més aclamats per la crítica.
Tot i que Díaz havia estat fent pel·lícules des de finals de la dècada de 1990, no va atreure gaire l'atenció del públic fora de les Filipines i el circuit de festivals fins a l'estrena de la seva pel·lícula Norte, the End of History (2013) , que va entrar a la secció Un Certain Regard del 66è Festival Internacional de Cinema de Canes i va rebre molts elogis de la crítica.
Les pel·lícules posteriors de Díaz també han rebut una atenció positiva de la crítica i molts premis. From What Is Before (2014) va guanyar el Lleopard d'Or al Festival Internacional de Cinema de Locarno de 2014; A Lullaby to the Sorrowful Mystery (2016) va competir per l'Ós d'Or al 66è Festival Internacional de Cinema de Berlín i va guanyar el Premi Alfred Bauer; i The Woman Who Left (2016) van competir al 73è Festival Internacional de Cinema de Venècia i van guanyar el Lleó d'Or. Va rebre el premi a la trajectòria de FAMAS el 2018. Ha rebut el Natatanging Gawad Urian de 2021 (Premi Gawad Urian a la trajectòria).
Ha rebut una Beca Guggenheim el 2010 i un Premi Príncep Claus dels Països Baixos el 2014.
Filmografia
Premis i nominacions
Mostra Internacional de Cinema de Venècia
Premis Alfred Bauer
Premis Gawad Urian
World Premieres Film Festival (Filipines)
Mostra Internacional de Cinema de São Paulo
Premis del Young Critics Circle (Filipines)
Festival Internacional de Cinema de Friburg
Festival Internacional de Cinema de Singapur
Referències
Crítics de cinema asiàtics
Cinema de les Filipines
Directors de cinema asiàtics
Artistes filipins | {
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Against all odds —
Google "not optimistic" Apple will approve its upcoming iOS maps app
But the company is trying to build one anyway.
Chris Foresman - Nov 5, 2012 6:20 pm UTC
Google is still forging ahead with plans to make a native Google Maps iOS app, despite the fact that those inside the company are "not optimistic" Apple would approve it. At least one source reported to the Guardian that prospects might be better now that former SVP of iOS software Scott Forstall has effectively been shown the door, but the company appears to believe a Google Maps app appearing in the App Store anytime soon would be "an unlikely event."
When Apple released iOS 6 with a redesigned Maps app that uses Apple's own mapping data sources, we complained that the lack of built-in transit information and less-than-perfect data were a problem. And though most users haven't been put off by the issues, plenty of people had enough problems that Apple CEO Tim Cook apologized for the state of the app on release, and promised fixes were coming.
In the meantime, Apple has promoted several Maps alternatives in the App Store, as well as pointing to Web-based options such as those from Google and Nokia. According to the Guardian's sources, though, none of the apps Apple is promoting in the App Store use Google's APIs to access its store of location, routing, or point-of-interest data.
We tested a few of the Maps alternatives, and found Google's Web-based Google Maps to offer the missing data and transit directions that Apple's own Maps currently lacks. However, the interface is relatively cumbersome, and doesn't integrate with Siri or iOS 6's turn-by-turn navigation.
Though it hasn't publicly stated its plans, Google has apparently been working on its own mapping app for iOS since this summer and plans to have it ready to ship before the end of the year. However, inside the company there's little hope for Apple approval. Instead, Apple is expected to "keep moving forward in an effort to make its obviously inferior product better," and "save face" with the public.
Chris Foresman Chris is an Associate Writer at Ars Technica, where he has spent the last five years writing about Apple, smartphones, digital photography, and patent litigation, among other topics.
Email chris.foresman@arstechnica.com // Twitter @foresmac | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,388 |
\section{Introduction}
Thanks to recent advances in the observation of high-redshift
star-forming galaxies, and to the improved statistics in
low-redshift galaxy surveys, it is now possible to have a
quantitative view of the star formation history in the universe, in
a global, volume-averaged sense \citep{Hughes98, Steidel99,
Flores99, Glazebrook99, Yan99, Massarotti01, Giavalisco03}, but also
for each galactic halo, in a statistical, mass-averaged sense
\citep{Heavens04}.
In the standard picture of galaxy formation, small mass objects
collapse first, when gravitational instability enters the non-linear
regime. This so-called `hierarchical scenario' relies heavily on
the observation of primordial density fluctuations imprinted on the
Cosmic Microwave Background \citep{Spergel03}. These first
generation objects eventually form stars, the so-called Population
III stars, whose properties are likely to differ strongly from
present-day galactic stars. The contribution of Population III stars
to the global star formation history in the universe is currently a
matter of debate: they likely contribute to a prompt initial
enrichment of the intergalactic medium, as well as a possible early
reionization epoch, but they may also cause a strong negative
feedback on the star formation efficiency at this very early
epoch. In this paper, we only adress star formation occurring inside
galactic discs, concentrating on quiescent modes of star formation.
This means that we only consider atomic processes to drive the
cooling and subsequent catastrophic collapse of halo gas that
ultimately leads to the formation of a centrifugally supported
disc. Molecular processes that are relevant for first stars
formation, or for molecular clouds formation inside galactic discs
are neglected in this study.
Our main concern is therefore to follow the formation of the first
generation of gas rich, rotating discs, and the subsequent merging
hierarchy that leads to larger and larger discs, up to present day
spiral galaxies, as well as galaxy groups and clusters. The
standard approach in current galaxy formation scenario is to
consider that hot gas virializes first into extended dark matter
halo potential well. Depending on the Virial temperature of this hot
gas, it cools rather rapidly, looses pressure support and collapses
up to a point where centrifugal equilibrium sets in. It is inside
these cold, rotating discs that star formation takes place at the
end of a complex cascade of turbulent fragmentation and molecular
cloud formation. On cosmological scale, models of star formation
still rely on heuristic recipes, partially based on
observational constraints, and partially based on theoretical
arguments.
This rather simple picture is altered by feedback processes due to
these stars. First of all, massive stars create collectively a
strong UV background that photo-ionizes the intergalactic medium,
preventing small mass halos from building their virialized, hot gas
component, and altering their cooling efficiency. This means that,
after the universe is re-ionized, we have to introduce a mass
threshold (or equivalently a Virial temperature threshold) below
which star formation is suppressed. This threshold allows us to
distinguish between a diffuse component, defined by small mass dark
matter halos with a very low gas fraction, and a star forming
component, defined by high mass dark matter halos where disc and
star formation is possible. The diffuse component is often called
`the Lyman alpha forest' or `the smooth baryon background' in the
literature. In the present paper, we name the other component `the
star forming halos'. Following standard definitions in galaxy
formation studies, we further divide even more the baryons sitting
in these `galactic halos' into 3 different phases: hot gas, cold
discs and stars. The purpose of this paper is to compute as
precisely as possible the evolution of these 4 phases, for the
universe as a whole, as well as for individual dark matter halos,
thus studying the baryon logistics.
A lot of other different feedback processes are currently under
close examination in the literature, like supernovae-driven winds,
young protostellar jets, AGN-driven jets and associated buoyant
bubbles, and so on and so forth. Such outflows are actually observed
in high redshift galaxies \citep{Martin99}. They consist of
high-velocity fountains of ionized gas, expelled from a parent,
star-forming disc. These winds are likely to be caused by collective
outbreaks of supernovae bubbles \citep{DeAvillez01}. Other energy
sources are however possible, like a central massive black hole, or
collective jet ram pressure from star forming clouds. Regardless of
their physical origin, these outflows are fundamental in explaining
the metal enrichment in the intergalactic medium, as well as the
high-metallicity observed in rich galaxy cluster. The modeling of
such winds will also be addressed in this paper, in a simplified
way, and their impact on the star formation history in the universe
as a whole will be computed. We will investigate in more
detail the impact of winds on the star formation history of
individual star forming halos in a follow-up paper.
Several approaches have been used to investigate the evolution of
the baryons in the hierarchical model of galaxy formation. Numerical
simulations \citep{Cen92, Navarro93, Mihos94, Katz96, Gnedin97,
Thacker00, Springel03a} and semi-analytic models \citep{White91,
Somerville99, Kaufmann99, Cole00, Hatton03} have been the most
popular techniques. In a recent paper, an analytical approach has
been proposed by \cite{Springel03c} to compute the star formation
history of the universe. Here, we present also a simple
self-consistent analytical model. This model, validated on
simulations, allows us to quickly compute the evolution of the 4
baryons components, for the Universe as a whole, and also for an
individual average halo.
The paper is organized as follows: in the next section, we present
numerical simulations of star formation in a $\Lambda$CDM universe,
based on our Adaptive Mesh Refinement code named RAMSES. We then
describe a simple analytical model that allows us to compute the
star formation history inside individual halos and in the
universe as a whole. In the section~4, we compare our model to
simulation results, showing that, once calibrated to our
simulations, the model works very well in predicting the evolution
of the 4 different baryon phases. We finally compare our model to
several observational constraints. Our analytical approach allows
us to explore efficiently the parameter space, which is in our case
limited to a 2-dimensional space: star formation time $t_*$ versus
wind efficiency $\eta_{\rm w}$. We finally confront our predictions to
current observational constraints, and found a rather narrow
parameter range.
\section{Simulations}
Our simulations were performed using the Adaptive Mesh Refinement
code called RAMSES and described in detail by
\cite{Teyssier02}. The N-body solver is very similar to the ART
code \citep{Kravtsov97} and the hydrodynamical solver is based on a
second-order Godunov-type method, called the MUSCL-Hancock scheme
\citep{Toro97}. We evolve the collisionless dark matter particle
distribution by solving the Vlasov-Poisson equation, and the
baryonic component by solving the Euler equation with gravity source
terms. More detail about our hydrodynamical solver and our
refinement strategy are given in appendix \ref{method_num}. We also
solve for a heating and cooling source terms in the energy equation,
assuming primordial H and He plasma photo-ionized by the
\cite{Haardt96} UV background. In dense and cold regions of the
flow, we turn a fraction of the gas into collisionless `star'
particles. This numerical approach is widely used in current galaxy
formation studies \citep{Cen92, Navarro93, Mihos94, Katz96,
Gnedin97, Thacker00}. We will briefly recall here the properties of
our own implementation. We then describe cosmological parameters,
box sizes and mass resolution we use in our three main simulation
series.
\subsection{Star formation recipe}
\begin{table*}
\begin{tabular}{|c|c|c|c|c|}
\hline
Name & L10N64S30 & L10N128S30 & L10N256S30 & L10N512S30 \\
\hline
$L(h^{-1}\textrm{Mpc})$&10&10&10&10\\
\hline
$n_{\rm 0}$(cm$^{-3}$) &0.036&0.036&0.036&0.036\\
\hline
$t_{\rm 0}$(Gyr) &30&30&30&30\\
\hline
$\alpha_{\rm 0}$ &3&3&3&3\\
\hline
$N_{\rm cell}$& $64^3$&$128^3$&$256^3$&$512^3$\\
\hline
$\ell_{\rm max}$&5&5&5&5\\
\hline
$z_{\rm end}$&3&3&3&3\\
\hline
$m_{\rm DM}(h^{-1}\textrm{M}_{\odot})$&$2.8\times 10^8$&$3.4\times 10^7$
&$4.3\times 10^6$&$5.4\times 10^5$\\
\hline
$\Delta x(h^{-1}\textrm{kpc})$&4.9&2.4&1.2&0.6\\
\hline
\end{tabular}
\cpt{Runtime parameters for the `convergence study' simulation
suite. $L$ is the box length, $n_{\rm 0}$ is the density threshold at $z=0$, $t_{\rm 0}$ is the star formation time scale for density $n_{\rm 0}$, $\alpha_{\rm 0}$ gives the evolution of the star formation time scale with redshift, $N_{\rm cell}$ is the number of cells at the coarse level, $\ell_{\rm max}$ is the level of refinement, $z_{\rm end}$ is the final redshift, $m_{\rm DM}$ is the dark matter particle mass and $\Delta_{\rm x}$ is the spatial resolution.}
\label{table1}
\end{table*}
We now describe our method for implementing star formation in the
RAMSES cosmological code. It is based on an heuristic approach
adopted in many, if not all, cosmological studies. For a complete
review of various implementations, see \cite{Kay02}. Basically, one
considers that star formation proceeds at a given time scale,
written here $t_*$, in region where one or several physical criteria
are fullfiled. In this paper, as well as in many previous other
papers in the literature, we adopt a simple scheme to turn gas mass
into star particles, adding a source term in the continuity equation
\begin{eqnarray}
\left( \frac{D\rho}{Dt} \right)_* = & -\frac{\rho}{t_*}
& \mbox{~if~} \rho > \rho_{\rm 0} (1+z)^{\alpha_{\rm 0}},
\label{starrateeq} \\
\nonumber
\left( \frac{D\rho}{Dt} \right)_* = & 0 & \mbox{~otherwise},
\end{eqnarray}
\noindent where the threshold density may depend on redshift through
index $\alpha_{\rm 0}$. The star formation time scale $t_*$ is proportional
to the local free-fall time
\begin{equation}
t_* = t_{\rm 0 }\left( \frac{\rho}{\rho_{\rm 0}} \right)^{-1/2}.
\end{equation}
\noindent In the literature, one can find basically two different
approaches: a density threshold constant in physical units,
corresponding here to $\alpha_{\rm 0}=0$, and a density threshold constant
in comoving units, corresponding here to $\alpha_{\rm 0}=3$.
This simple model of star formation has been discussed and
criticized extensively in the literature \citep{Kay02}: we recall
here briefly its possible physical and observational origins.
Spiral galaxies in our nearby universe are seen to form stars at a
rate given by the Kennicutt law \citep{Kennicutt98}, similar to
equation~\ref{starrateeq}, with volume densities $\rho$ and $\rho_{\rm 0}$
replaced by average disc surface densities.
\begin{equation}
\Sigma_{\rm 0} = 10 \mbox{~M}_{\odot} /\mbox{pc}^2 \mbox{~and~} t_{\rm 0}=2 ~\mbox{Gyr}.
\end{equation}
The physical origin of such a behavior is not clearly identified
yet. One good candidate is the sustained, interstellar,
self-gravitating turbulent cascade, which controls the mass flux
between large scale filaments in the disc and small scale star
forming molecular clouds, at a rate given by the local (on large
scale, though) free-fall time \citep{Elmegreen02}. Such a precise
description of star formation is completely irrelevant for our
present cosmological study. Using a `sub-cell approach', similar
in spirit to the one developed by the fluid mechanics community and
often named the $k-\epsilon$ method, our current modeling of star
formation try to mimic in a statistical sense the complex behavior
of the interstellar medium.
Along the same lines, \cite{Yepes97} and more recently
\cite{Springel03a} have developed a multiphase model of the
interstellar medium, based on \citet{McKee77} early work on
molecular clouds evaporation within supernovae remnants hot bubbles.
This multiphase model offers an interesting alternative to
the previous turbulent model. It also gives the possibility to
computes self-consistently the star formation parameters, which are
\begin{equation}
n_{\rm 0} = 0.1 \mbox{~cm}^{-3} \mbox{~and~} \alpha_{\rm 0}=0 \mbox{~and~} t_{\rm 0}=2.1 \mbox{~Gyr}.
\end{equation}
Recall that in this case, star formation is allowed in region whose
gas density lies above a {\it physical} density threshold. In
cosmological simulations, however, the {\it comoving} density is
usually preferred to define collapsed, high-density clumps where
star formation is likely to occur. In \cite{Springel03b} for
example, equation~\ref{starrateeq} is augmented with the requirement
that the local {\it overdensity} should exceed 200. This guarantees
that star formation cannot occur in smooth regions of the
cosmological flow, but only within collapsed, virialized halos.
In the present paper, we would like to explore a different option
than the one used in \cite{Springel03b}. We use the following star
formation density threshold
\begin{equation}
n_{\rm 0} = 0.036 \mbox{~cm}^{-3} \mbox{~and~} \alpha_{\rm 0}=3.
\end{equation}
\noindent This corresponds to a baryon overdensity threshold of $1.6
\times 10^5$. This approach was also adopted in earlier works on the
star formation history in the universe \citep{Cen92, Kay02,
Nagamine04}, with threshold overdensities ranging from $5$ to
$10^5$, depending on the authors. In our case, this is motivated by
our AMR refinement scheme: $n_{\rm 0}(1+z)^3$ corresponds exactly to the
density threshold triggering the maximum level of refinement
$\ell_{\rm max}=5$. When baryons cool and settle down at the centre of
their host dark matter halos, the density does increase
dramatically. Our AMR code describes this collapse accurately, by
adding recursively new cells at the centre of the halo. At the
point when the maximum level of refinement has been reached, the
numerical description of the collapse is not valid anymore. We then
turn on star formation, as our sub-cell modelling of this uncomplete
collapse, that would have lead ultimately to the formation of one or
several star forming molecular clouds.
As soon as star formation is active, we create collisionless star
particles of constant mass $m_*$. As we explain in
appendix~\ref{sectrefine}, this constant mass is chosen to be
equal to the initial mass resolution in the gas distribution, in
order to prevent spurious refinement or de-refinement triggered by
star formation. Using a constant mass for our star particles has
also the advantage of controlling the {\it maximum number} of star
particles created at the end of the simulation. In order to solve
for the star formation source term (Eq.~\ref{starrateeq}) with
constant star particles mass, we need to adopt a stochastic
approach, similar to the one proposed in \cite{katz92}. In star
forming cells, we generate $N$ equal mass star particles, where $N$
is drawn from a Poisson process, with probability
\begin{equation}
P(N) = \frac{\lambda^N}{N!}\exp{(-\lambda)},
\end{equation}
\noindent and with parameter (or mean value)
\begin{equation}
\lambda = \left( \frac{\rho \Delta x^3}{m_*}\right) \frac{\Delta t}{t_*}.
\end{equation}
\noindent These star particles are created at each time step, at the
very centre of their parent cell. They are given a velocity equal to
the local fluid velocity, plus a random component that we take equal
to the local sound speed in the gas. The corresponding mass,
momentum and internal energy is of course removed from the parent
cell's conservatively.
In the simulation presented here, the only free parameter is the
star formation time scale at {\it comoving} threshold density, $t_{\rm 0}$.
\subsection{Parameters}
\begin{table*}
\begin{tabular}{|c|c|c|c|}
\hline
Name &L100N256S3 &L10N256S3 &L1N256S3\\
\hline
$L(h^{-1}\textrm{Mpc})$&100&10&1\\
\hline
$n_{\rm 0}$(cm$^{-3}$)&0.036&0.036&0.036\\
\hline
$t_{\rm 0}$(Gyr)&3&3&3\\
\hline
$\alpha_{\rm 0}$ &3&3&3\\
\hline
$N_{\rm cell}$&$256^3$&$256^3$&$256^3$\\
\hline
$\ell_{\rm max}$&5&5&5\\
\hline
$z_{\rm end}$&0&2.5&5.5\\
\hline
$m_{\rm DM}(h^{-1}\textrm{M}_{\odot})$&$4.3\times10^9$&$4.3\times10^6$&$4.3\times10^3$\\
\hline
$\Delta x(h^{-1}\textrm{kpc})$&12&1.2&0.12\\
\hline
\end{tabular}
\cpt{ Runtime parameters for the `high efficiency' simulation
suite. The meaning of the symbols is the same as table \ref{table1}.}
\label{table2}
\end{table*}
\begin{table*}
\begin{tabular}{|c|c|c|c|}
\hline
Name &L100N256S30&L10N256S30&L1N256S30\\
\hline
$L(h^{-1}\textrm{Mpc})$&100&10&1\\
\hline
$n_{\rm 0}$(cm$^{-3}$)&0.036&0.036&0.036\\
\hline
$t_{\rm 0}$(Gyr)&30&30&30\\
\hline
$\alpha_{\rm 0}$&3&3&3\\
\hline
$N_{\rm cell}$&$256^3$&$256^3$&$256^3$\\
\hline
$\ell_{\rm max}$&5&5&5\\
\hline
$z_{\rm end}$&0&2.5&5.5\\
\hline
$m_{\rm DM}(h^{-1}\textrm{M}_{\odot})$&$4.3\times10^9$&$4.3\times10^6$&$4.3\times10^3$\\
\hline
$\Delta x(h^{-1}\textrm{kpc})$&12&1.2&0.12\\
\hline
\end{tabular}
\cpt{Runtime parameters for the `low efficiency' simulation
suite. The meaning of the symbols is the same as table \ref{table1}.}
\label{table3}
\end{table*}
We assume throughout this paper a single cosmological model, the so
called `concordance model', with a cold dark matter component with
$\Omega_{\rm m}=0.3$, a baryon component with $\Omega_{\rm b}=0.04$ and a dark
energy component with $\Omega_{\Lambda}=0.7$. The Hubble constant
was set to $h=0.7$. The initial power spectrum is assumed to be
Harrisson-Zeldovich with $n=1$ and a normalization constrained
by $\sigma_{\rm 8}=0.93$. The exact functional form we use for the
transfer function is given in \cite{sugiyama95}.
We performed several AMR simulations, varying the box size, the
number of particles and the star formation time scale $t_{\rm 0}$. The
density threshold for star formation was set to $n_{\rm 0}=0.036$
cm$^{-3}$, and was held constant in {\it comoving units}. In our
notation, this translates into $\alpha_{\rm 0}=3$. This star formation
threshold corresponds to the overdensity threshold triggering our
last level of refinement $\ell_{\rm max}=5$.
Our simulation suite can be organized in 3 different sets. The first
set was designed to perform a convergence study. We used a box size
of 10$h^{-1}$ Mpc, with identical physical and cosmological
parameters. We vary only the initial number of particles and grid
cells, from $64^3$ to $512^3$. Runtime parameters for this
`convergence study' simulations are summarized in
Table~\ref{table1}.
The second set of simulations was designed to explore a `high
efficiency' SFR scenario with the maximum dynamical range we can
afford (Table~\ref{table2}). We used for that purpose 3 different
box size: 1, 10 and 100 h$^{-1}$ Mpc. All these simulations were
performed using $256^3$ particles and the same number of grid cells.
This translates into a dark matter particle mass of $4.3\times
10^3$, $4.3\times 10^6$ and $4.3\times 10^9 $ h$^{-1}$M$_\odot$ .
The third set of simulations is similar to the second one but for a
`low efficiency' SFR scenario (Table~\ref{table3}).
The 2 last simulation sets span a huge dynamical range in mass, but
each simulation is valid for a limited range of redshifts. When the
typical scale of non linearity reaches the box size, the simulation
has to be stopped: the simulated volume is not statistically
representative of the universe as a whole, and spurious effects due
to periodic boundary conditions become visible. The stopping
redshift was chosen to be 5, 2.5 and 0, for a box size of 1, 10 and
100 h$^{-1}$ Mpc respectively.
We have not implemented feedback processes in the RAMSES code: this
is currently under development. As explained in the introduction,
galactic winds are a key ingredient in computing the star formation
history in the universe. In this paper, we use SPH simulations
results obtained by \cite{Springel03b} to estimate the influence of
winds in the overall baryon budget.
Three set of AMR simulations performed by us using the RAMSES code
(`convergence study', `high efficiency' and `low efficiency')
and one set of SPH simulations \citep{Springel03b} compose the data
we analyze and discuss intensively in this paper.
\subsection{Results}
\label{results}
We present now general results obtained by the RAMSES code, for a
typical cooling and star formation run. We would like to outline
that current high-resolution numerical simulations reproduce
qualitatively the global picture of galaxy formation: fast cooling
gas builds up centrifugally supported discs at the center of dark
matter haloes, in which star formation quietly proceeds. We will
analyze our results more quantitatively in the next sections.
\begin{figure*}
\centering \includegraphics[width=\hsize]{megamap512bis.ps}
\cpt{Gas density (up) and stellar density (down) in our
highest resolution run L10N512S30. The first map on the left of
each row shows a projection of the full periodic volume. The
squares in each map delimits the zoom region where the next
image in the row is defined. For color map definitions, see
text.}
\label{zoom512}
\end{figure*}
Figure~\ref{zoom512} presents the simulated density field projected
along one principal axis of the comoving periodic box. Run
L10N512S30 is shown, at redshift $z=2.8$. The projected gas density
is shown with a logarithmically scaled color table. The 4 images
show a zooming sequence, starting from the whole periodic box with
the typical large scale filamentary structure, down to a particular
region where 2 spiral discs are clearly visible.
The second set of images in Figure~\ref{zoom512} shows the same
zooming sequence for the projected stellar density. These images
were computed using a `true color' color scheme: stars are divided
into 3 families according to the formation redshift of each star
particle ($3<z_{\rm f} < 4$, $4<z_{\rm f}<5$ and $5<z_{\rm f}$). 3 projected density
maps are computed for each family. The 3 images are then combined
using the RGB color scheme: `Red' stands here for old stars,
`Blue' for young stars and `Green' for intermediate formation
redshift stars.
These rather spectacular images compare favorably with observed
high redshift galaxies observed for example with the Hubble Space
Telescope. We have to be careful in making definitive conclusions:
the spiral discs we observe in our simulations are highly
underresolved. Figure~\ref{zoom512} illustrates this point: the AMR
grid structure clearly shows up in the highest resolution figure,
demonstrating that these runs are not meant to resolve the internal
structure of galactic discs. We are however confident in the
computed baryon fraction inside each halo, as soon as the halo mass
is greater than a few hundred particles.
The traditional way of describing the baryon density field is to
divide it into 4 phases. This phase separation is apparent in
Figure~\ref{phases}, which shows the density-temperature histogram
at $z=0$ for our large box run (100 h$^{-1}$ Mpc) L100N256S30. The
first regime occurs at low density ($\rho < 200 \bar \rho$) and low
temperature ($T < 10^5$ K) in the phase space diagram. This is the
diffuse intergalactic medium, also known as the Lyman alpha forest.
The diffuse intergalactic UV flux is responsible for preventing this
warm gas from collapsing into their parent dark matter halo.
The second regime occurs at high density ($\rho > 10^5 \bar \rho$)
and low temperature ($T < 10^5$ K), and corresponds in our
simulations to cold, centrifugally supported discs. This gas is
termed here `cold gas'. It is shown in Figure~\ref{phases} as the
rightermost box. In this region, the quasi-neutral gas follows a
very tight $\rho$-T relation, typical of high-density HI discs.
The remaining gas corresponds to shock heated gas into virialized
halos. We call it `hot gas' although, when cooling is efficient,
its temperature never exceed a few $10^4$ K. This gas, in rough
hydrostatic equilibrium, spans a large density range, from $\rho
\simeq \bar \rho$ in the outskirts of dark matter halos up to $\rho
\simeq 10^5 \bar \rho$ in the X ray emitting cores. This ionized
gas is also rapidly cooling from the inside out, leading to the
formation of the cold phase.
The last component in the baryon budget is the stellar phase. Stars
originate from the highest density tail in the phase space diagram.
In our model, star formation occurs above the overdensity threshold
$\rho > 10^5 \bar \rho$, which corresponds to our cold phase.
Molecular cooling as well as supernovae feedback are not modeled
here. The cold neutral gas is in reality decomposed into several
components, typical of the interstellar medium: molecular, very cold
clouds embedded into hot, supernovae driven bubbles. The correct
description of this multiphase interstellar medium is far beyond the
possibilities of current cosmological simulations. The only
possibility is to rely on sub-cell modeling, along the lines
described for example in \cite{Yepes97} and \cite{Springel03a}.
These authors noticed that their multiphase modeling does not
affect the overall baryon budget between the hot halo gas, the cold
disc component and the stellar fraction. The main effect of this
multiphase approach was to modify the internal structure of the
gaseous discs, altering the effective equation of state of the dense
gaseous component.
The most important feature that is likely to affect the computed
baryon budget is the mass resolution of the code. We have to
carefully assess its effect on our results. Let us first
examine Figure~\ref{resolmap}, which shows a sequence of images of
the projected gas density fields in a chosen region of the
`convergence study' suite. Initial conditions were generated
self-consistently for these 4 simulations, in order to recover the
same large scale distribution. From the lowest resolution run with
$64^3$ particles to start with, up to the highest resolution run
with $512^3$ dark matter particles, one clearly sees the spectacular
increase in small mass haloes along the filaments and in the voids
in between.
If we now examine in Figure~\ref{resolmap} the corresponding stellar
density distributions, we see that small haloes appear devoid of
stars. Small haloes are indeed unsufficently resolved to
reach the high density contrast required to allow star formation.
Cooling is also very likely affected by the poor resolution in these
small halos. We will see in the next sections that for our
simulations the minimum mass for a halo to host stellar particles
lies around 400 dark matter particles.
Another effect of mass resolution is also visible in the same
figure. Large galactic discs obtained at a given resolution tend to
fragment at higher resolution, leading to smaller discs with several
orbiting satellites. These satellites are the remnants of progenitor
halos, which form earlier at smaller mass. Insufficient resolution
also affects the history of mass assembly inside galactic halos.
Figure~\ref{resolmap} shows that the color of a given galaxy is
affected by the finite mass resolution. At low resolution, star
formation is artificially delayed to late times and the galaxy
appears blue, while at higher resolution, the correct star formation
history is recovered and the same galaxy appears red. The purpose
of this paper is to carefully estimate the effect of finite mass
resolution on our predictions.
\begin{figure}
\centering
\includegraphics[width=\hsize]{phases.eps}
\cpt{The different baryon phases in the $\rho-T$ diagram. Gray
contours show a mass-weighted histogram: the baryon mass fraction
at a given density and temperature. Each region corresponds to a
given phase (diffuse background, hot or cold gas), as defined in
the text.}
\label{phases}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=\hsize]{megamapres.ps}
\cpt{Projected density maps (top) and projected stellar
density (bottom) of our `convergence study' simulations series
($L = 10$ h$^{-1}$ Mpc). The image size is 1.25 h$^{-1}$ Mpc
wide on a side. From left to right, the number of dark matter
particles is increased from $64^3$, to $128^3$, $256^3$ and
$512^3$. }
\label{resolmap}
\end{figure*}
\section{Model}
In this section, we present a simple analytical model to compute the
evolution of the baryon budget in the universe. The purpose of this
analytical model is to shed light on the complex behavior of our
numerical simulations. Our analytical treatment of the cosmic star
formation history is, of course, unaffected by finite resolution
effect. It is however a crude model, and a careful comparison
with numerical simulations is required to validate our approach.
This model differs with the `semi-analytic' modeling of galaxy
formation, an approach pursued by several teams \citep{White91,
Somerville99, Kaufmann99, Cole00, Hatton03}. These models are based
on a quite sophisticated treatment of the physics of galaxy
formation: cooling, star formation and spiral discs evolution are
few examples among the numerous ingredients in semi-analytical
modeling.
In this paper, our goal is to compute the evolution in the mass
fraction of the 4 different components in the baryon distribution:
diffuse background, hot virialized plasma, cold neutral discs and
stars. We will make predictions for the universe as a whole, but
also for individual halos of a given mass. Semi-analytical models,
when coupled to N body simulations, can predict the baryon history
for individual halo, based on the specific merging history at the
origin of the halo hierarchical mass assembly. Our simple model
allows us to compute only the average baryon history for halos of a
given mass range.
In a recent paper, an analytical approach has been proposed to
compute the star formation history in the universe
\citep{Springel03c}. The proposed method was to use the
Press-Schechter theory for the dark matter halos statistics
together with the star formation rate as a function of halo mass, as
measured in SPH numerical simulations. In the present paper, we
develop a fully self-consistent analytical model, slightly more
complex than the one proposed by \cite{Springel03c}, but based on a
similar approach. This self-consistency allows us to compute the
mass fraction evolution of the 4 baryon components, for the
universe as a whole, and also for an individual {\it average} halo.
\subsection{Halo Model}
The method we use in this paper to compute the star formation
history is based on what is usually called the halo model
\citep{Cooray02}. The idea is to decompose dark matter and baryon
density fields into a collection of virialized halos, whose
distribution is described by the \cite{Press74} mass function. This
approach was introduced to describe galaxy and dark matter
clustering, using two additional ingredients: linear halo biasing
theory and the Navarro, Frenk \& White (1995) density profile
\citep{Ma00, Seljak00}. Later on, several authors used similar
tools to compute the Sunyaev-Zeldovich power spectrum, as well as
the mean Comptonization parameter \citep{Cooray00, Refregier02}.
A feature common to most of these earlier works is the rather static
picture they give to dark matter halos evolution. We propose here
to use a similar approach, in order to compute the history of star
formation within each dark matter halo. Our methodology differs
somewhat from earlier works, since it is based on a two step
approach.
In the first step, using static halo model, we compute the mass
transfer rates between each phase. In the second step, using
these computed mass transfer rates, we solve for the mass fraction
evolution equations, a system of first order Ordinary Differential
Equations (ODE). Our analytical model is therefore based on the
computation of the baryon logistics. In order to compute the baryon
mass fraction locked up in stars for example, we need to solve the
complete set of ODE from a large redshift ($z$ = 200 say) down to
$z$=0.
In this paper, we define the halo mass $M$ as $M_{\rm 200}$, the total
mass enclosed in radius $R_{\rm 200}$, where the mean overdensity
(relative to the background density) is $\Delta = 200$.
\begin{eqnarray}
M=M_{\rm 200}=\frac{4\pi}{3} \bar \rho(z) \Delta R_{\rm 200}^3.
\end{eqnarray}
This choice differs from earlier definitions, where $\Delta$ was
either defined relative to the critical density, or $\Delta$ was a
function of redshift, as suggested by the spherical collapse model.
As shown by \cite{Jenkins01} and \cite{White02}, these earlier
definitions are not suited to the Press \& Schechter approach.
Although they are based on physical principles, they destroy the
self-similarity of the Press \& Schechter mass function.
In this paper, we therefore adopt $\Delta = 200$ (relative to the
mean background density) independent of redshift. We also use very
often the halo circular velocity $V_{\rm 200}$ and the halo Virial
temperature $T_{\rm 200}$, instead of the halo mass. The Virial
temperature must not be considered as a true physical temperature,
but rather as yet another mass parameterization. In this paper, the
Virial temperature is defined as
\begin{eqnarray}
k_{\rm B}T_{\rm 200}=\frac{\mu m_{\rm H}}{2}\frac{GM_{\rm 200}}{R_{\rm 200}},
\label{virial}
\end{eqnarray}
(with $\mu$ the mean molecular weight) and the circular velocity as
\begin{eqnarray}
V_{\rm 200}= \sqrt{\frac{GM_{\rm 200}}{R_{\rm 200}}}.
\label{circular}
\end{eqnarray}
In the 5 following sections, we are going to compute the cosmic rates between the 4 phases. If necessary the reader can go directly to section \ref{chain}.
\subsection{Minimal Mass}
The first component in the baryon budget is the diffuse background.
This may be the most important one, since it is the reservoir of
fresh gas that will eventually feed star forming halos at all
epochs. It is usually called the Inter Galactic Medium (IGM) or the
Lyman Alpha Forest. We need to give a precise definition of what we
call `diffuse background' in this paper.
The diffuse background is the baryon component associated to dark
matter halo with masses lower than the minimal mass $M_{\rm min}$,
below which cooling is inefficient and pressure forces prevent
baryons to collapse in their potential wells. This minimal mass is
therefore fundamental because it is the transition between `star
forming halos' and `diffuse' one.
This Minimal Mass is taken to be the maximum between the Filtering
Mass and the Minimal Cooling Mass. The filtering mass $M_{\rm F}$ is the
minimal halo mass above which baryons are able to fall into their
dark matter halo potential wells (see Appendix~\ref{Filtering}). The
minimal cooling mass $M_{\rm cool}$ is the mass above which the gas is
able to cool and therefore to form stars (see
Appendix~\ref{coolingmodel}).
We implement this using a smooth function of both Virial temperatures
\begin{eqnarray} T_{\rm min} = T_{\rm F} + T_{\rm cool}. \end{eqnarray}
\begin{figure}
\centering \includegraphics[width=\hsize]{tf.eps}
\cpt{Time evolution of the Minimal Mass $M_{\rm min}$ for two
reionization scenarios: $z_{\rm r}$=20 (solid line) and $z_{\rm r}$=6
(dashed line). In both case, the background temperature after
reionization was set to $T_{\rm r}=6\times 10^3$ K. The Minimal
Cooling temperature was also set to $T_{\rm cool}=6\times 10^3$ K.}
\label{figtf}
\end{figure}
\noindent The Minimal Mass is finally computed using the Virial
relation (Eq.~\ref{virial}). We plot in Figure~\ref{figtf}
$M_{\rm min}$ as a function of redshift, for our two extreme
reionization scenarios. At early times, this mass remains roughly
constant, independent of redshift. As reionization proceeds, this
Minimal Mass increases steadily, up to a rather large value $M_{\rm min}
\simeq 10^{11}$ h$^{-1}$M$_{\odot}$ today.
\subsection{Cosmic Accretion Rate}
Using the Press \& Schechter formalism, we now compute the mass fraction in
the diffuse background and the mass fraction in star forming halos.
Since the Minimal Mass is considered here as the mass threshold
between these two components, and assuming that the baryon fraction
in each halo is equal to the universal one, we have
\begin{eqnarray}
f_{\rm hot} = f(M>M_{\rm min}) = f_{\rm b}
\int^\infty_{\nu_{\rm min}} \sqrt{\frac{2}{\pi}}\exp(-\nu^2/2)d\nu,
\end{eqnarray}
with
\begin{eqnarray}
\nu_{\rm min} = \frac{\delta_{\rm c}(t)}{\sigma(M_{\rm min})}
\mbox{~~~and~~~}
\delta_{\rm c}(t) = \frac{1.686}{D^+(t)},
\end{eqnarray}
where $D^+$ is the linear growth factor and $\sigma(M_{\rm min})$ the
variance of the density field smoothed at the Minimal Mass
scale. The rate at which baryons are transferred from the diffuse
background to star forming halos is computed by taking the time
derivative of the previous equation
\begin{eqnarray}
\dot{f}_{\rm acc} = \frac{df_{\rm hot}}{dt} = - \frac{df_{\rm back}}{dt}
= - f_{\rm b}\dot{\nu}_{\rm min}\sqrt{\frac{2}{\pi}}
\exp(-\nu_{\rm min}^2/2),
\end{eqnarray}
We then define the Cosmic Accretion Rate of fresh diffuse gas into
star forming halos by
\begin{eqnarray}
\dot{f}_{\rm acc} = \omega_{\rm acc}f_{\rm back},
\label{accmass}
\end{eqnarray}
where the accretion rate, in units of Gyr$^{-1}$, is given by
\begin{eqnarray}
\omega_{\rm acc} = - \dot{\nu}_{\rm min}\sqrt{\frac{2}{\pi}}
\frac{\exp(-\nu_{\rm min}^2/2)}{\mbox{erfc}(\nu_{\rm min}/\sqrt 2)}.
\label{accrate}
\end{eqnarray}
This last equations give the mass accretion rate of diffuse gas into
star forming halos {\it in the general case}, for which the mass
fraction in the diffuse component is allowed to vary from its
canonical value. Note that this accretion rate has nothing in common
to the traditional mass accretion rate on a given halo
\citep{Lacey93}. This rate gives the fraction of fresh diffuse gas
dispatched among all star forming halos. This fresh gas in assumed
to be transferred exclusively to the hot plasma component. The two
variables $f_{\rm back}$ and $f_{\rm hot}$ refer therefore to the total mass fraction
in the background and the total mass fraction in the hot component,
both integrated over the PS distribution.
It is also possible to compute the Cosmic Accretion Rate on a halo
by halo basis, using the Extended Press Schechter theory
\citep{Bond91, Lacey93}. This theory allows to compute the
progenitors mass distribution as a function of time, for a given
parent halo mass $M_{\rm 0}$, up to the `halo formation time' $t_{\rm 0}$.
The individual Cosmic Accretion Rates are very similar to the one
computed for the whole universe. We follow the same procedure,
computing first the mass fraction in star forming halos, assuming
that each progenitor hosts a baryon fraction equal to the universal
one.
\begin{eqnarray}
f_{\rm hot}(M_{\rm 0},t_{\rm 0}) = f_{\rm b}
\int^\infty_{\nu_{\rm min}} \sqrt{\frac{2}{\pi}}\exp(-\nu^2/2)d\nu,
\end{eqnarray}
with this time
\begin{eqnarray}
\nu_{\rm min}(M_{\rm 0},t_{\rm 0}) = \frac{\delta_{\rm c}(t) -\delta_{\rm c}(t_{\rm 0})}
{\sqrt{\sigma(M_{\rm min})^2-\sigma(M_{\rm 0})^2}}.
\label{extnuf}
\end{eqnarray}
The accretion rate is then computed exactly as for the previous
case, using Equations~\ref{accmass} and \ref{accrate}, with however
a different value for $\nu_{\rm min}$ given by Equation~\ref{extnuf}. We
want to stress again that we do not consider accretion of satellite
halos on the most massive progenitor, which is the traditional way
of computing the accretion rate. Here, we consider accretion of
diffuse gas on all star forming progenitors of the final halo.
\begin{figure}
\centering \includegraphics[width=\hsize]{wacc.eps}
\cpt{Cosmic Accretion Rate for the $\Lambda$CDM cosmology with
$z_{\rm r}=20$ for the universe as a whole (thick solid line) and for
various halo masses (thin lines). Halo mass are, from top to
bottom, $M_{\rm 0} = 10^{13}$, $10^{12}$, $10^{11}$, $5 \times
10^{10}$, $2.5 \times 10^{10}$ and $10^{10}$ h$^{-1}$M$_{\odot}$. The
halo formation redshift is set to $z_{\rm 0}=0$.}
\label{figwacc}
\end{figure}
This fresh gas contributes to fill up dark matter halos with hot,
virialized gas. Hot gas coming from satellite halos is
automatically accounted for in our formalism. If we assume very
short cooling rates and instantaneous star formation, this Cosmic
Accretion Rate is nothing but the Star Formation History in the
universe. In a realistic case, star formation and cooling introduce
a delay in the curve.
This cosmic accretion rate depends on the thermal history of the
background gas, on the density power spectrum through $\sigma(M)$,
and on the cosmological model through $D^+(t)$.
Figure~\ref{figwacc} shows the accretion rates (in Gyr$^{-1}$) as a
function of redshift, for a $\Lambda$CDM universe, and for different
halo masses. The halo formation redshift was fixed to $z_{\rm 0} = 0$.
For small mass halos, accretion stops abruptly as the Minimal Mass
reaches the parent halo mass. This means that star forming
progenitors are not present anymore in the halo. For halos more
massive than $M_{\rm min}$, accretion remains active up to the halo
formation redshift. The accretion rate actually diverges as $z
\rightarrow z_{\rm 0}$, as all the remaining diffuse gas is accreted into
the final virialized halo. This diffuse mass accretion rate remains
however finite for $M_{\rm 0} > M_{\rm min}$, and we can compute its value as
$z \rightarrow z_{\rm 0}$ \begin{eqnarray} \dot{f}_{\rm acc} = - f_{\rm b}
\sqrt{\frac{2}{\pi}} \frac{\dot{\delta}_{\rm c}(t_{\rm 0})}
{\sqrt{\sigma(M_{\rm min})^2-\sigma(M_{\rm 0})^2}}. \label{asymptoticaccrate}
\end{eqnarray}
\subsection{Cosmic Cooling Rate}
Our third baryon component (namely cold atomic gas in centrifugally
supported discs) is progressively built up by accreting cooling gas
into the very center of their parent dark matter halo. We need to
estimate the global rate at which hot gas is transferred into this
dense and cold component. Using EPS theory, we compute this rate
for individual halo mass ($M_{\rm 0}$, $z_{\rm 0}$). Note that we recover the results for the Universe as a whole by taking the limit $M_{\rm 0} \rightarrow +\infty$ and $z_{\rm 0} \rightarrow -1$.
We follow our basic methodology, assuming this time that all baryons
are in the hot halo phase. Using our simple cooling model detailed
in Appendix~\ref{coolingmodel}, we compute the total amount of gas
cooling from the hot halo component during a unit time interval by
integrating the instantaneous cooling rate over the PS mass
distribution from $M_{\rm min}$ to $M_{\rm max}$, the Maximal Cooling Mass.
Indeed, above this mass (see Appendix~\ref{coolingmodel}), the halo
enter in the slow regime cooling. We therefore neglect this
contribution. It gives
\begin{equation}
\dot f_{\rm cool} = f_{\rm b}\frac{1}{t_{\rm orb}}
\int^{\nu_{\rm max}}_{\nu_{\rm min}} \sqrt{\frac{2}{\pi}}\exp(-\nu^2/2)d\nu,
\end{equation}
where $\nu_{\rm min}$ is defined by Equation~\ref{extnuf},
$\nu_{\rm max}$ corresponds to the Maximum Cooling Mass $M_{\rm max}$
\begin{eqnarray}
\nu_{\rm max}(M_{\rm 0},t_{\rm 0}) = \frac{\delta_{\rm c}(t) -\delta_{\rm c}(t_{\rm 0})}
{\sqrt{\sigma(M_{\rm max})^2-\sigma(M_{\rm 0})^2}},
\label{extnumax}
\end{eqnarray}
and $t_{\rm orb}$ is the orbital decay timescale (see Appendix~\ref{coolingmodel}).
We then define the Cosmic Cooling Rate of hot halo gas into
cold gaseous discs by
\begin{eqnarray}
\dot{f}_{\rm cool} = \omega_{\rm cool}f_{\rm hot},
\label{cool}
\end{eqnarray}
where the cooling rate, in units of Gyr$^{-1}$, is given by
\begin{eqnarray}
\omega_{\rm cool} = \frac{1}{t_{\rm orb}}
\frac{\mbox{erfc}(\nu_{\rm min}/\sqrt 2) - \mbox{erfc}(\nu_{\rm max}/\sqrt 2)}
{\mbox{erfc}(\nu_{\rm min}/\sqrt 2)}.
\end{eqnarray}
The Cosmic Cooling Rate depends on the cosmological model, on the
thermal history of the background and on the details of the cooling
model. A similar model has been proposed by \cite{VanDenBosch02}, in a
different context.
\subsection{Star formation models\label{SF}}
\label{starformmodel}
In this simple analytical model, we completely discard the
description of the gaseous discs. Predicting the disc sizes and
surface density profiles obtained in the hierarchical scenario of
structure formation is beyond the scope of this paper. We are only
interested in the global baryon budget, and more precisely in the
global star formation history.
We therefore consider star formation in a dark matter halo as a
function of the total amount of cold gas in that halo. The star
formation rate in each halo is simply given by $\dot{M}_* =
\omega_* M_{\rm cold} $ with $\omega_*$ is the average star formation rate
in that halo. In order to compute this average time scale from first
principles, one needs to integrate the local, density dependent,
star formation rate over the cold gas density PDF.
In this analytical model, however, we consider star formation
models inspired by star formation recipes used in numerical
simulations and by semi-analytical models \citep{Somerville01}. The
halo star formation rate is parameterized by
\begin{eqnarray}
\omega_* = \frac{1}{t_*} (1+z)^{\alpha_*/2},
\end{eqnarray}
where $t_*$ is the present day star formation time scale and
$\alpha_*$ is the acceleration parameter. In the literature, two basic
quiescent models are usually discussed in galaxy formation studies.
The first model, usually referred to as a `constant efficiency'
model, assumes that the halo star formation time is a constant
$\alpha_* = 0$. This model corresponds to numerical simulations
with a constant star formation density threshold.
The second model assumes that $\alpha_* = 3$. It is usually called
an `accelerated efficiency' model. The star formation
time scale decreases with redshift (as the mean density of the
Universe increases). This model corresponds to numerical
simulations with a constant star formation {\it overdensity}
threshold. It is also used in semi- analytical models to mimic
starbursts triggered by mergers \citep{Somerville01}.
We compute now the global star formation rate, using our basic
methodology. Since the halo star formation rate, in our simple
scenario, does not depend on halo mass, we can integrate over the
EPS mass function, and obtain the Cosmic Star Formation Rate as
\begin{eqnarray} \dot{f}_{*} = \omega_* f_{\rm cold}. \label{star}
\end{eqnarray}
\begin{figure}
\centering \includegraphics[width=\hsize]{rates.eps}
\cpt{Cosmic Accretion Rate (solid line), Cosmic Cooling Rate
(dotted line), Cosmic Star Formation Rate (dashed line) and
Cosmic Outflow Rate (unbound fraction, dot-dashed line) for the
$\Lambda$CDM cosmology with the following model parameters:
$z_{\rm r}=20$, $t_* = 3$ Gyr, $\alpha_* = 0$, $T_{\rm w} = 2\times 10^6$ K
and $\eta_{\rm w} = 1.5$. These rates were computed using
$M_{\rm 0}=+\infty$ and $z_{\rm 0}=-1$, and therefore corresponds to the
universe as a whole.}
\label{figrates}
\end{figure}
\subsection{Cosmic Winds}
The last ingredient in our model, but not the least, is the
contribution of galactic winds to the overall baryon budget. It is a
well known issue in current models of galaxy formation that without
feedback processes, most baryons would end up into cold gas or
stars, in contradiction with several observational constraints.
This problem is known as the `overcooling problem' \citep{Blanchard92}.
As discussed in the introduction, the exact nature of the dominant
feedback process is still unknown. It is most likely that various
processes are in competition, and their impact on baryons may vary
as a function of halo mass. Following \cite{Springel03b}, we assume
in our model that winds occur during star formation events, probably
related to supernovae. We therefore assume that cold gas is ejected
from the disc with a typical wind velocity $u_{\rm w}$ and with a typical
outflow rate \begin{eqnarray} \dot{M}_{\rm wind}=\eta_{\rm w} \dot{M}_*.
\end{eqnarray} The two additional parameters are $\eta_{\rm w} \simeq
1-5$, the wind efficiency, and $u_{\rm w} \simeq 200-500$ km s$^{-1}$, the
wind velocity \citep{Springel03b}. These wind parameters are
typical of observed outflows in star forming galaxies
\citep{Martin99}.
The fate of this ejected gas depends on the halo mass. If the wind
velocity exceeds the escape velocity of the halo, the ejected gas
leaves the halo into the diffuse background, from where, eventually,
it will be accreted again. Such winds are referred to as
`unbound'. If, on the other hand, the halo is too massive, the
ejected gas remains in the hot halo component, from where it will
eventually cool again. Such winds are referred to as `bound'.
Assuming again that all baryons are locked up into the cold
component, we can compute the global wind outflow rate by
integrating over the EPS mass function. We finally obtain the
following equations, valid in the general case \begin{eqnarray}
\dot{f}_{\rm wind} = \omega_{\rm w} f_{\rm cold}. \label{wind} \end{eqnarray} where
the outflow rate, in units of Gyr$^{-1}$, is given by
\begin{eqnarray} \omega_{\rm w} = \eta_{\rm w} \omega_*. \end{eqnarray} We first
compute the unbound fraction. It corresponds to winds emitted by
halos whose escape velocity is smaller than the wind velocity. Using
the EPS distribution, we get \begin{eqnarray} \zeta_{\rm w} =
\frac{\mbox{erfc}(\nu_{\rm min}/\sqrt 2) - \mbox{erfc}(\nu_{\rm w}/\sqrt 2)}
{\mbox{erfc}(\nu_{\rm min}/\sqrt 2)}, \end{eqnarray} where $\nu_{\rm w}$ is
defined by the `Wind Mass' \begin{eqnarray} \nu_{\rm w}(M_{\rm 0},t_{\rm 0}) =
\frac{\delta_{\rm c}(t) -\delta_{\rm c}(t_{\rm 0})}
{\sqrt{\sigma(M_{\rm w})^2-\sigma(M_{\rm 0})^2}}. \label{nuwind} \end{eqnarray}
This `Wind Mass' (the halo mass above which winds are bound) is
related to the wind velocity by noticing that for typical dark
matter halos, $v_{\rm esc} \simeq 3 V_{\rm 200}$. Using the standard Virial
relation (Eq.~\ref{virial}), we obtain the wind Virial temperature
\begin{eqnarray} k_{\rm B} T_{\rm w} = \frac{1}{18} \mu m_{\rm H} u_{\rm w}^2. \label{twind}
\end{eqnarray} The bound fraction is just $1-\zeta_{\rm w}$. The fate of
the unbound gas depends now on the parent halo mass $M_{\rm 0}$. If $M_{\rm 0}
> M_{\rm w}$, the gas is recycled into the halo diffuse component, and
ultimately into the halo hot component as $z \rightarrow z_{\rm 0}$. If
$M_{\rm 0} < M_{\rm w}$, the gas is lost into the intergalactic medium, outside
the boundaries of the parent halo, and never come back. Note that
in the latter case, $\zeta_{\rm w}$ is always equal to one. This very
crude model turns out to be surprisingly accurate in predicting
results obtained by numerical simulations (see the following
sections).
\begin{figure}
\centering \includegraphics[width=\hsize]{frac.eps}
\cpt{History of the mass fraction in the diffuse background
(solid line), in the hot halo gas (dotted line), in the cold
discs (dashed line) and in stars (dot-dashed line) for the
$\Lambda$CDM cosmology with the following model parameters:
$z_{\rm r}=20$, $t_* = 3$ Gyr, $\alpha_* = 0$, $T_{\rm w} = 2\times 10^6$ K
and $\eta_{\rm w} = 1.5$. These fractions were computed using
$M_{\rm 0}=+\infty$ and $z_{\rm 0}=-1$, and therefore corresponds to the
universe as a whole.}
\label{figfrac}
\end{figure}
\subsection{The Baryon Supply Chain}
\label{chain}
We are now in a position to compute the baryon budget history. The
last sections were devoted to computing mass transfer rates
between our 4 baryon components. The mass fraction in each component
are the independent variables in our problem: $f_{\rm back}$, $f_{\rm hot}$,
$f_{\rm cold}$ and $f_*$, referring respectively to diffuse background,
hot gas, cold discs and stars.
The methodology we follow in this paper allows us to compute in
advance 3 important mass transfer rates. This rates are
$\omega_{\rm acc}$, $\omega_{\rm cool}$ and $\omega_{*}$, referring
respectively to the Cosmic Accretion Rate, the Cosmic Cooling Rate
and the Global Star Formation Rate. Our very crude wind model
provides us with an additional parameter, namely the unbound
fraction $\zeta_{\rm w}$. These various rates are plotted in
Figure~\ref{figrates} for our fiducial model and our notations are summarized in table \ref{table4}.
\begin{table}
\begin{tabular}{|c|c|}
\hline
$z_{\rm r}$&reionization redshift\\
\hline
$t_*$&star formation time scale\\
\hline
$\eta_{\rm w}$&wind efficiency\\
\hline
$\zeta_{\rm w}$&unbound wind fraction\\
\hline
$f_*$&stellar fraction\\
\hline
$f_{\rm cold}$&cold gas fraction\\
\hline
$f_{\rm hot}$&hot gas fraction\\
\hline
$f_{\rm back}$&background gas fraction\\
\hline
$\omega_*$&star formation rate\\
\hline
$\omega_{\rm cool}$&cooling rate\\
\hline
$\omega_{\rm acc}$&accretion rate\\
\hline
\end{tabular}
\label{table4}
\cpt{Main notations}
\end{table}
We have to solve a set of ordinary differential equations, with
pre-computed transition rates between each component of our baryon
supply chain. \begin{eqnarray} \frac{df_{\rm back}}{dt} &=&
\zeta_{\rm w} \eta_{\rm w} \omega_* f_{\rm cold} - \omega_{\rm
acc}f_{\rm back}, \label{fback}\end{eqnarray} \begin{eqnarray}
\frac{df_{\rm hot}}{dt} &=& \omega_{\rm acc}f_{\rm back} -
\omega_{\rm cool}f_{\rm hot} + (1-\zeta_{\rm w}) \eta_{\rm w}
\omega_* f_{\rm cold}, \end{eqnarray} \begin{eqnarray} \frac{df_{\rm
cold}}{dt} &=& \omega_{\rm cool}f_{\rm hot} - \omega_* f_{\rm cold}-
\eta_{\rm w} \omega_* f_{\rm cold}, \end{eqnarray} \begin{eqnarray}
\frac{df_*}{dt} &=& \omega_* f_{\rm cold}. \end{eqnarray} If
$M_{\rm 0} > M_{\rm w}$, one sees that the total baryon mass is
conserved. Note that Eq. \ref{fback} should be modified if $M_{\rm 0}<M_{\rm w}$: the wind contribution is set to $0$ and therefore the total baryon mass in the parent halo is not conserved anymore. This is as expected, since winds are now escaping outside the parent halo boundaries.
Using any time integration method of sufficient accuracy, one can
finally solve for the previous set of differential equations. We
used in this work a Backward Euler scheme. Interestingly, one can
solve formally the latter system using matrix exponentials. This
type of equations are typical of galaxy formation studies, like the
early work of \cite{Tinsley80}. More recently, \cite{Pei99} have
designed a similar approach, based on the observed galaxy luminosity
functions, while here, our equations are based on EPS theory.
Figure~\ref{figfrac} shows the baryon budget evolution for our
fiducial model. Before applying this analytical model to
cosmological observations, we need to determine its validity range
using high resolution numerical simulations.
\section{Simulations versus Model}
We now compare our analytical predictions to the baryon budget
history obtained in our high-resolution hydrodynamical simulations.
Recall that we can compute analytically the baryon history for the
universe as a whole, but also on a halo by halo basis. In order to
make a careful comparison, we need to extract halos from the
simulated density field. We use for that purpose a halo detection
code based on the Spherical Overdensity algorithm \citep{Lacey93}.
We also need to carefully define the effective mass resolution of
our simulations, with respect to star formation. This additional
mass scale is a pure numerical artifact, that can be accounted for
explicitly in our analytical model. Using this modified model, we
will estimate how our results converge (or not) to the correct halo
model predictions.
We need also to estimate the halo star formation time, (as defined
in the previous section) in our numerical simulations. The star
formation algorithm is based on a simple Schmidt law, with a
specified (over-) density threshold. It is however more complex than
the approach used in the halo model. We will show that both
approaches can be related to each other, with a `shape factor'
reflecting the probability distribution function of the cold phase
density.
We finally need to estimate the unique free parameter in our
analytical model: the orbital decay timescale. This sort of `model
calibration' will be performed directly using our simulation
results. We will also extend this `model versus simulation'
comparison to other numerical data kindly provided to us by
\cite{Springel03b}, for which galactic winds were included.
\subsection{Halo Detection}
\label{halodetect}
It is an absolute necessity to define a halo in a numerical
simulation as it is defined in the theoretical model. As explained
before, in order for the Press \& Schechter approach to be valid,
the halo mass is the mass enclosed in radius $R_{\rm 200}$, enclosing an
overdensity 200 times larger than the average background density. As
noted by several authors \citep{Jenkins01,White02}, this rather
large region encloses dark matter particles which are not completely
relaxed yet, and also several large satellites flying by. As a
consequence, the halo mass turns out to be highly dependent on the
exact algorithm used to detect automatically dark matter halos in
the simulation.
These same authors suggest the following strategy: in a first step,
halos are detected with a high density contrast ($\Delta = 600$ is
our choice here) by any classical algorithm (we use Spherical
Overdensity in this paper). Since the density contrast is high, the
detected region is in a well relaxed state and all algorithms agree
more or less on the mass and on the number of detected halos. The
halo center is defined as the center of mass of this high density
region only. In a second step, the halo radius is increased up to
$R_{\rm 200}$, in order to obtain the large halo mass required by the
Press \& Schechter prediction.
We then compute for each individual halo the total stellar
mass within $R_{\rm 200}$. Since we have stored the formation epoch of
each individual star particle, we can also compute the complete star
formation history of the parent halo. This last point is very
important: we do not compute star formation rates and gas content of
individual galaxies. Since each halo can host several galaxies (one
central and several satellites), the galaxy baryon budget will be
somewhat different than the halo baryon budget. Within the halo
radius, we also compute the fraction of cold gas, defined as $T <
10^5$ K, and $\rho > 10^5 \bar \rho$. The remaining gas is
considered as `hot gas', even though its temperature can be lower
than the Virial temperature of the halo.
The effective star formation rate of each halo is computed by simply
dividing the total amount of star created during the last 10\% of
the halo age by the elapsed time. The results presented in this
section will be based on this analysis. Each individual halo baryon
budget are averaged into mass bins, in order to compare with the
halo model predictions.
\subsection{Mass Resolution}
Results of cosmological simulations depend strongly on the mass
resolution of the code. The two main numerical limitations are the
box length and the number of particles. As shown in
Figure~\ref{resolmap}, the more particles we start with, the more
small mass halos and galaxy satellites we obtain in the
simulations. Since we are interested here in the global baryon
budget, each individual halo must be able to allow gas to cool down
and condense, and ultimately form stars. This is a more stringent
requirement than just reach the Virial overdensity of $\Delta =
200$.
Since star formation occurs at the high end of the density
distribution, we take the star formation density threshold as the
limiting factor that defines our mass resolution. Let us assume for
sake of simplicity that each halo is a pure isothermal sphere. The
gas density profile is given by
\begin{eqnarray}
\rho = \frac{\Delta}{3}\bar \rho \left( \frac{r}{R_{\rm 200}} \right)^{-2}.
\end{eqnarray}
The radius above which star formation occurs is given by $\rho(r) >
\rho_{\rm 0} (1+z)^{\alpha_{\rm 0}}$. If we require that within this radius, we
have at least 10 dark matter particles of mass $m_{\rm p}$, we obtain the
minimal halo mass as
\begin{eqnarray}
M_{\rm resol} = 10 \left[
\frac{\rho_{\rm 0}(1+z)^{\alpha_{\rm 0}}}{\Delta/3\bar \rho(z)}
\right]^{1/2} m_{\rm p}.
\label{massresol}
\end{eqnarray}
One clearly sees that the higher the density threshold for star
formation, the larger the minimal mass will be. For star formation
density thresholds constant in comoving units ($\alpha_{\rm 0}=3$), this
mass scale is a constant in time. This is the case for the AMR
simulations presented here, for which, using
Equation~\ref{massresol}, we obtain $M_{\rm resol} \simeq 400 m_{\rm p}$. On
the other hand, for star formation density threshold constant in
physical units, the mass resolution scales as $(1+z)^{-1.5}$. SPH
simulations presented in \cite{Springel03b} were based on this
second approach. The mass resolution we obtain in this case is
$M_{\rm resol} \simeq 1000(1+z)^{-1.5} m_{\rm p}$. At high redshift, $z
\simeq 20$, the corresponding mass resolution can be as low as
$M_{\rm resol} \simeq 10 m_{\rm p}$.
Simulated halos with mass lower than $M_{\rm resol}$ will not be able
to form stars or, equivalently, condensed cool gas. Therefore, they
will be part of the simulated diffuse background. This new mass
scale is a pure numerical artifact, that strongly affects our
results. We take this mass scale into account in our analytical
model by setting the Minimal Mass for star forming halos $M_{\rm min}$
as the maximum between the true physical Minimal Mass and
$M_{\rm resol}$. As we will see later in this section, this trick is a
very powerful tool to account for finite resolution effect in the
simulation, and to assess the convergence properties of our
numerical results.
\subsection{Halo Star Formation Time}
\label{halotstar}
\begin{figure}
\includegraphics[width=\hsize]{frac_evol.ps}
\cpt{Time evolution of the various baryon phases in our
highest resolution run L10N512S30. Symbols are mass fraction
measured in the simulation (squares: diffuse background,
crosses: hot gas, diamonds: cold gas and triangles:
stars). Lines are the predictions of our analytical model with
$M_{\rm resol} \simeq 2\times10^8$ h$^{-1}$M$_{\odot}$ (solid: diffuse
background, dot-dot-dashed: hot gas, dot-dashed: cold gas and
dashed: stars).}
\label{frac_evol}
\end{figure}
\begin{figure}
\includegraphics[width=\hsize]{beta.ps}
\cpt{ Average star formation rate of simulated halos in
various Virial temperature bins, in unit of
$M_{\rm cold}/t_*(\rho_t)$. In our framework, this is a direct measure of
the `shape factor' $F(\mu)$ of the underlying cold gas density
distribution. Numerical data suggest a constant value
represented here as the dashed line $F(\mu) \simeq 3$. Diamonds
are for run L1N256S30 at $z=5.5$, triangles for run L10N256S30
at $z=2.5$ and squares for run L100N256S30 at $z=0$.}
\label{beta}
\end{figure}
\begin{figure} \includegraphics[width=\hsize]{alpha.ps}
\cpt{ Average star formation rate of simulated halos in
various Virial temperature bins, in unit of $M_{\rm hot}
R_{\rm 200}/V_{\rm 200}$ for the high efficiency runs. This is a direct
measure of the cooling rate of hot gas into dense cold discs. At
high temperature ($T > 10^7$ K), we observe a sudden drop due to
inefficient Bremstrhalung cooling. At lower temperatures, the
cooling rate has a plateau around $1/3$. In our framework, this
suggests an orbital decay timescale $t_{\rm orb} \simeq 3
R_{\rm 200}/V_{\rm 200}$ for infalling gas clumps. Diamonds are for run
L1N256S3 at $z=5.5$, triangles for run L10N256S3 at $z=2.5$ and
squares for run L100N256S3 at $z=2.5$ also.} \label{alpha}
\end{figure}
The methodology we use in this paper is to describe star formation
on a halo by halo basis. We completely discard the detailed
modeling of exponential gas discs and nuclear bursts. This is
usually performed in semi-analytical models of galaxy formation. In
our AMR simulations, we do have however a higher level of complexity
than in the analytical model. Visual inspection of density maps
shows the presence of gas discs in centrifugal equilibrium, as well
as several small satellites orbiting a central galaxy (see
Fig.~\ref{zoom512}). We are aware of the fact that many physical
ingredient are probably missing in our current numerical solution
of galaxy formation. Nevertheless, we need to establish the link
between our analytical model and our numerical implementation of star
formation.
Since star formation in the code is based on a Schmidt law (see
Eq.~\ref{starrateeq}), we can compute the instantaneous star
formation rate in any halo by integrating over the entire cold gas
present in that particular halo.
\begin{eqnarray}
\dot{M}_* = M_{\rm cold} \int_{\rho_{\rm t}}^\infty
\frac{\mu(\rho) d\rho}{t_*(\rho)},
\end{eqnarray}
where $\rho_{\rm t} = \rho_{\rm 0}(1+z)^{\alpha_{\rm 0}}$ is the star formation density
threshold and $\mu(\rho)$ is the mass fraction of cold gas with
density $\rho$. The exact form of the cold gas distribution function
$\mu$ is beyond the scope of this paper. It is likely to be
determined by the global surface density as well as the small scale
turbulence inside rotating discs. We will make here the very crude
approximation that $\mu$ is self-similar in the variable $\rho /
\rho_{\rm t}$, so we can simplify the last equation further more into
\begin{eqnarray}
\dot{M}_* = \frac{M_{\rm cold}}{t_*(\rho_{\rm t})} F(\mu),
\label{shapefac}
\end{eqnarray}
where $F(\mu)$ is a dimensionless `shape factor' that depends on
the exact form of the cold gas density distribution
\begin{eqnarray}
F(\mu) = \int_{\rm 1}^\infty \mu(x) x^{1/2} dx.
\end{eqnarray}
We are now in a position to make a direct link between our
analytical model and numerical simulations. We recognize in the last
equation the halo star formation rate as defined in
section~\ref{starformmodel}, with star formation parameters given by
$\alpha_* = \alpha_{\rm 0}$ and $t_* = t_{\rm 0} / F(\mu)$. The shape factor $F$
has the effect of reducing the effective halo star formation time
scale, relative to the reference time $t_{\rm 0}$. Indeed, if very high
density gas is present, the star formation rate is likely to
increase accordingly.
Since we can't predict the value of this shape factor, we have to
measure it directly in the simulations. We plot in Figure~\ref{beta}
the halo star formation rate, in units of $M_{\rm
cold}/t_*(\rho_{\rm t})$ (see Eq.~\ref{shapefac}), and averaged over
halos of similar mass. This should be equal to the shape factor
$F(\mu)$. For 3 different box sizes and at 3 different redshift,
this factor is not exactly a constant, although it varies slowly
with mass. This illustrates that our approach is only a first order
approximation of our simulation results. Nevertheless, we
approximate this by taking $F(\mu) \simeq 3$, as suggested by the
dashed line in Figure~\ref{beta}. This specifies how star formation
in the simulations and star formation in the model are connected to
each other.
\subsection{Halo orbital decay timescale}
\begin{figure*} \begin{tabular}{cc}
\includegraphics[width=0.5\hsize]{sfrdez.ps}
\includegraphics[width=0.5\hsize]{sfrnowind.ps} \\
\includegraphics[width=0.5\hsize]{sfrresolution.ps}
\includegraphics[width=0.5\hsize]{sfrdezSpringel.ps}
\end{tabular} \cpt{Global comoving star formation rate as a
function of redshift. In each plot, symbols are for numerical
simulations, while lines are for our corresponding analytical
prediction. \emph{Upper left plot: high efficiency series and
low efficiency series} with run L1N256S3 (black squares), run
L1N256S30 (grey squares), run L10N256S3 (black triangles), run
L10N256S30 (grey triangles), run L100N256S3 (black diamonds) and
run L100N256S30 (grey diamonds) . \emph{Lower left plot:
convergence study series} with run L10N512S30 (diamonds), run
L10N256S30 (triangles), run L10N128S30 (squares) and run L10N64S30
(crosses). \emph{Upper right plot: \cite{Springel03b}
simulations} with run 03 without winds (triangles) and Q3 including
winds (diamonds). \emph{Lower right plot: \cite{Springel03b}
simulations} including winds with run Q5 (stars), run Q4
(diamonds), run Q3 (triangles), run Q2 (squares) and run Q1
(crosses). The solid line is the `fully converged' model
prediction. } \label{sfrdez} \end{figure*}
The only unknown parameter in our analytical model is the orbital
decay timescale of infalling gas clumps (see appendix
\ref{coolingmodel}), before they reach the high-density disc in the
halo center. When cooling is very fast, for halos with Virial
temperature $T_{\rm min} < T_{\rm 200} < T_{\rm max}$, we have assumed that the
accretion rate into the disc is controlled by the orbital time scale
of infalling satellites. Computing this time scale is beyond the
scope of this paper. It is probably determined by details in the
gravitational dynamics and satellite dynamical friction. These
aspects are all key ingredients of semi-analytical models of galaxy
formation.
In order to determine this orbital decay timescale, we perform again
a direct analysis of our numerical simulations. Let us consider the
case of very fast star formation $t_{\rm 0} = 3$ Gyr and $\alpha_{\rm 0}=3$. In
this case, the halo star formation rate is almost equal (within
10\%) to the halo cooling rate. This can be later confirmed by the
analytical model. We plot in Figure~\ref{beta} the average star
formation rate of halos within different mass range, in units of
$M_{\rm hot} R_{\rm 200}/V_{\rm 200}$. In our framework, this quantity is
exactly equal to the ratio $(R_{\rm 200}/V_{\rm 200})/t_{\rm orb}$. Here again,
this ratio is not perfectly a constant, illustrating the fact that
our model is only a first order approximation, but for the 3
different box sizes and at 3 different redshifts, the curve exhibits
a plateau around $t_{\rm orb} \simeq 3 R_{\rm 200}/V_{\rm 200}$. We take this
value as our canonical value in the analytical model.
\subsection{Global Baryon Budget}
We now present in greater details our simulation results, starting
with the baryon history for the universe as a whole.
Figure~\ref{frac_evol} shows the baryon history in our highest
resolution run L10N512S30. The run parameters correspond to a low
efficiency star formation model. Each phase is defined by well
defined limits in the $\rho$ - $T$ diagram, as defined in
Section~\ref{results}. The various symbols in Figure~\ref{frac_evol}
refer to baryon fractions in different snapshots of the simulation,
while lines refer to the analytical model predictions, with
$M_{\rm resol} \simeq 2 \times 10^8$ h$^{-1}$M$_{\odot}$, as given by
Equation~\ref{massresol}. The other parameters of the model are set
to their standard values ($F(\mu)=3$ and $R_{\rm orb}=3R_{\rm 200}$). The
agreement between the simulation and the model is very good (within
a factor of 2), given the simplicity of the latter.
\begin{figure*} \begin{tabular}{cc}
\includegraphics[width=0.5\hsize]{sfrdem.ps}
\includegraphics[width=0.5\hsize]{sfrdemSpringel.ps} \\
\includegraphics[width=0.5\hsize]{sfrdemc001.ps}
\includegraphics[width=0.5\hsize]{sfrdemresol.ps}
\end{tabular} \cpt{Halo star formation rate, in units of
$M_{\rm 200}/t$, as a function of the halo Virial temperature. In
each plot, symbols are for numerical simulations, while lines are
for our corresponding analytical prediction. \emph{Upper left
plot: high efficiency series} with run L1N256S3 at $z=5.5$
(diamonds), run L10N256S3 at $z=2.5$ (triangles) and run
L100N256S3 at $z=0$. \emph{Lower left plot: low efficiency series}
with run L1N256S30 at $z=5.5$ (diamonds), run L10N256S30 at
$z=2.5$ (triangles) and run L100N256S30 at $z=0$. \emph{Lower
right plot: convergence study series} at $z=2.8$ with L10N64S30
(diamonds), run L10N128S30 (triangles), run L10N256S30 (squares)
and L10N512S30 (crosses). The solid line is the `fully converged'
model prediction. \emph{Upper right plot: \cite{Springel03b}
simulations} including winds, with run R4 (diamonds) at $z=6$, run
Q4 at $z=3$ (triangles) and run D4 at $z=1.5$ (squares).}
\label{sfrdem} \end{figure*}
We now examine more closely the global star formation rate as a
function of redshift, measured in all our simulations, and compare
our various results to the analytical model. This quantity is a key
prediction of hierarchical model of galaxy formation. It translates
more or less directly into galaxy colors and luminosities, and
provides a stringent test of the current cosmological theory. As
star particles are created during the course of a simulation, we
keep track of their birth epoch. It is straightforward to compute,
using the last output only, the star formation epoch histogram
(history).
Figure~\ref{sfrdez} shows this global star formation history for our
`convergence study' simulation suite: L10N64S30, L10N128S30,
L10N256S30 and L10N512S30. Numerical results are shown as symbols,
while analytical predictions are shown as lines. The analytical
model predictions are computed with a Minimal Mass corresponding to
the mass resolution of each run, as given by
Equation~\ref{massresol}. Since our star formation recipe is based
on a constant overdensity threshold, the mass resolution is
$M_{\rm resol} \simeq 400 m_{\rm p}$. The solid line stands for the analytical
prediction, without any finite resolution effects ($M_{\rm resol}=0$).
This gives an indication on how our results have converged to the
`true' star formation history in this particular model.
Around $z \simeq 3-5$, our highest resolution run L10N512S30 is
close to the correct value. At very high redshift however, the star
formation rate is lower than the expected value by a factor of
ten. The mass resolution $M_{\rm resol}$ is indeed significantly higher
than $M_{\rm min}$ at redshift $z > 10$: this explains the origin of
this discrepancy. As illustrated by the Figure~\ref{resolmap}, low
resolution runs do miss the formation of dwarf galaxies that
contribute significantly to the global star formation history.
This first series of simulations was performed for a box length
$L=10$ h$^{-1}$ Mpc. By $z \simeq 3$, the non-linear scale becomes
comparable to the box size. We have to stop the simulations, as our
realizations are not representative of the universe as a whole
anymore.
\begin{figure*} \begin{tabular}{cc}
\includegraphics[width=0.5\hsize]{fstar.ps}
\includegraphics[width=0.5\hsize]{fc.ps} \\
\includegraphics[width=0.5\hsize]{fh.ps}
\includegraphics[width=0.5\hsize]{fbar.ps} \end{tabular}
\cpt{Mass fraction in each baryon phase as a function of halo
Virial temperature for the \emph{high efficiency simulation
series}. In each plot, diamonds refer to run L1N256S3 at $z=5.5$,
triangles to run L10N256S3 at $z=2.5$ and squares to run
L100N256S3 at $z=0$. Lines are for the corresponding analytical
model. The \emph{upper left plot} shows the star mass fraction;
the \emph{lower left plot} shows the hot gas mass fraction; the
\emph{lower right plot} shows the total baryon fraction and the
\emph{upper right plot} shows the cold gas fraction.}
\label{fracdem} \end{figure*}
\begin{figure*} \begin{tabular}{cc}
\includegraphics[width=0.5\hsize]{fstarc001.ps}
\includegraphics[width=0.5\hsize]{fcc001.ps} \\
\includegraphics[width=0.5\hsize]{fhc001.ps}
\includegraphics[width=0.5\hsize]{fbarc001.ps} \end{tabular}
\cpt{Mass fraction in each baryon phase as a function of halo
Virial temperature for the \emph{low efficiency simulation
series}. In each plot, diamonds refer to run L1N256S30 at $z=5.5$,
triangles to run L10N256S30 at $z=2.5$ and squares to run
L100N256S30 at $z=0$. Lines are for the corresponding analytical
model. The \emph{upper left plot} shows the star mass fraction;
the \emph{lower left plot} shows the hot gas mass fraction; the
\emph{lower right plot} shows the total baryon fraction and the
\emph{upper right plot} shows the cold gas fraction.}
\label{fracdemc001} \end{figure*}
\begin{figure*} \begin{tabular}{cc}
\includegraphics[width=0.5\hsize]{fstarresol.ps}
\includegraphics[width=0.5\hsize]{fcresol.ps} \\
\includegraphics[width=0.5\hsize]{fhresol.ps}
\includegraphics[width=0.5\hsize]{fbarresol.ps} \end{tabular}
\cpt{Mass fraction in each baryon phase as a function of halo
Virial temperature for the \emph{convergence study simulation
series} at $z=2.8$. In each plot, diamonds refer to run
L10N64S30, triangles to to run L10N128S30, squares to run
L10N256S30 and crosses to run L10N512S30. Lines are for the
corresponding analytical model. The \emph{upper left plot} shows
the star mass fraction; the \emph{lower left plot} shows the hot
gas mass fraction; the \emph{lower right plot} shows the total
baryon fraction and the \emph{upper right plot} shows the cold gas
fraction. The solid lines are the `fully converged' model
predictions.} \label{fracdemresol} \end{figure*}
Our second series of simulations was designed to explore larger
($L=100$ h$^{-1}$ Mpc) and smaller ($L=1$ h$^{-1}$ Mpc) scales, as
well as another, more efficient, star formation scenario with $t_{\rm 0} =
3$ Gyr. The corresponding Schmidt law is therefore ten times more
effective than in the first case. The resolution was fixed to
$256^3$ particles. All six results are shown in Figure~\ref{sfrdez}
with symbols, while the corresponding analytical predictions are
overploted with lines. Note that the smaller box ($L=1$ h$^{-1}$
Mpc) has a mass resolution smaller than $M_{\rm min}$. It has therefore
converged to the `true' star formation history. The analytical
model is indeed in good agreement with our numerical results. This
very small scale simulations have to be stopped even earlier than
the previous ones ($z \simeq 5$). The largest box size ($L=100$
h$^{-1}$ Mpc), on the other hand, is strongly affected by finite
mass resolution effect. The star formation history is drastically
different than the other two. Star formation begins very late,
around $z \simeq 5$, and the peak value is one order of magnitude
lower than the `true' star formation rate. Our analytical model
provides again a good fit to numerical data, when the poor mass
resolution is taken into account to define the diffuse background.
Let us now compare the two different star formation scenario
(inefficient with $t_{\rm 0}=30$ Gyr and efficient with $t_{\rm 0}=3$ Gyr). Both
star formation history are parallel to each other, but with a factor
of two difference only. For the very efficient scenario, star
formation is limited by accretion and cooling, rather than by the
Schmidt law. As discussed in the next section, very efficient star
formation models give SFR quasi independent of the physical
parameters. This is confirmed by our numerical simulations. The
agreement between the simulation and the model is extremely good,
except at low redshift ($z <3$), for our large box runs, where the
model significantly overestimates the star formation rate. This
disagreement might be reduced by improving the analytical model,
along several lines we have outlined in this paper. Note that
finite volume effect might also have an additional effect on the
numerical simulation predictions, but we do not try to include those
subtleties in the analytical model.
From now on, we have analyzed AMR simulations for which star
formation is triggered in region where the gas density exceeds an
{\it overdensity threshold}. In this case, we naturally compare our
numerical results to the `accelerated efficiency' analytical
model. We now test the model predictions for the `constant
efficiency' analytical model, using SPH results kindly provided to
us by \cite{Springel03b}. Our interest to these SPH results is
twofold: first, star formation is triggered in regions where the gas
density exceeds a {\it physical density threshold}. This scenario,
as already discussed in Section~\ref{halotstar}, corresponds to a
`constant efficiency' star formation model. Second,
\cite{Springel03b} have included galactic winds in their numerical
modeling, giving us the opportunity to test our simple feedback
model.
In Figure~\ref{sfrdez} are shown \cite{Springel03b} results when
galactic winds are turned off. The only difference with our
simulations, apart from the overall numerical techniques,
comes from the star formation details. They have considered the
following parameters $t_{\rm 0} = 2.1$ Gyr, $n_{\rm 0}=0.1$ cm$^{-3}$ and
$\alpha_{\rm 0} = 0$. For the analytical model, we use the same `shape
factor' $F(\mu)=3$, as determined earlier using our AMR results.
This translates into the following parameters for the analytical
model: $t_* = 0.7$ Gyr and $\alpha_* = 0$. Taking into account the
finite mass resolution (using Equation~\ref{sfrdez}), we can now
compare our analytical prediction to SPH results. The agreement is
clearly very good: this is rather encouraging for our model, since
our main unknown parameter, the mean orbital length, was kept fixed
to its canonical value $R_{\rm orb}=3R_{\rm 200}$, calibrated on our AMR
results.
We now analyze SPH results when galactic winds are turned on. Recall
that winds are assumed to eject cold gas from the rotating discs at
a typical wind velocity $u_{\rm w} \simeq 250-500$ km/s and with an
efficiency parameterized by $\eta_{\rm w}$. This approach, directly
inspired by \cite{Springel03b}, allows a straightforward comparison
between our analytical model and SPH results with winds. This
comparison is shown in Figure~\ref{sfrdez}. The analytical model
parameters were set to $T_{\rm w} = 2\times 10^6$ K and $\eta_{\rm w} = 3$. The
Virial temperature $T_{\rm w}$ (below which winds are unbound from their
parent halo) corresponds closely to the wind velocity $u_{\rm w} \simeq
500$ km/s used by \cite{Springel03b} in their SPH simulation (see
Eq.~\ref{twind}). The wind efficiency parameter, however, was chosen
50\% higher than the value $\eta_{\rm w} = 2$ used in the SPH simulation.
As suggested by \cite{Springel03b}, the global wind efficiency (at
the halo scale) is higher than the local wind efficiency (at the star
forming regions scale) due to gas entrainement: additional cold gas,
lying outside star forming regions from which winds originate, can
be expelled by the ram pressure of the outflow. Figure~\ref{sfrdez}
shows again a good agreement between SPH simulation results and our
analytical model.
We consider now SPH results from the `Q series' in
\cite{Springel03b} paper, in order to study finite mass resolution
effects. Wind parameters are fixed to $T_{\rm w}=2\times 10^6$ K and
$\eta_{\rm w} = 3$. Note that in this case, due to a star formation
strategy based on a constant physical density, $M_{\rm resol}$ is now
varying with time $M_{\rm resol} \simeq 1000(1+z)^{-1.5}m_{\rm p}$. In
Figure~\ref{sfrdez} are plotted \cite{Springel03b} simulation
results together with our model predictions, when mass resolution
effects are taken into account. At high redshift, we recover the
correct convergence towards the asymptotic ($M_{\rm resol}=0$) converged
curve. At intermediate redshift, however, the agreement is getting
worse, although both curves remain close to each other within a
factor of 2. One possible reason is that Equation~\ref{massresol}
is not accurate enough to estimate the effective halo mass
resolution of SPH simulations, especially in presence of winds.
We conclude that the global baryon history we obtain in numerical
simulations is correctly described by our analytical model, {\it
when finite mass resolution effects are taken into account}. This
was tested for 2 different numerical technics (AMR and SPH), two
different star formation scenarios (`constant' and `accelerated'
efficiency), with and without galactic winds. The main effect of
insufficient mass resolution is to artificially decrease the average
age of stars in the universe and to lower the star formation rate at
peak value.
\subsection{Halo Baryon Budget}
We present now our simulation results concerning the baryon budget
in individual halos. Halos are detected in the simulations
according to the method explained in Section~\ref{halodetect}. The
Extended Press \& Schechter (EPS) theory give us the opportunity to
apply our analytical method to an `average' halo of mass $M_{\rm 0}$ at
redshift $z_{\rm 0}$. The model predictions can then be compared to the
{\it average} baryon fractions (in each of the 4 different baryon
phases), the average being taken over all halos in a given mass
range, centered around $M_{\rm 0}$.
Using all star particles found within $R_{\rm 200}$, we compute the star
formation history within each halo. We then compute the star
formation rate as explained in Section~\ref{halodetect}.
Figure~\ref{sfrdem} shows our various numerical results, including
the SPH simulations with winds provided to us by \cite{Springel03b},
together with the analytical model predictions. We have plotted the
halo star formation rate as a function of the halo temperature
$T_{\rm 200}$, in units of $M_{\rm 200}/t$. This definition corresponds
closely to the `specific star formation rate' defined by
\cite{Springel03b}.
Each curve shows a sharp cut-off at the low mass end, corresponding
to the Minimal Mass for each run. The `convergence study' series
clearly illustrates that this Minimal Mass is in fact equal to the
mass resolution $M_{\rm resol}$, except for our smallest box
size. Another non trivial effect of finite mass resolution is to
increase the halo star formation rate. This is due to a higher
Cosmic Accretion Rate ($\dot{f}_{\rm acc} \propto \sigma(M_{\rm min})^{-1}$,
see Eq.~\ref{asymptoticaccrate}), as $M_{\rm min}$ is increased. Our
analytical model predictions are in good agreement with the halo
star formation rates, even when winds are present, as soon as finite
mass resolution is explicitly accounted for in the model. The role
of winds is to remove cold gas from small mass halos $T_{\rm 200} < 2
\times 10^6$ K, so that the halo star formation rate decrease
accordingly.
When we compare the high star formation efficiency series with $t_{\rm 0} =
3$ Gyr with the low star formation efficiency series with $t_{\rm 0} = 30$
Gyr, we see that the former has a higher halo star formation rate at
high redshift, but a much lower halo star formation rate at low
redshift, as cold gas is almost completely consumed. This complex
behavior is well reproduced by the analytical model. Both series
show a sharp decline of the halo star formation rate at the high
mass end: the cooling efficiency decreases for high mass halos and the
mass accretion rate on cold discs vanishes. Note that we assumed
here a zero metallicity plasma. Gas cooling is likely to be more
efficient around $T_{\rm 200} \simeq 10^7$ K if metals are present.
We now present the baryon budget inside dark matter halos, as a
function of the halo Virial temperature. In each halo, we compute
the total baryon fraction, which can be further decomposed into hot
gas, cold gas and stars, using the definitions of
Section~\ref{results} and \ref{halodetect}. Figure~\ref{fracdem}
shows this halo baryon budget for the `high efficiency' simulation
series, Figure~\ref{fracdemc001} for the `low efficiency'
simulation series and Figure~\ref{fracdemresol} for the
`convergence study' simulation series.
In each case, there is good agreement between numerical and
analytical results. The total baryon fraction is close to the
average value $f_{\rm b}$ for halos more massive than $M_{\rm min}$ and
vanishes for smaller halos. Note that the total baryon fraction is
actually slightly {\it above} the universal value for massive halos.
Dissipative collapse of baryons condense more mass than
collisionless collapse of dark matter. The analytical model predicts
a sharp transition for $M < M_{\rm resol}$ where the total baryon
fraction vanishes. In the simulations, this transition is much
smoother, but its location is correctly predicted around
$M_{\rm resol}$. On the other hand, when the mass resolution is small
enough, the low mass end of the analytical curve has also a smooth
transition. This comes from halos whose mass was greater than
$M_{\rm min}$ at early time, but as reionization heats up the background
temperature, these same halos end up with a mass smaller than
$M_{\rm min}$. The total baryon fraction at the final redshift is
therefore smaller than the universal value but greater than zero.
The hot gas fraction is also strongly affected by resolution effect.
As we resolve smaller and smaller mass progenitors, more and more
gas is condensed into cold discs and eventually
stars. Figure~\ref{fracdemresol} shows the hot gas fraction for the
`convergence study' series and one clearly sees that our poorest
resolution run overestimates the hot gas fraction by a factor of 3.
As a consequence, the cold gas and star mass fraction gradually
increase as the mass resolution is improved. The curve showing the
cold gas fraction as a function of Virial temperature is very
similar to the halo star formation rate, as expected from our
analytical model, although they have been computed completely
differently in the simulation analysis. In the `high efficiency'
series, our largest box run L100N256S3 has almost entirely consumed
the cold gas of the most massive halos, where cooling has virtually
stopped (see Fig.~\ref{fracdem}). The corresponding run with a
`low efficiency' star formation scenario still shows some surviving
cold gas at the high mass end of the halo distribution.
We conclude at the end of this section that our analytical model has
proven to successfully reproduce the complex behavior observed in
our simulations and in simulations of \cite{Springel03b} (with
winds) {\it if finite resolution effects are taken into account}. We
note however a slight tendency of the model to overestimate the star
formation rate at low redshift. The baryon budget was analyzed in
great details on a halo by halo basis (and in an average sense). We
emphasized the important role played by the Minimal Mass $M_{\rm
min}$ in controlling the baryon history in each mass range. We also
described how cooling and winds introduce characteristic features in
the various plots showing the baryon evolution as a function of the
halo Virial temperature.
\section{Observations versus Model}
\begin{figure} \includegraphics[width=\hsize]{sfrdez_zr.eps}
\cpt{Cosmic Star Formation Rate for different model
parameters: $t_*=0$ and $\eta_{\rm w}=0$, $z_{\rm r}=20$ (solid
line), $z_{\rm r}=12$ (dotted line), $z_{\rm r}=6$ (dashed
line).} \label{sfrdez_zr}
\end{figure}
\begin{figure*} \begin{tabular}{ccc}
\includegraphics[width=0.5\hsize]{sfrdez_tstar.eps}
\includegraphics[width=0.5\hsize]{sfrdez_etawind.eps}
\end{tabular} \cpt{Cosmic Star Formation Rate for different
model parameters. Left plot: $z_{\rm r}=20$ and $\eta_{\rm w}=0$,
$t_*=0$ (solid line), $t_*=0.1$ Gyr (dotted line), $t_*=1$ Gyr
(dashed line), $t_*=3$ Gyr (dot-dashed line), $t_*=10$ Gyr
(dot-dot-dashed line). Right plot: $z_{\rm r}=20$ and $t_*=3$ Gyr,
$\eta_{\rm w}=0$ (solid line), $\eta_{\rm w}=0.1$ (dotted line),
$\eta_{\rm w}=1$ (dashed line), $\eta_{\rm w}=3$ (dot-dashed
line), $\eta_{\rm w}=10$ (dot-dot-dashed line).}
\label{sfrdezparams} \end{figure*}
From now on, we assume that our analytical model gives an accurate
(within a factor of 2) modeling of the baryon history in a
hierarchical universe. The careful comparison to numerical
simulations we performed in the previous section was a necessary
step to calibrate our model and to estimate its level of
accuracy. We are now in a position to use this model as a tool to
analyze current observational constraints put on the baryon history
in the universe.
\subsection{Model parameters study}
We briefly recall how the analytical predictions depend on the model
parameters. We consider that all cosmological parameters are fixed
to their `concordance' $\Lambda$CDM value, as we did throughout
this paper. We also assume that the reionization temperature is
given by $T_{\rm r} \simeq 6 \times 10^3$ K. We finally use the cooling
model of Section~\ref{coolingmodel}, valid for a primordial, zero
metallicity, H and He plasma and a wind velocity $u_{\rm w} \simeq 500$
km/s. We end up with 3 main parameters that we allow to vary in our
model: the reionization redshift $z_{\rm r}$, the star formation time
scale (or consumption time scale) $t_*=M_{\rm cold}/\dot{M}_*$ and the
wind efficiency $\eta_{\rm w}=\dot{M}_{\rm wind}/\dot{M}_*$. We consider in this section only
`constant efficiency' star formation models $\alpha_*=0$. The
alternate scenario with `accelerated efficiency' star formation
and $\alpha_*=3$ gives similar, though slightly poorer, results.
Using current observational constraints, it is not easy to
discriminate between these two models.
We now present the model predictions for the
Cosmic Star Formation Rate, for different values of $z_{\rm r}$, $t_*$ and
$\eta_{\rm w}$. Figure~\ref{sfrdez_zr} corresponds to an `infinite
efficiency' star formation model ($t_*=0$ Gyr), for various
reionization redshifts. In this extreme case, the cosmic star
formation rate (CSFR) is equal to the mass accretion rate of cold
discs, since star formation is instantaneous. At reionization, the
background gas is promptly heated to $10^4$ K and the star formation
rate drops. We notice that at redshifts lower than 6, all models
agree with one another, leading us to the conclusion that, as soon
as reionization proceeds early enough, this parameter is in fact
irrelevant to the low redshift universe.
Figure~\ref{sfrdezparams} shows the influence of the two
remaining (and really interesting) parameters. Star formation inside
cold, centrifugally supported, galactic discs acts merely as a delay
with respect to diffuse gas accretion. The epoch of peak star
formation is delayed from $z \simeq 4$ for $t_*=0$ down to $z \simeq
1$ for $t_*=10$ Gyr. It is worth noticing that the amplitude of the
global star formation rate is mainly determined by the cosmological
accretion rate, and that the small scale, poorly known, physics of
star formation introduces a rather small modification to the curve.
When winds are included in the model, they lower significantly the
amplitude of the CSFR. They also have the non trivial effect of
{\it advancing} the epoch of peak star formation, from $z \simeq 1$
for $\eta_{\rm w} = 0$ up to $z \simeq 3$ for $\eta_{\rm w} = 10$. They play an
important role at low redshift whereas at high redshift the CSFR
seems independent of $\eta_{\rm w}$. Indeed, for the `constant
efficiency' recipe the star formation rate at high redshift is so
small compared to the accretion rate that winds (which are
proportional to the star formation rate) doesn't affect the amount
of cold gas.
Computing the baryon history using our analytical model is a very
fast operation. This gives us the opportunity to compute various
observational quantities for a grid of model parameters. Using
various observations, we will now try to constrain our parameter
space and shed light to these 2 important galaxy formation
ingredients: $t_*$ and $\eta_{\rm w}$.
\subsection{Cosmic Star Formation Rate}
One of the main goal of this paper is to compute the star formation
history in a hierarchical universe. We will now compare the model
predictions to the observed star formation rate. Figure~\ref{sfrobs}
shows the observational data points, usually referred to as the
`Madau plot', compiled by \cite{Elbaz05} and uniformally corrected
from cosmological distances, incompletness and dust
absorption. Original data points came from \cite{Hughes98,
Steidel99, Flores99, Glazebrook99, Yan99, Massarotti01,
Giavalisco03}. These data points indicates an epoch of peak star
formation rate at $z \simeq 2$, followed by a rapid fall off (by a
factor around ten) between $z=1$ and $z=0$.
\begin{figure*} \begin{tabular}{cc}
\includegraphics[width=0.46\hsize,height=0.37\hsize]{sfrobs.ps} &
\includegraphics[width=0.46\hsize,height=0.37\hsize]{sfr2d.ps}\\
\includegraphics[width=0.46\hsize,height=0.37\hsize]{omega.ps} &
\includegraphics[width=0.46\hsize,height=0.37\hsize]{omega2d.ps}\\
\includegraphics[width=0.46\hsize,height=0.37\hsize]{fcolddez.ps} &
\includegraphics[width=0.46\hsize,height=0.37\hsize]{omegacold2d.ps}\\
\end{tabular} \cpt{\emph{Upper left plot}: Cosmic Star Formation Rate
as a function of redshift. Data points, uniformally corrected
from various observational biases, come from \cite{Hughes98,
Steidel99, Flores99, Glazebrook99, Yan99, Massarotti01,
Giavalisco03}. The solid line is our canonical model ($t_*=3$
Gyr, $\eta_{\rm w}=1.5$). The dashed line is our best fit model for the
SFR ($t_*=1.5$ Gyr, $\eta_{\rm w}=1$). \emph{Upper right plot}: contour of
constant star formation rate (solid contours: $z=0$, dashed
contours: $z=3$) as a function of $t_*$ and $\eta_{\rm w}$. Bold
contours are the observed value, while others correspond to a
factor of 2 above and below the observed central value. The cross
is the position of our canonical model. \emph{Middle left plot}: Cosmic
Stellar Density as a function of redshift, from
\cite{Dickinson03}. The solid line is the prediction of our
canonical model, while the dashed line is our best fit model
($t_*=10$ Gyr, $\eta_{\rm w}=0.5$). \emph{Middle right plot}: contour of
constant stellar density (solid contours: $z=0$, dashed contours:
$z=3$). \emph{Lower left plot}: cosmic gas density in damped Lyman alpha
systems, from \cite{Somerville01}. The solid and dashed lines
corresponds to the mass fraction in cold gas, as predicted by our
canonical model. \emph{Lower right plot}: contour of constant cold gas
density (solid contours: $z=0$, dashed contours: $z=3$).}
\label{sfrobs} \end{figure*}
The shape and the normalization of the Madau plot are well
reproduced by the parameter choice $t_* \simeq 1.5$ Gyr and $\eta_{\rm w}
\simeq 1$. It is worth noticing that this star formation time scale
corresponds roughly to the value one can infer from the consumption
timescale in local galaxies \citep{ Kennicutt94, Kennicutt98}. The
wind efficiency we obtain from this fitting exercise is close to the
value inferred from $H_\alpha$ observations of high-SFR galaxies
\citep{Martin99}. Galactic winds are required in order to reproduce
the rapid fall off of the star formation rate at low redshift.
The observational constraints on our two main star formation
parameters are summarized in the upper right plot of
Figure~\ref{sfrobs}, which represents contours of constant star
formation rates at $z=0$ and $z=3$ in the $\eta_{\rm w}$ - $t_*$ plane.
The observed central value is shown as the bold lines. The two other
contours correspond to a SFR value a factor of 2 higher and lower
than the central value. As we see, the main constraint given by the
observed SFR is that $t_*$ must be lower than 5 Gyr. If not, the SFR
will be underestimate at high redshift by a factor more than 2. The
contours also indicate that winds are required to reproduce the low
SFR observed at low redshift. More qualitatively, winds are also
required to prevent from `the overcooling problem' and to remove
baryons from the cold gas phase.
\begin{figure*}
\begin{tabular}{cc}
\includegraphics[width=0.48\hsize]{heavens.ps}&
\includegraphics[width=0.48\hsize]{sfrdezdeM0obs.ps}\\
\includegraphics[width=0.48\hsize]{fhobs.ps}&
\includegraphics[width=0.48\hsize]{fhobsmegawind.ps}\\
\end{tabular}
\cpt{\emph{Top}: average star formation history as a function of
lookback time for different final stellar masses. The observed
values from \cite{Heavens04} are plotted on the top left panel.
As in the article, the curves are offset vertically successively
by 0.5 in log except for the most massive galaxies which are
offset by an additional 1.0. The top right panel shows the
average star formation history of our canonical model ($t_*=3\
Gyr$ and $\eta_{\rm w}=1.5$). For the most massive galaxies, a
second model with metal-rich cooling and `superwinds' is also
plotted, in better agreement with the data. \emph{Bottom}: mass
fraction in the hot X-ray emitting gas in several observed
groups and clusters \citep{Sanderson03a}. The lower left solid
line is the prediction of our canonical model, while the lower
right solid line is the prediction of our `superwind' model.}
\label{sfrdezdeM0obs}
\end{figure*}
\subsection{Cosmic Stellar Density}
A complementary method to investigate the history of the baryon
assembly in the universe is to observe the evolution of the global
stellar mass density $\Omega_*(z)$. This is an independent way of
constraining the model, since the SFR is related to young stars,
while cosmic stellar observations focus on old, red and low mass
stars. The middle left plot of Figure~\ref{sfrobs} shows the
observational data points of $\Omega_*$ compiled by
\cite{Dickinson03} from various near-infrared and optical
observations \citep{Cole01,Brinchmann00,Cohen02,Dickinson03}.
In order to perform an accurate comparison, we need to take into
account the death of short-lived stars during the history of the
universe. Since most of the stars are formed when the universe is a
few Gyr young, this effect is likely to be important. We use here
the stellar Initial Mass Function (IMF) proposed by \cite{Kroupa93}
in order to estimate the amount of stars still alive at a given
redshift. The corresponding predicted stellar density is plotted in
the middle left plot of Figure~\ref{sfrobs}. Our best fit model for
this observations is now $t_*=10$ Gyr and $\eta_{\rm w}=0.5$, in complete
disagreement with our best-fit model for the Cosmic SFR. This rather
surprising result is due entirely to the high redshift observations
of $\Omega_*$. Low redshift observations, on the other hand, are
compatible with the SFR constraints.
This puzzle was identified as a possible `missing galaxy problem'
\citep{Nagamine04}. This indicates that, if the IMF remains
universal, the star formation rate should fall off at high redshift
more strongly than in Figure~{\ref{sfrobs}}. In the model, this can
be obtained for $t_*=10$ Gyr, but then, as can be seen in
Figure~\ref{sfrdezparams}, the SFR is too high at low redshift. The
possible inconsistency between the observed SFR and the observed
stellar density is discussed in great details in \cite{Dickinson03}.
The mass-luminosity relation might be poorly estimated at high
redshift because the galaxy luminosity function is not well
constrained. Another solution is to invoke an evolving IMF, with
more high mass stars at high redshift. Finally, the last solution
is to invoke a new physical process, not included in the model,
which inhibits star formation at high redshift.
\subsection{Extragalactic Background Light}
Another strong observational constraint is the Extragalactic
Background Light (EBL), integrated from the UV to the IR. The EBL is
an estimate of the total amount of energy emitted by stars and AGN
in the history of the universe. Consequently, the cumulative
luminosity of all stars created by the model cannot exceed the value
of this background. \cite{Madau00} compute the value of the
observed EBL, integrating from 0.2 to 2000 $\mu m$, and found
$I_{\rm EBL} = 55$ nW/m$^2$/sr. Following the method presented in
\cite{Madau00}, we compute the EBL corresponding to our canonical
model \begin{equation} I_{\rm EBL}=\frac{c}{4\pi} \int_{\rm
0}^{+\infty} \frac{\rho_{\rm bol}(t)}{1+z}
\left|\frac{dt}{dz}\right| {\rm d}z, \end{equation} with the
bolometric emissivity at epoch t \begin{equation} \rho_{\rm
bol}(t)=\int_{\rm 0}^t L(\tau) \dot{\rho}_*(t-\tau) {\rm
d}\tau. \end{equation} In this expression, $L(\tau)$ is the
bolometric luminosity of a single stellar cluster of unit mass, as a
function of the cluster age $\tau$. We use for $L(\tau)$ the
analytical approximation given by \cite{Madau00} for a solar
metallicity stellar population. We find for our canonical model
$I_{\rm EBL} \simeq 100$ nW/m$^2$/sr, larger than the expected
value, but within observational uncertainties (for example optical
data from \cite{Bernstein99} and IR data from \cite{Chary01} and
\cite{Fixsen98} lead to $I_{\rm EBL} \simeq 80$ nW/m$^2$/sr). Our
best fit model for the Cosmic SFR ($t_* \simeq 1.5$ Gyr and
$\eta_{\rm w} \simeq 1$) gives $I_{\rm EBL} \simeq 150$ nW/m$^2$/sr,
and can be therefore considered as being ruled out by the
observational constraint.
\subsection{Cosmic Baryon Budget}
The observed Baryon Budget is discussed in great details in
\cite{Fukugita98}. They infer from various observations at $z=0$
baryon mass fractions of $\Omega_{\rm back} \simeq 0.002$, $\Omega_* =
0.0035$, $\Omega_{\rm cold} = 0.00063$ and $\Omega_{\rm hot} = 0.017$. We
note immediately that, if one considers the most recent WMAP
estimate \citep{Spergel03} of $\Omega_{\rm b} \simeq 0.04$, roughly 50\%
of the baryons are missing. The most striking disagreement between
these estimates and our model is the very low baryon fraction in the
background (or Lyman alpha forest) at $z=0$. As it is now admitted,
Lyman alpha observations at low redshift are very uncertain, so we
consider here that most of the missing baryons are in fact in the
diffuse background. Recent observations from \cite{Penton04} seems
to confirm that at least 30\% of the baryons lie in the Lyman alpha
forest.
Another consequence of the analysis performed by \cite{Fukugita98}
is that baryons in the condensed phase (cold gas + stars) are only
10\% of the total amount of baryons in the universe ($\Omega_{\rm b}
\simeq 0.04$). This strongly supports the presence of galactic
winds, in order to overcome the `overcooling problem'. Additional
support is given by the fact that a large fraction ($\simeq$ 40\%)
of baryons are today in the hot gas phase that is to say plasma in
groups and clusters. Our canonical model (with winds) predicts
fractions (relative to $f_{\rm b}$) of $f_{\rm back} \simeq$ 50\%,
$f_{\rm hot} \simeq$ 30\%, $f_{\rm cold} \simeq$ 1\% and $f_*
\simeq$ 20\% (10\% with the IMF of \cite{Kroupa93}), in rough
agreement with present day observations. Without winds, the same
model is much more difficult to reconcile with observations, since
we obtain in this case $f_{\rm back} \simeq$ 40\%, $f_{\rm hot}
\simeq$ 20\%, $f_{\rm cold} \simeq$ 4\% and $f_* \simeq$ 40\%.
Another strong observational constraint come from the evolution of
the cold gas density, deduced from the observations of Damped Lyman
Alpha Systems (DLAS) in distant quasars spectra. Following
\cite{Somerville01}, we use observations performed by
\cite{StorrieLombardi96} and \cite{Zwaan97} to estimate the
evolution of $\Omega_{\rm cold}$ as a function of redshift. These
observations of DLAS are lower limit estimates of $\Omega_{\rm
cold}$ (see \cite{Somerville01} for a discussion). Dust absorption
is likely to have a strong effect in biasing these results. We use
the method proposed by \cite{Pei99} to correct from these effects.
Observational data points are shown in the lower left plot of
Figure~\ref{sfrobs}. The curve reaches its peak value $\Omega_{\rm
cold} \simeq 0.004$ at a redshift $z \simeq 2$, in good agreement
with our canonical model, which, in this case, is also the best fit
model. It is worth noticing that the observed $\Omega_{\rm cold}$
curve is proportional to the observed SFR curve. This can be
considered as a nice consistency check in the observational data and
provides support to our simple star formation model.
The lower right plot of Figure~\ref{sfrobs} shows contours of
constant $\Omega_{\rm cold}$ at $z=0$ and $z=3$ in our model
parameter space. Contours appear as straight lines in the
$t_*$-$\eta_{\rm w}$ plane. This is consistent with the fact that
the cold gas fraction depends mainly on the gas depletion time scale
which can be estimated to be $t_* / (\eta_{\rm w}+1)$ (see
Section~\ref{chain}). The low observed value of $\Omega_{\rm cold}$
favors a small depletion time scale with $1 < t_* / (\eta_{\rm w}+1)
< 3$ Gyr, a rather tight constraint.
\subsection{Halo Star Formation History}
We now analyze recent observations of individual galaxies star
formation history. Using Extended Press \& Schechter theory, we can
apply our model to predict the baryon history within individual
halos, in an average sense. Using the Sloan Digital Sky Survey
(SDSS), \cite{Heavens04} infer from $10^5$ nearby galaxy optical
spectra the age distribution of their stellar population. For each
galaxy, they deduce its individual star formation history (using a
Salpeter initial mass function). Finally, they compute the average
SFR of all galaxies in a given {\it stellar} mass range. This
quantity can be directly compared to the prediction of the model, if
one converts the halo Virial mass into a halo star mass, using our
model `star-to-mass ratio'.
Interestingly, the observed average star formation histories (upper
left plot of Figure~\ref{sfrdezdeM0obs}) seem to indicate that large
galaxies ($M_*>10^{12} M_{\odot}$) form stars earlier than small
ones ($M_*<10^{10}\ M_{\odot}$). This behavior was called
`anti-hierarchical' by \cite{Heavens04} and was identified as a
potential problem for the hierarchical scenario of structure
formation. The upper right plot of Figure~\ref{sfrdezdeM0obs} shows
our canonical model predictions ($t_* = 3$ Gyr and $\eta_{\rm w} =
1.5$). The exact quantity we plot is \begin{eqnarray} \dot{\rho}_* =
\dot{f}_*(M_{\rm 0},z_{\rm 0})M_{\rm 0} n(M_{\rm 0}) \Delta M,
\end{eqnarray} where $n(M_{\rm 0})$ is the Press \& Schechter halo
mass function. We see that the same `anti-hierarchical' behavior
(downsizing effect) is reproduced by our model, within the
hierarchical collapse framework . There are two explanations. First,
low mass halos have a total mass $M_{\rm 0}$ very close to the
Minimal Mass $M_{\rm min}$, so that the mass fraction in `star
forming progenitors' is smaller than for high mass galaxies. This
can be seen in Figure~\ref{figwacc}, where the diffuse gas mass
accretion rate vanishes as $M_{\rm 0} \rightarrow M_{\rm min}$.
Second, for large mass halos, the cooling efficiency drops at low
redshifts, as more and more progenitors reach $T_{\rm max}$, the
maximum cooling temperature. At this point, no more fresh gas is
accreted onto the cold disc and star formation is slowing down.
The observed individual star formation histories are therefore
successfully reproduced by the model {\it on a qualitative level.}
If one looks carefully into Figure~\ref{sfrdezdeM0obs}, one sees
that for large galaxies, the predicted star formation history
disagree with the observed one. The sudden drop in the star
formation rate (interpreted as the end of gas cooling into cold
discs) happens earlier in the model (look-back time between 7 and 8
Gyr) than in the data (look-back time around 2 Gyr). Moreover, after
gas cooling ends, the decrease in the star formation rate is much
faster in the data than in the model.
The first feature can be accounted for if one takes into account
metal enrichment into the cooling gas. For a metallicity of one
tenth or one third solar, cooling can be significantly higher than
for a primordial H and He plasma. We have tried a new model with an
increased Maximum Cooling temperature $T_{\rm max}=2 \times 10^6$
K. In this case, the position of the knee in the star formation rate
curve agrees perfectly with observational data. On the other hand,
the strong decrease after the knee is difficult to explain. Note
however that the measure of the SFH from the optical spectrum is a
very complex operation and the uncertainties are important. For
example, a large part of the star formation is enshrouded and may be
missed. Moreover, the IMF is probably not a Salpeter one.
Consequently, the decrease might be less accentuated than
presented. Nevertheless, if the decrease is confirmed, one solution
is to invoke very strong winds that remove most of the cold gas
accumulated before the end of disc accretion. To illustrate this,
we further modify the canonical model, introducing what we call here
`superwinds', with parameters $T_{\rm w} \simeq 10^7$ K and
$\eta_{\rm w} = 15$. This last model (see Fig.~\ref{sfrdezdeM0obs})
is finally able to reproduce the observed star formation history in
large galaxies.
This `superwinds' scenario is completely ruled out by the observed
global baryon history (see previous section). We have therefore to
consider 2 coexisting wind models of very different nature:
`galactic winds', driven by supernovae bubbles, for normal and dwarf
galaxies, and `superwinds', possibly driven by a massive central
black hole or AGN for massive galaxies. Building a self-consistent
model along those lines is beyond the scope of this paper, but
several attempts of `superwinds' models can already be found in the
literature \citep{Springel05}.
\subsection{Hot Gas Fraction in X-ray Clusters}
The observed amount of hot gas in groups and clusters ($\Omega_{\rm
hot} = 0.017$ in \cite{Fukugita98}) put a strong constraint on the
hierarchical scenario, mostly by requiring galactic winds to solve
the `overcooling problem'. \cite{Sanderson03a} have analyzed
various X-ray observations of groups and clusters and computed the
fraction of hot gas ($f_{\rm hot}$) as a function of the observed
X-ray, emission weighted, temperature. These data points are
plotted with their corresponding error bars in
Figure~\ref{sfrdezdeM0obs}.
First, we learn from these observations that most baryons in
clusters have to be in the hot phase, unless we are ready to face a
serious crisis with the WMAP constraint $\Omega_{\rm b} = 0.04$.
Second, there is a clear correlation between the fraction of hot gas
in each halo and the X-ray temperature. Moreover, this correlation
$f_{\rm hot} \propto T_{\rm X}$ is often used as a possible
explanation for the observed $L_{\rm X} - T_{\rm X}$ relation in
clusters \citep{Neumann01}.
We have plotted on the same figure our canonical model prediction
for the hot gas fraction as a function of the Virial temperature. In
small galaxies, where both galactic winds and gas cooling are
important, the hot gas fraction have its minimum around $f_{\rm hot}
\simeq 3\%$. For large halos, cooling is less and less efficient,
and in the same time, winds can not escape the halo potential
well. This double mechanism (cooling + winds) has the important
consequence of refilling with hot gas the parent halo. This
qualitative picture is interesting, but when one compares our
canonical model predictions with the X-ray data, the result is quite
disappointing. The refilling mechanism we have just explained
occurs at too low temperature ($T \simeq 0.1$ keV), while data
suggests a significantly higher transition temperature. Note however
that the observations concern the center of the clusters and the
extrapolation to larger radii (using $\beta$ model) is not safe
\citep{Neuman05}.
One solution might come again from our `superwinds' scenario. We
modify our canonical model by first increasing the Maximum Cooling
temperature up to $T_{\rm max}=2 \times 10^6$ K, as it should be for
the case of a realistic, metal-rich, plasma. We then modify our
wind parameters to $T_{\rm w} \simeq 10^7$ K and $\eta_{\rm w} =
15$, as the `superwinds' model we use in the last section. We plot
the hot gas fraction we obtain for this new model in
Figure~\ref{sfrdezdeM0obs}. The transition from `gas poor' to `gas
rich' regime occurs now at a much more realistic temperature ($T
\simeq 1$ keV). This transition is however still much sharper in
our model than it is in the data. As suggested in the last section,
we could therefore improve our wind model by explicitly introduce 2
different feedback scenarios: `supernovae driven' and `AGN driven'.
\cite{Kay04} have performed hydrodynamical simulations of galaxy
clusters with a kind of feedback which is very close to our
superwind scenario. Indeed, they heat the dense and cold gas of
their clusters at a temperature of $17$ keV. This corresponds in
our notations to $T_{\rm w} \simeq 2\times 10^7$ K (compared to
$T_{\rm w} \simeq 10^7$ K in our superwind model). Using this
strong feedback, they reproduce the observed cluster $L_{\rm
X}-T_{\rm X}$ relation with the correct level of entropy in the ICM
core. Such a strong feedback seems therefore essential to reproduce
both global fraction of hot gas and entropy profile.
\subsection{Stellar and HI mass functions}
\begin{figure}
\centering
\includegraphics[width=\hsize]{ndembar.ps}
\cpt{Predicted and observed stellar and HI mass function. The
thick dashed line is the stellar mass function predicted by
our standard model, whereas the thin dashed line is for our
superwind model. Shown as diamonds is the Schechter fit to the
observed stellar mass function \citep{Cole01}. Similarly, the
thick dot-dashed line is the cold gas mass function predicted
by our standard model whereas the thin dot-dashed line is for
our superwind model. The Schechter fit to the HI mass
function from \cite{Zwaan05} is shown as triangles. For
comparison, the thick continuous line is the Press-Schechter
mass function, assuming a universal baryon fraction
everywhere.}
\label{ndembar}
\end{figure}
In this final part, we investigate the stellar and cold gas mass
function at z=0. We reach here the limit of our model because, as
mentionned before, our analytical predictions are for baryon mass
fractions as a function of halo masses, whereas observed baryon mass
fractions are usually given as a function of individual galaxy
masses. The conversion between galaxy and halo masses can be performed
using the 'halo occupation number' \citep{Kravtsov04b}, which gives
the average number of galaxy satellites within $R_{\rm 200}$ as a
function of the parent halo mass. We intend to apply this correction
to our model predictions in a future paper. Nevertheless, we compare
our predicted stellar and cold gas halo mass function to the observed
stellar \citep{Cole01} and HI galaxy mass function \citep{Zwaan05}.
As presented in figure~\ref{ndembar} the global normalisation of both
stellar and cold gas mass function is in good agreement with the
observations. Indeed the integral of this mass function multiplied by
the mass is nothing else that the z=0 cosmic baryon budget for this 2
phases. As for the observations, the profile shape shows a fall-off
for high mass and a flatening for lower masses. This is easily
explained by our predicted stellar and cold gas mass fraction. For the
high mass tail, the fall off is due to the fast decline of
Press-Schechter mass function, where as for low mass the flatening is
due to the minimal halo mass $M_{\rm min}$ above which the fraction of
baryons become weaker and weaker.
If we look in more details, we see an encouraging agreement for
intermediate mass. The decline of the high mass tail is however
steeper in the observations, both for the HI mass function $n(M_{\rm
HI})$ and the stellar one $n(M_*)$. We have identified 2 reasons for
that. First, as highlighted before, the halo occupation number should
be taken into account, because it would decompose each large halos in
a collection of lower mass halos. As a consequence, the high-mass
decline may be more steep. Second, superwinds may also lower the cold
and stellar mass fraction in high mass halos \citep{Springel05}. To
illustrate this point, we have plotted the effects of our superwind
scenario on the mass function. Note that a more complex model with 2
different winds would result in more realistic predictions. For low
mass galaxies, the model tends to underestimate the cold gas mass
function. Here again the halo occupation number would explain part of
this discrepancy by increasing the number of low mass halos. In our
model the transition between the star forming halos and the diffuse
background occurs abruptly at $M_{\rm min}$. As seen in our
simulations, this transition is in fact much smoother: this is likely
to improve the agreement between the predicted and the observed mass
function.
\section{Conclusion}
We have studied the baryon budget evolution in the framework of the
hierarchical scenario of structure formation, using 4 different
phases: diffuse background (Ly$\alpha$ Forest), hot gas (plasma in
the halo of galaxies, groups and cluster), cold discs and stars. We
have paid particular attention to the star formation rate, as it is
a key observational constraint for our current cosmological model.
We have analyzed the baryon history for the universe as a whole, but
also on a halo-by-halo basis. For that purpose, we have developed a
fully self-consistent (though simple) analytical model. These last
two point are the most original aspects of our work. Our analytical
model has proven to be an efficient tool to quickly compute accurate
predictions for the baryon budget history. It is currently
available as a set of IDL routines, and can be provided by the
authors upon request.
In order to validate this model, we have performed numerical
simulations of galaxy formation using the AMR code RAMSES. Our
highest resolution run reach $512^3$ dark matter particles and half
a billion AMR cells, which is among the largest galaxy formation
simulations performed so far. We have also analyzed the simulation
results of \cite{Springel03b} based on the SPH code GADGET. We found
in all cases a good agreement between simulations and our model.
This cross-validation has allowed us to use our model to analyze
observational data.
We have explored our physical parameter space $t_*-\eta_{\rm w}$
(star formation time scale and wind efficiency) and compared the
model results to the cosmic observations of the comoving star
formation rate, the evolution of the comoving density of stars and
cold gas, and the intensity of the integrated extragalactic
background. The conclusion is that the parameters $t_*=M_{\rm
cold}/\dot{M}_*=3\ {\rm Gyr}$ and $\eta_{\rm w}=\dot{M}_{\rm
wind}/\dot{M}_*=1.5$ are favored. It means that winds with ejection
rates around 1 or 2 times the star formation rate are required to
prevent the overcooling problem.
Comparisons with individual halo properties, such as the age
distribution of stars in galaxies and the hot gas fraction in
clusters seems to indicate that high velocity and high intensity
outflows (`superwinds') are required in massive galaxies. The
origin of these violent outflows could come from a central AGN
\citep{Springel05}. The modeling of such winds and their exact role
in the metal enrichment of the IGM are currently under
investigation.
\begin{acknowledgements}
The simulations presented here were performed at `Centre de
Calcul pour la Recherche et la Technologie' (CCRT). We thank
Alexandre Refregier for providing us IDL routines to compute the
cosmological evolution of the halo model (ICOSMO package \cite{Refregier05}). We thank Volker Springel
for providing us his simulations results. We thank David Elbaz for
his help on analyzing observational data points of the Cosmic Star
Formation Rate. We thank Alastair Sanderson for providing us hot
gas fraction in clusters.We also thank Pierre-Alain Duc for
stimulating discussions.
\end{acknowledgements}
\newpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,044 |
Survey Says …
Who doesn't enjoy a survey about sex? These surveys are not representative of the population as a whole, or even of Christians as a whole; however, some of the results are instructive. The surveys were originally posted on facebook with links in from Twitter. The audience taking these surveys are primarily Christian, married, and "sex positive". The surveys are all still active, and we will update results from time to time.
Note: This page shows our most recent surveys. Older surveys here.
Anal Sex: Sin, okay, unhealthy, or what?
300 Women and 595 men have answered
A note on gender and age: In the 20-29 age range a slight majority of those answering were female (53% to 47%) while men outnumbered women from 2 to 1 to as high as 4 to 1 in the other age ranges. This resulted in some answer differences in the 20's which were more about gender than age.
Are you opposed to anal sex for any of the following reasons? (Check all that apply)
Are you Opposed? Female Male
No, not opposed. 37% 69%
I think it is sin/immoral. 12% 5%
I think it is unhealthy/harmful. 42% 25%
I think it's gross. 42% 17%
I think it is disrespectful/unloving. 21% 11%
It hurts. 30% 7%
Women in the 20-29 age range were most likely to find anal sex gross, with 60% giving this answer. The same group was very likely to see anal sex as unhealthy or harmful at 53%.
Acceptance (not opposed) based on age and gender:
Gender 20-29 40-49 40-49 50-59 60 or better
Male 58% 71% 76% 67% 39%
Female 25% 41% 40% 50% 72%
The lowest acceptance for women was for those in their 20's. Acceptance rose with age for women, to a high in the 60's. Low numbers of women over 50 answering mean this may be skewed. For men acceptance of anal sex was lower for those over 60, but was second lowest for those in their 20's.
How often is anal sex a part of your sex life?
Frequency Female Male
Never 60% 72%
1-3 times a year 20% 17%
5-10 times a year 8% 5%
about once a month 6% 3%
2-3 times a month 3% 2%
4-5 times a month 2% 0.5%
more than once a week 1% 0.7%
Contrary to what was expected, those in their 20's were very unlikely to have anal sex. A full 75% said it was never a part of their sex life. Only those over 60 were less likely to have anal sex, with 801% saying it was never a part. This was true for both men and women. Twenty somethings who did have anal sex did it less often than those between 30 and 59.
Have you ever tried heterosexual anal intercourse?
Have you? Female Male
No 39% 53%
Yes, but not with my spouse 10% 8%
Yes, only with my spouse 43% 32%
Yes, with my spouse and someone else 9% 7%
61% of women, but only 47% of men have tried anal sex. This discrepancy is likely due to the fact that those who answer a survey like this tend to be more sexually open than those who do not.
Of those women who have tried it, how many have had trouble?
Tried it never got him all the way in. 15%
Done it once, it hurt too much. 14%
Done it 1-3 times, it always hurts. 16%
Done it 4-6 times, it always hurts. 7%
Done it more than 6 times, times, it always hurts. 11%
Did not hurt (much or at all) from the first time. 14%
Did not hurt (much or at all) after the 2nd or 3rd time. 14%
Did not hurt (much or at all) after the 4th or 5th time. 9%
18% have never been able to do anal without pain, despite multiple attempts. Regardless of why this is so, it shows that some women (1 in 6) cannot have anal sex without pain.
Of those who have tried it, has anal sex ever been pleasurable?
No 33%
Mildly, on occasion. 16%
Mildly some or all the time. 4%
Very much so on occasion. 15%
Very much so some or all the time. 14%
I orgasm from anal alone occasionally. 6%
I orgasm from anal alone regularly. 3%
I orgasm from anal plus clitoral stimulation occasionally. 14%
I orgasm from anal plus clitoral stimulation regularly. 16%
One third of those who have tried it never get any pleasure out of it. More than half the women who never have had pleasure can do it without pain, and some who experience pleasure on occasion also still experience pain every time.
Of the men who have done it, how much do they enjoy it?
No, did not enjoy it. 5%
It was/is okay. 21%
It was/is good. 36%
It was/is great. 40%
My favourite sex act 5%
A couple of women and one man said she is the one who asks for it.
A couple of women and men said they don't do it because of his size. This is likely a factor in what some women have experienced. In this case, smaller, especially girth, would probably be better.
A couple of women have given in on occasion because of their husband's regular requests, and one was forced by her ex.
What colour lingerie do men prefer?
268 men have answered
Several indicated naked is their choice, and a couple said they don't care.
Color doesn't matter as much as the fact that she's wearing lingerie.
If she'd just wear anything that was meant to turn me on I wouldn't give a rip about the color.
Ladies, what does your husband do that makes sex really great for you?
77 Women responding
The answers are roughly grouped here by the number of women who gave each answer.
11 Going slow – Not moving to breasts and genitals too fast. (Some wanted it all slow, others wanted to start slow and get aggressive)
10 Oral sex (Two specifically mentioned sucking of the clitoris)
8 Kissing
8 Makes sure I climax first (as often as I want)
7 Shows he desires me (as opposed to just wants sex)
7 An intimate connection outside the bedroom
6 Talking during sex (feedback, loving words, narrative of what he is doing or is going to do)
6 Sensitive to my moods and desires (knowing when to go slow, when to be aggressive, etc.)
6 Tells me how much he likes my body (several said they have body issues, but he convinces them he loves their body)
6 He is clearly all about my pleasure/Selfless
5 Multiple orgasms
4 Manual stimulation of genitals
4 Expressing his love
4 He is tender and loving
3 Kiss behind ears or back of neck
3 Variety – especially during foreplay
3 Not making me feel rushed
2 Nipple Stimulation
2 Anal stimulation (manual)
2 Initiating sex
1 Stop start during intercourse
1 Let me take control
1 Stimulation of lips (face) by hand
1 Making sure I orgasm after intercourse
1 What he does outside the bedroom (helping around the house)
1 Orgasm by stimulating clitoris and g-spot at the same time
1 How he smells
1 Makes sure I climax even if he does not
What wives do to make sex great
First and foremost, men want their wives to be actively involved in and enjoying sex. Half a dozen men said something to the effect of "Sex is best for me when it is great for her." In addition to the one in four men who specifically said something about her being "involved, enthusiastic, passionate, active participation, or enjoying herself" nearly as many hinted at the same thing without saying it outright.
Receiving oral sex was mentioned by 22% of the men. A couple wanted it as foreplay, most wanted it to orgasm. Three specified 69, one deep throat. Only three (1.3%) mentioned swallowing.
The wife initiating sex on occasion was specifically mentioned by 16% of the men, and hinted at by even more.
Lingerie or some other special clothing was mentioned by 8% of the men. Several mentioned an outfit or type of clothing that signalled a desire for sex, or signalled they were going to have sex.
Replies, in order of how many men listed them:
84 Showing she wants me sexually
47 Give me oral
35 Initiate
18 Lingerie or other specific clothing
14 Vocal about her pleasure
11 Teasing: Sex play/flirting/sex talk/texts during the day
10 Aggressive during sex
10 Slow down, make it last
10 Let go/Be uninhibited
9 Receives oral (most indicated to climax)
9 TALKING: Dirty or graphic, talk about what she is feeling, telling me what she is doing or what she is going to do
9 Committed to my pleasure
8 Touch non-sexual parts of the body during sex
8 Her on top
8 Manual stimulation of penis
7 Playing with my testicles
7 Variety of positions, or a special one
6 Communicate what she wants during sex
6 Undressing while I watch/Striptease
6 Saying yes
5 Admire my penis
5 Play with herself – nipples and/or genitals
4 Hand job to climax
4 Have multiple orgasms
4 Planning for sex/setting scene
4 Sex just for me (Gift sex)
4 Edging (taking him to the brink repeatedly before orgasm)
4 Let me bring her to orgasm
4 Eye contact during oral
3 Play with my nipples
3 Great relationship outside of sex
3 Rub breasts on me
3 Back massage
3 Swallowing
3 Bring herself to orgasm while with me
3 Walk in naked
3 Sexually adventurous
3 Lights on
3 Sex in rooms other than the bedroom
2 Play Bliss (a game)
2 Frequent sex
2 Expressing love during sex
2 Quality time and non-sexual touch first
2 Getting made up for sex
2 Pubic hair removal
2 Anal finger play on her
2 Anal finger play on me, and/or prostate stimulation
2 Anything other than a quickie
2 Gets her lubricant on my face
1 Hotel sex
1 Sex in a jacuzzi
1 Role playing
1 Stimulates herself to climax with the head of my penis
1 Kegels
1 Female ejaculation
Birth Control Survey
Full list at the bottom.
Vasectomy was the most common, at 27%.
Condoms were a close second at 26%.
Hormonal methods were third, at 15%.
Methods that require knowing when a woman is fertile (FAM & NFP) were used by 13%.
Withdrawal was used by 11%.
Nine percent were using nothing.
In six percent the woman had been sterilised – either her tubes tied of insertion of the Ensure device.
Five percent were sexless – most against their will, a couple because of separation due to deployment.
Withdrawal/Pulling Out
Almost half who used this method also used condoms some of the time, and 13% were also using some form of hormonal contraception. Thirty percent, who used this method were practicing FAM. Twenty-two percent who pulled out used nothing else. This means only 2.5% of the total group were using withdrawal as their only form of birth control. Withdrawal was most common in those under 40.
FAM & NFP
Usage was about the same for those in their 20's (20%) and 30's (18%), but fell off a great deal for those in their 40's (5%).
FAM + some other method during fertile time was the most common, at 6%.
FAM + sex other than intercourse during fertile time was used by 3.1%.
NFP was used by 2.8%
FAM with abstinence during fertile time was used by 1.1%
Note: Many who give the last three, all of which avoid intercourse during the fertile time, also indicated using condoms. 47% of those who used NFP also used condoms in the last 3 months. Likewise 47% who used NFP + some other form of sex during the fertile time used condoms . And 71% of those who use NFP and abstain during the fertile time used condoms over the last 3 months. Some of this could be switching what they use recently, but clearly many struggle to avoid intercourse during fertile times.
Fifteen percent use some form of hormonal birth control. The combination pill is the most common, followed by the progesterone only pill, extended cycle pills, vaginal rings, and Depro and the patch and Plan B. Hormonal contraception is far more common for younger women. It was used by 36% of those in their 20's, but only 16% of those in their 30's
In the last 3 months, which of the following forms of birth control have you and your spouse used? Check all that apply.
Condoms 25.8%
Female Condom 0.5%
Combination Pill 4.2%
Progestin-only Pill 1.6%
Extended-cycle pill (Lybrel, Seasonale, Seasonique ) 1.1%
Pill (Don't know what kind) 6.8%
Plan B 0.3%
Patch (Ortho Evra ) 0.5%
Shot/Injection (Depo-Provera) 0.5%
Implant (Implanon, Norplant ) 0.6%
Vaginal Ring (NuvaRing) 1.0%
Diaphragm or Cervical Cap 0.3%
Spermicide 2.1%
Sponge 0.3%
IUD 7.8%
Withdraw (Pulling out) 11.2%
FAM + some other method during fertile time (Fertility awareness method) 6.0%
FAM + sex other than intercourse during fertile time 3.1%
FAM with abstinence during during fertile time 1.1%
NFP (Natural Family Planning – abstinence during fertile time) 2.8%
Vasectomy 27.1%
Hysterectomy, ablation, or ovaries removed 1.0%
Tubes Tied or Ensure 6.2%
On Demand Breast Feeding 0.5%
Currently Pregnant 4.7%
Infertility 1.0%
Nothing 9.2%
Too old to need it 2.4%
Not having sex 5.2%
What Is Most Lacking in Your Marriage?
88 men and 88 women have answered
Which of the following are lacking in your marriage?
For women:
69% Romance
55% Communication
52% Non-Sexual Touching
49% Time Together
49% Sex
38% Affirmation
35% Neat Home
34% Spirituality
33% Income
32% Appreciation
31% Help Around the home
25% Friendship
25% Respect
15% Kindness
13% Self-control with Spending
9% Children under control
Several women added "Trust"
For Men:
10% Children under control
9% Friendship
7% Help Around the home
Please rate how important each of these is in your marriage with 1 being the most important 16 being least important.
Average Ratings for women – 1 is most important, 16 is least:
4.8 Time together
5.6 Respect
6.0 Spirituality
6.1 Kindness
6.5 Romance
6.8 Sex
8.0 Non-sexual touching
8.1 Appreciation
8.3 Friendship
9.0 Affirmation
11.2 Help around the house
11.8 Income
13.7 Self-control with Spending
13.7 Neat Home
13.8 Children under control
Average Ratings for men – 1 is most important, 16 is least:
Infidelity Survey
Cheating, who is doing what:
One interesting thing hidden in the numbers is that very few people engage in just non-intercourse cheating. Those who cheat usually have intercourse in addition to whatever else they do. For example, 85% of the women and 90% of the men who had oral sex outside marriage also had intercourse. For most categories, fewer than 25% engaged in that form of cheating and did not also engage in intercourse outside marriage.
There were a couple of exceptions to this:
61% of the women who engaged in phone sex without orgasm had not had intercourse.
56% of men who engaged in phone sex without orgasm did not have intercourse.
61% of men who engaged in phone sex with orgasm did not have intercourse.
Half the men and women who had homosexual contact also had engaged in intercourse with a member of the opposite sex, half had not.
The average age of the women answering the survey was about ten years younger than average for the men. (This is the norm for our surveys.) This means the men have had ten more years to cheat, making their numbers higher. To separate the age and gender differences, we looked at those who have cheated in some way in any way by age.
Never Cheated
Based on this we still have men more likely to cheat. The lower numbers for the older ages suggest there is less cheating in older generations, but we have relatively few answering in the highest age range so this may be a result of insufficient data.
Does your spouse know?
Yes, I confessed
Yes, s/he found out
S/he has asked, I lied
I think s/he suspects
Has no clue
Have you ever faked orgasm with your spouse?
63% of women, and 33% of men have faked orgasm at least once. The figure for men is somewhat higher than the one in four figures from recent studies on the issue.
14% of women fake orgasm at least 45% of the time.
A couple of women said they don't fake it, but do not correct him when he assumes they have.
Why do you fake orgasm? (Choose all that apply)
I don't want my spouse to feel bad
I know I'm not going to climax.
To end sex.
I'm embarrassed to admit I struggle.
It's not worth the hassle of telling the truth.
Just too tried.
I don't want my spouse to think I am broken.
Trying to act how I think it is supposed to be.
I'm in pain.
To keep my spouse from being mad at me.
Comments from men:
Mostly when I am tired and don't have the Stamina to keep going. I always give her at least 2 orgasms before I fake mine!
I'll admit I've faked it but only on really rare occasions when I knew it would hurt her feelings but I couldn't finish for some reason. It was at a period in our relationship when we were not doing very well.
There were several occasions where I just didn't have the energy and so I acted like I had.
I only did it to see if she was paying attention early on when she was a refuser. I don't do it anymore.
In a PhD program – I have a lot of stress, and not always in the mood as research is more often on my mind than sex; however, the spouse's needs must be met!
Do you think your spouse knows you fake it?
Sometimes, not always.
Do you think your spouse has ever faked orgasm with you?
I know s/he has, but I've never said anything.
I know s/he has, I said something, s/he denied it.
I know s/he has, I said something, s/he admitted it.
I strongly suspect s/he has.
I suspect s/he has.
I think s/he did in the past, but no more.
I am very sure s/he has never faked it.
How could he do that?
In the comments, several women asked how a man could fake an orgasm. It is fairly easy if the couple is using a condom; without a condom it is difficult. One men's website actually posted an article on how to do this! Among other things, they suggested getting into a non-face-to-face position. Not to be outdone, another website wrote an article for women on how to tell if a guy is faking orgasm.
Attitudes about breast size
First let us address some of the questions about this survey:
Q. What good can come from this survey?
A. To show that not all me are obsessed with huge breasts.
Q. Isn't the only thing that matters how a woman's husband feels about her breasts?
A. Yes, but some women think their husband lies to spare their feelings. Seeing other men saying the same thing anonymously can help.
Q. Why didn't you ask about cup sizes instead of small, medium, and so on?
A. We have people from around the world answering these surveys, and cup sizes are not the same in all places.
Cup sizes are not what many think. Cup is based on the measurement below the breast subtracted from the measurement at the fullest part of the breasts (usually at the nipples). A petite woman with a two inch difference and a larger woman with a two inch difference have the same cup size (B in the us, C in the UK and Australia) but the petite woman will seem to have much larger breasts.
Small, medium and so on are perceptions of the individual, but the survey is about perceptions so that works.
What size are your or your wife's breasts?
Very small
Men were less likely to say very small or very large than women were.
If your/her breasts size could be changed with the wave of a hand, with no pain, cost, or problems, would you want them to be:
A bit Bigger
Much Bigger
A bit Smaller
Much Smaller
Men were less likely to want a size change than women, and 12% actually would like smaller. Only 2% of men wanted much bigger.
83% of women who said they are "extra-large" would like to be smaller.
52% of women who said they are large would like to be smaller.
6% of women who said they are medium would like to be smaller, while 67% would like to be larger.
88% of women who said they are small would like to be larger.
87% of women who said they are very-small would like to be larger.
Many women mentioned their breasts were oddly shaped, were different size or shape than each other, or that they were not perky enough.
Several women said they'd had a reduction, and were glad they did.
Several women have had augmentation, and none was unhappy with the results.
Several men said they were fine with their wife's breasts, but wanted her to get augmentation for her self-image.
One man complained his wife's augmentation left her with less sensation.
Several men said something along the lines of "a handful is perfect".
Do You Pray as a Couple?
79 women and 75 men have answered
Note: The age difference between male and female respondents was greater than usual for this survey. While half the women were in their 20's, and 84% were under 40, only 16% of the men were in their 20's, and 52% were 40 or older. This and low numbers overall mean any gender based differences are suspect.
Several men and women said the only real prayer they do together is when they pray before meals.
One man summed up a common male problem well saying, "This is something we'd like to do as a couple, but we've had difficulty remembering to do it. We are both highly spiritual people and lifelong Christians in full time ministry. As a man, I've struggled with making my spirituality that vulnerable with her."
Are you "getting enough" sex in your marriage?
How does the sexual frequency in your marriage (any form of sex with your spouse) compare to your ideal frequency?
Half of the men said they have way less than they want.
A third of the women said way less than they want.
Only 20% of men and 34% of women are having the amount of sex they want.
11% of the women and a .3% of men are having more sex than they want.
No one said they are having way more sex than they want.
For men, age had little effect on the numbers. Way less was selected by each age range as follows:
Age 20-29 51%
Age 50 & up 48%
For women age had a greater effect. Way less was selected by each age range as follows:
In women the 30-39 age range has the most who said they were having way less than they wanted, and the least who said they were having the amount they wanted.
In an average week, how often do you and your spouse have some form of sex?
Sexless marriages by age range :
Age 20-29 6%
The 30's seems to be a low frequency time for many, while older couples are having plenty of fun – Frequencies of six or more times a week were most common among the 50 and up group. One couple in their 70 have sex daily!
Sex more than twice a week by age range :
Frequencies of six or more times a week were most common among the 50 and up group.
Several women said they had little sex due to their husband's health and/or medications.
Several women said they have limited sex because of his work schedule – either away from home or changing shifts.
Pregnancy and kids were given as reasons for limited sex by half a dozen women.
Several women said they have come to understand the importance of sex or learned to enjoy it as they have gotten older. One woman said they were happy with the current frequency, but expected it to go up as the last of the kids
How do you feel about having sex while staying at someone else's home?
Very few said absolutely not – only 7% of women and 1.5% of men.
More than half of men, and a quarter of women, find sex at someone else's home exciting.
Women were more likely to put conditions on having sex.
More men than women said they were willing but their spouse was not – 28% to 7%.
By age, those in their 40's were most likely to say they would.
The biggest concern was being heard, followed by the mess. Over a third of women, and 15 of men felt it was disrespectful!
Those in their 20's were the most concerned with being heard, and the most likely to feel it was disrespectful.
One woman said only at her parent's house.
Several people said, "Only if the bed is quiet".
Only if the kids are in another room was also mentioned by several.
Two said they have done it quietly in the dark while in the same room with the kids.
Several said they will have a quickie, and most of these indicated it was "just for him".
Image Source: © Sue Harper | Dreamstime.com | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 303 |
The 2019/2020 undergraduate summer vacation program is open to students studying Forest Science, Forest Engineering, Environmental Sciences or other related disciplines who have a keen interest in the plantation forest industry.
The program will run from November 2019 to February 2020 with positions available in Launceston, Tasmania and Mount Gambier, South Australia.
Applications close 31 July 2019.
For more information, email Gayle Quin.
Positions are available for recent graduates in Forest Science, Forest Engineering, Environmental Sciences or other related disciplines.
The successful applicant will commence a program that provides them with a practical and broad understanding of our business and the Industry. They will bring excellent communication skills, a high degree of self-motivation and a keen interest in a career in the plantation forestry industry.
Positions are available in Launceston, Tasmania and Mount Gambier, South Australia.
Applications close 30 May 2019 for a July 2019 start or 31 August for a January 2020 start.
If you have a passion for growing crops, science, mathematics, planning, R&D, or IT and wish to apply your skill set in a practical and measurable way, we'd love to hear from you. Please send your CV to enquiries@tppl.com.au.
Timberlands Pacific is a very diverse forestry company that offers career opportunities across the entire forestry and business management spectrum. We provide a very positive and dynamic working environment and promote a team-focused and social culture. Many of our staff members are sports minded and enjoy the outdoors.
Respect that everyone needs quality time away from work.
Roles within the organisation range from management of land preparation and planting activities, ongoing forest health monitoring, management and coordination of harvesting activities, resource modelling and data analytics, forest practices planning, to business management and administration support. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,383 |
//Implementation of EBNF-to-BNF converter.
#include <algorithm>
#include <cctype>
#include <functional>
#include <iostream>
#include <memory>
#include <sstream>
#include <stdexcept>
#include <string>
#include <vector>
#include "action.h"
#include "action_factory.h"
#include "bnf.h"
#include "commons.h"
#include "conversion.h"
#include "concrete_bnf.h"
#include "converter.h"
#include "descriptor.h"
#include "descriptor_type.h"
#include "ebnf__imp.h"
#include "ebnf_builder_res.h"
#include "ebnf_extension__imp.h"
#include "ebnf_visitor__imp.h"
#include "types.h"
#include "util.h"
#include "util_mptr.h"
#include "util_string.h"
namespace ns = synbin;
namespace conv = ns::conv;
namespace ebnf = ns::ebnf;
namespace types = ns::types;
namespace util = ns::util;
using std::unique_ptr;
using util::MContainer;
using util::MRoot;
using util::MHeap;
using util::MPtr;
using util::String;
///////////////////////////////////////////////////////////////////////////////////////////////////
//BNF Traits, etc.
///////////////////////////////////////////////////////////////////////////////////////////////////
namespace {
typedef MPtr<const ns::Action> ActionMPtr;
typedef ns::ConcreteBNF BnfGrm;
typedef BnfGrm::Sym BnfSym;
typedef BnfGrm::Nt BnfNt;
typedef BnfGrm::Tr BnfTr;
typedef BnfGrm::Pr BnfPr;
typedef ns::BnfGrammarBuilder<ns::ConcreteBNFTraits> BnfBld;
}//namespace
//
//ConvSym
//
//Instead of using BnfSym and its subclasses, wrappers like ConvSym are used - in order to hide
//the dependency from BNF.
class conv::ConvSym {
NONCOPYABLE(ConvSym);
const BnfSym* const m_bnf_sym;
protected:
ConvSym(const BnfSym* bnf_sym)
: m_bnf_sym(bnf_sym)
{
assert(bnf_sym);
}
public:
virtual ~ConvSym(){}
const BnfSym* get_sym() const {
return m_bnf_sym;
}
virtual MPtr<const SymDescriptor> get_descriptor() const = 0;
};
//
//ConvNt
//
class conv::ConvNt : public ConvSym {
NONCOPYABLE(ConvNt);
public:
ConvNt(const BnfNt* bnf_nt) : ConvSym(bnf_nt){}
const BnfNt* get_nt() const {
return static_cast<const BnfNt*>(get_sym());
}
MPtr<const SymDescriptor> get_descriptor() const override {
return get_nt()->get_nt_obj();
}
};
//
//ConvTr
//
class conv::ConvTr : public ConvSym {
NONCOPYABLE(ConvTr);
public:
ConvTr(const BnfTr* bnf_tr) : ConvSym(bnf_tr){}
const BnfTr* get_tr() const {
return static_cast<const BnfTr*>(get_sym());
}
MPtr<const SymDescriptor> get_descriptor() const override {
return get_tr()->get_tr_obj();
}
};
//
//ConvPr
//
class conv::ConvPr {
NONCOPYABLE(ConvPr);
const BnfPr* const m_bnf_pr;
const ActionMPtr m_action;
public:
ConvPr(const BnfPr* bnf_pr) : m_bnf_pr(bnf_pr){}
const BnfPr* get_pr() const {
assert(m_bnf_pr);
return m_bnf_pr;
}
};
//
//ConvPrBuilder : definition
//
namespace synbin {
class ConvPrBuilder;
}
class ns::ConvPrBuilder : public ns::ConvPrBuilderFacade {
NONCOPYABLE(ConvPrBuilder);
ns::Converter* const m_converter;
std::vector<const BnfSym*> m_elements;
ActionMPtr m_action;
public:
ConvPrBuilder(ns::Converter* converter);
void add_element(util::MPtr<conv::ConvSym> sym) override;
void set_action_factory(ns::ActionFactory& action_factory) override;
const std::vector<const BnfSym*>& get_elements() const;
ActionMPtr get_action() const;
};
//
//(Misc.)
//
namespace {
const std::string g_auto_nt_name_prefix = "A_";
const std::string g_user_nt_name_prefix = "N_";
const std::string g_str_tr_name_prefix = "S_";
const std::string g_name_tr_name_prefix = "T_";
const String g_auto_empty_nt_name(g_auto_nt_name_prefix + "Empty");
struct TrDeclIndexFn {
std::size_t operator()(const ebnf::TerminalDeclaration* tr) const {
return tr->tr_index();
}
};
struct NtDeclIndexFn {
std::size_t operator()(const ebnf::NonterminalDeclaration* nt) const {
return nt->nt_index();
}
};
}
//
//ConversionResult
//
ns::ConversionResult::ConversionResult(
GrammarBuildingResult* building_result,
unique_ptr<const ConcreteBNF> bnf_grammar,
unique_ptr<const std::vector<const ConcreteBNF::Nt*>> start_nts,
unique_ptr<const std::vector<const NtDescriptor*>> nts,
unique_ptr<const std::vector<const NameTrDescriptor*>> name_tokens,
unique_ptr<const std::vector<const StrTrDescriptor*>> str_tokens,
unique_ptr<const std::vector<const PrimitiveTypeDescriptor*>> primitive_types,
util::MPtr<const PrimitiveTypeDescriptor> string_literal_type,
std::size_t class_type_count)
: m_common_heap(std::move(building_result->m_common_heap)),
m_bnf_grammar(std::move(bnf_grammar)),
m_start_nts(std::move(start_nts)),
m_nts(std::move(nts)),
m_name_tokens(std::move(name_tokens)),
m_str_tokens(std::move(str_tokens)),
m_primitive_types(std::move(primitive_types)),
m_string_literal_type(string_literal_type),
m_class_type_count(class_type_count)
{}
const ns::ConcreteBNF* ns::ConversionResult::get_bnf_grammar() const {
return m_bnf_grammar.get();
}
const std::vector<const ns::ConcreteBNF::Nt*>& ns::ConversionResult::get_start_nts() const {
return *m_start_nts;
}
const std::vector<const ns::NameTrDescriptor*>& ns::ConversionResult::get_name_tokens() const {
return *m_name_tokens;
}
const std::vector<const ns::StrTrDescriptor*>& ns::ConversionResult::get_str_tokens() const {
return *m_str_tokens;
}
const std::vector<const ns::PrimitiveTypeDescriptor*>& ns::ConversionResult::get_primitive_types() const {
return *m_primitive_types;
}
//
//Converter : definition
//
namespace {
typedef std::map<String, MPtr<const ns::PrimitiveTypeDescriptor>> PrimitiveTypeMap;
}
class ns::Converter : public ConverterFacade {
NONCOPYABLE(Converter);
unique_ptr<MHeap> m_managed_heap;
BnfBld m_bnf_builder;
std::size_t m_auto_nt_name_index;
MPtr<conv::ConvNt> m_empty_conv_nt;
PrimitiveTypeMap m_system_primitive_type_map;
PrimitiveTypeMap m_user_primitive_type_map;
std::map<String, MPtr<conv::ConvTr>> m_str_tr_to_bnf_map;
util::IndexedMap<const ebnf::TerminalDeclaration*, MPtr<conv::ConvTr>, TrDeclIndexFn> m_tr_to_bnf_map;
util::IndexedMap<const ebnf::NonterminalDeclaration*, MPtr<conv::ConvNt>, NtDeclIndexFn> m_nt_to_conv_map;
std::map<String, MPtr<const ClassTypeDescriptor>> m_class_type_map;
const types::Type* const m_string_literal_type;
MPtr<const VoidTypeDescriptor> m_void_type;
MPtr<const TypeDescriptor> m_string_literal_type_desc;
unique_ptr<std::vector<const ConcreteBNF::Nt*>> m_start_bnf_nts;
unique_ptr<std::vector<const NtDescriptor*>> m_nts;
unique_ptr<std::vector<const NameTrDescriptor*>> m_name_tokens;
unique_ptr<std::vector<const StrTrDescriptor*>> m_str_tokens;
const unique_ptr<MContainer<conv::ConvSym>> m_managed_conv_syms;
const MPtr<MContainer<Action>> m_managed_actions;
const MPtr<MContainer<SymDescriptor>> m_managed_sym_descriptors;
const MPtr<MContainer<PrDescriptor>> m_managed_pr_descriptors;
const MPtr<MContainer<TypeDescriptor>> m_managed_types;
public:
Converter(
unique_ptr<MHeap> managed_heap,
MHeap* managed_heap_ref,
const types::Type* string_literal_type,
std::size_t tr_count,
std::size_t nt_count)
: m_managed_heap(std::move(managed_heap)),
m_string_literal_type(string_literal_type),
m_auto_nt_name_index(0),
m_tr_to_bnf_map(tr_count),
m_nt_to_conv_map(nt_count),
m_start_bnf_nts(new std::vector<const ns::ConcreteBNF::Nt*>()),
m_nts(new std::vector<const ns::NtDescriptor*>()),
m_name_tokens(new std::vector<const NameTrDescriptor*>()),
m_str_tokens(new std::vector<const StrTrDescriptor*>()),
m_managed_conv_syms(new MContainer<conv::ConvSym>()),
m_managed_actions(managed_heap_ref->create_container<Action>()),
m_managed_sym_descriptors(managed_heap_ref->create_container<SymDescriptor>()),
m_managed_pr_descriptors(managed_heap_ref->create_container<PrDescriptor>()),
m_managed_types(managed_heap_ref->create_container<TypeDescriptor>())
{}
MPtr<conv::ConvNt> get_empty_nonterminal() override;
MPtr<conv::ConvSym> cast_nt_to_sym(MPtr<conv::ConvNt> conv_nt) const override;
MPtr<conv::ConvSym> convert_expression_to_nonterminal(ebnf::SyntaxExpression* expr, MPtr<const TypeDescriptor> type) override;
MPtr<conv::ConvNt> convert_nonterminal(ebnf::NonterminalDeclaration* nt);
void convert_terminal_init(ebnf::TerminalDeclaration* tr);
MPtr<conv::ConvTr> convert_terminal(ebnf::TerminalDeclaration* tr);
void convert_expression_to_production(MPtr<conv::ConvNt> conv_nt, const ebnf::SyntaxExpression* expr) override;
MPtr<conv::ConvSym> convert_symbol_to_symbol(ebnf::SymbolDeclaration* sym) override;
MPtr<conv::ConvSym> convert_expression_to_symbol(ebnf::SyntaxExpression* expr) override;
MPtr<conv::ConvSym> convert_string_to_symbol(const syntax_string& str) override;
MPtr<conv::ConvNt> create_auto_nonterminal(MPtr<const TypeDescriptor> type) override;
void create_production(
MPtr<conv::ConvNt> conv_nt,
const std::vector<MPtr<conv::ConvSym>>& conv_elements,
ActionFactory& action_factory) override;
MPtr<const TypeDescriptor> convert_type(const types::Type* type) override;
MPtr<const ListTypeDescriptor> create_list_type(MPtr<const TypeDescriptor> element_type) override;
void convert_primitive_type_init(const types::PrimitiveType* primitive_type);
MPtr<const PrimitiveTypeDescriptor> convert_primitive_type(const types::PrimitiveType* primitive_type) override;
MPtr<const PrimitiveTypeDescriptor> convert_string_literal_type(const types::Type* type);
MPtr<const ClassTypeDescriptor> convert_class_type(const types::ClassType* class_type) override;
MPtr<const PartClassTypeDescriptor> convert_part_class_type(
MPtr<const ClassTypeDescriptor> class_type,
const PartClassTag& part_class_tag) override;
MPtr<Action> manage_action(std::unique_ptr<Action> action) override;
MPtr<const TypeDescriptor> manage_type(TypeDescriptor* type) override;
MPtr<const VoidTypeDescriptor> get_void_type() override;
MPtr<const TypeDescriptor> get_symbol_type(MPtr<conv::ConvSym> conv_sym) const override;
ActionMPtr create_action(ActionFactory& action_factory, const std::vector<const BnfSym*>& elements);
unique_ptr<ConversionResult> convert_EBNF(bool verbose_output, GrammarBuildingResult* building_result);
private:
MPtr<conv::ConvNt> create_nonterminal(const String& name, MPtr<const NtDescriptor> descriptor);
void create_production0(
MPtr<conv::ConvNt> conv_nt,
const std::vector<const BnfSym*>& elements,
MPtr<const Action> action);
void create_implicitly_casted_production(
MPtr<conv::ConvNt> conv_nt,
const std::vector<const BnfSym*>& elements,
MPtr<const Action> action,
MPtr<const TypeDescriptor> nt_type,
MPtr<const TypeDescriptor> pr_type);
MPtr<const Action> create_implicit_cast_action(
MPtr<const TypeDescriptor> cast_type,
MPtr<const TypeDescriptor> source_type);
MPtr<const NtDescriptor> manage_nt(NtDescriptor* nt);
String generate_auto_nt_name();
MPtr<const TypeDescriptor> get_string_literal_type_desc();
};
//
//ConvPrBuilder : implementation
//
ns::ConvPrBuilder::ConvPrBuilder(ns::Converter* converter)
: m_converter(converter)
{}
void ns::ConvPrBuilder::add_element(MPtr<conv::ConvSym> conv_sym) {
assert(conv_sym.get());
assert(!m_action.get());
m_elements.push_back(conv_sym->get_sym());
}
void ns::ConvPrBuilder::set_action_factory(ns::ActionFactory& action_factory) {
assert(!m_action);
m_action = m_converter->create_action(action_factory, m_elements);
assert(!!m_action);
}
const std::vector<const BnfSym*>& ns::ConvPrBuilder::get_elements() const {
return m_elements;
}
ActionMPtr (ns::ConvPrBuilder::get_action)() const {
assert(m_action.get());
return m_action;
}
//
//Converter : implementation
//
MPtr<conv::ConvNt> ns::Converter::get_empty_nonterminal() {
if (!m_empty_conv_nt.get()) {
MPtr<const VoidTypeDescriptor> void_type = get_void_type();
MPtr<const NtDescriptor> descriptor = manage_nt(new AutoNtDescriptor(void_type, g_auto_empty_nt_name));
m_empty_conv_nt = create_nonterminal(g_auto_empty_nt_name, descriptor);
ActionMPtr action = m_managed_actions->add(new VoidAction(void_type));
std::vector<const BnfSym*> elements;
create_production0(m_empty_conv_nt, elements, action);
}
return m_empty_conv_nt;
}
MPtr<conv::ConvSym> ns::Converter::cast_nt_to_sym(MPtr<conv::ConvNt> conv_nt) const {
return conv_nt;
}
MPtr<conv::ConvSym> ns::Converter::convert_expression_to_nonterminal(
ebnf::SyntaxExpression* expr,
MPtr<const TypeDescriptor> type)
{
assert(!!type);
//Convert.
String nt_name = generate_auto_nt_name();
MPtr<const NtDescriptor> descriptor = manage_nt(new AutoNtDescriptor(type, nt_name));
MPtr<conv::ConvNt> conv_nt = create_nonterminal(nt_name, descriptor);
expr->get_extension()->get_conversion()->convert_nt(this, conv_nt);
return conv_nt;
}
void ns::Converter::convert_expression_to_production(MPtr<conv::ConvNt> conv_nt, const ebnf::SyntaxExpression* expr) {
ConvPrBuilder pr_builder(this);
expr->get_extension()->get_conversion()->convert_pr(this, &pr_builder);
ActionMPtr action = pr_builder.get_action();
const std::vector<const BnfSym*>& elements = pr_builder.get_elements();
create_production0(conv_nt, elements, action);
}
MPtr<conv::ConvSym> ns::Converter::convert_symbol_to_symbol(ebnf::SymbolDeclaration* sym) {
class SymConvertingVisitor : public ns::SymbolDeclarationVisitor<MPtr<conv::ConvSym>> {
Converter* const m_converter;
public:
SymConvertingVisitor(Converter* converter) : m_converter(converter){}
MPtr<conv::ConvSym> visit_SymbolDeclaration(ebnf::SymbolDeclaration* sym) override {
throw err_illegal_state();
}
MPtr<conv::ConvSym> visit_TerminalDeclaration(ebnf::TerminalDeclaration* tr) override {
MPtr<conv::ConvTr> conv_tr = m_converter->convert_terminal(tr);
return conv_tr;
}
MPtr<conv::ConvSym> visit_NonterminalDeclaration(ebnf::NonterminalDeclaration* nt) override {
MPtr<conv::ConvNt> conv_nt = m_converter->convert_nonterminal(nt);
return conv_nt;
}
};
SymConvertingVisitor visitor(this);
MPtr<conv::ConvSym> conv_sym = sym->visit(&visitor);
return conv_sym;
}
MPtr<conv::ConvSym> ns::Converter::convert_expression_to_symbol(ebnf::SyntaxExpression* expr) {
ns::SyntaxExpressionExtension* ext = expr->get_extension();
MPtr<conv::ConvSym> conv_sym = ext->get_conversion()->convert_sym(this);
return conv_sym;
}
namespace {
bool is_str_name_start(int k) {
return 0 != std::isalpha(k) || '_' == k;
}
bool is_str_name_part(int k) {
return 0 != std::isalnum(k) || '_' == k;
}
bool is_str_name(const ns::syntax_string& str) {
const std::string& s = str.str();
assert(!s.empty());
bool is_name = is_str_name_start(s[0]);
for (std::size_t i = 1, n = s.size(); i < n; ++i) {
if (is_name != is_str_name_part(s[i])) {
throw ns::raise_error(str, "Mixing identifier and non-identifier in a string literal");
}
}
return is_name;
}
}//namespace
MPtr<conv::ConvSym> ns::Converter::convert_string_to_symbol(const syntax_string& synstr) {
const String& str = synstr.get_string();
MPtr<conv::ConvTr> conv_tr = m_str_tr_to_bnf_map[str];
if (!conv_tr.get()) {
std::size_t idx = m_str_tr_to_bnf_map.size() - 1; //The size has just been incremented by the operator[].
const String name(g_name_tr_name_prefix + std::to_string(idx));
MPtr<const TypeDescriptor> type = get_string_literal_type_desc();
std::size_t token_id = m_str_tokens->size();
bool is_name = is_str_name(synstr);//TODO Determine is_name (and report mixing error) earlier.
MPtr<const StrTrDescriptor> descriptor = m_managed_sym_descriptors->add(new StrTrDescriptor(type, str, token_id, is_name));
m_str_tokens->push_back(descriptor.get());
const BnfTr* bnf_tr = m_bnf_builder.create_terminal(name, descriptor);
conv_tr = m_managed_conv_syms->add(new conv::ConvTr(bnf_tr));
m_str_tr_to_bnf_map[str] = conv_tr;
}
return conv_tr;
}
MPtr<conv::ConvNt> ns::Converter::convert_nonterminal(ebnf::NonterminalDeclaration* nt) {
MPtr<conv::ConvNt> conv_nt = m_nt_to_conv_map.get(nt);
if (!conv_nt.get()) {
//Determine the type.
MPtr<const types::Type> concrete_type = nt->get_extension()->get_concrete_type();
assert(!!concrete_type);
MPtr<const TypeDescriptor> type = convert_type(concrete_type.get());
//Create BNF nonterminal.
const String& original_name = nt->get_name().get_string();
String name = String(g_user_nt_name_prefix + original_name.str());
MPtr<const NtDescriptor> descriptor = manage_nt(new UserNtDescriptor(type, name, original_name));
conv_nt = create_nonterminal(name, descriptor);
//The new nonterminal must be put into the map before the conversion of the expression, to
//avoid infinite recursion.
m_nt_to_conv_map.put(nt, conv_nt);
//Convert the expression.
ebnf::SyntaxExpression* expr = nt->get_expression();
ns::SyntaxExpressionExtension* expr_ext = expr->get_extension();
expr_ext->get_conversion()->convert_nt(this, conv_nt);
if (nt->is_start()) m_start_bnf_nts->push_back(conv_nt->get_nt());
}
return conv_nt;
}
void ns::Converter::convert_terminal_init(ebnf::TerminalDeclaration* tr) {
MPtr<conv::ConvTr> conv_tr = m_tr_to_bnf_map.get(tr);
assert(!conv_tr);
const util::String& original_name = tr->get_name().get_string();
//Define type.
MPtr<const TypeDescriptor> type;
const types::PrimitiveType* concrete_type0 = tr->get_type();
if (concrete_type0) {
MPtr<const types::Type> concrete_type = MPtr<const types::Type>::unsafe_cast(concrete_type0);
type = convert_type(concrete_type.get());
} else {
type = get_void_type();
}
//Generate BNF name.
std::ostringstream name_out;
name_out << g_name_tr_name_prefix;
name_out << original_name;
const String name(name_out.str());
//Create BNF terminal.
MPtr<const NameTrDescriptor> descriptor =
m_managed_sym_descriptors->add(new NameTrDescriptor(type, original_name));
m_name_tokens->push_back(descriptor.get());
const BnfTr* bnf_tr = m_bnf_builder.create_terminal(name, descriptor);
conv_tr = m_managed_conv_syms->add(new conv::ConvTr(bnf_tr));
m_tr_to_bnf_map.put(tr, conv_tr);
}
MPtr<conv::ConvTr> ns::Converter::convert_terminal(ebnf::TerminalDeclaration* tr) {
MPtr<conv::ConvTr> conv_tr = m_tr_to_bnf_map.get(tr);
assert(!!conv_tr);
return conv_tr;
}
MPtr<conv::ConvNt> ns::Converter::create_auto_nonterminal(MPtr<const TypeDescriptor> type) {
const String name = generate_auto_nt_name();
const MPtr<const NtDescriptor> descriptor = manage_nt(new AutoNtDescriptor(type, name));
MPtr<conv::ConvNt> conv_nt = create_nonterminal(name, descriptor);
return conv_nt;
}
void ns::Converter::create_production(
MPtr<conv::ConvNt> conv_nt,
const std::vector<MPtr<conv::ConvSym>>& conv_elements,
ActionFactory& action_factory)
{
std::vector<const BnfSym*> bnf_elements;
bnf_elements.reserve(conv_elements.size());
for (MPtr<conv::ConvSym> conv_sym : conv_elements) bnf_elements.push_back(conv_sym->get_sym());
ActionMPtr action = create_action(action_factory, bnf_elements);
create_production0(conv_nt, bnf_elements, action);
}
MPtr<const ns::TypeDescriptor> ns::Converter::convert_type(const types::Type* type) {
class ConvTypeVisitor : public TypeVisitor<MPtr<const TypeDescriptor>> {
Converter* const m_converter;
public:
ConvTypeVisitor(Converter* converter) : m_converter(converter){}
MPtr<const TypeDescriptor> visit_Type(const types::Type* type) override {
throw err_illegal_state();
}
MPtr<const TypeDescriptor> visit_PrimitiveType(const types::PrimitiveType* type) override {
return m_converter->convert_primitive_type(type);
}
MPtr<const TypeDescriptor> visit_ClassType(const types::ClassType* type) override {
return m_converter->convert_class_type(type);
}
MPtr<const TypeDescriptor> visit_VoidType(const types::VoidType* type) override {
return m_converter->get_void_type();
}
MPtr<const TypeDescriptor> visit_ArrayType(const types::ArrayType* type) override {
MPtr<const TypeDescriptor> element_type_desc = type->get_element_type()->visit(this);
return m_converter->m_managed_types->add(new ListTypeDescriptor(element_type_desc));
}
};
ConvTypeVisitor visitor(this);
MPtr<const TypeDescriptor> type_desc = type->visit(&visitor);
return type_desc;
}
MPtr<const ns::ListTypeDescriptor> ns::Converter::create_list_type(MPtr<const TypeDescriptor> element_type) {
return m_managed_types->add(new ListTypeDescriptor(element_type));
}
void ns::Converter::convert_primitive_type_init(const types::PrimitiveType* primitive_type) {
PrimitiveTypeMap& map = primitive_type->is_system() ? m_system_primitive_type_map : m_user_primitive_type_map;
const String& name = primitive_type->get_name();
MPtr<const PrimitiveTypeDescriptor> result = map[name];
assert(!result);
result = m_managed_types->add(new PrimitiveTypeDescriptor(primitive_type));
map[name] = result;
}
MPtr<const ns::PrimitiveTypeDescriptor> ns::Converter::convert_primitive_type(const types::PrimitiveType* primitive_type) {
PrimitiveTypeMap& map = primitive_type->is_system() ? m_system_primitive_type_map : m_user_primitive_type_map;
const String& name = primitive_type->get_name();
MPtr<const PrimitiveTypeDescriptor> result = map[name];
assert(!!result);
return result;
}
MPtr<const ns::PrimitiveTypeDescriptor> ns::Converter::convert_string_literal_type(const types::Type* type) {
if (type->is_void()) return MPtr<const ns::PrimitiveTypeDescriptor>();
const types::PrimitiveType* primitive_type = type->as_primitive();
assert(primitive_type);
MPtr<const PrimitiveTypeDescriptor> string_literal_type = convert_primitive_type(primitive_type);
return string_literal_type;
}
MPtr<const ns::ClassTypeDescriptor> ns::Converter::convert_class_type(const types::ClassType* class_type) {
const String& name = class_type->get_class_name();
std::size_t next_index = m_class_type_map.size();
MPtr<const ClassTypeDescriptor> type = m_class_type_map[name];
if (!type) {
type = m_managed_types->add(new ClassTypeDescriptor(next_index, class_type->get_class_name()));
m_class_type_map[name] = type;
}
return type;
}
MPtr<const ns::PartClassTypeDescriptor> ns::Converter::convert_part_class_type(
MPtr<const ClassTypeDescriptor> class_type,
const PartClassTag& part_class_tag)
{
//TODO Cache part class type descriptors, do not create a new one every time.
return m_managed_types->add(new PartClassTypeDescriptor(class_type, part_class_tag.get_index()));
}
MPtr<ns::Action> ns::Converter::manage_action(std::unique_ptr<Action> action) {
return m_managed_actions->add(action.release());
}
MPtr<const ns::TypeDescriptor> ns::Converter::manage_type(TypeDescriptor* type) {
return m_managed_types->add(type);
}
MPtr<const ns::VoidTypeDescriptor> ns::Converter::get_void_type() {
if (!m_void_type) m_void_type = m_managed_types->add(new VoidTypeDescriptor());
return m_void_type;
}
MPtr<const ns::TypeDescriptor> ns::Converter::get_symbol_type(MPtr<conv::ConvSym> conv_sym) const {
MPtr<const SymDescriptor> descriptor = conv_sym->get_descriptor();
MPtr<const TypeDescriptor> type = descriptor->get_type();
assert(!!type);
return type;
}
MPtr<conv::ConvNt> ns::Converter::create_nonterminal(const String& name, MPtr<const NtDescriptor> descriptor) {
const BnfNt* bnf_nt = m_bnf_builder.create_nonterminal(name, descriptor);
MPtr<conv::ConvNt> conv_nt = m_managed_conv_syms->add(new conv::ConvNt(bnf_nt));
return conv_nt;
}
MPtr<const ns::Action> ns::Converter::create_action(
ActionFactory& action_factory,
const std::vector<const BnfSym*>& elements)
{
class ConvContainer : public ActionFactory::Container {
Converter* const m_converter;
public:
ConvContainer(Converter* converter) : m_converter(converter){}
MPtr<const VoidTypeDescriptor> get_void_type() override {
return m_converter->get_void_type();
}
MPtr<const Action> manage_action(std::unique_ptr<Action> action) override {
return m_converter->m_managed_actions->add(action.release());
}
};
class ConvTypeProduction : public ActionFactory::TypeProduction {
const std::vector<const BnfSym*>& m_elements;
public:
ConvTypeProduction(const std::vector<const BnfSym*>& elements) : m_elements(elements){}
std::size_t size() const override {
return m_elements.size();
}
MPtr<const TypeDescriptor> get(std::size_t index) const override {
const BnfSym* bnf_sym = m_elements[index];
MPtr<const SymDescriptor> descriptor;
if (const BnfNt* bnf_nt = bnf_sym->as_nt()) {
descriptor = bnf_nt->get_nt_obj();
} else {
const BnfTr* bnf_tr = bnf_sym->as_tr();
assert(bnf_tr);
descriptor = bnf_tr->get_tr_obj();
}
MPtr<const TypeDescriptor> type = descriptor->get_type();
return type;
}
};
ConvContainer conv_container(this);
ConvTypeProduction conv_production(elements);
ActionMPtr action = action_factory.create_action(&conv_container, &conv_production);
return action;
}
void ns::Converter::create_production0(
MPtr<conv::ConvNt> conv_nt,
const std::vector<const BnfSym*>& elements,
MPtr<const Action> action)
{
MPtr<const TypeDescriptor> pr_type = action->get_result_type();
assert(!!pr_type);
MPtr<const TypeDescriptor> nt_type = conv_nt->get_descriptor()->get_type();
assert(!!nt_type);
if (!pr_type->is_void() && !nt_type->equals(pr_type.get())) {
//Result type of the production differs from the type of the nonterminal.
//Create a helper nonterminal which and a production which will adapt the original production's result
//to the target nonterminal's type.
create_implicitly_casted_production(conv_nt, elements, action, nt_type, pr_type);
} else {
//Result type of the production is the same as the type of the nonterminal (or compatible).
MPtr<const PrDescriptor> pr_descriptor = m_managed_pr_descriptors->add(new PrDescriptor(action));
m_bnf_builder.add_production(conv_nt->get_nt(), pr_descriptor, elements);
}
}
void ns::Converter::create_implicitly_casted_production(
MPtr<conv::ConvNt> conv_nt,
const std::vector<const BnfSym*>& elements,
MPtr<const Action> action,
MPtr<const TypeDescriptor> nt_type,
MPtr<const TypeDescriptor> pr_type)
{
MPtr<const PrDescriptor> pr_descriptor = m_managed_pr_descriptors->add(new PrDescriptor(action));
MPtr<conv::ConvNt> temp_nt = create_auto_nonterminal(pr_type);
const BnfNt* temp_bnf_nt = temp_nt->get_nt();
m_bnf_builder.add_production(temp_bnf_nt, pr_descriptor, elements);
MPtr<const Action> cast_action = create_implicit_cast_action(nt_type, pr_type);
assert(nt_type->equals(cast_action->get_result_type().get()));
std::vector<const BnfSym*> cast_elements;
cast_elements.push_back(temp_bnf_nt);
MPtr<const PrDescriptor> cast_pr_descriptor = m_managed_pr_descriptors->add(new PrDescriptor(cast_action));
m_bnf_builder.add_production(conv_nt->get_nt(), cast_pr_descriptor, cast_elements);
}
MPtr<const ns::Action> ns::Converter::create_implicit_cast_action(
MPtr<const TypeDescriptor> cast_type,
MPtr<const TypeDescriptor> source_type)
{
const ClassTypeDescriptor* class_source_type = source_type->as_class_type();
assert(class_source_type);
const ClassTypeDescriptor* class_cast_type = cast_type->as_class_type();
assert(class_cast_type);
assert(class_cast_type == cast_type.get());
MPtr<const ClassTypeDescriptor> class_cast_type_ptr = MPtr<const ClassTypeDescriptor>::unsafe_cast(class_cast_type);
assert(class_source_type == source_type.get());
MPtr<const ClassTypeDescriptor> class_source_type_ptr = MPtr<const ClassTypeDescriptor>::unsafe_cast(class_source_type);
return m_managed_actions->add(new CastAction(class_cast_type_ptr, class_source_type_ptr));
}
MPtr<const ns::NtDescriptor> ns::Converter::manage_nt(NtDescriptor* nt) {
m_nts->push_back(nt);
return m_managed_sym_descriptors->add(nt);
}
String ns::Converter::generate_auto_nt_name() {
std::ostringstream outs;
outs << g_auto_nt_name_prefix;
outs << m_auto_nt_name_index++;
return String(outs.str());
}
MPtr<const ns::TypeDescriptor> ns::Converter::get_string_literal_type_desc() {
if (!m_string_literal_type_desc) m_string_literal_type_desc = convert_type(m_string_literal_type);
return m_string_literal_type_desc;
}
//
//ConcreteBNFPrinter
//
namespace {
class ConcreteBNFPrinter {
std::ostream& m_out;
private:
ConcreteBNFPrinter(std::ostream& out);
public:
static void print_concrete_bnf(std::ostream& out, const ns::ConcreteBNF* bnf);
private:
void print_concrete_bnf0(const ns::ConcreteBNF* bnf) const;
void print_nt(const BnfNt* nt) const;
void print_pr(const BnfPr* pr) const;
void print_sym(const BnfSym* sym) const;
};
}
ConcreteBNFPrinter::ConcreteBNFPrinter(std::ostream& out) : m_out(out){}
void ConcreteBNFPrinter::print_concrete_bnf(std::ostream& out, const ns::ConcreteBNF* bnf) {
ConcreteBNFPrinter printer(out);
printer.print_concrete_bnf0(bnf);
}
void ConcreteBNFPrinter::print_concrete_bnf0(const ns::ConcreteBNF* bnf) const {
typedef std::vector<const BnfNt*> NtVector;
const NtVector& nts = bnf->get_nonterminals();
for (const BnfNt* nt : nts) print_nt(nt);
}
void ConcreteBNFPrinter::print_nt(const BnfNt* nt) const {
m_out << nt->get_name() << " { ";
MPtr<const ns::NtDescriptor> desc = nt->get_nt_obj();
desc->get_type()->print(m_out);
m_out << " }\n";
const std::vector<const BnfPr*>& prs = nt->get_productions();
for (std::size_t i = 0, n = prs.size(); i < n; ++i) {
m_out << "\t" << (i == 0 ? ":" : "|");
const BnfPr* pr = prs[i];
print_pr(pr);
m_out << '\n';
}
m_out << '\n';
}
void ConcreteBNFPrinter::print_pr(const BnfPr* pr) const {
const std::vector<const BnfSym*>& elems = pr->get_elements();
for (const BnfSym* elem : elems) {
m_out << " ";
print_sym(elem);
}
m_out << " { ";
pr->get_pr_obj()->get_action()->print(m_out);
m_out << " }";
}
void ConcreteBNFPrinter::print_sym(const BnfSym* sym) const {
if (sym->as_nt()) {
m_out << sym->get_name();
} else {
const BnfTr* tr = sym->as_tr();
assert(tr);
tr->get_tr_obj()->print(m_out);
}
}
//
//convert_EBNF_to_BNF()
//
namespace {
typedef std::vector<const ns::PrimitiveTypeDescriptor*> PrimitiveTypeVec;
void fill_primitive_types(const PrimitiveTypeMap& map, PrimitiveTypeVec& vector) {
for (const PrimitiveTypeMap::value_type& value : map) vector.push_back(value.second.get());
}
}//namespace
unique_ptr<ns::ConversionResult> ns::Converter::convert_EBNF(
bool verbose_output,
GrammarBuildingResult* building_result)
{
MPtr<ebnf::Grammar> ebnf_grammar = building_result->get_grammar();
//Convert primitive types (everyone must be converted, even if it is not referenced).
const std::vector<const types::PrimitiveType*>& primitive_types = building_result->get_primitive_types();
for (const types::PrimitiveType* type : primitive_types) convert_primitive_type_init(type);
//Convert every terminal declaration (every declaration must be converted, even if it is not referenced).
const std::vector<ebnf::TerminalDeclaration*>& ebnf_trs = ebnf_grammar->get_terminals();
for (ebnf::TerminalDeclaration* tr : ebnf_trs) convert_terminal_init(tr);
//Convert every nonterminal.
const std::vector<ebnf::NonterminalDeclaration*>& ebnf_nts = ebnf_grammar->get_nonterminals();
for (ebnf::NonterminalDeclaration* nt : ebnf_nts) convert_nonterminal(nt);
//Create BNF Grammar.
std::unique_ptr<const BnfGrm> bnf_grammar(m_bnf_builder.create_grammar());
if (verbose_output) {
std::cout << "*** BNF GRAMMAR ***\n";
std::cout << '\n';
ConcreteBNFPrinter::print_concrete_bnf(std::cout, bnf_grammar.get());
std::cout << '\n';
}
//Convert the type of custom token.
MPtr<const types::Type> string_literal_type0 = building_result->get_string_literal_type();
MPtr<const PrimitiveTypeDescriptor> string_literal_type =
convert_string_literal_type(string_literal_type0.get());
//Create ConversionResult.
building_result->get_common_heap()->add_heap(std::move(m_managed_heap));
unique_ptr<PrimitiveTypeVec> primitive_type_descriptors = make_unique1<PrimitiveTypeVec>();
primitive_type_descriptors->reserve(m_system_primitive_type_map.size() + m_user_primitive_type_map.size());
fill_primitive_types(m_system_primitive_type_map, *primitive_type_descriptors);
fill_primitive_types(m_user_primitive_type_map, *primitive_type_descriptors);
unique_ptr<ConversionResult> result = make_unique1<ConversionResult>(
building_result,
std::move(bnf_grammar),
util::const_vector_ptr(std::move(m_start_bnf_nts)),
util::const_vector_ptr(std::move(m_nts)),
util::const_vector_ptr(std::move(m_name_tokens)),
util::const_vector_ptr(std::move(m_str_tokens)),
util::const_vector_ptr(std::move(primitive_type_descriptors)),
string_literal_type,
m_class_type_map.size());
return result;
}
unique_ptr<ns::ConversionResult> ns::convert_EBNF_to_BNF(
bool verbose_output,
unique_ptr<GrammarBuildingResult> building_result)
{
unique_ptr<MHeap> managed_heap = make_unique1<MHeap>();
MHeap* managed_heap_ref = managed_heap.get();
MPtr<ebnf::Grammar> ebnf_grammar = building_result->get_grammar();
std::size_t tr_count = ebnf_grammar->get_tr_count();
std::size_t nt_count = ebnf_grammar->get_nt_count();
const types::Type* string_literal_type = building_result->get_string_literal_type().get();
Converter converter(std::move(managed_heap), managed_heap_ref, string_literal_type, tr_count, nt_count);
return converter.convert_EBNF(verbose_output, building_result.get());
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,426 |
{"url":"http:\/\/www.solutioninn.com\/use-the-data-given-in-exercise-547-a-find-the-probability-that","text":"# Question\n\nUse the data given in Exercise 5.47.\na. Find the probability that a randomly chosen person was female given that the person said Yes. In other words, what percentage of the people who said Yes were female?\nb. Find the probability that a randomly chosen person who reported being Unsure was female. In other words, what percentage of the people who were Unsure were female?\nc. Find the probability that a randomly selected person from the entire group was a female who said she was Unsure.\n\nSales0\nViews31","date":"2016-10-26 01:18:56","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\": 0.9627231955528259, \"perplexity\": 308.93535975461685}, \"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-2016-44\/segments\/1476988720471.17\/warc\/CC-MAIN-20161020183840-00217-ip-10-171-6-4.ec2.internal.warc.gz\"}"} | null | null |
package de.newsarea.homecockpit.connector;
import de.newsarea.homecockpit.connector.event.ValueChangedEventListener;
import org.apache.commons.lang3.event.EventListenerSupport;
public abstract class AbstractGeneralConnector<E> implements GeneralConnector<E> {
private EventListenerSupport<ValueChangedEventListener> eventListeners;
public AbstractGeneralConnector() {
eventListeners = EventListenerSupport.create(ValueChangedEventListener.class);
}
public void addEventListener(ValueChangedEventListener<E> eventListener) {
eventListeners.addListener(eventListener);
}
protected void fireEvent(E data) {
eventListeners.fire().valueChanged(data);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,030 |
\section{Introduction}
The conditions under which planetary formation is initiated in the
protoplanetary disk remain poorly understood despite several decades of
astrophysical research. Furthermore, the continued
discovery of extrasolar planets (over two hundred currently) further
emphasizes the need to understand the formation of not only our own solar
system, but of these numerous strange and diverse systems as well.
At the forefront still remains the basic question of
how growth occurs from dust to planet. The key properties of protoplanetary
nebulae remain controversial, yet grain growth from dust to larger
agglomerates
($\sim $ cm-m), and from planetesimals ($\gtrsim$ km) to planets must have
occurred in some manner. While numerous studies of $N$-body dynamics have
addressed the growth of terrestrial planets from asteroid-size planetesimals
which are largely decoupled from the gas \citep[e.g.,][]{lei05,bro06},
it is the transition from agglomerates to
planetesimals which continues to provide the major stumbling block in
planetary origins \citep{wei97,wei02,wei04,wei93,ste97,dul05,cuz06,dom07}.
Grain growth begins with sticking of sub-micron sized grains, which are
dynamically coupled to the nebula gas and collide at
low relative velocities which are
size-dependent \citep{wei93}. These relative velocities can be caused
by a variety of mechanisms, such as Brownian motion for the smallest grains,
differences in coupling to eddies in a
turbulent nebula \citep{vol80,wei84,mar91,wei93,cuz03,orm07},
and/or vertical, radial, and azimuthal drift driven by global gas pressure
gradients
\citep{nak86,wei97}. For the
smallest sized grains and aggregates, sticking is caused by weak van der
Waals interaction forces \citep[although stronger forces may also act; see,
e.g.,][]{dom07}. This sticking forms larger and larger aggregates
until particles grow large enough to decouple from the gas and
settle to the midplane where they may, barring strong turbulence
\citep[although see][for an exception]{joh07}, grow into
larger objects, even planetesimals \citep{saf69,wei93,cuz93,wei97}.
There is considerable observational evidence supporting dust growth
in protoplanetary disks, at least from sub-micron to millimeter scales, based
on observations of the dust continuum emission originating from protoplanetary
disks. The wavelength dependence of dust emission from IR to radio wavelengths
suggests, in many cases, that typical grain sizes are in the millimeter range,
which is orders of magnitude larger than interstellar grains
\citep{bec91,dul05,dom07}.
Measurements of the contrast of the Orion trapezium
silicate emission feature at 10 $\mu$m relative to the local continuum implies
an increase in size from interstellar dust (sub-micron) to micron sizes in the
disk photospheres \citep{van03,pry03,kes06}.
Theoretical approaches to the study of dust coagulation normally involve
solving the cumbersome collisional coagulation equation (see Eq. 1) at each
spatial location ($R,\phi,Z$) in the disk.
The collisional kernel $K(m,m^\prime)$ for particles of mass $m$ and $m^\prime$
usually involves some sort of sticking efficiency, and particle cross-sectional
area, in addition to the particle-particle
relative velocities. Although the {\it systematic} particle-particle
velocities that arise from
gas pressure gradients \citep{nak86} are
somewhat easy to incorporate into the collisional kernel,
relative velocities in turbulence are more complicated.
If one assumes that nebula turbulence follows a Kolmogorov inertial range
\citep{vol80} in which energy
cascades from the largest, slowly rotating eddies, down to some minimum
lengthscale where the intrinsic molecular viscosity can dissipate the
macroscopic gas motions (and turbulence ceases), one can obtain closed-form
expressions for the particle-particle, and particle-gas
turbulent relative velocities \citep{cuz03}, even for particles
of different sizes \citep{orm07}. Numerical
models of particle growth in the inner regions of protoplanetary disks
indicate that inclusion of these turbulence-induced relative
velocities is essential \citep{dul05}, as it provides a
source of energetic collisions that, in the presence of fragmentation
and destruction, leads to the
ongoing existence of small grains even after several million years.
A full scale solution to the problem of dust coagulation for a size spectrum
at every vertical and radial location in the protoplanetary disk, as a
function of time,
is thus prohibitive even with today's computational capabilities
\citep{wei02,wei04}. So it is
advantageous to find alternative methods of extracting information from the
coagulation process, such as time scales of growth, the largest particle
size, the size that dominates the area (radiative transfer), and the size
that dominates
the mass (migration and redistribution) without having to track the behavior
of the entire mass distribution.
In
this paper, we describe a method that utilizes the moments of the coagulation
equation to accomplish this task, in particular when
the functional form of the mass distribution can be assumed. In {\S} 2,
we derive the
moment equations from the coagulation equation, and compare
direct integration of the coagulation equation with solutions using
the moments approach in the case of a simple turbulent coagulation kernel.
We also present
two different approaches to obtaining solutions using a realistic kernel,
when the form of the size distribution is assumed to be a powerlaw.
In {\S} 3, we evaluate the different approaches developed in {\S} 2
for realistic kernels by numerical integration of the moments equations
and compare these moments to those obtained by direct integration of the
coagulation equation.
We also
compare the moments solution to an alternate analytical approach by
\citet{gar07}. In {\S} 4, as a demonstration of the practical application
of
the moments method, we calculate the wavelength-independent opacity, and
discuss generalization to wavelength-dependent Planck or Rosseland
mean opacities. In {\S} 5, we indicate how porosity may be included.
In {\S} 6, we present our conclusions.
\section{Solution Techniques Using the Moments Method}
The moments method approach to modeling particle growth allows an attractive
alternative to solving the full coagulation equation when it is only necessary
to know a particular property, or properties of the evolving particle size
distribution. The coagulation equation \citep{smo16} in its
integro-differential form is given by
\begin{eqnarray}
\frac{df(m,t)}{dt} = \frac{1}{2}\int_{0}^{m} K(m-m^{\prime},
m^{\prime}) f(m-m^{\prime},t) f(m^{\prime},t) dm^{\prime} - \nonumber \\
\int_{0}^{\infty}
K(m,m^{\prime})f(m,t) f(m^{\prime},t) dm^{\prime} \,\,\,+ \,\,\,
\frac{df}{dt}\Big|_{+}\,\,\,+\,\,\, \frac{df}{dt}\Big|_{-}\,\,,
\end{eqnarray}
\noindent
where $f(m,t)$ is the particle number density per unit mass at mass $m$, and
$K(m,m^{\prime})$ is the collision kernel connecting the properties of
interacting
masses $m$ and $m^\prime$, which typically takes into account mutual particle
cross section $\sigma(m,m^\prime)$, relative velocity $\Delta V(m,m^\prime)$,
and a particle sticking efficiency $S(m,m^\prime)$. The last two terms on
the RHS of Eq. (1) represent sources and sinks such as particle
erosion, fragmentation and subsequent redistribution of the fragmented
population, and gravitational growth. Although we will not specifically
address these mechanisms, we will indicate how they may be treated, and
save implementation for a forthcoming paper.
The
motivation behind using the moments method is to avoid the computational cost
inherent in Eq. (1) which entails solving the convolution integral (the first
term on the RHS of Eq. 1) at every location, and at every time step.
Depending on the functional form of the kernel, the second integral on the
RHS may also need to be integrated repeatedly. Given
that the typical mass spectrum may involve $10^2-10^3$ bins to acquire the
desired accuracy, the computational burden required becomes a detriment to any
study involving a wide range of parameter space. The
so-called brute force solution of the coagulation equation thus becomes the
primary bottleneck in running 2D evolutionary models over extended periods of
time \citep{wei04,dul05}.
We define the $p$-th moment $M_p$ of the distribution, where $p$ need not be
an integer, as
\begin{equation}
M_p = \int_{0}^{\infty} m^p f(m,t) dm,
\end{equation}
\noindent
where the units of $f(m,t)$ are such that $M_0=\int f(m,t)\,dm$ represents
the total number
density of particles, and $M_1 = \int mf(m,t)\, dm = \rho$ is the
total volume mass density
of solids. The essence of the moments method is as follows. We multiply both
sides of the coagulation equation (Eq. 1) by $m^k$, where $k$ is an integer,
and then integrate both sides over mass $m$:
\begin{eqnarray}
\frac{dM_k}{dt} = \int_{0}^{\infty} m^k \frac{df}{dt} dm =
\frac{1}{2} \int_{0}^{\infty} m^k\,dm \int_{0}^{m} K(m-m^{\prime},
m^{\prime}) f(m-m^{\prime},t) f(m^{\prime},t) dm^{\prime} - \nonumber \\
\int_{0}^{\infty} m^k f(m,t)\,dm \int_{0}^{\infty}
K(m,m^{\prime}) f(m^{\prime},t) dm^{\prime}.
\end{eqnarray}
\noindent
We introduce a step function $H(m-m^\prime)$, such that $H = 0$ for
$m -m^\prime < 0$ and $H = 1$ otherwise, to extend the limits of the
integral over $m^\prime$ from ($0,m$) to ($0,\infty$). The convolution
integral (the first integral on the RHS of Eq. 3) then becomes
\begin{eqnarray}
\frac{1}{2}\int_{0}^{\infty} m^k\,dm \int_{0}^{\infty}
H(m-m^\prime) K(m-m^\prime,m^\prime) f(m-m^\prime,t) f(m^\prime,t)\,
dm^\prime = \nonumber \\
\frac{1}{2} \int_{0}^{\infty} f(m^\prime,t)\,dm^\prime \int_{0}^{\infty}
(u+m^\prime)^k K(u,m^\prime) f(u,t)\,du,
\end{eqnarray}
\noindent
where on the RHS of Eq. (4) we have switched the order of integration and
made the substitution $u = m - m^\prime$. The RHS side of Eq. (4) may then be
combined with the last double integral in Eq. (3) to yield the set of
ordinary differential equations (ODEs)
for the integer moments \citep{mar01}:
\begin{equation}
\frac{dM_k}{dt} = \int_{0}^{\infty} \int_{0}^{\infty} \left[\frac{1}{2}
(m + m^\prime)^k - m^k\right] K(m,m^\prime) f(m,t) f(m^\prime,t) dm dm^\prime,
\end{equation}
\noindent
where we have substituted $m$ for $u$ with no loss of generality.
Depending on the mass dependence of the kernel (as illustrated below)
the right hand side can readily be
expressed as products of moments of order $k$ or less, leading to a closed
system of equations which may be solved using standard techniques.
In this definition of the coagulation equation in which it is assumed there
are no sources and sinks, the first
moment $M_1 \equiv \rho$ is constant in time, that is
$dM_1/dt = 0$\footnote{We note that under realistic protoplanetary nebula
conditions, $\rho$ will not be constant due to, e.g., size-dependent advection
terms in the equations. Such effects can be treated separately from the
``coagulation'' step. See {\S} 2.2.2.}.
Exact solutions to the coagulation equation have been obtained for
some specific choices
of the kernel \citep[e.g.,][]{smo16,tru71}, the most simple
being that of constant kernel $K(m,m^\prime) = \beta_0$. Although the exact
solution for $f(m,t)$ cannot be obtained from the moments equations, the
time rate of
change of the zeroth moment ($k=0$), can easily be obtained from Eq. (5)
which reduces to $dM_0/dt = - (1/2)\beta_0M_0^2$ and has the trivial solution
\citep[see, e.g.,][]{sil79}
\begin{equation}
M_0(t) = \frac{M_0(0)}{1 + (1/2)\beta_0M_0(0)t},
\end{equation}
\noindent
independent of the initial choice of distribution $f(m,0)$. Likewise, the
ODE for $M_2$, which can be associated with the density-weighted mean particle
size ($\left<m\right> = M_2/M_1$), yields the simple solution
$M_2(t)=M_2(0) + \beta_0\rho^2t$. Despite not
knowing the exact solution, we are able
to understand the behavior of general properties of the
mass distribution with time through the moments equation without
tracking the behavior of the full mass spectrum. Thus,
if it is only desired to know, for example, the time evolution of the
particle representing most of the {\it area} (first moment) or most of
the {\it mass} (second moment) of the distribution, then the advantage of
the moments method becomes clear. A small
number of moments is all that is necessary to determine the behavior of the
system. In particular, we will be interested in the size of the largest
particle $m_L(t)$ in the entire mass distribution as a function of time
(which we show below can be computed from the integer moments),
because these are usually the most rapidly drifting and most violently
colliding particles \citep{cuz06}.
\subsection{Example: Saffman and Turner Turbulent Coagulation Kernel}
We can illustrate the moments method approach using a very simple turbulent
coagulation kernel
where $K(m,m^\prime) = \gamma_0 (m^{1/3} + m^{\prime 1/3})^3$ \citep{saf56},
and no sources or sinks. Physically, this represents the
product of $(r + r^{\prime})^2$ for
area, and $(r + r^{\prime})$ for relative velocity of two particles in a
laminar shear flowfield. If a sticking coefficient were desired then we
would specifically have $K(m,m^\prime) = \gamma_0(m^{1/3}+m^{\prime 1/3})^3
S(m,m^\prime)$ where $0\leq S(m,m^\prime) \leq 1$. Here, $\gamma_0$ is a
constant that depends on the Reynolds number of the gas flow.
Inserting this kernel into Eq. (5), we find the set of equations
\begin{eqnarray}
\frac{dM_0}{dt} = -\gamma_0(M_0M_1 + 3M_{1/3}M_{2/3}), \,\,\,\,\,
\frac{dM_1}{dt} = 0 \nonumber \\
\frac{dM_2}{dt} = 2\gamma_0(M_2M_1 + 3M_{4/3}M_{5/3}).
\end{eqnarray}
\noindent
We note that the physical derivation of kernels in terms of particle radius $r$
leads to {\it fractional}
moments in terms of $m$. These fractional moments look complicated, but can
be solved for by simple
interpolation using Lagrange polynomials in terms of the more familiar integer
moments \citep{log79,pre92,mar01}.
Thus, any
fractional moment $M_p$ may be expressed compactly in the normalized form
$M^\prime_p(t) = M_p(t)/M_p(0)$:
\begin{equation}
M^\prime_p(t) = \prod_{j=k}^{k+n} \left[M^\prime_j(t)\right]^{L^n_j(p)},
\end{equation}
\noindent
where $n$ is the number of integer moments, $k = 0,1,...,n$,
and the exponent $L^n_j$ is defined as
\begin{equation}
L^n_j(p) = \frac{1}{n!}{\prod}_{n+1}(p)\frac{(-1)^{n-j}C^j_n}{p-j},
\end{equation}
\noindent
with ${\prod}_{n+1}(p) = p(p-1)...(p-n)$, and the $C^j_n$
are the binomial coefficients $n!/j!(n-j)!$. The
important thing to
note here is that the order of the moments must remain less than or equal
to the largest moment in order for the system to remain closed. In general,
this will be true for realistic collision kernels (see {\S} 2.2).
To test the accuracy of the moments method, we integrated equations (7) using a
fourth order Runge-Kutta scheme, and compared the results to a brute force
integration of the coagulation equation (Eq. 1). For the latter, we
integrated the distribution function $f(m,t)$ at each timestep
to determine the moments of the distribution $M_0$, $M_1$, and $M_2$. We chose
$f(m,0) = c_0m^{-q}$ as our initial distribution for simplicity. Since the
form of the mass distribution is only assumed at $t=0$, we consider this to
be an example of an implicit assumption (see {\S} 2.2.2).
The results of this calculation
are presented in Figure 1. We have used two different resolutions for the
brute force calculation in order to demonstrate how the higher resolution
(and much more computationally expensive) case (solid symbols) approaches the
moments approach solution. Notice that the first moment $M_1 = \rho$ remains
constant as expected. Given that
general numerical errors can arise from the finite mass grid and coarse
timesteps ($\Delta t = 10$ years) used for
the calculation, and that systematic errors may be
introduced by the Lagrangian interpolation, the fit is quite good.
It is important
to point out here that while
the coagulation calculation required as many as $20-30$ hours of CPU time to
complete on a 2 GHz machine, the moments calculation of Eq. (7) is
essentially instantaneous.
\subsection{Realistic Coagulation Kernels}
The Saffman-Turner turbulent coagulation kernel we used as an example in
{\S} 2.1
is simple to utilize and is expressible explicitly in powers of the mass $m$.
In practice, however, the realistic coagulation kernels that we will be using
will be more complicated than this simple example. A realistic
coagulation kernel will be, at the very least, a product of a mutual cross
section $\pi (r + r^\prime)^2$ and a relative velocity
\begin{equation}
\Delta V(m,m^\prime) = \sqrt{ (U - U^\prime)^2 + (V - V^\prime)^2 +
(W - W^\prime)^2 + v_T^2(m,m^\prime)},
\end{equation}
\noindent
where $U(m)$, $V(m)$, and $W(m)$ are the $x,y,z$ systematic (pressure
gradient driven) particle
velocities, and $v_T$ is the turbulent velocity contribution
which in general is not separable into distinct functions of
$m$ and $m^\prime$ \citep{cuz03,orm07}. The
systematic velocities,
which are derived for a discretized particle size distribution in the
appendix, are complicated functions of the particle size through the
stopping time $t_s$,
\begin{equation}
t_s = \frac{m\Delta V_{pg}}{F_D},
\end{equation}
\noindent
where $\Delta V_{pg} = |\vec{\bf{V}} - \vec{\bf{v}}|$ is the relative
velocity between the particle and the gas, and
$F_D$ is the drag force on the particle of size $r$ which depends on the size
of the particle relative to the mean free path $\lambda$ of the gas molecule
\citep[e.g., see][]{cuz93}. Depending on whether $r \gtrsim
\lambda$ or $r \lesssim \lambda$ determines whether the stopping time itself
depends on the instantaneous relative velocity $\Delta V_{pg}$
between the particle and the gas (Stokes regime) or does not (Epstein regime).
In the latter case, calculation of the systematic velocities ($U,V,W$) and
gas velocities ($u,v,w$) for a particle size distribution is
straightforward. In the former case, iterations are required to correctly
calculate $\Delta V_{pg}$ (see appendix).
The turbulent velocities are less straightforward to implement than their
systematic counterparts because of the different coupling that exists
between particles and eddies of different sizes (and thus stopping and decay
times, respectively). It has not been until very recently that closed form
expressions for the turbulence-induced velocities (for a particle size
distribution) were derived \citep{orm07}. These expressions can be written
separately for the so-called ``class I'' and ``class II''
eddies\footnote{The concept of eddy classes were introduced by \citet{vol80}
to distinguish between slowly and rapidly varying eddies. Class I eddies
are defined as those in which the eddy fluctuates slowly enough that a
particle's
stopping time $t_s$ is much shorter than the eddy decay time and the time
it takes
to cross the eddy; thus, they will align themselves with the gas motions of
the eddy prior to its decay.
Class II eddies then are defined as ones in which the eddy decay time is fast
compared to the particles $t_s$, and thus only provide a
small perturbation to the particle motion.} as:
\begin{equation}
\Delta V_I^2 = v_g^2\frac{{\rm{St}}_i-{\rm{St}}_j}{{\rm{St}}_i+{\rm{St}}_j}
\left(\frac{{\rm{St}}^2_i}{{\rm{St}}^*_{ij}
+{\rm{St}}_i} - \frac{{\rm{St}}^2_i}{1+{\rm{St}}_i} - \frac{{\rm{St}}^2_j}
{{\rm{St}}^*_{ij}+{\rm{St}}_j} +
\frac{{\rm{St}}^2_j}{1+{\rm{St}}_j}\right),
\end{equation}
\begin{equation}
\Delta V^2_{II} = v_g^2\left( 2({\rm{St}}^*_{ij} - {\rm{Re}}^{-1/2}) +
\frac{{\rm{St}}^2_i}{{\rm{St}}_i+{\rm{St}}^*_{ij}} - \frac{{\rm{St}}^2_i}
{{\rm{St}}_i+
{\rm{Re}}^{-1/2}} + \frac{{\rm{St}}^2_j}{{\rm{St}}_j+{\rm{St}}^*_{ij}} -
\frac{{\rm{St}}^2_j}{{\rm{St}}_j+{\rm{Re}}^{-1/2}}\right),
\end{equation}
\noindent
where $v_T^2 = \Delta V^2_I+\Delta V^2_{II}$, ${\rm{Re}}$ is the Reynolds
number, $v_g = \alpha^{1/2}c_g$ is the turbulent gas velocity with $c_g$ the
sound speed \citep[e.g.,][]{nak86,cuz06}, and the particle Stokes numbers
${\rm{St}}_i = t_{{\rm{s}}i}/t_L$, where $t_L$ is the turnover time of the
largest eddy, typically
taken to be $\Omega^{-1}$. The boundary between class I and class II is
defined by the ``combined'' Stokes number
${\rm{St}}^*_{ij} = {\rm{max}}({\rm{St}}^*_i,{\rm{St}}^*_j)$
and the values of the ${\rm{St}}^*_k$ are obtained from the
equation \citep{orm07}:
\begin{equation}
\frac{2}{3}y_k^*(y_k^*-1)^2 - \frac{1}{1+y_k^*} = -\frac{{\rm{St}}_k}
{1+{\rm{St}}_k} + \frac{1}{{\rm{St}}_k}\frac{\Delta V^2_{pg}}{v^2_g},
\end{equation}
\noindent
where $y^*_k = {\rm{St}}^*_k/{\rm{St}}_k$.
In our calculations, we will make use of these expressions when comparing
cases with turbulence-induced velocities.
Given that the moments method requires that we
express the kernel in terms of fractional or integer moments, the problem
becomes expressing the relative velocity (Eq. 10) in a form that satisfies
this requirement. As an
example, the systematic velocities $U$, $V$, and $W$ can each be fit easily
by a finite series in fractional powers $p_i$ of $m$
as $U(m) = \sum^N_i a_i m^{p_i}$ where the
coefficients $a_i$ can be found by fitting $N$ points to the function
$U(m_l) = U_l$ ($l$ an integer). This leads to the system of equations
\begin{equation}
U_l = \sum_{i=1}^N a_i m_l^{p_i}.
\end{equation}
\noindent
The coefficients $a_i$ then
follow from matrix inversion. Similarly, one can find expressions
$V(m) = \sum_i^N b_i m^{p_i}$, and $W(m) = \sum_i^N c_i m^{p_i}$
so that we may
construct the laminar expression for $(\Delta V)^2$ in terms of the variables
$m$ and
$m^\prime$ by multiplying out the individual terms, e.g.,
$U(m)U(m^\prime) = \sum_{i,j} a_ia_j m^{p_i}m^{\prime p_j}$ and so on. We
find then that we can express the square of Eq. (10) as
\begin{equation}
(\Delta V(m,m^\prime))^2 = \sum_{i,j} A_{ij} \left[m^{p_i+p_j} -
2m^{p_i}m^{\prime p_j} + m^{\prime p_i+p_j}\right],
\end{equation}
\noindent
where $A_{ij} = a_ia_j + b_ib_j + c_ic_j$, and $p_i+p_j\le 2$ for closure.
This determines the matrix of
coefficients $A_{ij}$ which operate on the finite power series in
$m$ and $m^\prime$. Although we cannot use the same approach for solving
for $\Delta V$ in
Eq. (10) because of the radical, we were motivated to express $\Delta V$ in
a similar form, i.e., as
$\Delta V(m,m^\prime) = \sum_{i,j} A^\prime_{ij} [m^{p_i+p_j} -
2m^{p_i}m^{\prime p_j} + m^{\prime p_i+p_j}]$, and $p_i+p_j\le 1$.
The problem reduces to solving for the
coefficients $A^\prime_{ij}$ directly, by similar matrix
inversion techniques. Unfortunately, the number of points needed to obtain
an accurate representation of the two-variable function
$\Delta V(m,m^\prime)$ this way
exceeds the limitations of the inversion. In
practice we found that we could solve a system for $\Delta V$
with at most
$5-6$ points ($25-36$ matrix elements),
whereas Eq. (16) carries much less restriction in the
number of points needed because we could construct $(\Delta V)^2$
by the product of accurate representations of single variable functions.
Furthermore, due to the coupling evident in
the turbulence induced velocities (Equations 12 and 13), the turbulent
velocities are not separable functions of the masses $m$ and $m^\prime$,
further complicating the analysis.
{\it Powerlaw assumption}: Fortunately, it turns out that, although
the direct calculation approach to $\Delta V$ described above would
be mathematically appealing in the sense that the moment equations would
remain implicit (i.e., the form of the mass distribution would only
be assumed at $t=0$), it is not necessary. In the next sections
we describe two alternative approaches to dealing with realistic coagulation
kernels that employ the moments method under the assumption that the
form of the mass distribution $f(m,t)$ is a powerlaw.
Such an assumption is
not entirely unfounded. A number of detailed models by
\citet{wei97,wei00,wei04}
have shown that powerlaw size distributions result, which have
nearly constant mass per decade radius to an upper limit $m_L(t)$ which
grows with time until the frustration limit of around a meter is reached, and
(under turbulent conditions, at least)
growth stalls \citep{cuz06}. Similar trends are found by \citet{dul05} in
which different assumptions about collisional ejecta are made. These authors
do find minor fluctuations in the distribution, but if one is primarily
interested in general properties of the distribution, such as
how the largest particle size changes with time, and not the fine
structure of the
mass distribution itself, the approach is quite advantageous. There are reasons
to believe that a powerlaw is a natural end-state, especially those
with equal mass per decade, because they have self-preserving properties for
the collision kernels of interest \citep{cuz06}.
The significance of the upper mass cutoff $m_L$ depends on the assumed slope
of the powerlaw. In all real distributions, there will be a rapidly decreasing
abundance of particles for masses exceeding some threshold, even though the
abundance may not drop immediately to zero as in our assumed model. The rare,
extremely large particles might be of interest for some applications, but
our focus will be the particle size carrying most of the area, or most of
the mass (the first or second moments). For powerlaw distributions
$f(m) \propto m^{-q}$ with $q < 2$, $m_L$ is itself the mass of the particle
carrying most of the mass and is independent of the selection of a lower
particle size cutoff. For $q=2$, the distribution contains equal mass per
decade, and for $q>2$, most of the mass is in the small particles and the
value of $m_L$ depends on the lower particle size chosen. Most realistic
distributions, and those most commonly treated in the literature, have $q<2$;
here we assume $q=11/6$, a widely used fragmentation powerlaw. In this case,
a strict cutoff at $m_L$ represents a well-defined distribution with
easily-understood moments where most of the mass is at $m_L$ and the area
is nearly equally distributed per decade with a mass dependence $m^{-1/6}$.
Although we do not include either imperfect sticking or fragmentation at this
stage in the model, we believe that the moment equations as expressed in
Eq. (5) will remain valid up to a "fragmentation barrier", which may be
defined as that size for which the typical disruption energy of a particle
is on the same order as the energy of identical colliding particles.
The fragmentation barrier will also depend on one's choice of
nebula parameters. This treatment (up to the fragmentation size) is
consistent with recent work by \citet{bra08} (their Fig. 13) which shows
a constant powerlaw mass distribution up to a cutoff size which then falls
off abruptly. This ``knee'' in the distribution represents that efficient
fragmentation size.
Once $m_L$ reaches the fragmentation barrier, growth beyond this
stage would need to be treated in a different manner, for
example, by simple sweepup of small, less disruptive particles by large
ones \citep{cuz93}. Creation and disruption of these
particles can be handled as part of the source and sink terms in which
for example, disrupted particles are assumed to be fragmented back into
a powerlaw distribution (which is suggested by experimental evidence and
widely assumed by other modelers) as opposed to monomers. The model as
presented within this paper, however, should be useful for the early
stages of protoplanetary nebula particle growth relevant to spectral
energy distributions and MRI suppression. We leave the incorporation of
growth stages beyond the efficient fragmentation stage for a later paper.
\subsubsection{Approach 1: Explicit Assumption}
Motivated by our discussion of the previous section, if we assume that the
form of the particle mass distribution function remains a powerlaw
at all times, we may express $f(m,t) = c(t)m^{-q}$, such that
$m_L(t)$ is the growing upper limit of the distribution, $q$ is the slope
which is assumed to be constant (although see below), and
$c(t)$ is a normalization coefficient. Taking the lower limit of the mass
distribution to be $m_0$,
then moments as expressed in Eq. (2) are explicitly given for $q-p\neq 1$ by
\begin{equation}
M_p(t) = \frac{c(t)}{1 + p -q}\left(m_L(t)^{1+p-q} - m_0^{1+p-q}\right).
\end{equation}
\noindent
We then take the time derivative of Eq. (17) for the zeroth and second
moments, and substitute these expressions on the LHS of the corresponding
moment equations in Eq. (5) to get
\begin{equation}
\frac{m_L^{1-q} - m_0^{1-q}}{1-q}\frac{dc}{dt} + cm_L^{-q}\frac{dm_L}{dt} =
-\frac{1}{2}c^2\Gamma_0(m_L),
\end{equation}
\begin{equation}
\frac{m_L^{3-q} - m_0^{3-q}}{3-q}\frac{dc}{dt} + cm_L^{2-q}\frac{dm_L}{dt} =
c^2\Gamma_2(m_L),
\end{equation}
\noindent
where Eq. (18) is valid for $q\neq 1$, and $\Gamma_0$ and $\Gamma_2$ are the
integrals on the RHS of Eq. (5) for $k=0$ and $k=2$, respectively. That is
\begin{equation}
\Gamma_0(m_L) = \int_{m_0}^{m_L(t)}\int_{m_0}^{m_L(t)}
K(m,m^{\prime})m^{-q}m^{\prime -q}\,dm\,dm^\prime,
\end{equation}
\begin{equation}
\Gamma_2(m_L) = \int_{m_0}^{m_L(t)}\int_{m_0}^{m_L(t)}
K(m,m^{\prime})m^{1-q}m^{\prime 1-q}\,dm\,dm^\prime,
\end{equation}
\noindent
where the kernel is given by
$K(m,m^\prime) = \sigma(m,m^\prime)\Delta V(m,m^\prime) S(m,m^\prime)$, with
$\sigma(m,m^\prime) = K_0(m^{1/3}+m^{\prime 1/3})^2$,
$K_0 = \pi(3/4\pi\rho_s)^{2/3}$, and $\rho_s$ is the particle material
density, assumed
constant. Note that for the special case of $q=1$, Eq. (18) has a slightly
different form which depends on $\ln{(m_L/m_0)}$.
After some simple algebra, Eq.'s (18) and (19) can be written as
\begin{equation}
\frac{dm_L}{dt} = c\left[\frac{(3-q)(m_L^{1-q}-m_0^{1-q})\Gamma_2 +
\frac{1}{2}(1-q)(m_L^{2-q}-m_0^{2-q})\Gamma_0}{(3-q)(m_L^{1-q}-m_0^{1-q})
m_L^{2-q} - (1-q)(m_L^{3-q}-m_0^{3-q})m_L^{-q}}\right],
\end{equation}
\begin{equation}
\frac{dc}{dt} = -c^2\left[\frac{(3-q)(1-q)(m_L^{-q}\Gamma_2 +
\frac{1}{2}m_L^{2-q}\Gamma_0)}{(3-q)(m_L^{1-q}-m_0^{1-q})
m_L^{2-q} - (1-q)(m_L^{3-q}-m_0^{3-q})m_L^{-q}}\right],
\end{equation}
\noindent
which we integrate using a fourth order Runge-Kutta scheme. Equation (17) then
gives the moments $M_0$ and $M_2$ as a function of time, which can be
directly compared with direct integration of the same conditions
using the coagulation equation (Eq. 1).
The advantage of this approach (in which a powerlaw is assumed at all
times) is its transparency; that is, the variables being sought
($m_L$ and $c$) are solved
for directly. Furthermore, the change in the coagulation kernel as the
particle size distribution changes is included because the kernel is
updated and explicitly integrated into $\Gamma_0$ and $\Gamma_2$ with every
time step. This will prove
advantageous when additional effects such as sticking
are included in the kernel. In addition, source and sink terms need not
parameterized in terms of integer moments and can be implemented directly.
An unfortunate disadvantage of this approach is
that because the kernel must be integrated (in fact several times) over
both $m$ and $m^\prime$ every
time step to get $\Gamma_0$ and $\Gamma_2$, the CPU time involved is
significantly longer than a fully
implicit case (e.g., {\S} 2.1; also see {\S} 2.2.2); however, it
remains a much faster approach (orders of magnitude) than solving Eq. (1)
directly since the
cumbersome convolution has been eliminated.
\subsubsection{Approach 2: Semi-implicit Assumption}
Here, we present an alternative moment-based approach to solving the realistic
coagulation kernel in which the integer moments appear directly.
Unlike the previous case of {\S} 2.2.1, where the double integral of
Eq. (5) is used to get the functions of $m_L$, $\Gamma_0$ and $\Gamma_2$,
we only integrate over one mass
variable (i.e., only integrate one of the integrals), defining the
functions
\begin{equation}
C_0(m,t) = \int_{m_0}^{m_L} K(m,m^\prime)f(m^\prime,t) \,\,
dm^\prime,
\end{equation}
\begin{equation}
C_2(m,t) = \int_{m_0}^{m_L} m^\prime K(m,m^\prime)f(m^\prime,t)
\,\,dm^\prime,
\end{equation}
\noindent
where the form of the kernel $K(m,m^\prime)$ is the same that in {\S} 2.2.1.
We then fit $C_0$ and $C_2$ with a finite series in fractional
powers of $m$ in the same manner as given in Eq. (15). Substituting these
functions in place of one of the integrals (say over $m^\prime$), we may
integrate over $m$ to get
\begin{eqnarray}
\frac{dM^\prime_0}{dt} = -\frac{1}{2M_0(0)}\int_{m_0}^{m_L(t)}
f(m,t)C_0(m,t)\,dm = -\frac{1}{2}\sum_i a_i \mu_{p_i}M^\prime_{p_i} \nonumber \\
\frac{dM^\prime_2}{dt} = \frac{1}{M_2(0)}\int_{m_0}^{m_L(t)} m f(m,t)C_2(m,t)
\,dm = \sum_i b_i \nu_{p_i+1}M^\prime_{p_i+1}, \nonumber \\
M^\prime_{p_i} = \left[M^\prime_0\right]^{\frac{1}{2}(p_i-1)
(p_i-2)}\left[M^\prime_1\right]^{-p_i(p_i-2)}\left[M^\prime_2\right]^
{\frac{1}{2}p_i(p_i-1)},
\end{eqnarray}
\noindent
where we have made use of equations (8) and (9) to express the solution in
terms of the integer moments $k = 0,1,2$. Here, $\mu_{p_i} = M_{p_i}(0)/
M_0(0)$, $\nu_{p_i+1} = M_{p_i+1}(0)/M_2(0)$, $M^\prime_k = M_k(t)/M_k(0)$,
and $0\leq p_i \leq 1$. The $C_k$ are fairly smooth functions over a large
range of particle radii;, however, the accuracy of fitting a single series in
fractional moments over a very broad range of particle sizes
(i.e., over many orders of magnitude) may drop off significantly as the
broadness of the range increases. This issue may be circumvented by
employing a piecewise fit to the integrated kernels $C_k$.
It is interesting to note that by this definition of $C_k$, we have
effectively accomplished what we set out to do in
our discussion at the beginning of {\S} 2.2, that is, defining the
coagulation kernel in terms of finite series in powers of the mass $m$.
The difference
here is that we have done so through the first integral of the kernel, and
not the kernel itself. This means that the mass distribution $f(m,t)$ is
expressed in the calulation of the $C_k$, but remains
implicit in the definition of the ODEs (Eq. 26). Thus, the method is
semi-implicit, because the RHS of the equations above can be
expressed in terms of the moments (as defined in Eq. 2). Equations (26)
are then integrated using the fourth order Runge-Kutta method, and may be
compared with the results of {\S} 2.2.1 and the direct integration of
Eq. (1).
This semi-implicit approach tracks
the evolving kernel through the integration of Equations (24) and (25)
after every timestep, thus the computational time involved is similar
to the explicit case. In order to update the kernel,
one may solve for the new $m_L$ after each $\Delta t$
using the equation ($q\ne 1$)
\begin{equation}
\frac{m_L-m_0}{m_L^{2-q}-m_0^{2-q}} - \frac{M_q}{(2-q)M_1} \simeq 0.
\end{equation}
\noindent
The new
normalization coefficient $c$ can then be found from the definition of
$M_1 = \rho$.
We then reintegrate Equations (24) and (25) under the powerlaw assumption,
and then proceed to fit the $C_k(m,t)$ with a finite series in fractional
powers of $m$. Although, in principle, any other two moments could be
used to obtain $m_L$, $M_q$, which lies between $M_2$ and $M_1$, and because
it roughly characterizes the evolution of the largest particle (see discussion
at the end of {\S} 2.2), seems the most
consistent choice. The $q$-th moment is calculated using the Lagrange
polynomial interpolation scheme (Eqs. 8 and 9).
The advantage of the moments method lies in the ability to express the
differential equations in terms of the moments of the
distribution (i.e., their integrated properties).
If a more explicitly
mass-dependent approach is adopted (as is the case in {\S} 2.2.1, and the
semi-implicit approach described here),
then the computational time significantly increases.
One can improve the speed of computation by calculating the $C_k$
periodically, or in the extreme case, only at $t=0$ which would make
the approach truly {\it implicit} (e.g., {\S} 2.1). The advantage of an
implicit approach
is that it becomes fully general
(the form of $f$ is only assumed at the onset), and
also in the time it takes to solve ($< 1$ minute). The bulk of the time is
spent in the
integration of equations (24) and (25) which would occur only once.
The disadvantage, of course, is that the particle velocity distribution is
not updated as it changes with time (due to, e.g., changes in the bounds
of the size distribution). We present examples of both extremes in {\S} 3.
If one wanted to implement a mass- or velocity-dependent sticking
coefficient $S$, it can readily be included in the integration to
obtain the $C_k$. The additional inclusion of source and sink terms
due to erosion, fragmention, or
gravitational growth in this semi-implicit formalism would require that we fit
these terms in a similar manner to the $C_k$ so that their subsequent
integration over all $m$ will yield sums over integer moments weighted by
different sets of coefficients. Caution must be exercised in fitting, e.g.,
the gravitational growth term to ensure
that the system of equations remains closed. Under such circumstances, a
fully explicit approach such as that of {\S} 2.2.1 may be preferred.
Alternatively, these effects may be included in particle-histogram space
in between coagulation interations.
Finally,
we should point out that allowing for other parameters (such as the index
of the powerlaw $q$)
to vary with time, does not pose a problem in either of
the approaches we have presented. In both cases, one would simply need an
additional moment, e.g. $M_3$, to determine $q(t)$. Similarly, a bifurcated
distribution in which the powerlaw exponent changes at a particular particle
size \citep[see, e.g.,][]{ken99} may also be studied.
In {\S} 3, we will compare the
two approaches to the direct integration of the coagulation equation for
cases in which there are
only systematic velocities ($v_T = 0$), as well as cases in which the
velocity differences are induced by turbulent motions.
\section{Numerical Results}
We carried out several calculations in order to demonstrate the accuracy
of each alternative method compared to the brute force integration of the
collisional coagulation equation. For the purposes of comparison,
unless otherwise noted, we chose the initial conditions to be a minimum mass
solar nebula at 1 AU at a height of $z=10^3$ km above the midplane, and a
particle size distribution with a minimum initial radius of 1 cm and
a maximum initial radius of 10 cm. The
powerlaw exponent $q = 11/6$, which is assumed to be constant in these
calculations, is representative of a fragmentation population.
Standard integrations were carried out with a timestep of $\Delta t = 10$
years.
\subsection{Laminar Case ($v_T = 0$)}
In Figure 2, we compare the case in which there are only systematic
(pressure-gradient driven) velocities
between particles ($v_T=0$). The solid curves represent the explicit
assumption (invariant powerlaw slope $q$ assumed in integration of
Equations 22 and 23), while the dashed curves represent the
implicit assumption. That is, in the integration of Eq. 26, the form of
the mass distribution $f$ is assumed only at $t=0$. As before, the symbols
represent the brute force calculation for two different grid resolutions
(solid = 100 bins, open = 1000 bins). Since with the explicit assumption we do
not solve for the moments specifically, the values for the solid curves were
obtained by substituting the time integrated values of $c(t)$ and $m_L(t)$
back into Eq. (17) for $k = 0,2$ only. This is because
although we have used the moment equations to obtain these results, the
explicit assumption of an invariant powerlaw size distribution means that
only the equations for the moments $M_0, M_2$ are needed (although see
{\S} 3.2). This is not true, however, for the implicit (and semi-implicit)
assumption, where
all the moments $M_0,M_1,M_2$ appear in the differential equations.
We see that there is excellent agreement between both approaches and the
numerical values obtained with the highest resolution case. In fact, the
agreement between the explicit and implicit assumptions is also
quite good. However, at this stage there has not been a great deal of
growth (the largest particle size in the distribution has only grown
to $r_L = 11$ cm in size by the end of the simulation for the explicit case).
Note that even though
in the implicit case the $C_k$ are calculated only once, the estimate for the
largest mass $m_L$ for the evolving distribution $f(m,t)$ is still calculated
using Eq. (27). At least in the case of minimal or slow growth, it appears
that an implicit approach, or even periodic calculation of the $C_k$, may be
sufficient.
Numerical glitches in the low resolution brute force case arise from the
interpolation
scheme for sampling the kernel. In particular, these glicthes are
likely enhanced because the systematic
relative velocities quickly approach zero
for identical particle sizes (with the effect much more prominent for larger
particles, hence not appearing so much in $M_0$). The higher resolution case
contains enough points to smooth out this effect.
Note that (in all simulations) the direct integration of the coagulation
equation gives a constant $M_1$
as is to be expected given that we found $dM_1/dt = 0$ in the
derivation of the moment equations in the absence of sources and sinks; this
further validates the numerics
of the brute force solution even
for the complicated collisional kernel being utilized, and also provides
a posteriori validation of our result that $M_1=\rm{constant}$ from,
e.g., Eq. (5), which further validates derivation of equations (24)
and (25) which was based on symmetry of the kernel.
\subsection{Turbulence Case ($v_T \neq 0$)}
Next, we explored cases in which the systematic velocities were set to zero
so that velocity differences between particles are due only to those induced
by turbulence. Depending on the magnitude of the turbulence parameter
$\alpha$, these induced velocities can be either large or small relative to the
systematic velocities. To demonstrate, we ran cases for three
different values $\alpha = 10^{-6},10^{-5}$, and $10^{-4}$. In the absence
of any mechanism to counter growth (e.g., fragmentation), larger $\alpha$
translates to faster rates of growth (due to larger relative
velocities).
In Figure 3 we plot the results for $\alpha = 10^{-4}$,
for the explicit (solid curves) and implicit (short dashed curves)
approaches. The growth rate is more rapid than in the laminar case, with
the second scaled moment $\sim 70$\% larger (compared to Fig. 2) at the
end of the run, meaning that the
largest particle size achieved is $r_L \sim 13.5$ cm. We note
that, as was the case for the $M^\prime_2$ curves in Fig. 2, the explicit
and implicit cases are very similar; however, this is not the case
for the $M^\prime_0$ curves. The explicit approach overestimates the the
value of $M^\prime_0$ compared to the highest resolution brute force
calculation, whereas the implicit approach significantly underestimates
the zeroth moment. Both approaches understimate the value of $M^\prime_2$.
When we compare integrations of $M^\prime_2$ for smaller
values of $\alpha$ as we do in Figure 4, we find that both approaches fit
the coagulation calculation quite well. The growth rate for these two
values of $\alpha$ are more in line with growth rate found when there are
only systematic velocities, so the agreement should not be surprising.
The long-dashed curves in Figures 3 and 4
are the implementation of the semi-implicit approach ({\S} 2.2.2) in which
the $C_k$ are updated at every time step. In this case, the semi-implicit
approach provides a much a much better fit to the brute force calculation,
indicating that the kernel is evolving fast enough that an implicit
approach cannot capture this effect.
We consider the slight discrepencies between the
semi-implicit approach and the brute force calculation to be as much a
result of grid resolution as inaccuracy in using the Lagrange polynomial
fits to the fractional moments. The explicit and fully implicit approach
values of $r_L$ apparently understimate the largest particle size
relative to the semi-implicit approach (which gives a value of
$r_L \simeq 14$ cm).
Finally, we explored a variation of the explicit approach in which
the condition $M_1 = \rho$ was strictly enforced (recall that
$M_1$ does not appear in Equations 18-19). This amounts to replacing
Eq. (18) (derived for $M_0$) with the corresponding equation for $M_1$.
As a result, we cannot simultaneously fit both $M_2$ and $M_0$.
Alternatively, we can use only Eq. (19) for the second moment, by substituting
$c = (2-q)\rho/(m_L^{2-q}-m_0^{2-q})$ (from the definition of $M_1$, Eq. 17)
in Eq. (19) so that Eq. (19) alone
determines the growth of the largest particle. The differential
equation for $m_L$ then becomes
\begin{equation}
\frac{dm_L}{dt} = \left[\frac{(3-q)(2-q)\rho\Gamma_2}
{(3-q)(m_L^{2-q}-m_0^{2-q})m_L^{2-q} - (2-q)(m_L^{3-q}-m_0^{3-q})
m_L^{1-q}}\right].
\end{equation}
\noindent
The results of the integration of Eq. (28) are shown as the dotted curves
in Figure 3 and for the $\alpha = 10^{-4}$ case in Figure 4. It is clear that
this modification provides a much better fit to $M_2$, perhaps even better
than the semi-implicit case above. However, using
$m_L(t)$ calculated from Eq. (28), and solving for $c(t)$ does a poor job
fitting $M_0$ (see Figure 3). Thus we conclude that although the
modification of the explicit approach matches the brute force calculation of
$M_2$ quite well, only the semi-implicit case is able to provide a
simultaneous fit to both the zeroth and second moments (under the
condition that $M_1=\rho = {\rm{constant}}$).
\subsection{A Model Comparison}
\citet{gar07} has developed a simplified analytical approach for dealing
with the
growth of particles in a turbulent regime, in which the particle size
distribution is parameterized by a powerlaw in particle mass with the same
exponent we have used in this paper ($q = 11/6$).
Similar to what we have presented in previous sections, the underlying
assumption of \citet{gar07} is that collisions between particles occur
frequently enough that a steady-state balance is reached in the form of
the particle
size distribution, but with an upper size cutoff that varies with time.
Thus, the model of \citet{gar07} is {\it explicit} and follows only two
parameters: the growth
of the largest particle $m_L(t)$, and the normalization factor $c(m_L,t)$.
Given the similarity of the underlying assumptions for the Garaud model
and the examples we have presented, it is a
useful exercise to compare the results of the two approaches directly.
The growth of the largest particle $r_L$ in the Garaud model (her Eq. 36),
expressed in terms of the notation used in this paper, is given by
\begin{equation}
\frac{dr_L}{dt} = \frac{\rho \Omega H}{\rho_s} \sqrt{\frac{\alpha
\gamma {\rm{St}}_L}{1 + 64{\rm{St}}^2_L(2 + 5{\rm{St}}_L^{-0.1})^{-2}}}.
\end{equation}
\noindent
In Eq. (29) ${\rm{St}}_L$ is the Stokes number for the largest particle
$r_L = (3m_L/4\pi\rho_s)^{1/3}$, $H=c_g/\Omega$ is the scale height of the gas,
$\gamma$ is the adiabatic index of the gas, and
in this expression it is assumed that the sticking coefficient $S = 1$, and
that $m_L>>m_0$. The above
equation is similar to the formula for grain growth proposed by \citet{ste97}
to factors of order unity. Finally, we point out that
the Garaud model for particle growth
is restricted to the value $q = 11/6$ in order to preserve its completely
analytical nature.
As a means of a fair comparison, we chose to compare the Garaud model to
our explicit case (Eq. 28) for reasons explained below. In the limit of
$m_L >> m_0$ and $q=11/6$, Eq. (28) becomes
\begin{equation}
\frac{dr_L}{dt} \simeq 0.05 \frac{\rho}{\rho_s}\frac{\Gamma_2}{m_L}.
\end{equation}
We present
the results of our comparison in Figure 5 for the same initial conditions
as described at the beginning of {\S} 3.2 (upper curves).
We find quite generally that our explicit calculation (Eqs. 28 and 30)
leads to a faster growth rate than what is predicted from the Garaud model,
initially. We note that
the Garaud expression steepens quickly for $r_L \gtrsim 13$ cm,
suggesting that the growth rates of the two approaches are more comparable at
later times.
However, the minor ``kink'' in
the long-dashed curve is due to the shift from Epstein to Stokes flow. A
much more subdued kink is visible in the explicit approach
which uses the full expressions for the turbulent velocity, whereas in the
derivation of Eq. (29), it is
assumed that the stopping time in the Stokes regime is defined
by some mean characteristic velocity which leads to a much more
noticeable discontinuity.
Regardless, the overall more subdued growth rate elicited by Eq. (29) (despite
the steepening at later times due to a shift in flow regimes) is apparent
from the lower set of curves in Fig. 5 where
the initial conditions were chosen with $r_L(0) = 1$ cm. For this case, growth
occurs only in the Epstein regime, but still, the curves for the explicit
approach
and that calculated from Eq. (29) begin to diverge. Thus,
the Garaud expression apparently underestimates the growth rate relative to
our approach.
We emphasize that the treatment of particle growth by \citet{gar07} requires
several approximations in order to derive a purely analytical expressions for
$dr_L/dt$. Besides the aforementioned restriction of $q=11/6$, Garaud
approximates the full expressions for the turbulent velocities that we
use here by partitioning $v_T$ into seperate cases dependent on
the particles' stopping time relative to the turnover times of the smallest
and largest scale eddies. Furthermore, some question may be raised as to the
comparability of the moment equations used here to derive Eq. (28), versus
the particle growth equation used by \citet[her Eq. 29]{gar07}.
The equation used by \citet{gar07}
is more akin to a ``sweep-up'' equation, with no sources or sinks, than to a
formal coagulation equation. Because
it bears some resemblance to the equation for $M_2$ (with some algebra,
to factors of order unity), we concluded that our Eq. (28) is the appropriate
analog. Despite the differences in growth rate, we find the agreement in
the general trend of growth of the two approaches reassuring.
\section{Opacity Calculations}
Evolutionary models of protoplanetary nebulae, giant planet atmospheres, etc.
must somehow treat the escape of thermal radiation \citep{pol96,hub05,dur07}.
In most cases, particles provide the primary opacity for these
models. Observations of these and similar objects often rely on Spectral Energy
Distributions (SEDs) which can be compared to a model once the model's internal
temperature distribution is known; clear evidence is seen for grain growth in
many cases \citep[see review by][]{nat07}. Because of the nearly
insurmountable
computational burden involved with performing a fully self-consistent
calculation of particle growth by coagulation along with an already difficult
fluid dynamical calculation, most modelers simply assume some invariant
particle size distribution, such as the MRN interstellar grain distribution, or
make arbitrary assumptions about particle growth \citep{hub05}.
In the simplest regime (monodisperse particle radius $r$ larger than a
wavelength), the particle opacity can be written as the area per unit
mass:
\begin{equation}
\kappa = \frac{3}{4 \rho_s r} \hspace{0.1 in} {\rm cm}^2\,{\rm g}^{-1};
\end{equation}
\noindent
thus growth in radius from 0.1$\mu$ to 1 mm leads to a factor of $10^4$ change
in opacity. To the degree that this wavelength-independent regime holds,
including particle size evolution by the moments method in one's
evolutionary models would allow a very simple way to track particle growth and
decreasing opacity. For instance, equation (31) above is easily generalized to
the area per unit mass integrated over the size distribution:
\begin{equation}
\kappa = \frac{\int \pi r^2 f(m)\, dm}{\int m f(m)\, dm}
= \left(\frac{9 \pi}{16 \rho_s^2}\right)^{1/3}\frac{M_{2/3}}{M_1}.
\end{equation}
\noindent
As an application of the moments method in Figure 6, we have calculated
the decrease
in opacity (given by Eq. 32) with time using the semi-implicit approach.
An initial particle size distribution with a lower bound of $r_0=0.1$ cm
and $q=11/6$ was used. Both the pressure gradient driven systematic
velocities and the
turbulence-induced velocities were used. In the absence of any mechanism to
hinder particle growth, larger values of $\alpha$ lead to more steeply
decreasing opacities with time.
In a regime where the particle extinction efficiency $Q(r,\lambda)$ is
wavelength-dependent (say, if the particles are comparable to or smaller than
the wavelength), one simply integrates $Q(r,\lambda)$ over the powerlaw mass
distributions resulting from the moments model. For example,
\begin{eqnarray}
\kappa_{\lambda} = \frac{\int_{m_0}^{m_L} \pi r^2 Q(r,\lambda) f(m)\, dm}
{\int_0^{m_L} m f(m)\, dm}
= \frac{1}{M_1}\int_{m_0}^{m_L} \pi r^2 Q(r,\lambda) c(m_L)m^{-q}\, dm
\nonumber \\
= \frac{c(m_L)}{M_1}\left(\frac{9 \pi}{16 \rho_s^2}\right)^{1/3}
\int_{m_0}^{m_L} Q(r,\lambda)m^{2/3 -q } dm.
\end{eqnarray}
\noindent
These opacities $\kappa_{\lambda}$ can be used to calculate Planck or Rosseland
(wavelength-averaged) means for use in radiative transfer models.
Recall that the powerlaw slope
$q$ can be freely adjusted within a small but plausible range to explore
different growth regimes.
\section{Porosity}
Fractal growth of particles by low-velocity sticking of small solid monomers
with radius $r_o$ and mass $m_o$ (and/or aggregates of such monomers) causes
them to have a density much less than the material density of the monomers
\citep{bec00,dom07,orm08}. These porous
particles can be described as fractals with dimension $D$, such that the
particle mass $m$ increases proportionally to $r^D$ where $r$ is some effective
radius and $D$ is the fractal dimension. Thus the particle's internal density
is a function of particle size:
\begin{equation}
\rho(r) = \frac{3 m }{4 \pi r^3} \sim \frac{3 m_o (r/r_o)^D}
{4 \pi r^3} = \frac{3 m_o}{4 \pi r_o^3}(r/r_o)^{D-3} = \frac{3 \rho_o}
{4 \pi}(r/r_o)^{D-3}.
\end{equation}
\noindent
For a typical situation where $D \sim 2$ \citep{dom07},
$\rho(r) \propto r^{-1}$ and thus the
product $r \rho(r)$ is a constant across a wide range of particle sizes (until
compaction sets in). These more complex but quite plausible particle
density-size relationships complicate the expressions for particle stopping
time and Stokes number ({\S} 2.2, Eq. 11; also, see appendix).
Because the particle stopping times enter in through the collisional kernel,
the fractal nature of particles can be accounted for using the method of
moments in a straightforward manner while maintaining the criterion for
closure of the system. We can verify this by noting that the mutual particle
cross section
\begin{equation}
\sigma (m,m^\prime) = \frac{\pi r_o^2}{m_o^{2/D}}\left(m^{1/D} +
m^{\prime 1/D}\right)^2,
\end{equation}
\noindent
is proportional to $m^{2/D}$, whereas the stopping time (Eq. 11) in the Epstein
regime (the regime that would apply to fluffy fractal aggregates) where
the drag force $F_D = (4/3)\pi r^2 c_g \rho_g \Delta V_{pg}$
\citep[e.g.,][]{cuz93} is given by
\begin{equation}
t_s = \frac{3m}{4\pi r^2 c_g\rho_g} = \frac{3m_o^{2/D}}{4\pi r_o^2 c_g\rho_g}
m^{1-2/D}.
\end{equation}
\noindent
In the limit of small Stokes number
($\rm{St}\ll 1$), both the systematic and turbulence-induced velocities
are proportional to $\rm{St}$, so that $\Delta V \propto t_s \propto
m^{1-2/D}$, and the entire kernel is proportional to $m$. For
larger $\rm{St}$, the kernel has a shallower powerlaw
dependence. Thus, in general, the dependence of $K(m,m^\prime)$ on the
mass is $\lesssim m$, preserving closure of the system, even for fractal
particles. Even though the variation in the properties of the evolving
distribution due to porous particles are incorporated directly into the
kernel, and are folded into the integration of the explicit approach, the
effects of fractal aggregates can, nonetheless, affect
which moments characterize what properties in the semi-implicit (or implicit)
approach. For
example, the wavelength-independent opacity expressed in Eq. (32) takes
the form $\kappa = (\pi r_o^2/m_o^{2/D})M_{2/D}/M_1$.
It should be noted that the value $D=2$ represents a special case in that
the particle-to-particle relative velocities in the Epstein regime do not
depend on the mass of the fractal particle. Indeed, this would seem to
indicate that fractal growth can proceed unabated with the corresponding
stopping time of the fluffy aggregate remaining the same as that of a single
monomer, which would have a significant effect on other particle properties. In
particular, the wavelength-independent opacity for $D=2$ is constant. However,
impacts will eventually lead to compaction or even fragmentation depending
on the relative velocities \citep[e.g.,][]{orm07b}.
Both fractal grains and non-fractal particles in the same mass distribution
can be treated in the explicit approach without any modifications, while a
piecewise fit to
the integrated kernel $C_k(m)$ ({\S} 2.2.2) can be used in order to
account for the change in regimes in the semi-implicit approach.
\section{Conclusions}
We have demonstrated an approach to solving the collisional coagulation
equation with an arbitrary collisional kernel which should be useful in cases
when it is only necessary to keep track of general
properties of the distribution. This approach involves solving a finite set of
coupled differential equations in terms of the integer moments of the particle
size distribution. The number of equations (and thus moments) needed depends
on the number of properties being tracked. The advantage of the moments method
approach is that it allows for considerable savings in computational
time compared to direct integration of the coagulation equation,
which requires keeping track of every particle size at every spatial location
and timestep.
In this paper we have specifically
studied the growth of the largest particle under the assumption that
the particle size distribution is a powerlaw; however, the technique can be
extended to track other properties of the distribution that may change with
time. There are many reasons to believe that a powerlaw size distribution is
a natural end-state of particle growth, especially those with equal mass
per decade, because they have self-preserving properties \citep{cuz06}.
With the assumption of a powerlaw distribution, we
have provided two different approaches to solving the moment equations,
one explicit in which the powerlaw assumption is enforced rigorously at
all times, and a semi-implicit approach in which the kernel is integrated
over one of the mass variables as much as once every time step. The latter
approach can be made fully implicit by only assuming the form of the mass
distribution at $t=0$. These approaches
are significantly faster than solutions of the coagulation equation
because, in particular, the convolution integral (first term on RHS of Eq. 1)
has been eliminated. In realistic evolutionary models, intermediate steps
performed in particle ``histogram'' or ``size-distribution'' space may be
interleaved with moments-based coagulation steps in order to account for,
e.g., advective/transport terms.
We have compared
these alternate approaches to the brute force integration of the full
coagulation equation for cases in
which there are only systematic velocities, and cases in which the differences
in velocity between particles is induced by turbulence. If the growth rate
is gradual, the explicit and implicit
approaches match the brute force calculation well (Fig. 2),
whereas
faster growth rates are more difficult to model (Fig. 3). We find that we
are able to use the semi-implicit approach in which the $C_k$ are updated
at every timestep, and a modification to the explicit approach, in which
we solve the equation for $M_2$ only with the assumption
of $\rho=M_1$ strictly enforced,
in order to compensate for the faster growth rates.
The modification to the explicit approach is useful if
one is not particularly interested in following the evolution of the number
density of particles, or other properties which may be decribed by
(fractional) moments $< 1$ (e.g., see {\S} 4). Our results also suggest
that a fully implicit approach is probably most useful under circumstances in
which the kernel depends on the mass in a straightforward manner (e.g, the
Saffman-Turner kernel, {\S} 2.1).
We have compared the approaches developed in this paper to an alternative
model for particle growth in a turbulent nebula
\citep{gar07}. We have found that there is fairly good agreement in general,
but that the curves diverge as time proceeds. This does not appear to be
particle-size dependent, or due to a shift in the flow regime.
The Garaud expression (Eq. 29) underestimates the growth rate of
the largest particle size relative to our approach by only $\sim 20-30$\%
in our comparison. We note that the advantage
of the method of \citet{gar07} is
that it is purely analytical; however, preservation of her analytical
approach requires, amongst other things, the powerlaw
exponent be restricted to $q=11/6$. Our approach has no such restriction
on the choice of exponent $q$, nor for $q$ to even be a constant.
As a sample application, we show how the moments method can
be used in one's evolutionary model to track
particle growth and opacity. As a specific case, we calculated
the change in wavelength-independent opacity with time for an initial
particle size distribution with upper and lower bounds of $0.1-1$ cm.
Both systematic and turbulent velocities were included. In the absence of
any mechanism to counter growth, the opacity decreases sharply for higher
choices of the turbulent parameter $\alpha$. Extension to cases in
which the extinction efficiency is wavelength dependent is straightforward.
Such opacities can be used to calculate Planck or Rosseland
wavelength-averaged means for use in radiative transfer codes.
Finally, we indicate how porous particles with fractal dimension $D$ can
be accounted for in the moments method. The particle-size density
relationships that arise affect the particle stopping times and Stokes
number, both of which appear only in the collisional kernel. Thus
implementation is straightforward. We can treat mass distributions composed
of only fractal grains, or both fractal and non-fractal particles.
In either case, relatively little modification is needed in the explicit
approach, whereas with the semi-implicit (and implicit) approach, a piecewise
fit to the integrated kernel $C_k$ using the method as described in {\S} 2.2.2
can be used to account for the change in particle growth regime when both
types of particles are included.
The computational burden of directly solving the coagulation equation
makes it
quite prohibitive to explore large regions of parameter space, and thus
it serves
as the primary bottleneck in evolutionary growth models. The approach
demonstrated herein is intended to obtain robust, quantitative results for
disk properties such as particle growth
timescales and ``typical'' particle sizes that may be used in modeling
efforts that are
focused more on the larger problem of planetesimal formation.
Although we have not included sticking or fragmentation in this paper, we
believe that the moment equations (Eq. 5) will remain valid at least up
to the fragmentation barrier. This size will depend on the
assumed particle strengths and choice of nebula parameters. The treatment
of growth up to the fragmentation size is consistent with recent work by
\citet{bra08} for the case in which additional effects such as radial
drift are not included. The model as presented in this paper, however,
should be useful for the early stages of protoplanetary nebula particle
growth relevant to spectral energy distributions and MRI suppression.
In a forthcoming paper we will explore the effects of a
variable sticking coefficient $S(m,m^\prime)$. Furthermore, we will explore the
addition of source and sink terms such as gravitational growth,
erosion, sublimation, and condensation in addition to fragmentation. The
ultimate goal will be to apply this methodology to
a global model that studies the evolution of both the gas and solids in
nebular and subnebular (giant planetary) environments.
\acknowledgements{We wish to thank Sandy Davis, Fred Ciesla, and Olenka
Hubickyj for internal reviews which improved the exposition of this paper,
and an anonymous reviewer for his or her careful analysis of the manuscript.
This work was supported by a grant from NASA's
Origins of Planetary Systems Program.}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,149 |
{"url":"http:\/\/math.stackexchange.com\/tags\/banach-spaces\/new","text":"Tag Info\n\n1\n\nIn order to invoke Baire you have to assume the closedness of $C$. Otherwise your proof is fine.\n\n2\n\nLet $h'(x) := \\max(f'(x), g'(x))$. $h'$ is continuous. Get a primitive function $h$ with $h(0) = \\max(f(0), g(0))$. This should do it.\n\n2\n\nIf isomorphic means isomorphic as rings then no, they're not isomorphic. Keen fact: Suppose $K_1$ and $K_2$ are compact Hausdorff spaces. Then $C(K_1)$ is isomorphic to $C(K_2)$ if and only if $K_1$ and $K_2$ are homeomorphic. (Sketch: $C(K)$ is a Banach algebra with maximal ideal space $K$.) This is a keen thing, because it shows that the ...\n\n0\n\nAlternative answer:your operator induces a continuous bijection between quotient space and image subspace. By the open mapping theorem if it were compact then the unit ball in both spaces were compact, thus finite dimensional by Riesz\n\n1\n\nFirst, note that $Z:=Im(K)$ is a closed subspace of a Banach space and thus, itself a Banach space. Thus, $K: X\\to Z$ is onto. By the open mapping theorem, $K$ is open and hence, $K$ is mapping open sets to open sets. Now, assume that $K$ is compact and take the image of the open unit ball $C:=K(B_X^\\circ)$ which is open in $Z$ and relatively compact in $Y$...\n\n2\n\nRegarding convergence and completeness: For $n\\in N$ let $f_n(x)=0$ for $x\\leq 1\/2-1\/(n+2)$ and $f_n(x)=1$ for $x\\geq 1\/2.$ Let $f_n(x)$ be linear for $x\\in [1\/2-1\/(n+2),1\/2].$ Then $(f_n)_{n\\in N}$ is a Cauchy sequence with respect to the norm $\\|f-g\\|=[\\int_0^1|f(x)-g(x)|^2\\;dx]^{1\/2}.$ Let $h(x)=0$ for $x\\leq 1\/2$ and $h(x)=1$ for $x>1\/2.$ The ...\n\n1\n\nHere is an alternative proof, using as a starting point that $\\ell^{\\infty}$ is a Banach space. As $c$ is a subspace of $\\ell^{\\infty}$, it suffices to show that $c$ is closed in $\\ell^{\\infty}$. To see this let $\\mathbf{x}=(x_1,x_2,\\dotsc)\\in\\ell^{\\infty}$, and suppose that $\\{\\mathbf{x}^n\\}_{n=1}^{\\infty}$ is a sequence in $c$ converging to $\\ell^{\\infty}$...\n\n3\n\nThe function $t\\cdot \\chi_{[-1,1]}(t)$ is orthogonal to each $t^{2k}$ in $L^2[-1,3],$ hence is orthogonal to the the linear span of $\\{t^{2k} : k=0,1,\\dots \\}$ in $L^2[-1,3],$ hence is orthogonal to the closure of this linear span in $L^2[-1,3].$ Therfore this closure cannot be all of $L^2[-1,3].$\n\n2\n\nLet $A$ denote the linear span of $\\{t^{2k}\\}_{k \\in \\mathbb{N}}$. Then, if instead of $[-1,3]$ the domain was $[1,3]$ instead, we could use the Stone-Weierstrass theorem to conclude that $A$ is dense ins $C([1,3])$ and thus in $L^2([1,3])$ since $t^2$ separates the points of $[1,3]$ and $1 \\in A$. However, for $[-1,3]$, we have $t^{2k}(-1) = t^{2k}(1)$ ...\n\n0\n\nUnless I am missing something, $Y$ is not even closed (so it cannot be complemented either). Indeed, $Y$ is the image of the compact operator that multiplies elements of $E$ by $(1\/n)$. Compact operators cannot have closed range unless they are finite-rank operators.\n\n1\n\nI will make the following assumption: (2'): $\\exists (x_n)\\subset E$ and $c>0$ such that $x_n$ converges weakly to $0$ and $\\|x_n\\|>c$ for all $n$. This follows from (2), since non-convergence in norm implies the existence of a subsequence bounded away from $0$. (As a side note, this also implies (2).) Let $(x_n)$, $c$ satisfy (2'), and let $x\\in ... 2 One such proof is via the so-called Kuratowski embedding. 2 Your sequence of equalities should have been $$\\|A_\\lambda f\\|^2=\\lambda\\int_0^1f(\\lambda t)^2dt= \\int _0^\\lambda f(t)^2=\\int _0^1f(t)^2\\chi_{[0,\\lambda]}^2\\leq \\|f\\|\\|\\chi_{[0,\\lambda]}\\|=\\lambda \\|f\\|.$$ Now, I don't know what inequality you are using, but I cannot make sense of it. And you get$\\|f\\|$instead of$\\|f\\|^2$, which is a bad sign. The ... 3 If the basis is unconditional (with, as David Mitra points out, the unconditional basis constant also bounded by something independent of$n$) then yes, at least with some constant independent of$n$. Using$c$to denote anything not depending on$n$: Of course$|a_k|\\le c||\\sum a_je_j||$just because it's a normalized basis. On the other hand, if$-1\\le ...\n\n1\n\nThat's right. Reflexive spaces are both $M$-embedded and $L$-embedded.\n\n2\n\nIn this context, $\\cong$ is to be interpreted as \u201cisometrically isomorphic.\u201d This means that there exists a map $f:X\\to J(X)$\u00a0such that $f$\u00a0is linear; $f$\u00a0is bijective; and $f$\u00a0preserves norms: $\\|f(x)\\|_{J(x)}=\\|x\\|_X$\u00a0for each $x\\in X$. The natural candidate for such a function $f$\u00a0is the canonical mapping $J:X\\to J(X)$ itself. By construction, it is ...\n\n1\n\nFor each $x\\in I=[-1,1]$, define $T_{x}(y):=\\frac{1}{2}\\ln(1+y^{2})+\\sin(\\ln(x+2))$. Observe that $T_{x}'(y)=\\frac{y^{2}}{1+y^{2}}\\leq \\frac{1}{2}$ on $I$, so we can show $|T_{x}(y')-T_{x}(y)|\\leq \\frac{1}{2}|y'-y|$ and $T_{x}$ has a unique fixed point for each $x$ by Banach fixed point theorem. Define $f(x)$ as fixed point of $T_{x}$ for each $x\\in I$. To ...\n\n1\n\nYou should indeed use the Banach fixed-point theorem. The \"obvious\" map to consider is $T : C[-1,1] \\to C[-1,1]$ $$(Tf)(x) = \\frac{1}{2}\\ln(1+f(x)^2) + \\sin(\\ln(x+2))$$ Check that $T$ is a contraction mapping. The inequality $$\\vert \\ln(1+x^2)-\\ln(1+y^2) \\vert \\leq \\vert x - y \\vert$$ might prove useful. (The latter inequality follows e.g. from the mean-...\n\n4\n\nBen, the answer is no. Note that if $X$ is reflexive, then $B(X)$ is isometric to $B(X^*)$ via $T\\mapsto T^*$. Note that this map is an anti-isomorphism of Banach algebras. As for less trivial examples, $B(\\ell_p)$ is Banach-space isomorphic to $B(L_p)$ as well to $B(X)$ for any other separable, infinite-dimensional $\\mathscr{L}_p$-space and $\\ell_p$ is ...\n\n2\n\nNo. Not that that makes any sense to me, but no. In fact it turns out that $\\kappa_{X^{**}}=\\kappa_X^{**}$ if and only if $X$ is reflexive. In fact the truth behind my erroneous feeling that the two must be equal in general is this: $$\\kappa_{X^{**}}\\kappa_X=\\kappa_X^{**}\\kappa_X.$$ Proofs: Writing $x\\in X$, $x^*\\in X^*$, etc.: If you unpack two ...\n\n1\n\nTwo answers. First, you can easily derive the general case from the case where $Z$ is closed. Second, and I think more important, $Z$ being closed is really irrelevant in the first place! (i) You can derive the general case from the case where $Z$ is closed: If $d=0$ you can just set $\\Lambda=0$ and you're done. Suppose $d>0$. Then $y\\notin\\overline Z$...\n\n1\n\nLet $P(x)=\\inf_{z\\in Z}\\|x-z\\|$. It is easy to show that $P(ax)=aP(x),a>0$ and $P(x+y)\\leq P(x)+P(y)$. Meanwhile, $P(z)=0, \\forall z\\in Z$ and $P(y)=d$. We can show that $P(x)\\leq \\|x\\|$ by choosing $z=0$ in the definition. Next, we define another function $L(x)$. $L(z)=0, \\forall z\\in Z$ and $L(y)=d$. Let $L$ be defined on the subspace $Y=\\{by+z|z\\in Z,... 2 The quantity$\\|f\\| = |f(a)| + \\|f\\|_{TV}$is a natural norm on$BV[a,b].$If$f\\in AC,$then this norm equals$|f(a)| + \\int_a^b|f'|.$Suppose$f_n$is a sequence in$AC$that is Cauchy in this norm. Then$f_n(a)$is a Cauchy sequence in$\\mathbb R,$and$f_n'$is Cauchy in$L^1[a,b].$Thus$f_n(a) \\to c$in$ \\mathbb R$and$f_n'\\to g$in$L^1[a,b].$We ... 1 By definition,$f: [a,b] \\to B$is Riemann integrable on$[a,b]$with integral$L \\in B$if for each$\\epsilon > 0$there is$\\delta > 0$such that for every tagged partition$x_0,\\ldots,x_n$,$t_0, \\ldots, t_{n-1}$of$[a,b]$with mesh$<\\delta$, $$\\left\\| \\sum_{i=0}^{n-1} (x_{i+1}-x_i) f(t_i) - L \\right\\| < \\epsilon \\tag{1}$$ I claim that$...\n\n3\n\nWe give the proof the OP wants, also an example showing that the result fails for functions on the line. The standard proof via the vector-valued Cauchy Integral Formula seems like the \"right\" proof, because that vector-valued CIF is going to be fundamentally important soon anyway, when we get to Banach algebras and operator theory and so on. But, if we ...\n\n2\n\nLet $(h_n)$ a sequence in $V$ converging in $\\mathbb{C}$. Let $u_n = \\frac{f(z_0+h_n)-f(z_0)}{h_n}$ The bounded $\\sup$ condition implies $||u_p - u_q|| \\leq M |h_p-h_q|$ Hence $(u_n)$ is a Cauchy sequence in $X$ so it converges in $X$ by completeness. Therefore $\\lim \\frac{f(z_0+h_n)-f(z_0)}{h_n}$ exists in X.\n\n2\n\nYes, take a subsequence $(x_{n_k})_k$ such that $\\|x_{n_k}\\|\\leq 2^{-k}$. Then $$\\|\\sum_{k=1}^{m_2}x_{n_k}-\\sum_{k=1}^{m_1}x_{n_k}\\|=\\|\\sum_{k=m_1+1}^{m_2}x_{n_k}\\|\\leq\\sum_{k=m_1+1}^{m_2}\\|x_{n_k}\\|\\leq\\sum_{k=m_1+1}^{m_2}2^{-k}=2^{-m_1}-2^{-m_2}\\to0$$ as $m_1,m_2\\to0$. Thus the partial sums of $x_{n_k}$ are Cauchy.\n\n0\n\nLet $d$ be a metric for $X.$ Let $E$ be a dense subset of $X$. For any $x\\in X$ and $r>0,$ there exists $q\\in (0,r\/3)\\cap \\mathbb Q$ and $e\\in E$ such that $d(x,e)<q.$ $$\\text { This gives }\\quad x\\in B_d(e,2q)\\subset B_d(x,r).$$ (Because $y\\in B_d(e,2q)\\implies d(x,y)\\leq d(x,e)+d(e,y)< q+2q<r$.) Therefore $B=\\{B_d(e,2q):q\\in \\... 2 Let$D\\subset X$be countable and dense and$Z\\subset X$. Assume there is no$z_0\\in Z$with a sequence$(z_n)_{n=1}^\\infty$of points in$Z$converging to$z_0$. Then for every$z\\in Z$there exists$r=r(z)>0$such that$B_r(z)\\cap Z=\\{z\\}$. By decreasing$r(z)$if necessary, we may assume that$r(z)=\\frac1{n(z)}$with$n(z)\\in\\Bbb N$. For$m\\in \\Bbb N$, ... 3 Yes, and we do not even need to assume completeness. Suppose not, towards a contradiction. Then we can find open balls$U_z=B_{\\epsilon_z}(z)$around each$z\\in Z$such that$U_z\\cap Z=\\{z\\}$. Let$q>0$be such that uncountably many$z\\in Z$have$\\epsilon_z>q$(Exercise: why does such a$q$exist?). Let$Z'=\\{z\\in Z: \\epsilon_z>q\\}$. Now consider ... 2 My first thought was surely not. But actually the answer is yes. Say$f$is a non-zero bounded linear functional on$X$. Let$N$be the null space of$f$. Now$X\\setminus N$certainly spans$X$, so there exists a Hamel basis$B\\subset X\\setminus N$. So$f(b)\\ne0$for every$b\\in B$; now modify$B$, replacing every element by an appropriate scalar multiple ... 1 The following proof is from \"On the convergence of sample probability distributions\" of Varadarajan. As you may not have access to jstor, i write it down here. Remark that it is important that we lie in a probability space. So that it is a particuliar case of your question. Let$f:M \\to X$measurable, suppose$\\mu(M)=1$, let$\\mu_f$the \"law\" of$f$, ie:$\\...\n\n1\n\nMore generally, if $\\{T, S_1, S_2, \\ldots\\}$ is a uniformly continuous family of maps from metric space $X$ to metric space $Y$, then $\\{x \\in X: \\lim_{n \\to \\infty} S_n(x) = T(x)\\}$ is closed. This does not require completeness of either metric space.\n\n0\n\nYour argument is fine, besides the typo where it should be $(\\|S_n\\|+\\|T\\|)$ instead of $\\|S_n+T\\|$. And I don't think you need for $X$ to be complete.\n\n2\n\nAny $*$-homomorphism between C$^*$-algebras is contractive. This is standard (i.e., it appears in every book on the subject) and is due to three things: The C $^*$-identity $\\|a\\|^2=\\|a^*a\\|$, which reduces the problem to norms of positives; The equality $\\|a\\|=\\text {spr}\\, (a)$ for $a$ positive; The fact that a $*$-homomorphism reduces the spectral ...\n\nTop 50 recent answers are included","date":"2016-07-28 01:03:11","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\": 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\": 0.9887935519218445, \"perplexity\": 175.92008154919685}, \"config\": {\"markdown_headings\": false, \"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-2016-30\/segments\/1469257827781.76\/warc\/CC-MAIN-20160723071027-00081-ip-10-185-27-174.ec2.internal.warc.gz\"}"} | null | null |
Tonnes of carbon reduced
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Since 2001, Mozambique has lost more than 3.5 million hectares of forest. That's an area more than a third the size of Portugal.
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An area of degraded mangrove forest measuring 506 hectares along the coast of Biak Island in northeastern Indonesia. Biak Island is a small island situated north of mainland Papua (New Guinea).
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© 2023 Our Forest. All rights reserved. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
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These Waterproof Lip Cards definitely caught our eyes! These are lip color that comes in the form of a card, for you to put between your lips, press your lips against it, and voila – coloured lips! It's bright, lasting, and glossy, with a nice shiny finish. Enjoy interesting, convenient lip makeup at a reasonable price anytime, anywhere by carrying it in your pouch!
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This product can be used 3 times depending on the method of use. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,884 |
\section{Introduction}
The cosmic censorship conjecture remains one of the most important unsolved
problems in classical general relativity. In physical terms the conjecture
states that singularities arising in generic gravitational collapse are
hidden behind an event horizon. The lack of a general approach for
investigating censorship has lead to studies of simplified collapse models,
and in particular, study of spherical gravitational collapse. The best
understood spherical collapse is that of dust, where it has been shown that
for certain initial data the central singularity which forms in collapse is
naked, whereas for other initial data it is covered \cite{dust}. A more
realistic collapse model would invoke pressure. While there are general
existence arguments suggesting the occurence of naked singularities when
pressure is included \cite{dj}, explicit examples highlighting the role of
initial data are rare. It is thus not known how the naked singularity
arising in dust collapse is modified, when pressure is introduced. An
important exception is the numerical study of self-similar collapse of a
perfect fluid, where it was shown that naked singularities result
generically in such a collapse \cite{ori}.
The energy-momentum tensor for spherically collapsing matter is diagonal in
the comoving frame, the components being the energy density, the radial
pressure, and the tangential pressure. Evidently, the general analytical
solution of the Einstein equations for this system is not known, even after
an equation of state is specified. Major simplification of the system of
equations results if the radial pressure is set to zero, and only the
tangential pressure retained. This is the approximation studied in the
present paper. Admittedly, it is rather unphysical to consider a collapsing
system in which there is only tangential pressure. However, our purpose is
to explicitly study how the introduction of pressure modifies the dust
scenario, and the present analysis should be considered a small step in that
general direction.
We assume that the tangential pressure $p_{\theta}$ is related to the energy
density by a linear equation of state $p_{\theta}=k\rho$ where $k$ is a
constant. As a result, the Einstein equations can be reduced to a non-linear
ordinary differential equation for the evolution of the area radius of a
fluid element. The initial density and velocity profiles of the collapsing
object enter as free parameters in this equation, which has a one parameter
family of solutions, labelled by the parameter $k$.
The equation can be solved exactly for some specific values of $k$, and can
be integrated numerically for a general, given, $k$. However, our aim is to
check whether any singularities that form are naked, and that can be done
without resorting to numerical study. The behaviour that we find is quite
surprising, and quite unlike the evolution of dust. For positive $k$, we
find there is a region near the center which necessarily moves outwards, if
started from rest, and just cannot collapse. Depending on how far the star
extends, this outgoing region could be surrounded by a region which
collapses inwards, and could develop a singularity. However, the singularity
will not be naked, but covered by a horizon. Beyond this collapsing region,
there will in general be a second region of expansion. For negative $k$, the
entire object collapses inwards, unlike for positive $k$, but any
singularities that arise are covered and not naked. Thus the naked
singularities in dust collapse are removed when a non-zero value of $k$ is
switched on, howsoever small. Of course, this may not be the case when the
radial pressure is included, or when the tangential pressure obeys a
different equation of state. It does however suggest the possibility that
when collapse to a naked singularity occurs, the stability of the collapse
in the presence of small perturbations should be examined.
In Section 2 we describe the relevant field equations, and in Section 3 the
evolution for positive $k$ (including the case $k=1/4$ which can be solved
exactly). In Section 4 we describe the evolution for negative $k$.
Spherically symmetric static and quasi-static solutions with only tangential
pressure have been studied before, notably by Lemaitre \cite{lem}, as well
as by others \cite{oth}. Those results do not appear to have any direct
bearing on or connection with the results reported here.
\section{The Field Equations}
In comoving coordinates $(t,r,\theta ,\phi )$ the spherically symmetric
line-element is given by
\begin{equation}
\label{metric}ds^2=e^\sigma dt^2-e^\omega dr^2-R^2d\Omega ^2
\end{equation}
where $\sigma $ and $\omega $ are functions of $t$ and $r$. The area radius $%
R$ also depends on both $t$ and $r.$ In comoving coordinates the
energy-momentum tensor for a spherically symmetric object takes the diagonal
form $T_k^i=(\rho ,p_r,p_\theta ,p_\theta )$. The quantities $p_r$ and $%
p_\theta $ are usually referred to as the radial and tangential pressure,
respectively. The Einstein field equations for this system are
\begin{equation}
\label{mprime}m^{\prime }=4\pi \rho R^2R^{\prime },
\end{equation}
\begin{equation}
\label{mdot}\dot m=-4\pi p_rR^2\dot R,
\end{equation}
\begin{equation}
\label{sigpri}\sigma ^{\prime }=-\frac{2p_r^{\prime }}{\rho +p_r}+\frac{%
4R^{\prime }}{R(\rho +p_r)}(p_\theta -p_r),
\end{equation}
\begin{equation}
\label{omedot}\dot \omega =-\frac{2\dot \rho }{\rho +p_r}-\frac{4\dot R(\rho
+p_\theta )}{R(\rho +p_r)},
\end{equation}
and
\begin{equation}
\label{energy}m=\frac 12R\left( 1+e^{-\sigma }\dot R^2-e^{-\omega }R^{\prime
^2}\right) .
\end{equation}
Here, $m(t,r)$ is a free function arising out of integration of the Einstein
equations. Its initial value, $m(0,r),$ is interpreted as the mass interior
to the coordinate $r$.
We now make the approximation that the radial pressure is identically zero.
Eqn. (\ref{mprime}) remains unchanged, whereas Eqn. (\ref{mdot}) gives the
important result that the mass-function does not change with time, and
remains constant at its initial value $m(0,r)\equiv m(r)$. This is what
simplifies the analysis very considerably. Eqn. (\ref{energy}) remains as
above, while Eqns. (\ref{sigpri}) and (\ref{omedot}) simplify to
\begin{equation}
\label{sigpri2}\sigma ^{\prime }=\frac{4R^{\prime }}R\frac{p_\theta }\rho ,
\end{equation}
and
\begin{equation}
\label{omedot2}\dot \omega =-\frac{2\dot \rho }\rho -\frac{4\dot R}R\left( 1+%
\frac{p_\theta }\rho \right) .
\end{equation}
Next we assume, for simplicity, that the equation of state is $p_\theta
=k\rho $, where the constant $k$ is chosen to lie in the range $-1\leq k\leq
1$, in accordance with the weak energy condition, and the dominant energy
condition. Eqn. (\ref{sigpri2}) then has the solution
\begin{equation}
\label{sigsol}e^{\sigma (t,r)}=R^{4k}
\end{equation}
where a pure function of time has been set to unity. (This choice merely
reflects a rescaling of the time coordinate in the metric). Eqn. (\ref
{omedot2}) has the solution
\begin{equation}
\label{omesol}e^{-\omega (t,r)}=\psi (r)\rho ^2R^{4(1+k)}.
\end{equation}
Here, $\psi (r)$ is an aribtrary function of the position coordinate $r$
which we retain because we would like to use this freedom later. Using Eqn. (%
\ref{mprime}) we can write this as
\begin{equation}
\label{omesol2}e^{-\omega (t,r)}=\frac{\psi (r)m^{\prime ^2}}{16\pi ^2}\frac{%
R^{4k}}{R^{\prime ^2}}\equiv C(r)\frac{R^{4k}}{R^{\prime ^2}}.
\end{equation}
Note that $C(r)$ is a positive function. Using the solutions (\ref{sigsol})
and (\ref{omesol2}) in Eqn. (\ref{energy}) yields the following equation for
the evolution of the area radius:
\begin{equation}
\label{reqn}\dot R^2+R^{4k}-2m(r)R^{4k-1}-C(r)R^{8k}=0.
\end{equation}
This non-linear differential equation, which can be integrated in principle,
gives a one-parameter solution of Einstein equations. While the function $%
m(r)$ is determined by the initial density profile, the function $C(r)$
becomes known once the initial velocity profile is also given. Note that
this simplified equation for $R$ results because of the simplifying
assumptions that the radial pressure is identically zero and that the
tangential pressure obeys the linear equation of state. If $R(t,r)$ could be
solved for from this equation, the evolution of the density and the metric
components follows from the earlier equations. We note that the special case
of dust can be recovered from the above equation by setting $k=0$. That
yields the familiar equation for dust collapse,
\begin{equation}
\label{dust}\dot R^2=\frac{2m(r)}R+f(r),
\end{equation}
where we have set $C(r)\equiv 1+f(r)$.
We shall now study the properties of Eqn. (\ref{reqn}) and its implications
for collapse and singularities, treating the cases of positive and negative $%
k$ separately.
\section{The evolution for positive k}
Evidently, Eqn. (\ref{reqn}) cannot be solved analytically for an arbitrary
value of $k$, though numerical integration for a given value of $k$ would be
straightforward. Our interest here, however, is not so much in the exact
solution of this equation, but rather in those qualitative features which
would help us determine whether singularities form in collapse, and if so,
whether they are naked. To this end, we note that for a fixed $r$ Eqn. (\ref
{reqn}) can be thought of as the equation for the one-dimensional motion of
a zero-energy particle, along coordinate $R$, in the potential $V(R,r)$
given by
\begin{equation}
\label{potn}V(R,r)=R^{4k}-2m(r)R^{4k-1}-C(r)R^{8k}.
\end{equation}
Let us analyse the shape of this potential, as a function of $R$. By writing
$V(R,r)$ as
\begin{equation}
\label{dub}V(R,r)=R^{4k-1}(R-2m-CR^{4k+1})\equiv R^{4k-1}W(R,r)
\end{equation}
we note that $V(R,r)$ will be positive if and only if the function $W(R,r)$
is positive. Consider first the case of positive $k$. The extremum of $%
W(R,r) $ is at the point $R_0$ given by
\begin{equation}
\label{ext}R_0^{4k}(r)=\frac 1{C(r)(1+4k)}.
\end{equation}
It is easily checked that this extremum is a maximum. It follows from (\ref
{dub}) that in order for $W(R_0,r)$ to be positive, the following condition
should be satisfied :
\begin{equation}
\label{max}(2m)^{4k}C<\frac{(4k)^{4k}}{(1+4k)^{(1+4k)}}.
\end{equation}
We will assume that collapse begins from rest - it then follows from Eqn. (%
\ref{reqn}) that $C(r)$ gets fixed as
\begin{equation}
\label{cee}C(r)=\frac{1-\frac{2m}r}{r^{4k}},
\end{equation}
and Eqn. (\ref{max}) becomes
\begin{equation}
\label{ineq}\left( \frac{2m}r\right) ^{4k}\left( 1-\frac{2m}r\right) <\frac{%
(4k)^{4k}}{(1+4k)^{(1+4k)}}.
\end{equation}
Note that $0\leq 2m/r\leq 1$. It is easily shown that the function $(2m/r)$
always satisfies this inequality, except when $(2m/r)=4k/(1+4k)\equiv k_{*}$%
. In the latter case, the two sides of Eqn. (\ref{ineq}) are actually equal,
and the maximum value $W(R_0,r)$ is exactly zero. It follows that the
maximum $W(R_0,r)$ is necessarily positive, except when $(2m/r)$ takes the
special value $k_{*}$. Further, since $W(R,r)$ goes to $(-2m)$ as $R$ goes
to zero, and it goes to negative infinity as $R$ goes to infinity, we
conclude that $W$ has two zeroes. These are also zeroes of the potential
function $V(R,r)$. In the region bounded by these two zeroes, $V(R,r)$ will
be positive. Thus, as can be seen from Eqn. (\ref{reqn}), this region will
be forbidden, by the requirement of positivity of $\dot R^2.$ Outside of
this forbidden region, both $W(R,r)$ and $V(R,r)$ will be negative - that is
the allowed region.
In the region to the left of the lower zero, where $V(R,r)$ is negative, the
shape of the potential depends on $k$. For $k<1/4,$ $V(R)$ monotonically
decreases as $R$ decreases, ultimately going to $-\infty $ as $R$ goes to
zero. For $k=1/4$ the potential again monotonically decreases with
decreasing $R$, going to $-2m(r)$ at $R=0$. For $k>1/4$ the potential will
have at least one turning point (which is a minimum) and goes to zero at $%
R=0 $. In the region to the right of the larger zero, where $V(R)$ is again
negative, it will go to $-\infty $ as $R\rightarrow \infty $, for all $k$.
While there are no turning points in this region for $k\geq 1/4$, there may
be turning points for $k<1/4$, depending on the functions $C(r)$ and $m(r)$.
The shape of $V(R)$ for the various cases is shown schematically in Fig. 1.
We now show that this shape of the potential function has interesting
implications for the nature of the motion. A fluid element labelled by $r$
can lie either to the right of the larger zero, or to the left of the
smaller zero. For convenience of analysis, we shall assume that at the time $%
t=0$, which marks the start of collapse, we have the scaling $R=r$. (Recall
that such a freedom of labelling is available). Since the motion starts from
rest, it is evident that the potential $V(R,r)$ is zero at the beginning.
Hence one of the two zeroes of the potential is given exactly by $R=r$. It
is important to settle whether this is the larger or smaller of the two
zeroes. For this purpose, we note that if the starting point $R=r$ lies to
the left of the maximum $R_0$, it will satisfy $r^p<R_0^p$, i.e.
\begin{equation}
\label{less}\frac{4k}{1+4k}<\frac{2m}r,
\end{equation}
as follows after using Eqns. (\ref{ext}) and (\ref{cee}). Similarly, if the
starting point lies to the right of the maximum, it will satisfy
\begin{equation}
\label{more}\frac{4k}{1+4k}>\frac{2m}r.
\end{equation}
\begin{center}
\leavevmode\epsfysize=4 in\epsfbox{fa.ps}
\end{center}
Hence for all those points $r$ for which (\ref{less}) holds $R=r$ is the
smaller of the two zeroes, and all points for which (\ref{more}) holds, $R=r$
is the larger of the two zeroes. Now consider a neighborhood of the center.
The ratio $(2m/r)$ is equal to zero at the origin, and hence there will be a
neighborhood of the origin for which the inequality (\ref{more}) will
necessarily hold. All the fluid elements in this neighborhood will have
their motion begin at the larger of the two zeroes, and such elements will
necessarily move away from the origin, towards infinity. They simply cannot
collapse. Hence, there will be a neighborhood of the origin which will
necessarily move outwards, starting from rest. If the condition (\ref{more})
is satisfied all over the star, then all of it will move outwards. For $%
k\geq 1/4$ the shape of the potential implies that everything will escape to
infinity. For $k<1/4$ some or all of the fluid elements could get trapped at
a possible minimum the potential could have.
The ratio $2m(r)/r$ is zero at the origin, and near $r=0$ increases with
increasing $r$. Hence it is possible that the above-mentioned neighborhood
of the center (for which (\ref{more}) holds) is surrounded by a region in
which (\ref{less}) holds. If so, this outer region will have the left zero
of the potential as its starting point. As a result, this region will
collapse inwards, and will intersect the inner region which is moving
outwards. This appears to be a shell-crossing singularity. We will assume
here that the inner and outer regions simply cross each other. In other
words, the ingoing region does not see the outgoing inner region,
effectively.
This infalling region will necessarily shrink to $R=0$ for $k\leq 1/4$. It
could do so for $k > 1/4$ as well - i.e. it could overshoot the minimum of $%
V(R)$ for suitably chosen velocity and density profiles. $R=0$ possibly
corresponds to a curvature singularity - this would happen if the energy
density defined by Eqn. (\ref{mprime}) blows up. In order to actually check
whether this happens we have to integrate Eqn. (\ref{reqn}) to find out the
behaviour of $R^{\prime}$ as $R$ goes to zero. However, for the present, our
intention is not really to check whether singularities do form, but to show
that if they form, they are not naked.
We show this by analysing the null geodesic equation for radial geodesics.
This method has been developed in earlier papers (see, for instance \cite
{djprd}), and here we briefly recall the equation we need to use. If the
coordinate $r_0$ develops a singularity, the geodesic equation for possible
geodesics emerging from this singularity can be written as
\begin{equation}
\label{geo}X=\lim_{r\rightarrow r_0,R\rightarrow 0}\frac{R^{\prime }}{\alpha
|r-r_0|^{(\alpha -1)}}\left( 1-\sqrt{\frac{f+2m/R}{1+f}}\right) .
\end{equation}
The various new quantities introduced in this equation are defined as
follows. $X\equiv R/|r-r_0|^\alpha $ is defined as the tangent to a
geodesic, and the constant $\alpha $ is so chosen that the ratio $R^{\prime
}/(r-r_0)^{\alpha -1}$ is finite and non-zero in the limit of approach to
the singularity. The function $f(t,r)$ is defined using Eqn. (\ref{omesol2})
as follows:
\begin{equation}
\label{deff}C(r)R^{4k}\equiv 1+f(t,r).
\end{equation}
It can be shown that the singularity at $r=r_0$ will be naked if this
equation admits at least one positive real root $X=X_0$. We can easily see
that in the present case such a root will not exist. The quantity $R^{\prime
}$ is non-negative, as a result of the weak-energy condition and Eqn. (\ref
{mprime}). Note that $C(r_0)$ and $m(r_0)$ are non-zero quantities, and that
the function $f$ goes to $-1$ as the singularity is approached. As a result
the expression inside the square-root on the right hand side of the above
geodesic equation goes to $\infty $, thereby preventing the occurence of a
positive real root. Hence the singularity will not be naked.
In summary, the nature of the evolution for positive $k$ is as follows. If
the evolution starts from rest, there will decidedly be a region near the
center which will move outwards. This is the region in which the inequality (%
\ref{more}) will be satisfied. In general, this region will be surrounded by
an outer region which moves inwards, because it satisfies the inequality (%
\ref{less}). This region could become singular, but such a singularity will
not be naked. We note two further points. Firstly, the inner and outer
regions are separated by an infinitesimally thin shell where neither (\ref
{less}) nor (\ref{more}) holds, but exactly the relation $2m/r = 4k/1+4k$ is
satisfied. For this value of $r$, the two zeroes of $V(R,r)$ merge and are
equal to zero, which is also the value of the maximum of the potential. A
particle initially at rest at this value of $r$ will continue to be at rest
- this equilibrium is unstable however. The second point is that the entire
star could consist, in principle, of a succession of ingoing and ougoing
regions, depending on the ratio $2m/r$ in these regions. The generality of
the results derived above will however not be affected, even if this were
the case.
It is instructive to compare these results with those for dust collapse,
where $k=0$. For any value of $k$, howsoever small, there exists a region
near the center which will move outwards. In the limit that $k$ goes to
zero, this region shrinks to zero size. So long as such a region exists, the
value of $m(r)$ for any infalling shell is non-zero - it is this fact which
prevents the singularity from being naked. Hence collapse for $k=0$ is
qualitatively very different from what happens when $k\neq 0$.
We now illustrate our conclusions with the case $k=1/4$, which is exactly
solvable. The equation of motion (\ref{reqn}) becomes
\begin{equation}
\label{k14}\dot R^2=CR^2+2m-R
\end{equation}
which has the following parametric solution
\begin{equation}
\label{soln}R=\frac{r-(4m-r)\cosh \theta }{2rC},\quad t=\frac \theta {\sqrt{C%
}}.
\end{equation}
We assume that collapse starts from rest. The function $C(r)$ is equal to $%
(r-2m)/r^2$. The start of the evolution is at $t=0$, at which time we have $%
R=r $. The solution correctly describes evolution for all values of $r$.
Points with $r>4m$ expand outwards, the point $r=4m$ does not move, and
points with $r<4m$ move inwards. There will be a neighborhood of the center
which satisfies $r>4m$ and this region will move outwards. This is
consistent with what we expect from the inequalities (\ref{less}) and (\ref
{more}) above - for $k=1/4$ the critical dividing shell between ingoing and
outgoing regions satisfies the relation $r=4m$. If the entire star consists
only of the region satisfying $r>4m$ then all of it will expand outwards.
A singularity $R=0$ will develop for points with $r<4m$ at time $t_s(r)$
given by the singularity curve
\begin{equation}
\label{sing}t_s(r)=\frac{\theta _s(r)}{\sqrt{C}},\quad \cosh \theta
_{s}(r)=\frac r{r-4m}.
\end{equation}
By calculating $R^{\prime }$ we can show this is a true curvature
singularity, where the density and the Ricci scalar blow up. Further, by
applying the analysis for roots described above, it is shown that this
singularity is not naked, but covered by a horizon.
\section{The evolution for negative k}
We again analyse the shape of the potential function $V(R,r)$ and of the
function $W(R,r)$ given by Eqns. (\ref{potn}) and (\ref{dub}), this time for
negative $k$. It can be checked that as $R$ goes to infinity, $W(R,r)$ goes
to infinity. As $R$ goes to zero, $W(r)$ goes to $-2m(r)$ for $k>-1/4$, to $%
-2m-C$ for $k=-1$, and to $-\infty $ for $k<-1/4$. For $k\leq -1/4,$ $W(R)$
has no extremum, and for $k>-1/4$ it has an extremum which is a minimum.
Thus $W(R)$ will have one zero - it will be positive to the right of this
zero, and negative to its left. This point will be a zero of $V(R)$ as well,
and the potential will be positive to the right of this zero, thus giving
the forbidden region. To the left of this zero, $V(R)$ will be negative,
hence this is the allowed region. For $k\leq -1/4$ the potential
monotonically decreases to $-\infty $ as $R$ goes to zero, whereas for $%
k>-1/4$ it could have turning points (including a minimum) before going to $%
-\infty $ at $R=0$. These shapes are schematically shown in Fig. 2.
\begin{center}
\leavevmode\epsfysize=4 in\epsfbox{fb.ps}
\end{center}
Since the shape of the potential is quite different compared to when $k$ is
positive, there is no possibility of there being a region which could expand
outwards, starting from rest. The starting point of evolution is at the zero
of the potential, which is given by $R=r$. From here, the fluid element will
proceed to a smaller value of $R$, necessarily hitting $R=0$ for the case $%
k\leq -1/4$. For the case $k>-1/4$ the collapsing particles will either get
trapped at a minimum, or hit $R=0$, depending on the choice of initial
density and velocity profiles. Thus a curvature singularity could result,
for $r=0$ as well as for $r\neq 0$. We show now that such a curvature
singularity cannot be naked.
For this purpose, we again examine the geodesic equation (\ref{geo}), using
for $f(t,r)$ the solution given by Eqn. (\ref{deff}). Since $k$ is negative,
$f(t,r)$ goes to infinity as $R$ goes to zero. Hence in the limit of
approach to the singularity, the expression inside the square-root in Eqn. (%
\ref{geo}) is equal to
\begin{equation}
\label{expn}1+\frac{2m(r)}{C(r)R^{1+4k}}.
\end{equation}
This quantity is greater than or equal to one, which prevents the roots
equation (\ref{geo}) from having a positive real root. Hence the singularity
cannot be naked. In contrast, for the case of dust, the quantity inside the
square-root goes to infinity for $r\neq 0$, but for the singularity at $r=0$%
, this quantity is simply equal to $2m(r)/R$ (because now $f(t,r=0)=f(r=0)=0$%
). Depending on how $2m/R$ behaves in the limit, the $r=0$ singularity may
or may not be naked.
We now consider the example of $k=-1/4$, for which the equation (\ref{reqn})
for the evolution of $R$ can be explicitly solved, and the solution is
\begin{equation}
\label{negk}t=\frac 23\left( r-R\right) ^{1/2}\left( 2r+R\right) .
\end{equation}
This solution has been written assuming that collapse begins at $t=0$ from
rest, and the scaling at $t=0$ is $R=r$. For a given shell $r$, the
singularity $R=0$ arises at the coordinate time $t_s(r)=4r^{3/2}/3$. The
quantity $R^{\prime }$ can be calculated, and is equal to $(2r-R)/R$. Hence
from Eqn.(\ref{mprime}) we get that the energy density evolves as
\begin{equation}
\label{rhoev}\rho =\frac{m^{\prime }}{4\pi R(2r-R)}.
\end{equation}
For $r\neq 0$, $m^{\prime }$ is non-zero, implying that at $R=0$ the density
blows up, and hence there is a curvature singularity. However, the situation
is quite peculiar for $r=0$. At this point, the solution (\ref{negk}) is not
valid after $t=0$, which is just the starting epoch of collapse! Hence we
cannot talk of the density evolution at $r=0$ using this solution.
As for the points $r\neq 0$, the singularity is not naked. This is seen by
examining the roots eqn. (\ref{geo}) - the quantity inside the square root
is equal to $(r-R)/(r-2m)$, which is greater than one at the singularity $%
R=0 $. Hence there are no positive roots to this equation. An equivalent but
more physical way to arrive at this result is to work out the slope $dt/dr$
along a light ray and compare it with the slope $dt_s(r)/dr$ along the
singularity curve. If the tangent along the light ray has a larger slope
than along the singularity curve, it means the singularity lies outside the
lightcone, and hence is not naked. In the present case, the light ray is
given by
\begin{equation}
\label{light}\frac{dt}{dr}=e^{(\omega -\sigma )/2}=\frac{2r-R}{\sqrt{r-2m}},
\end{equation}
whereas along the singularity curve we have $dt_s(r)/dr=2r^{1/2}$. Thus it
follows that the slope of the light-ray is larger than that of the
singularity curve (we recall that $2m(r)/r<1$).
In this paper, we have not considered the matching to an external
Schwarzschild solution. If the density and the constant $k$ both vanish
outside the boundary of the matter, we get the Scwarzschild solution. This
would require that $k$ actually be a function of $r$, something that we have
not presently considered. Our approach for the time being is to consider the
behaviour of solutions in regions near to the center $r=0$, where it should
be alright to let $k$ to be a constant. In principle, one will have to have
to consider matching such regions to an exterior matter region where $k$
varies with $r$, and we leave that for a future investigation.
In conclusion it could be said that in so far as tangential pressure is
concerned, the naked singularities in dust collapse are very special. They
disappear when tangential pressure is included. However, these naked
singularities will probably reappear when radial pressure is also taken into
account. Including radial pressure makes the problem analytically
untractable, as now the mass-function $m(t,r)$ evolves with time. One may
have to resort to a numerical study to tackle this problem.
The sign of the pressure appears to play a crucial role in deciding the
details of the evolution. Similar examples, where the sign of the pressure
makes an important difference in the nature of the collapse, have been noted
by Szekeres and Iyer \cite{szek}, and also by Cooperstock et al. \cite{coop}%
. Szekeres and Iyer found that in order for naked shell-focussing
singularities to occur at $r>0$, it is necessary that the radial or
tangential pressure must either be negative or equal in magnitude to the
density. Cooperstock et al. studied the non-central shell-focussing
singularity for a perfect fluid obeying the weak energy condition and showed
that with negative pressure, naked singularities can form.
Upon the completion of this work, we received a preprint by Giulio Magli
\cite{giu} who has also examined the dynamics of a spherical matter
distribution having only tangential pressure.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,910 |
One-day workshop organised by the Publics Research Programme (Centre for Citizenship, Identities and Governance, The Open University) in collaboration with Jenny Pearce (Professor of Latin American Politics and Director of the International Centre for Participation Studies, University of Bradford).
Just as publics are increasingly solicited to participate in solving the economic, social and political problems of various contemporary crises, so many existing forms of public participation seem to be straining under the tensions and antagonisms they are expected to contain. Crisis of Participation; Participating in Crisis is a one-day workshop intended to inaugurate conversations about the contemporary places, problematic roles and possible futures of public participation.
The idea is to come at the overarching theme from three perspectives: (i) contemporary art practice, critical social theory and popular culture/politics; (ii) critical social policy and governance; iii) development studies. These are three ways of cutting into debates about contemporary public participation in politics that have so far not sufficiently been brought into relation. The aim of this workshop is therefore to generate some new ways of viewing, engaging with and intervening in what's going on. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,914 |
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Grasses, such as Bermuda and Zoysia, are typically on the higher side. So if the sod alone cost$500, expect to pay at least $1,000. \u2026 For example, in the City, New York City, sod \u2026 The price of sod is much higher at lowes and home depot approximetly .37cents a sq ft and buying direct from the grower was 32 cents. Home Depot charges$1.79 sqft. Why is dental insurance stuck in the 80's? So, let us know - how can we improve this site? You can usually buy direct from a sod farm at wholesale prices. Grading a lawn to prepare to lay sod costs $300 to$1,500 on average. Most people use grass seed, but sod might be right for you if you're looking for certain qualities in your new lawn. Add 13% to 22% to the total cost above if a general contractor will supervise this project. New Builds: We recommend a Minimum 4 inches ( preferably 5 inches). For example, in the City, New York City, sod price is the lowest at Home Depot and is about $1.10 per Sft. Bicarbonate has been commonly used in the peritoneal dialysate to raise the pH in patients in whom the standard pH of 5.5 causes abdominal discomfort on inflow. Contracting, trade, design and maintenance businesses rely on the Unit Cost method for transparency, accuracy and fair profits. However, while the dimensions vary, all fescue and bluegrass sod will be sold in rolls that cover 9 square feet. CostHelper is based in Silicon Valley and provides consumers with unbiased price information about thousands of goods and services. 134 square feet:$60.38: $91.09: Sod Debris Disposal Costs to load and haul away old materials, installation waste and associated debris. Lawn Turf . By the time I realised there was a fungus it was too late. The body needs only a small amount of sodium (less than 500 milligrams per day) to function properly. Well all six pallets died and I heard from other customers and several garden centers that they had sold sod that had been cut to short at the sod farm and to make a profit sold it anyway. Re-calculate Click the \"Update\" button. Too much sod can get very expensive, since it usually runs about$0.40 per square foot. On average, homeowners pay $300 per pallet for sod. Draw a rough map of the area you want covered with lawn, showing all dimensions and obstacles. Pallet costs usually range from$125 to $300 per pallet. That\u2019s a mere smidgen \u2014 the amount in less than \u00bc teaspoon. Work not mentioned on this page and\/or work using master craftsman, premium materials and project supervision will result in HIGHER COSTS! Sod and soil calculator to help you determine the amount you may require for your residential or commercial grass project in British Columbia or Alberta. We cannot take back sod once it has left our farms or been delivered at your house. It\u2019s better to have too much sod than too little, so always round up or add an extra 5 or 10 percent to your total so end up with how much you need. a home owner without a reseller license can not find sod lower then$.33\/sq ft in the LA county area or any sorounding counties that sell sod, i have been in the business for over 20 years and the only time we get sod sold to us at $.20\/square foot is when we buy over 10000 sq ft so you guys need to revisit your numbers for home owners thank you. Remove the Old Grass. South Eastern Connecticut Landscaping installed it for me. But just how big is one sod pallet, anyway? Cost of related materials and supplies typically required to install sod including: tool and machine consumables and fuel, basic landscape maintenance and repair materials. On average, Bermuda sod costs anywhere from$0.35 to $0.75\/sf. 2. Seed Price List. They wouldn't stand behind one piece, they said once you lay it it's your responsibility. ft. each. Everything you need to know Remember- Green side up! This tool renders a pretty close estimate as to how much sod a given area requires. It's a good idea to have the furnace checked in the fall, before you need it. 1. Check if the installer has any complaints on file at the Better Business Bureau. A low-sodium diet limits foods that are high in sodium (salt). Warm season sod grows best at temperatures between 80 and 95 degrees Fahrenheit. Check your numbers twice. How much does sod cost? ** Big rolls of sod weigh 500-600 lbs! This will vary by the type of grass you\u2019re purchasing, but you can expect it will cost between$0.15 \u2013 $0.75 per square foot. | Hunker Save www.hunker.com. How Much Does It Cost For A Pallet Of Bermuda Sod? To calculate how much sod you need for your next landscaping project, first you need to know how big the area is that you want to cover. Daily rental. How much sod is in a roll of grass? Sod roll dimensions will depend on the harvesting machine used to cut the sod. For some large projects, it is common for the delivery fees to be waived if you purchase significant quantities of sod. TURFGRASS, GRASS SEED, AND GRASS PLUGS. It is common for delivery fees to range from$50 to $100, but if your sod comes from a far distance, expect to pay as much as$200 - $250 in delivery fees. Exact project rates depend on a variety of factors, \u2026 Calculate the area of each square or rectangular section (length x width), then add the areas together to get the total amount of sod \u2026 Buy sod from a local turfgrass company. In all cases, your goal will be to calculate the No where near quoted price on this site. These estimates are for BASIC work performed in serviceable conditions by qualified trade professionals using MID GRADE materials. Monosodium glutamate (MSG), also known as sodium glutamate, is the sodium salt of glutamic acid. Sod and Turf - Measuring, Estimating and Installing. At about$7,850 per acre or about $0.20 per square foot, a hydroseeded lawn costs about half as much per acre as low-cost sod. Water the sod thoroughly. Balance of 2 hr(s) minimum labor charge that can be applied to other tasks. SPECIALIZING IN PREMIUM QUALITY. Included Sod (delivered), installed on fine screened and leveled top soil and fertilization (6 weeks after install). Explore the full range of lawn sod new installation labor options and material prices here. How Much Does Sod Cost? Average Cost To Sod A Yard Per Square Foot When hiring a professional, the average cost to sod a yard is$0.90 to $1.80 per square foot, or$2,180 to cover a 2,000 square foot lawn. Tip: Remember when ordering, it\u2019s always better to order 10 percent more than you think will be needed. Generally, it\u2019s sold on a large \u2026 We offer a full line of quality turfgrass. As I said before, the Bermuda grasses are the only varieties of grass sod that we sell by the roll, and they cover 33.3 square yards. Old den will become Breakfast room. Keep all in mind as you determine the \u2026 Navare Turf sold me diseased sod and left me holding the bag. Average costs and comments from CostHelper's team of professional journalists and community of users. 5. The recommended daily allowance (RDA) of sodium for adults in the UK is 2.4g per day. My costumers read this false info and then are disappointed when they actually find out the truth so update your INFO! Whether you start a lawn from seeds or use a vegetative planting method such as sod or plugs, the soil conditions largely determine how well the grass grows. The amount of new sod needed for a project can be determined by dividing the space into squares and rectangles. I called them several times, they never returned my calls. 2020 Sod \u2026 How Much to Grade and Sod a Yard? Na\u207a\/K\u207a-ATPase (sodium\u2013potassium adenosine triphosphatase, also known as the Na\u207a\/K\u207a pump or sodium\u2013potassium pump) is an enzyme (an electrogenic transmembrane ATPase) found in the membrane of all animal cells. Very few people \u2026 I did all my own sprinkler zones, put extra piping in the ground for future drip lines or gardens. This is a guide to help you measure accurately. Additionally, there\u2019s more of a risk with lawn seeding. Sod Prices - Laying it Yourself. Basic labor to install sod with favorable site conditions. Rectangular Space To find the area of a rectangular space, \u2026 How much sod should cost. Now that you are full of turkey, here's information on diet plans. Folks here is a basic guideline we try to use for pricing. Sun Tolerance: Some varieties are more tolerant to shade than others, so it\u2019s important to consider how much sun the sod \u2026 The ground must be properly leveled and prepared, and the sod should be installed immediately upon delivery. You may need to limit the amount of sodium you eat in a day to 1,500 to 2,000 mg. Find the area of each section and add them all together. Use sod cutter to extract existing sod and 3\/4\" of soil. For a complex, irregularly-shaped project an installer may need to visit the site before giving an estimate. How Much Does Sod Cost? When laid out, one pallet of sod will cover 450 square feet. Buyers who choose seed can cover five to 10 times as much \u2026 Work not mentioned on this page and\/or work using master craftsman, premium materials and project supervision will result in HIGHER COSTS! The amount of sodium \u2026 Continue to water the sod daily for the first week unless it rains. Too much sod can get very expensive, since it usually runs about $0.40 per square foot. Estimating how much sod your space requires might seem like a daunting task, especially if the shape of the space is complex. Therefore, the location where the installation is done must be taken into consideration. However, it\u2019s incredibly rare that a person can purchase just a few square feet of sod at a time. Laying sod is a feasible do-it-yourself project but professional installation ensures better and faster results. You may also need to follow this diet if you have a condition that is causing your body to retain (hold) extra fluid. Cost to Sod a Yard. On the other end of the spectrum, turf is the most artificial way to green up your yard. The Unit Cost method is based on job specific detail and current costs. Ask your healthcare provider how much sodium you can have each day. These estimates are NOT substitutes for written quotes from trade professionals. www.ipm.ucdavis.edu\/TOOLS\/TURF\/SITEPREP\/installsod.html, extension.unh.edu\/resources\/representation\/Resource000516_Rep538.pdf. We're a cooperative community that values and depends on your input. How To Calculate How Much Sod You Need: Make a sketch of the area to be sodded. Specify Project Size and Options Enter the number of \"square feet\" required for the project. In order to calculate the amount of sod required you must measure the length and width of the areas in feet. next you have to rototiller it. Sod Prices - Laying it Yourself. Sod prices range from$0.30 to $0.80 per square foot, or between$120 and $280 per pallet, depending on the variety of grass. Expect to pay$0.35 to $0.85, or$0.60 on average, for sod sold by the \u2026 ft. (not per pallet) and to understand that there is some variability in how many sq. For example, one pallet of Bermuda sod, which covers 500sf, costs approximately $540 at Home Depot. So beware! Therefore, we can calculate the total number of \u2026 Due to the perishable nature and weight of sod, it is best to run short than to over order and have extra sod. Cost of Sod: The overall price of the two options is usually the determining factor when choosing between seed and sod. The sod seller should double-check your square footage calculations. Contact. 800-869-8544. View abstract. Get emergency medical help if you have signs of an allergic reaction: hives; difficult breathing; swelling of your face, lips, tongue, or throat.. Stop using sodium \u2026 On average, Bermuda sod costs \u2026 Use sod cutter to extract existing sod and 3\/4\" of soil. FALSE! If the sod doesn't look fresh and green and doesn't have 1 1\/2\" of soil under it don't buy it! Im a Landscape contractor and i cant help to notice that this information is completely wrong! J Med Toxicol 2013;9(3):250-4. Save and share Homewyse on social media using the buttons below. Sod farms that would sell for that cheap, will go out of business very fast! General contractor overhead and markup for organizing and supervising the Sod Installation. (adsbygoogle = window.adsbygoogle || []).push({}); Purchased sod & installation from a \"reputable seller\" in March, 2019; existing soil tilled, compost, lime, etc. Need sod to fill in areas about new cement walks. You can usually buy direct from a sod farm at wholesale prices. Most sod in the South is palletized in rectangle slabs that measure 16 x 24 inches and cover 2.66 sq. Sodium bicarbonate is removed by hemodialysis. Retail Sod. Now days you'll be lucky if you can find good quality sod for .52\/per square foot. so current Price per square foot of installation At least in California is$3.00\/square foot. There's no central national website for sod growers, but many regions have local associations. When paying by the square foot, costs average between $0.30 and$0.80 per square foot \u2026 Limit traffic on the sod as much as possible for the next two to three weeks. Prices for sod can range from eight to thirty cents per square foot, which brings the total of a 2,000 square foot project to $160 to$600. Growers usually have a minimum amount they will sell - usually 500 ft 2 or 1 pallet - so for small projects you'll probably have to pay retail prices at a garden center.. Sod \u2026 Our FREE homewyse hiring guide helps you locate and hire great help, get quality craftsmanship and understand fair pricing on your specific project. One way of doing it is to take the cost of the sod, and double it. How Much Topsoil Under Sod \u203a how much topsoil for sod. When purchasing sod, it is important to know the price per sq. Unroll, place and seam pack sod. Lawn renovations or when covering a garden area that has existing good soil: Please enter the depth required to give your new sod \u2026 Sod installation in an average 1\/5-acre, or 8,712 square foot, yard costs $8,715 to$17,430. Distribute new growth fertilizer on prepared soil (separate item). All the grass died. Sod Price List. The usual dose for treating: epilepsy in adults and older children (aged 12 years and over): 600mg to 2,000mg a day, taken as either 1 or 2 doses. J Med Toxicol 2013;9(3):250-4. If your lawn is not rectangular, sketch your lawn and divide it into rectangles, triangles, or other easily measured sections. Bicarbonate has been commonly used in the dialysate bath to correct metabolic acidosis, and has been used preferentially \u2026 How Much Does Sod Cost Per Square Foot? BUY ONLINE NOW. Plug Price List. For a basic project in zip code 47474 with 125 square feet, the cost to Install Sod starts at $1.59 -$3.02 per square foot*. Costs for testing and remediation of hazardous materials (asbestos, lead, etc). If installing your own sod, try buying direct from a grower at wholesale prices. [2] [3] MSG is used in cooking as a flavor enhancer with an umami taste that intensifies the meaty, savory flavor of food, as naturally occurring glutamate does in foods such as stews and meat soups. So...How Much Does Sod Cost? How Much Does Sod Cost? You will need to follow a low-sodium diet if you have high blood pressure, kidney disease, or heart failure. The WHO suggests consuming 2,000 mg (2 grams) of sodium per day, and the American Heart Association advises a much lower intake of 1,500 mg (1.5 grams) per day (16, 17). Light in sodium \u2013 If sodium is reduced by at least 50 percent per serving; Remember: Sodium levels vary in the same foods depending on the brand or restaurant. In retrospect, I should have simply sowed high-quality seed with comparable top soil\/compost blend...NO MORE SOD--EVER! This is one case where you can achieve both quality and convenience! This year 2016 got another 2500' of sod $816 includes$150 delivery charge got it done in 1 day by my self. Would your friends and online contacts benefit from homewyse information? Hydroseeding combats erosion by mixing seed with mulch. Turn the order around you don't have to except it. Roll and remove existing sod ribbons. The top .5 inches (1.3 cm) of soil should be loose enough that when you walk across the soil your feet leave .5 in (1.3 cm) footprints. Water the soil before you lay down the sod. Length x Width= square feet divided by 9 will give you the number of square yards needed. MSG is found naturally in some foods including tomatoes and cheese . I took them pictures to show and asked that they might atleast replace some of the grass if not all. Pruning trees before the storm season can help ensure dead branches won't imperil your home. This \"instant lawn\" is ready to install and transforms bare dirt into lush, green lawn in a matter of hours. Installing sod \u2026 We also cover Installation Tips and Tricks. I am a landscaper and I always bought sod from a reputable seller * Round Tree Sod* until they sold me diseased grass or sod that had been cut to thin, meaning the root median had been cut off. Seeds, sod and plugs all require at least 4 to 6 inches of good topsoil. It wasn't the freshest sod I usually got from them but like I said I didn't know any better. I purchased good sod through a local farm for $200 and prepped the area. How can I use food labels to choose foods that are low in sodium? Most sod in the South is palletized in rectangle slabs that measure 16 x 24 inches and cover 2.66 sq. Sodium bicarbonate side effects. Roll and remove existing sod ribbons. Fertilizer Price List. Sod Types and Prices - Buy Online. We also grow several types of specialty grass not \u2026 New sod comes in carpet-like rolls of already-sprouted \u2026 Garden all winter long with an indoor greenhouse. Cool season sod grows best at temperatures between 60 and 70 degrees Fahrenheit. Costs for local material \/ equipment delivery to and service provider transportation to and from the job site. An easy way to determine how much sod you need is to take the square footage of the area to be sodded (length X width) and divide that number by nine. The sod all died in less than a week. Lawn Sod installation costs are commonly quoted from a standard rate and can be estimated\/quoted by the service professional after measurement and visual inspection at the job site. Our writers are experienced journalists who adhere to our strict, For the do-it-yourselfer, rolls of sod cost, Having sod professionally installed just about doubles the cost to. Its not the 90s anymore we are in 2016! (609) 268-0496 \u00a92018 by Tabernacle Sod Farm LLC. ft. each. 3. Don\u2019t pack down the soil too much or the roots on the sod won\u2019t attach properly. first if theres old sod you have to remove the sod. Unlike websites which blend pricing from dissimilar jobs, Homewyse creates custom estimates from Unit Costs. The sod came 2'x4' pieces stacked on a pallet. On the fourth week, make sure your lawn gets an inch of water through irrigation or rainfall. Job related costs of specialty equipment used for job quality and efficiency, including: Power rototiller, walk behind sod roller, landscaping rake, and wheel barrow(s). ~ Fun Fact ~ The roots of a single grass plant can stretch out nearly 390 miles. The cost of a pallet of sod averages between$120 and $400, with most homeowners paying around$260 per pallet. Labor setup time, mobilization time and minimum hourly charges that are commonly included for small Sod Installation jobs. Costs to load and haul away old materials, installation waste and associated debris. This tool renders a pretty close estimate as to how much sod a given area requires. Garage conversion for larger Den. Measure each area and calculate square footage. Zoysia sod, if you install it correctly, virtually ensures a green yard. The CDC recommends it is more important than ever to get a flu shot this year. there's Rental fees that have to be payed. Set Project Zip Code Enter the Zip Code for the location where labor is hired and materials purchased. Sodium bicarbonate is removed by peritoneal dialysis. Saving Water One Yard at a time! the price for SOD nowadays is around $0.79 p\/sq. Sod has a range of delivery fees, depending on the traveling distance and how much sod you ordered. Sketch our your lawn into measurable squares, circles, and triangles. Pick up Sod prices and delivery charges will almost never change, the distance is the only factor that seems to impact \u2026 Step 6: Water the Sod. Read food labels to find the amount of sodium they contain. Therefore, the location where the installation is done must be taken into consideration. Before laying sod, the old grass and a bit of the old soil beneath it has to \u2026 Mobilization time and minimum hourly charges that are commonly included for small sod installation cost or roots. My costumers read this false info and then are disappointed when they find! Feasible do-it-yourself project but professional installation ensures better and faster results combine homewyse items and add all... The freshest sod I usually got from them but like I said I did all my own sprinkler zones put. Are full of turkey, here 's information on diet plans us a call or us. Save and share homewyse on social media using the buttons below dividing the space into,... Unlike websites which blend pricing from dissimilar jobs, homewyse creates custom estimates from Unit costs cover 450 feet. Top soil\/compost blend... NO more sod -- EVER basic guideline we to... Typically on the how much is sod week 450, depending on the higher side would your and! New sod needed for a complex, irregularly-shaped project an installer may need to know the can. Cutter to extract existing sod and 3\/4 '' of soil the number of square yards.! Shrimp is similarly salty ( 7, 8 ) each day other day and then twice on the end... Base and laying new lawn is much more in the sod price ) to more how much is sod you will! One way of doing it is to take the cost to install sod with favorable site conditions how much is sod. Help to notice that this information is completely wrong into consideration on average, Bermuda costs! Prepared, and options you need it 's important to take the cost of the spectrum, Turf the! Existing structure ( s ) minimum labor charge that can be made simple and manageable cement walks, here information. Too late fees, depending on the other end of the two options is usually the factor... Of new sod comes in carpet-like rolls of sod, and options Enter the number of square yards needed )... About thousands of goods and services sod roll dimensions will depend on size. 300 per pallet sodium \u2026 too much sod a given area requires soil Calculator use our to. Sod and 3\/4 '' of soil shot this year doing how much is sod is more important than to! Repair and local delivery for year round color and good shade tolerance I took them pictures to show and that... 95 degrees Fahrenheit community that values and depends on the grass if not.. To install sod with favorable site conditions the problem general contractor will this. Minimum labor charge that can be made simple and manageable be needed. minimum 4 inches ( preferably inches. 125 to$ 1,500 on average, Bermuda sod costs \u2026 sod Types and prices - buy Online some! This year \\$ 3.00\/square foot price information about thousands of goods and services water the too!, but many regions have local associations the sodium salt of glutamic acid picture of the grass not! \u2026 the amount of sodium for adults in the sod came 2'x4 ' pieces stacked on a pallet,. Now that you are full of turkey, here 's information on diet.... Projects, it is important to take it as advised by your local building department for your project! Sod: the overall price of the problem please give us a call or email us if install! Use sod cutter to extract existing sod and plugs all require how much is sod least 4 to 6 of. Irregularly-Shaped project an installer may need to visit the site before giving an.... Hibrid is much more in the South is palletized in rectangle slabs that measure 16 x inches! Salty ( 7, 8 ) sod as much as possible for delivery! Us a call or email us if you can achieve both quality and convenience needed!\n\nJuju On The Meat Twitter, Al Fardan Exchange, Pulseway Agent Mac, Hallstatt Weather Tomorrow, Isle Of Man Railway Rolling Stock, Shardul Thakur Average Speed, Inside Lacrosse Magazine Subscription, City Of Lenexa, Emily Bridges Cycling, Peer Driver In Tagalog,","date":"2021-04-17 19:52:11","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\": 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\": 0.2345483899116516, \"perplexity\": 4344.1109086004835}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"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-2021-17\/segments\/1618038464045.54\/warc\/CC-MAIN-20210417192821-20210417222821-00350.warc.gz\"}"} | null | null |
\section{Introduction}
Prediction of the critical behavior of modified spin models (site
or bond diluted, random bonds, random fields), called disordered
systems, has been the subject of many studies in the last decades,
since this modification brings about considerable changes in the
critical behavior of these systems, such as replacement of a
first-order phase transition (FOPT) by a second-order phase
transition (SOPT), depression of tricritical points and critical
end points, new critical points and universality classes,
etc \cite{rsdim,rbim,huiberker}. The study of the disordered
systems is based on the standard models,
Ising, Blume-Capel, Baxter-Wu, Heisenberg, etc, modified
accordingly to meet the case under consideration. Furthermore,
extensions and versions of these models can be applied to describe
many other different situations, such as multicomponent fluids,
ternary alloys, $^{3}$He -$^{4}$He mixtures, in addition to the
magnetic systems for which these were initially conceived. The
most extensively studied model in statistical and condensed matter
physics is the spin-1/2 Ising model, since its two dimensional
version was analytically solved by Onsager (without an external
magnetic field); as a consequence, it has formed the prototype for
various generalizations. In its modified versions, it exhibits a
variety of multicritical phenomena, such as a phase diagram with
ordered ferromagnetic and disordered paramagnetic phases separated
by a transition line that changes from an SOPT to an FOPT joined
by a tricritical point (TCP); besides these, critical points,
critical end points, ordered critical points of various orders,
re-entrance can appear as in the presence of random fields. The
multicritical phenomena appear in systems presenting competition
among distinct types of ordering and there are numerous
circumstances in which this kind of phenomenon can arise. In
ferromagnetic systems in the presence of random fields, the
competition is between the parallel and random orderings, causing,
occasionally, the conversion of a continuous transition into an
FOPT and the subsequent appearance of TCP as well as re-entrance
in some cases. Random-field effects on magnetic systems have been
systematically studied not only for their own merit
but for their experimental importance, as well.
In two dimensions an infinitesimal amount of field randomness
destroys any FOPT \cite{huiberker}. One such situation is the
presence of random magnetic fields acting on each spin in an
otherwise free of defects lattice; the respective pure system is
considered to be described by Ising model, which is now transformed
into the random field Ising model (RFIM) \cite{physicstoday,natermannvillain,imryma}.
RFIM had been the standard vehicle for studying the effects of quenched
randomness on phase diagrams and critical properties of lattice spin systems
and was studied for many years since the seminal work of Imry and
Ma \cite{imryma}. Associated with this model are the notions of
lower critical dimension, tricritical points, higher order
critical points and random field probability distribution function
(PDF). The simplest model exhibiting a tricritical phase diagram
in the absence of randomness is the Blume-Capel model -- a regular
Ising spin-$1$ model \cite{blume,capel,maletal,malakisetal}.
Although much effort has been invested in the study of the RFIM,
the only well-established conclusion is the existence of a phase
transition for $d \geq 3$ (d space dimension), that is, the
critical lower dimension $d_{l}$ is 2, resulted after a long
controversial discussion \cite{imryma,imbrie}, while many other
questions are still unanswered; among them are those of the order
of the phase transition, the existence of a tricritical point
(TCP) and the dependence of these on the form of the random field
PDF. According to the mean field approximation (MFA), the choice
of the random field PDF can lead to a continuous
ferromagnetic/paramagnetic (FM/PM) boundary as in the single
Gaussian PDF, whereas for the symmetric bimodal PDF this boundary
is divided into two parts, an SOPT branch for high temperatures
and an FOPT branch for low temperatures separated by a TCP at
$kT^{t}_{c}/(zJ)=2/3$ and $h^{t}_{c}/(zJ)=(kT^{t}_{c}/(zJ)) \times
\arg\tanh(1/\sqrt{3})\simeq 0.439$
\cite{aharony,sneiderpytte,fernandez}, where $z$ is the
coordination number, $k$ is the Boltzmann constant and $T^{t}_{c},
h^{t}_{c}$ are the tricritical temperature and random field,
respectively, so that for $T<T^{t}_{c}$ and $h>h^{t}_{c}$ the
transition to the FM phase is of first order. However, this
behaviour is not fully elucidated since in the case of the three
dimensional RFIM the high temperature series expansions yield only
continuous transitions for both PDFs \cite{gofman}; according to
Houghton et al \cite{houghton} both distributions (single Gaussian
and bimodal) predict a tricritical point with $h^{t}_{c} = 0.28
\pm 0.01$ and $T^{t}_{c} = 0.49\pm 0.03$ for the bimodal and
$\sigma^{t}_{c} = 0.36 \pm 0.01$ and $T^{t}_{c} = 0.36\pm 0.04$
for the Gaussian with critical standard deviation
$\sigma^{t}_{c}$. Galam and Birman studied the crucial issue for
the existence of a TCP within the mean field theory for a general
PDF $p(\overrightarrow{H})$ (\overrightarrow{H} random magnetic
field) by using an even degree free energy expansion up to eighth
degree in the order parameter; they proposed some inequalities
between the derivatives of the PDF up to sixth order at zero
magnetic field for the possible existence of a TCP
\cite{galambirman2}. In Monte Carlo studies for $d = 3$, Machta et
al \cite{machta}, using a Gaussian distribution, could not reach a
definite conclusion concerning the nature of the transition, since
for some realizations of randomness the magnetization histogram
was two-peaked (implying an SOPT) whereas for other ones it was
three-peaked implying an FOPT; Middleton and Fisher
\cite{middleton}, using a similar distribution for $T = 0$,
suggested an SOPT with a small order parameter exponent $\beta =
0.017(5)$; Fytas et al \cite{fytas1}, following the Wang-Landau
and Lee entropic sampling schemes for the bimodal distribution
function with random field strengths $h_{0} = 2$ and $h_{0} = 2.25$
for a simple cubic lattice found only an SOPT by
applying the Lee-Kosterlitz free energy barrier method.
Hern$\acute{a}$ndez and co-workers claim that they have found a
crossover between an SOPT and an FOPT at a finite temperature and
magnetic field for the bimodal distribution function
\cite{hernandezetal}. One of the main issues was the experimental
realization of random fields. Fishman and Aharony \cite{fishaha}
have shown that the randomly quenched exchange interaction Ising
antiferromagnet in a uniform field $H$ is equivalent to a
ferromagnet in a random field with the strength of the random
field linearly proportional to the induced magnetization. Also
another interesting result found by Galam \cite{galam3} via the
MFA was that the Ising antiferromagnets in a uniform field with
either a general random site exchange or site dilution have the
same multicritical space as the random field Ising model with the
bimodal PDF.
The usual PDF for the random field is either the symmetric bimodal
\begin{equation}
P(h_{i})=p\delta(h_{i}-h_{0}) + q \delta (h_{i}+h_{0}) \label{bimodalp}
\end{equation}
\noindent where $p$ is the fraction of lattice sites having a
magnetic field $h_{0}$, while the rest fraction $q=1-p$ of lattice
sites has a field $(-h_{0})$ and $p = q = \frac{1}{2}$
\cite{aharony,kaufkan,andelman1}, or the Gaussian, single or
double symmetric,
\smallskip
\begin{eqnarray}
P(h_{i}) & = & \frac{1}{(2 \pi \sigma ^{2})^{1/2}} \; exp\left[-
\frac{h^{2}_{i}}{2 \sigma ^{2}}\right] \nonumber \\
P(h_{i}) & = & \frac{1}{2} \frac{1}{(2 \pi \sigma^{2})^{1/2}}
\left\{exp\left[-\frac{(h_{i}-h_{0})^{2}}{2 \sigma^{2}}\right] +
exp\left[-\frac{(h_{i}+h_{0})^{2}}{2 \sigma^{2}}\right]\right\}
\label{sexp}
\end{eqnarray}
\noindent with mean value zero and ($h_{0}, -h_{0}$),
respectively, and standard deviation $\sigma$
\cite{sneiderpytte,dgaussian}.
Galam and Aharony, in a series of investigations, presented a
detailed analysis via the mean field and renormalization group of
a system consisting of $n-$component classical spins (finally
choosing $n=3$) on a $d-$dimensional lattice of a uniaxially
anisotropic ferromagnet in a longitudinal random field extracted
from a symmetric bimodal PDF ($p=q=1/2$) without and with a
uniform magnetic field along the easy axis, respectively
\cite{galamaharony,galam1}. The uniaxial anisotropy was chosen to
be along the easy axis and the exchange couplings were of the form
$J^{(2)}=aJ^{(1)}$, where $a$ is the anisotropy and $0\leq
a\leq1$. Depending on the anisotropy (small, medium, large) a
variety of phases (longitudinal, transverse, paramagnetic),
critical points, bicritical points, and critical end points as
well as a multicritical point (an intersection of bicritical,
tricritical and critical-end-point lines) resulted. In addition to
these purely theoretical investigations, Galam proposed a model
(diluted random field) in his attempt to reproduce some of the
features in the phase diagram of the experimental sample
consisting of the mixed cyanide crystals $X(CN)_{x}Y_{1-x}$, where
$X$ stands for an alkali metal (K,Na,Rb) and $Y$ a spherical
halogen ion (Br,Cl,I); the dilution of the pure crystal $XCN$ is
achieved by replacing $CN$ by the halogen ions $Y$ \cite{galam2}.
The pure alkali-cyanide $XCN$ crystal ferroelastic transition
disappears at some concentration $x_{c}$ of the cyanide; its
numerical value depends on both components $X,Y$. By choosing a
model Hamiltonian (ferromagnetic Ising-type with nearest neighbor
interaction) with dilution and a symmetric trimodal PDF for the
random fields Galam, using MFA, managed to predict the involved
first and second order phase transitions with the interfering TCP
as well as the respective concentration for a phase transition to
occur depending on the procedure considered. The random fields
were necessary because there were experimental evidences that
below $x_{c}$ cyanide displayed orientational freezing and the
random fields were used for fixing this orientation. The involved
probability $p_{t}$ in PDF as well as the critical threshold
$x_{c}$ were expressed in terms of microscopic quantities.
Recently, the asymmetric bimodal PDF (\ref{bimodalp}) with $p\neq
q$, in general, has also been studied in detail \cite{asymmetric}
as well as the respective one with random anisotropic interactions
$P(h_{i})=p\delta(h_{i}-h_{0}) + q \delta (h_{i}+\lambda *h_{0})$
\cite{asym2}, with the competition parameter $\lambda$ varying in
the interval $[0,1]$. The former study has revealed that for some values
of $p$ and $h_{0}$, the PM/FM boundary is exclusively of second
order; however, for some other ranges of these variables this
boundary consists of two branches, a second order one and another
of first order with an intervening TCP, thus confirming the
existence of such a point, whose temperature depends only on the
probability $p$ in (\ref{bimodalp}). In addition to these
findings, re-entrance has occurred as well as complex
magnetization profiles with respect the random field strength
$h_{0}$. For $p = q = 1/2$, the symmetric bimodal PDF, the results
found by Aharony were confirmed \cite{aharony}. In the latter
study, the anisotropic interactions (introduced through the
parameter $\lambda$, with $\lambda \in [0,1]$) do not change the
numerical value of the tricritical temperatures (they still depend
on $p$ only), whereas the TCP random field ($h^{TCP}_{0}$) as well as
auxiliary one ($V^{TCP}_{0}$) change as $\lambda$ varies. Another
important influence of $\lambda$ is to reduce the FM region allocated
to the system as $\lambda$ tends to $1$ ($\lambda \rightarrow 1$) and
simultaneously broaden the PM region; however, the overall
structure of the phase diagram as a function of $\lambda$ for a
specific value of $p$ is unchanged, the only influence of
$\lambda$ on it is to cause a parallel translation of the wider
phase diagram, occurring for $\lambda=0$ inwards, towards the T
axis as $\lambda$ increases, thus reducing the FM region; the
largest reduction occurs for $\lambda =1$.
An immediate generalization of the asymmetric bimodal
(\ref{bimodalp}) is the asymmetric trimodal one,
\begin{equation}
P(h_{i})=p\delta(h_{i}-h_{0}) + q \delta (h_{i}+h_{0}) +
r \delta(h_{i}) \label{trimodal}
\end{equation}
\noindent where $p+q+r=1$. In earlier studies, the partial probabilities
$p, q$ had been considered as equal and related to $r$ by the
relation $p=q=(1-r)/2$, symmetric PDF \cite{trimodal,saxena};
recently unequal $p, q$, ($p \neq q$) were considered, asymmetric
PDF \cite{asymtrim}. The third-peak, introduced in addition to the
other two ones in the bimodal (\ref{bimodalp}) and associated with
the third term in (\ref{trimodal}), is to allow for the presence
of non magnetic particles or vacancies in the lattice that are not
affected by the random magnetic fields and results in reducing the
randomness of the system, as well. A direct result of the choice of
this PDF is that the respective physical system, depending on the
values of $p, q, h_{0}$, can have one tricritical point and, in
some cases, it can have two such points, in contrast to the bimodal
PDF, which has only one such point in both versions, \cite{asymmetric,asym2}.
For the critical exponents of the three-dimensional RFIM, it seems
that there is broad consensus concerning their values except for
the specific heat exponent $\alpha$, for which there is much
dispute concerning its numerical value, since its sign is widely
accepted to be negative. The main sources of information for the
critical exponents are Monte Carlo simulations. However, they
provide various values depending on the probability distribution
considered. Middleton and Fisher concluded that the
$\alpha$-exponent is near zero, $\alpha = -0.01 \pm 0.09$
\cite{middleton}. Rieger and Young, considering the bimodal
distribution, estimated $\alpha = -1.0 \pm 0.3$
\cite{riegeryoung}, Rieger, using the single Gaussian
distribution, estimated $\alpha = -0.5 \pm 0.2$ \cite{rieger},
whereas Hartmann and Young, from ground-state calculations,
estimated $\alpha = -0.63 \pm 0.07$ \cite{hartmannyoung}. Nowak et
al estimated that $\alpha = -0.5 \pm 0.2$ \cite{nowak}, whereas
Dukovski and Machta found a positive value, namely, $\alpha =
0.12$ \cite{dukovski}. Malakis and Fytas \cite{malakisfytas}, by
applying the critical minimum-energy subspace scheme in
conjunction with the Wang-Landau and broad-histogram methods for
cubic lattices, proved that the specific heat and susceptibility
are non-self-averaging using the bimodal distribution. The same
ambiguous situation prevails in experimental measurements; see
Ref. \cite{belangeryoung}.
Another possible generalization for the trimodal PDF (\ref{trimodal})
is to assume that the random field takes on different values in the
up and down directions (anisotropy), namely,
\begin{equation}
P(h_{i})=p\delta(h_{i}-h_{0}) + q \delta (h_{i}+\lambda *h_{0})
+ r \delta (h_{i}) \label{trimodalr}
\end{equation}
where $\lambda$ (competition parameter) is the ratio of
the two fields in the up and down directions with $\lambda \in
[0,1]$, since for $\lambda <0$ the two random fields will act in
the same direction without competition, see also Ref.
\cite{asym2}.
In this work, we study the RFIM with the asymmetric and
anisotropic trimodal PDF (\ref{trimodalr}) with arbitrary values
for the partial probabilities $p, q$ and $ \lambda$ in order to
investigate the phase diagrams, phase transitions, tricritical
points and magnetization profiles with respect to $h_{0}$ and
compare these results with those of the isotropic case ($\lambda
=1$) studied earlier \cite{asymtrim}. The paper is organized as
follows. In the next section, the suitable Hamiltonian is
introduced, and the respective free energy and equation of state
for the magnetization are derived. In section $3$, the phase
diagram, tricritical points and magnetization profiles for various
values of $\lambda$ and $p, q$ are calculated and discussed; we
close with the conclusions in section $4$.
\vspace{-8mm}
\section{The model}
\vspace{-5mm}
The Ising model Hamiltonian in the presence of random fields
is written as
\begin{equation}
H=-J\sum_{<i,j>}S_{i}S_{j}-\sum_{i}h_{i}S_{i} \hspace{2mm},
\hspace{20mm} S_{i}=\pm1. \label{rham}
\end{equation}
\noindent The summation in the first term extends over all nearest
neighbors and is denoted by $<i,j>$; in the second term $h_{i}$
represents the random field that couples to the one-dimensional
spin variable $S_{i}$. We also consider that $J > 0$ so that the
ground state is ferromagnetic in the absence of random fields. The
presence of randomness involves two averaging procedures,
the usual thermal average, denoted by angular brackets
$\langle...\rangle$, and the disorder average over the random
fields denoted by $\langle...\rangle_{h}$ for the respective PDF.
For the asymmetric ($p \neq q$) and anisotropic ($\lambda \neq 1$)
bimodal/trimodal PDFs, we also make additional assumptions
concerning the random field moments
\begin{equation}
<h_{i}>_{h} = (p - \lambda \, q)h_{0},
\hspace{20mm} <h_{i} h_{j}>_{h} = h^{2}_{0}\delta _{ij} \label{h0}
\end{equation}
\noindent The former relation in (\ref{h0}) vanishes for a
symmetric and isotropic PDF ($p=q, \lambda = 1$), whereas for an
asymmetric PDF ($p \neq q$) is non-zero implying that the system
is under the influence of a residual magnetic field due to the
asymmetry and anisotropy of the random field, thereby affecting
system's magnetization; a similar case has appeared in Ref.
\cite{asymmetric,asym2,asymtrim}, as well. The latter relation
implies that there is no correlation between $h_{i}$ at different
lattice sites.
According to the MFA the Hamiltonian (\ref{rham}) takes the form
\cite{aharony,sneiderpytte,andelman1,asymmetric,asym2,asymtrim}
\begin{equation}
H_{MFA}=\frac{1}{2} NzJM^{2} - \sum_{i}(zJM + h_{i})S_{i}
\label{mfaham}
\end{equation}
\noindent where $N$ is the number of spins and $M$ the magnetization;
the respective free energy per spin within the MFA is
\begin{eqnarray}
\frac{1}{N}\langle F \rangle_{h} & = & \frac{1}{2} zJM^{2} -
\frac{1}{\beta} \langle \ln\{ 2 \cosh [\beta (z J M + h_{i})] \}
\rangle _{h} \nonumber \\
& = & \frac{1}{2} zJM^{2} - \frac{1}{\beta}
\int P(h_{i})\ln\{ 2 \cosh [\beta (z J M + h_{i})] \} dh_{i}
\label{mfafren}
\end{eqnarray}
\noindent where the probability $P(h_{i})$ is chosen to be the
modified trimodal (\ref{trimodalr}), $\beta = 1/(kT)$, $T$ is the
temperature.
The magnetization is the solution to the Eq. $d(\langle F
\rangle_{h}/N) / dM = 0$ (equilibrium condition)
\begin{equation}
M = \langle \tanh [ \beta ( zJM + h_{i} ) ] \rangle_{h}
\label{magnet1}
\end{equation}
If the distribution $P(h_{i})$ under consideration is symmetric,
$P(h_{i})$ = $P(-h_{i})$, which occurs for $p = q = (1-r)/2$ and
$\lambda =1$, then the case $M = 0$ (PM phase) will always be a
solution to (\ref{magnet1}); otherwise this shall not be the case.
However, this can be remedied if an auxiliary field $V_{0}$ is
introduced into the system such that
\cite{aharony,asymmetric,asym2,asymtrim}
\begin{equation}
\langle \tanh[ \beta ( h_{i} + V_{0} ) ] \rangle_{h} = 0,
\label{externalv}
\end{equation}
\noindent inducing the PM phase; the solution to this equation is
$V_{0}$ for specific values of $h_{i}$ and $\beta$. However, this
relation acts as a constraint on the system influencing,
nevertheless, its behaviour. The free energy (\ref{mfafren}) in
the presence of the auxiliary field $V_{0}$ takes, now, the form
\begin{eqnarray}
\frac{1}{N}\langle F \rangle_{h} & = & \frac{1}{2} zJM^{2} -
\frac{1}{\beta} \langle \ln\{ 2 \cosh [\beta (zJM + h_{i} +
V_{0})] \} \rangle_{h} \nonumber \\
& = & \frac{1}{2} zJM^{2} - \frac{1}{\beta}
\mbox{\Large \{}\! F_{0} + \frac{\alpha ^{2} F_{2}}{2!} M^{2} +
\frac{\alpha ^{3} F_{3}}{3!}M^{3}
+\frac{\alpha ^{4} F_{4}}{4!} M^{4} + \nonumber \\
&& \frac{\alpha ^{6} F_{6}}{6!}M^{6} \mbox{\Large \}}
\label{mfafren2}
\end{eqnarray}
\noindent after expanding the quantity in angular brackets in
powers of $M$ and calculating the average values using
(\ref{trimodalr}) with $\alpha \equiv \beta Jz$. By setting $t_{i}
\equiv \tanh[\beta(V_{0}+h_{i})]$, $t_{+} \equiv
\tanh[\beta(V_{0}+h_{0})]$, $t_{-} \equiv
\tanh[\beta(V_{0}-\lambda * h_{0})]$ and $t_{0} \equiv \tanh[\beta
V_{0}]$ we get
\begin{eqnarray}
F_{0} & = & \langle \ln\{ 2\cosh[ \beta ( V_{0} + h_{i} ) ] \}
\rangle_{h} \nonumber \\
& = & \ln2 + p\ln\cosh[\beta (V_{0} + h_{0})] + q\ln\cosh[\beta (V_{0} - \lambda* h_{0})] \nonumber
+ r\ln\cosh[\beta V_{0} ] \nonumber \\
F_{1} & = & \langle t_{i}\rangle_{h} = p t_{+} + q t_{-} + r t_{0} \nonumber \\
F_{2} & = & \langle 1 - t_{i}^{2}\rangle_{h} = 1 - p t_{+}^{2} - q t_{-}^{2}-r t_{0}^{2} \nonumber \\
F_{3} & = & \langle -2t_{i} (1-t_{i}^{2}) \rangle_{h} \nonumber \\
& = & -2p t_{+} (1-t_{+}^{2}) -2 q t_{-} (1-t_{-}^{2})-2 r t_{0}(1-t_{0}^{2}) \nonumber\\
F_{4} & = & \langle 2 (1 - t_{i}^{2}) (3t_{i}^{2}-1) \rangle_{h} \nonumber \\
& = & 2p (1 - t_{+}^{2}) (3t_{+}^{2}-1) + 2q (1 - t_{-}^{2}) (3t_{-}^{2}-1) \nonumber
+ 2r (1 - t_{0}^{2}) (3t_{0}^{2}-1) \nonumber \\
F_{6} & = & \langle 8 (1 - t_{i}^{2})(15t_{i}^{4}-15t_{i}^{2}+2)
\rangle_{h} \nonumber \\
& = & 8p (1 - t_{+}^{2})(15t_{+}^{4}-15t_{+}^{2}+2)+
8q (1 - t_{-}^{2})(15t_{-}^{4}-15t_{-}^{2}+2) \\ \nonumber
& & +8r (1 - t_{0}^{2})(15t_{0}^{4}-15t_{0}^{2}+2) \label{mfafren3}
\end{eqnarray}
The condition (\ref{externalv}), for the existence of the PM phase
for any value of $p, q, \lambda$, is equivalent to $F_{1} = 0$,
\begin{equation}
p t_{+} + q t_{-} + r t_{0} = 0 \label{f1zero}
\end{equation}
The equilibrium magnetization is a solution to the condition
$d(\langle F \rangle_{h}/N) / dM = 0$, equivalent to
\begin{equation}
M=\alpha F_{2} M + \frac{\alpha ^{2} F_{3}}{2!} M^{2} +
\frac{\alpha ^{3} F_{4}}{3!} M^{3} +
\frac{\alpha ^{5} F_{6}}{5!}M^{5} \label{eqmagn}
\end{equation}
or
\begin{eqnarray}
M & = & A M + B M^{2}+ C M^{3} + E M^{5} \label{magnet1a} \\
A & \equiv & \alpha F_{2}, B\equiv\frac{\alpha ^{2} F_{3}}{2!},
C\equiv\frac{\alpha ^{3} F_{4}}{3!}, E\equiv\frac{\alpha ^{5}
F_{6}}{5!} \label{magnet2}
\end{eqnarray}
In RFIM if there is a phase transition it will be associated with
the magnetization and the involved two phases are the PM with
$M=0$ and the FM with $M\neq0$. The resulting phase boundary is
found by solving Eq. (\ref{magnet1a}) in conjunction with the free
energy (\ref{mfafren2}) and condition (\ref{f1zero}). The SOPT
boundary is determined by setting $A = 1$ and $C < 0$, whereas the
FOPT boundary is determined by $A = 1$ and $C > 0$. These two
boundaries, whenever they appear sequentially for the same values
of the parameters $\lambda, p, q$, are joined at a tricritical point
determined by the condition $A = 1$ and $C = 0$
\cite{aharony,sneiderpytte,houghton,kaufkan,andelman1,asymmetric,asymtrim,khurana},
provided that $E<0$ (equivalently, $F_{6}<0$) for stability
\cite{asymmetric,asymtrim,crok2,crok3}. However, for the FOPT
boundary we shall also use the equality of the respective free
energies $F(M=0) = F(M \neq 0)$, where $F \equiv \langle F
\rangle_{h}/N$.
\section{Phase diagram. Tricritical Point. Magnetization profiles}
The TCP coordinates $(T^{TCP}, h^{TCP}_{0}, V^{TCP}_{0})$,
according to the definition of this point in the previous
paragraph, are solutions to the simultaneous equations
\begin{eqnarray}
pt_{+} + qt_{-} + rt_{0}& = & 0 \nonumber \\
pt_{+}^{2} + qt_{-}^{2} + rt_{0}^{2} + 1/\alpha & = & 1 \nonumber \\
4(pt_{+}^{2} + qt_{-}^{2} + rt_{0}^{2})- 3(pt_{+}^{4} + qt_{-}^{4}
+ rt_{0}^{4}) & = & 1 \label{simequstrim}
\end{eqnarray}
\noindent which do not lead to analytical formulas for these
coordinates for both forms of the trimodal PDF (anisotropic and
isotropic); on the contrary, the bimodal PDF (resulting from the
trimodal by setting $r=0$), even in the presence of anisotropy,
leads to analytical formulas; namely, the respective tricritical
temperature $T^{TCP}$ satisfies the second-degree equation
\begin{equation}
3(Q-1) \alpha ^{2}_{TCP} + 2(2-3Q)\alpha_{TCP} +3Q=0 \;\; \label{tricritequ}
\end{equation}
\noindent where $Q =(p^{3}+q^{3})/qp$, thus $T^{TCP}$ is a
function only of the probability $p$ and independent of the
competition parameter $\lambda$ \cite{asymmetric}. The relevant
discriminant of Eq. (\ref{tricritequ}) is real in the interval
$(13-\sqrt{13})/26 \cong 0.37... \leq p \leq (13+\sqrt{13})/26
\cong 0.63...$, so only for these values of $p$ there exist
tricritical points and both phase transitions take place for the
same $p$, but for different temperatures and $h_{0}$s. The two
solutions to Eq. (\ref{tricritequ}) determine the respective
tricritical temperatures (in units of ($Jz/k$)), the upper and
lower ones
\begin{equation}
\frac{kT^{TCP}_{\pm}}{Jz} = \frac{3(Q-1)}{3Q-2\pm(4-3Q)^{1/2}}
\\ \label{tctemp}
\end{equation}
We retain only the minus solution $T^{TCP}_{-}$, since
the plus one $T^{TCP}_{+}$ does not lead to physical results and
is thus neglected, see Ref. \cite{asymmetric}.
The remaining coordinates $h_{0}^{TCP}$ and $V_{0}^{TCP}$ for the
bimodal PDF are
\begin{eqnarray}
h_{0}^{TCP} & = & \frac{1}{2(1+\lambda)} \ln \left[
\left(\frac {1+z_{2}}{1-z_{2}} \right) \left(\frac {1+z_{1}}{1-z_{1}}\right)\right]
\frac{kT_{-}^{TCP}}{zJ} \nonumber\\
V_{0}^{TCP} & = & \frac{1}{2(1+\lambda)} \ln \left[
\left(\frac {1+z_{2}}{1-z_{2}} \right)^{\lambda} \left(\frac {1-z_{1}}{1+z_{1}}\right)\right]
\frac{kT_{-}^{TCP}}{zJ} \label{h0V0}
\end{eqnarray}
where $z_{1}=\sqrt{p(\alpha^{TCP}_{-}-1)/(q\,\alpha^{TCP}_{-})}$
and $z_{2}=\sqrt{q(\alpha^{TCP}_{-}-1)/(p\,\alpha^{TCP}_{-})}$;
$h_{0}^{TCP}$ and $V_{0}^{TCP}$ depend on both parameters $p$ and
$\lambda$ (unlike $T^{TCP}_{-}$) as well as on the tricritical
temperature itself $T^{TCP}_{-}$.
In order to examine the validity of the process under
consideration, we focus on the well-studied symmetric and
isotropic bimodal PDF (\ref{bimodalp}), resulting from
(\ref{trimodalr}) by setting $p=q=1/2, r=0$ and $\lambda=1$ with
respective $Q = 1$; the plus solution vanishes ($kT^{TCP}_{+}/(Jz)
= 0$), whereas the minus solution ($kT^{TCP}_{-}/(Jz)$) is
singular; however, this singularity can be removed either by using
the de L' H\^{o}pital rules in Eq. (\ref{tctemp}) or by setting in
(\ref{tricritequ}) $Q = 1$, so that ($kT^{TCP}_{-}/(Jz)$) equals
($2/3$) in agreement with the tricritical temperature in Ref.
\cite{aharony}, see also Refs. \cite{asymmetric,asym2,asymtrim}.
\begin{figure}[htbp]
\begin{center}
\includegraphics*[height=0.75\textheight]{fig1.eps}
\caption{\label{figa}(Color online) Representative graphs for the
variation of the tricritical temperature for various values of the
competition parameter $\lambda$ and probability $p$ against the
probability $q$. The labels (i) (black symbols) and (ii) (red
symbols) refer to the upper and lower TCP temperatures,
respectively, in case two such temperatures exist for the same
$\lambda$ and $p$. Panels (a) and (b) correspond to
$\lambda=0.25$, for $p=0.37$ and $p=0.38$, respectively; panel (c)
$\lambda=0.50$ and $p=0.38$; panel (d) $\lambda=0.75$ and
$p=0.01$. $T^{TCP}$ is in units of $(Jz/k)$.}
\end{center}
\end{figure}
For the current model, the system of Eqs. (\ref{simequstrim}) can
be solved only numerically for determining the TCP coordinates;
this is achieved only for a limited number of $p$'s and $q$'s for
specific $\lambda$ values; for the TCP point, in general, $p \in
[0.0,x]$, in this interval the numerical value of the upper bound
$x$ depends upon the specific $\lambda$ value, but, in any case, it
satisfies the inequality $x \leq (13+\sqrt{13})/26 \cong 0.63...$; the
respective $q$ values depend on the specific $p$ values, but they
also lie in same interval as $p$. When $p$ takes on the value $p =
(13-\sqrt{13})/26 \cong 0.37...$ (which is the lower bound of the
probability $p$ for the bimodal PDF to have tricritical points),
then $q$ takes on all the values within the interval $[0,
(13+\sqrt{13})/26 \cong 0.63...$]; for $(13-\sqrt{13})/26 \leq p
\leq (13+\sqrt{13})/26$, the maximum possible value for $q$ is
such that $p+q = 1$ so that $r=0$. The resulting tricritical
temperatures exhibit a variety of variations as functions of the
competition parameter $\lambda$ and site probabilities $p, q$;
such graphs appear in Fig.~\ref{figa}. However, for some $p$ and
$q$ values two tricritical temperatures occur, the upper ones
(shown in black) and the lower ones (shown in red) as in
Figs.~\ref{figa}(a,c) \cite{galam,weizenmann}. In panel (c) the
TCP temperatures are grouped into two sets, the left-hand side one
for small $q$'s and the right-hand side one for larger $q$'s;
also, in this panel, the lack of points in the interval
$[0.12,0.32]$ of the $q$-axis is due to the absence of tricritical
points in this interval, thus forming the existing gap; a similar
behavior is also observed for another values of $\lambda, p, q$.
\begin{figure}[htbp]
\begin{center}
\includegraphics*[height=0.75\textheight]{fig2.eps}
\caption{\label{figb}(Color online) Indicative modes of variation
for the random field strength $h_{0}$ with $q$ for specific values
of $\lambda$ and $p$ at the tricritical point; the labels (i)
(black symbols) and (ii) (red symbols) refer to the quantities
corresponding to the upper and lower TCP temperatures,
respectively, in case two such temperatures exist. Panel (a)
corresponds to $\lambda=0.25, p=0.37$; panel (b) $\lambda=0.25,
p=0.38$; panel (c) $\lambda=0.50, p=0.38$; panel (d)
$\lambda=0.75, p=0.01$. The random field $h_{0}$ exhibits
monotonic and non-monotonic behavior with $q$. $h_{0}$ is in units
of $(Jz)$, i.e., $h_{0} \equiv h_{0}/(Jz)$.}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics*[height=0.75\textheight]{fig3.eps}
\caption{\label{figc}(Color online) Modes of variation of the
auxiliary field $V_{0}$ with $q$ for specific values of $\lambda$
and $p$ at the tricritical point; the labels (i) (black symbols)
and (ii) (red symbols) refer to the quantities corresponding to
the upper and lower TCP temperatures, respectively, in case two
such temperatures exist. Panel (a) corresponds to $\lambda=$
$0.25, p=0.37$; panel (b) $\lambda=0.25, p=0.38$; panel (c)
$\lambda=0.50, p=0.38$; panel (d) $\lambda=0.75, p=0.01$. The
auxiliary potential $V_{0}$ exhibits monotonic behavior with $q$.
$V_{0}$ is in units of $(Jz)$, i.e., $V_{0} \equiv V_{0}/(Jz)$.}
\end{center}
\end{figure}
The variation of the random field values at the TCP,
$h_{0}^{TCP}$, resulting from Eqs. (\ref{simequstrim}) appears in
Fig.~\ref{figb}, where various modes of variation are shown,
displaying monotonic and non monotonic behavior. The gap in panel
Fig.~\ref{figb}(c) is due to the absence of tricritical points for
these values of $\lambda, p, q$ as in Fig.~\ref{figa}(c). A
similar picture appears for the auxiliary field at the tricritical
point, $V_{0}^{TCP}$, but its variation is not so abrupt as that
of $h_{0}^{TCP}$, see Fig.~\ref{figc}.
Another important quantity is the magnetization at the TCP. The
equilibrium Eq. (\ref{eqmagn}) at the tricritical point assumes
the form,
\begin{equation}
\frac{\alpha ^{2}F_{3}}{2!}M^{2} + \frac{\alpha ^{5}
F_{6}}{5!}M^{5} = 0 \label{tcpmagn1}
\end{equation}
or
\begin{equation}
F_{6} \omega^{5} + 60 F_{3} \omega^{2} = 0 \label{tcpmagn2}
\end{equation}
\noindent where $\omega \equiv \alpha M$ by taking into account
the conditions for the TCP. The latter equation has the solutions,
\begin{eqnarray}
\omega _{1}^{TCP} & = & 0 \label{tcpmagn0} \\
\omega _{2}^{TCP} & = & \mbox{\Large(}-60 F_{3} /F_{6}
\mbox{\Large)}^{1/3} \label{tcpmagn3}
\end{eqnarray}
\begin{figure}[htbp]
\includegraphics*[height=0.20\textheight]{fig4.eps}
\caption{\label{figd}(color online) Free energy of the zero
(\ref{tcpmagn0}) and non zero magnetization (\ref{tcpmagn3}) at
the tricritical point for $\lambda = 0.25$ and $p=0.37$ (panel
(a)), $p=0.40$ (panel (b)) and $p=0.50$ (panel (c)). In all
panels, graph (i) corresponds to the zero solution $M_{1}$ and
graphs (ii,iii) to the non zero solutions $M_{2}$ associated with
the upper and lower TCP temperatures, respectively; the zero
magnetization free energy (i) is higher than the one for the non
zero magnetizations (ii,iii), implying that the non zero solution
is the stable one.}
\end{figure}
\begin{figure}[htbp]
\includegraphics*[height=0.25\textheight]{fig5.eps}
\caption{\label{fige} The tricritical nonzero
magnetization $M_{2}= \mbox{\large(}-60 F_{3} /F_{6}
\mbox{\large)}^{1/3}$ with respect to q for $\lambda=0.50$,
$p=0.49$ panel (a), and $p=0.62$ panel (b).}
\end{figure}
from which the magnetizations $M_{1,2}^{TCP} = \omega
_{1,2}^{TCP}*(kT^{TCP}/(Jz))$ can be deduced. The non zero TCP
magnetization $M_{2}^{TCP}$ (\ref{tcpmagn3}) has a lower free energy
than the zero solution (\ref{tcpmagn0}), implying that this is the
stable solution at the tricritical point, see Fig.~\ref{figd} for
the respective free energies for $\lambda = 0.25$. In case two
tricritical points exist, then the free energy of the non zero
magnetization corresponding to the lower TCP temperature is
smaller than the respective one for the non zero magnetization
corresponding to the upper TCP temperature for the same $q$
values. The plot of the TCP magnetization $M_{2}^{TCP}$ appears
in Fig.~\ref{fige}, exhibiting significant variation; however, for
the symmetric probability distribution, $p = q = 0.50$ and
$\lambda = 1$, $M_{2}^{TCP}$ becomes identical with the zero
solution, so that the zero solution, now, is the only one and
becomes stable for these $p$ and $q$ values; this is, also,
a result of the elimination of the residual magnetic field implied
by the first relation in (\ref{h0}).
The stability of the non zero magnetization over
the zero one is a direct consequence of the existence of the
residual magnetic field due to the first relation in equation
(\ref{h0}), since for the general case $p \neq q$ the mean value
of the random field is non zero, which is equivalent to the
presence of an external magnetic field in the system so that the
magnetization at the tricritical point scales as $M_{t}\equiv
M(T=T^{TCP}) \sim h^{1/\delta_{t}}_{TCP}$, where $h_{TCP}$ is the
random magnetic field and the tricritical exponent $\delta _{t} = 5$
according to the Landau theory \cite{asymmetric,asymtrim,stanley,robertson,lawrie}.
In a previous communication \cite{asymmetric}, the PDF of the RFIM
was selected to be the asymmetric bimodal (\ref{bimodalp}); this
system displayed a symmetric behavior at the tricritical point
with respect to the probability $p$\,; especially, two distinct
tricritical points with respective probabilities $p_{1}$ and
$p_{2}$ such that $p_{1}+ p_{2} = 1$ have identical tricritical
temperatures and random fields, whereas the respective auxiliary
fields and non zero magnetizations are absolutely equal. A similar
symmetry is also observed in the present model with respect to the
probabilities $p$ and $q$ for a specific $\lambda$ value; if the
probabilities ($p_{1},q_{1}$) and ($p_{2},q_{2}$) of the modified
trimodal (\ref{trimodalr}) are such that $p_{1} + p_{2}= 1$, then
these two systems have the same TCP temperatures and random
fields, whereas the respective nonzero magnetizations
$M_{2}^{TCP}$ are absolutely equal; the auxiliary fields
$V_{0}^{TCP}$ are absolutely equal in case $\lambda = 1$.
Additionally, these cases have equal the respective free energies
for the zero magnetization ($F(p_{1},q_{1},M_{1}^{TCP}=0) =
F(p_{2},q_{2},M_{1}^{TCP}=0)$) as well as the nonzero ones
($F(p_{1},q_{1},M_{2}^{TCP}) = F(p_{2},q_{2},M_{2}^{TCP})$); the
latter result implies that the two magnetizations
$M_{2}^{TCP}(\lambda,p_{1},q_{1})$,
$M_{2}^{TCP}(\lambda,p_{2},q_{2})$ are equally probable, an
expected result, since the magnetizations have equal absolute
values and the only difference being in their sign so that no
direction is favored \cite{asymtrim}.
\begin{figure}[htbp]
\begin{center}
\includegraphics*[height=0.50\textheight]{fig6.eps}
\caption{\label{figf}(Color online) The phase boundaries for the
PM/FM phase transitions for $\lambda = 0.00(i),0.25(ii),
0.50(iii), 0.75(iv), 1.00(v)$, in all panels; $p=0.35, q=0.30
\, (a)$, $p=0.40, q=0.35 \, (b)$, $p=q=0.50 \, (c)$, $p=q=0.45 \,
(d)$. A continuous line represents an SOPT and a dashed one an
FOPT joined at a TCP, represented by a full circle. In panels (a,
b) the system is in the FM phase for low temperatures and high
$h_{0}$s; on the contrary, in panel (c) it is in the PM phase for
low temperatures due to re-entrance. In panel (c) the TCP temperatures
are equal for any value of $\lambda$ ($T^{TCP}=\frac{2}{3}$, see ref. \cite{aharony}),
whereas in panel (d) the upper TCP temperatures as well as the lower
ones are not equal among themselves; the difference between the
respective temperatures of two consecutive graphs is very small.
Re-entrance is seen in panels (c, d). The temperature $T$ is expressed
in units of $(Jz/k)$, i.e., $T \equiv kT/(Jz)$.}
\end{center}
\end{figure}
An important component in the study of magnetic or fluid systems
is their phase diagram in which the general behavior of the system
is shown; in the current case, it results as a solution to
Eq. (\ref{magnet1a}) by varying the parameters $p, q, \lambda$ and
appears in the Fig.~\ref{figf} as ($h_{0}-T$) plots labelled by
the individual $\lambda$ value for specific $p$ and $q$ values.
These plots are classified into two main groups: the first one
includes those plots not possessing a TCP and corresponding only
to an SOPT as in the Fig.~\ref{figf}(a) for $p=0.35$, $q=0.30$ for
any $\lambda$ value; the second group includes the plots
possessing at least one TCP, which joins the FOPT branch with the
SOPT branch of the phase diagram as in the Fig.~\ref{figf}(b, c,
d) for $p=0.40$, $q=0.35$, $p=q=0.50$ and $p=q=0.45$,
respectively. In the Fig.~\ref{figf}(b, d) the systems, described
by the corresponding plots, have two TCPs (except the ones for
$\lambda = 0.0$ Fig.~\ref{figf}(b,d) and
$\lambda = 0.25$ Fig.~\ref{figf}(d)); the single and
twin TCPs appear for another values of the parameters $\lambda, p
,q$, as well. In some cases, a random system can present
re-entrance that might be attributed to the competition between
the exchange interaction (ordering factor) due to the first term
in the Hamiltonian (\ref{rham}), on the one hand, and the random
field (disorder factor), on the other hand, as in the
Fig.~\ref{figf}(c, d); this effect is more pronounced in the
Fig.~\ref{figf}(c). In re-entrance a vertical line in the
$(h_{0},T)$-plane crosses the transition line at least twice, in
that, by lowering the temperature at constant $h_{0}$, one
observes firstly a $PM/FM$ transition and then, on further
lowering the temperature, an $FM/PM$ transition appears, so the
magnetization is zero although the temperature is low and the
system remains in the PM phase for low temperatures as in the
Fig.~\ref{figf}(c); occasionally, there can be another PM/FM
transition (additional crossing) with the system returning to the
FM phase for low temperatures and high $h_{0}s$,
Fig.~\ref{figf}(d). Re-entrance appears only in case an FOPT is
present and its direct effect is to limit drastically the extent
of the FM phase space as in the Fig.~\ref{figf}(c). However,
within the MFA re-entrance may lead to nonphysical values
(negative) for the specific heat, since energy will also present
re-entrant behavior as magnetization because energy, within MFA,
is proportional to the magnetization squared thus behaving
similarly. The choice $\lambda=0.00$ in the PDF (\ref{trimodalr})
is a distinct case for this probability distribution, since the
random magnetic field in the ($-z$) direction is eliminated and
it refers to a lattice system in which some of its sites are either
vacant or occupied by non-magnetic particles (site dilution, in this
case the two last terms in (\ref{trimodalr}) become similar and can
be combined to a single term, $P(h_{i})=p\delta (h_{i}-h_{0}) +(q+r)\delta
(h_{i})$), whereas the remaining sites are occupied by magnetic
particles (fraction p) exposed to the random field whose direction
is considered to be the positive z-direction without the presence
of another competing random field; for this case the mean
magnetization is expected to be higher than that for $\lambda \neq
0.00$ for the same $p, q$ as long as there is not competition
between random fields. In all panels of the Fig.~\ref{figf} the
respective phase boundary for $\lambda=0.00$ is the outermost
graph (i) with the widest FM phase region. However, as $\lambda$
is switched on (at constant $p$, $q$) taking on values greater
than zero, the random fields in the negative z direction appear
and oppose the initially prevailing random fields (in the positive
z direction), thus reducing the mean magnetization, causing the
phase boundary to move towards to the temperature axis and,
consequently, reducing the phase space allocated to the FM phase
but simultaneously broadening the PM phase space; the former phase
space attains its smallest extent for $\lambda = 1$, since the
reduction is larger the higher the value of $\lambda$. The plots
in the Fig.~\ref{figf}(a) correspond to systems in the FM phase
for low temperatures and high random fields for any value of
$\lambda$; the critical temperatures seem to tend asymptotically
to the limit $T_{cr} \cong 0.50$ for $\lambda = 0$ and $T_{cr}
\cong 0.35$ for $\lambda \neq 0$, as $h_{0} \rightarrow \infty$.
In the Fig.~\ref{figf}(b) the critical temperatures tend
asymptotically to the limit $T_{cr} \cong 0.35$ for $\lambda = 0$
and $T_{cr} \cong 0.25$ for $\lambda \neq 0$ as $h_{0} \rightarrow
\infty$; in the same limit in the Fig.~\ref{figf}(d) the critical
temperatures tend asymptotically to the limit $T_{cr} \cong
0.1818...$ for $\lambda = 0$ and $T_{cr} \cong 0.1$ for $\lambda
\neq 0$. The succession of phase transitions depends on the number
of the tricritical points present although the SOPTs appear always
for low fields and high temperatures: if there is only one TCP
then the FOPTs appear for high fields and low temperatures;
however, in case two TCPs are present the FOPTs appear for medium
fields and temperatures, whereas the concluding phase transition
is an SOPT for high fields and low temperatures. In the
Fig.~\ref{figf}(b,d) the concluding phase transitions for those
$\lambda$s with two TCPs are of second order and those with one
TCP is of first order as well as in the Fig.~\ref{figf}(c). The
aforementioned reduction of the FM-space due to the gradual
increase of $\lambda$ (control parameter) towards one ($\lambda
\rightarrow 1$) has counterpart in the density profile of a
spherical drop as a function of the inverse range parameter $R$ of
the strength of the attractive forces between the fluid particles;
the overall structure of the density profile remains unchanged as
a function of $R$ in comparison to that with $R=1$ except that the
density profile either shrinks for $R>1$ or widens for $R<1$ to
accommodate inside the drop the available particles
\cite{sphdrop}. A characteristic feature in panel (c), $p=q=0.50$,
is that the TCP temperatures, irrespective of the $\lambda$-value,
are all the same, namely $T^{TCP}=2/3$; this value is identical to
the one estimated by Aharony for the symmetric bimodal PDF
\cite{aharony}.
\begin{figure}[htbp]
\begin{center}
\includegraphics*[height=0.70\textheight]{fig7.eps}
\caption{\label{figg} (Color online) Tricritical point temperature
$T^{TCP}$ vs $h_{0}$ (panel (a)) as well as TCP coordinates ($T,
h_{0}, V_{0}$) as functions of the competition ratio $\lambda$
(panels (b,c,d)), for $p=0.40, q=0.35$; the labels (i) (black
symbols) and (ii) (red symbols) refer to the quantities
corresponding to the upper and lower TCP temperatures,
respectively.}
\end{center}
\end{figure}
As it is evident from the Fig.~\ref{figf}(b), both groups of TCP
temperatures regarded as functions of the TCP random field
strength ($h_{0}^{TCP}$) display a systematic behavior, in that,
they follow a decreasing route, which in the last stages (large
$h_{0}$) becomes exponential as is revealed by the graphs in the
Fig.~\ref{figg}(a) resulting by combining both groups of TCP
temperatures. A similar systematic behavior is also followed by
TCP temperature plotted with respect to the competition ratio
$\lambda$, Fig.~\ref{figg}(b); the upper group of temperatures
follow a decreasing route and the lower an increasing route
tending to approach each other as $\lambda$ tends to one, $
\lambda \rightarrow 1$. Similarly, in Fig.~\ref{figg}(c) the
random fields $h_{0}^{TCP}$ as $\lambda$ approaches $1$,
corresponding to the upper group of tricritical temperatures
$(i)$, decrease slowly, whereas those for the lower tricritical
temperatures $(ii)$ initially decrease abruptly but, in the last
stages, decrease slowly tending to approach the ones of the upper
temperatures. In addition, the TCP auxiliary fields $V_{0}^{TCP}$
follow the inverse route in comparison to $h_{0}^{TCP}$: the
$V_{0}^{TCP}$ corresponding to the upper temperatures $(i)$
follows a systematically increasing route, whereas those for the
lower TCP temperatures $(ii)$ are increasing very slowly, tending
to approach the other group of $V_{0}^{TCP}$s as $\lambda
\rightarrow 1$ Fig.~\ref{figg}(d).
An immediate connection of the Fig.~\ref{figa} (describing the
variation of the TCP temperature $T^{TCP}$ with respect to
$p,q,\lambda$) with the Fig.~\ref{figf} (phase diagram) is the
extent of the branches for the SOPTs and FOPTs with respect to these
parameters. According to Fig.~\ref{figa}(b), $T^{TCP}$ initially decreases
implying that the respective SOPT branch increases at the expense
of the FOPT branch acquiring its largest extent when the respective
$T^{TCP}$ has its minimum value for $\lambda=0.25, p=0.38, q=0.21$,
after this point the extent of the SOPT branch starts reducing and
that for the FOPT increasing as the TCP temperature increases;
the reverse behavior of the extent of the branches of SOPTs and FOPTs
occurs for the case of Fig.~\ref{figa}(d) wherein the $T^{TCP}$ initially increases.
In case two such temperatures appear in the phase diagram, as in
Fig.~\ref{figa}(a) and similar plots for other values of the
aforementioned parameters, then the FOPT branch initially
decreases acquiring its smallest extent when $T^{TCP}$ becomes
minimum in the upper branch and maximum in the lower branch, but
later it starts increasing as the two temperatures get farther
apart, whereas according to Fig.~\ref{figa}(c) in the respective
phase diagram the extent of the FOPT branch continuously increases
as the two TCP temperatures get farther apart with respect to $q$
from the beginning for the right hand part; as far as the left hand
one the SOPT branch, in general, increases.
Solving Eq. (\ref{magnet1a}) to determine the phase diagram, the
magnetization is also calculated for either phase transition. The
condition $A = 1$ or $\alpha F_{2}(\beta,V_{0},h_{0}) = 1$ leads
to
\begin{equation}
pt_{+}^{2} + qt_{-}^{2} + rt_{0}^{2} = \frac{\alpha -1}{\alpha} \;\;\;\; \;\; \label{t2a}
\end{equation}
and by setting $T_{2} \equiv p\,t_{+}^{2} + qt_{-}^{2} +
rt_{0}^{2}$, (\ref{t2a}) can be written
\begin{equation}
T_{2} = \frac{\alpha -1}{\alpha} \;\;\;\; \;\; \label{t2b}
\end{equation}
Inverting Eq. (\ref{t2b}) the respective temperature for either
phase transition can be determined, namely
\begin{equation}
\frac{kT}{Jz} = 1 - T_{2} \;\;\;\; \;\; \label{temp}
\end{equation}
In order to specify the type of the transition, the sign of $C
\equiv \alpha^{3} F_{4}/6$ is checked; however, to facilitate the
calculations, the quantity $C$ is rewritten as
\begin{equation}
C = \frac{\alpha^{3}}{3}[4T_{2} - 3T_{4} - 1]
=\alpha^{3}[1-T_{4}-\frac{4}{3\alpha}] \;\; \;\; \label{consc}
\end{equation}
using (\ref{t2b}) and setting $T_{4} = p\,t^{4}_{+} + q\,t^{4}_{-}
+ rt_{0}^{4}$. For an SOPT, $C$ is negative \cite{aharony}, then
(\ref{consc}) yields
\begin{equation}
T_{4} > 1 - \frac{4}{3\alpha} \;\;\;\; \;\; \label{sectrns}
\end{equation}
otherwise if
\begin{equation}
T_{4} < 1 - \frac{4}{3\alpha} \;\;\;\; \;\; \label{firsttrns}
\end{equation}
the resulting transition is an FOPT. In order to determine the
magnetization for an FOPT the expression (\ref{eqmagn}) is
combined with the equality of the respective free energies
\begin{equation}
F(M=0) = F(M\ne 0) \label{feeq1}
\end{equation}
\noindent or,
\begin{equation}
M^{2}=F_{2} \alpha M^{2}+\frac{F_{3}}{3}\alpha^{2}M^{3}+
\frac{F_{4}}{12} \alpha^{3} M^{4}+\frac{F_{6}}{360} \alpha^{5}
M^{6} \label{feeq2}
\end{equation}
\noindent Combining Eqs. (\ref{eqmagn}), (\ref{feeq2}) and using
the condition $\alpha F_{2} =1$, we get
\begin{equation}
F_{6}\omega^{3} + 10F_{4}\omega = 0 \label{fopt2}
\end{equation}
Eq. (\ref{fopt2}), is broken up into two equations; the first is
$\omega _{1} = 0$ or, equivalently, $M_{1} = 0$ for the PM phase,
whereas the other is
\begin{equation}
F_{6}\omega^{2} + 10 F_{4}= 0 \label{fopt3}
\end{equation}
\noindent from which the nonzero solutions result, FM phase,
\begin{equation}
\omega_{2,3}= \pm \sqrt{-10 F_{4}/F_{6}} \label{foptsols}
\end{equation}
\noindent with $F_{6} < 0$ as long as $F_{4} > 0$ for an FOPT; the
value for $F_{3}$, consistent with (\ref{eqmagn}) and
(\ref{feeq2}), is $F_{3} = - (F_{4}/6) \sqrt{-10 F_{4}/F_{6}}$ for
the positive root in (\ref{foptsols}) and $F_{3} = (F_{4}/6)
\sqrt{-10 F_{4}/F_{6}}$ for the respective negative root. From the
solution of this equation we can extract the magnetization
$M = \omega*(kT/(Jz))$, since $\alpha=zJ/kT$ is already known from (\ref{temp}).
\begin{figure}[htbp]
\begin{center}
\includegraphics*[height=0.70\textheight]{fig8.eps}
\caption{\label{figh} Magnetization profile vs. $h_{0}$ for an FOPT.
The first row corresponds to $\lambda = 0.25, p=0.40, q=0.50$ panel (a) and
$\lambda = 0.50$, $p=0.25, q=0.55$ panel (b). The second
row corresponds to $\lambda = 0.50, p=0.30, q=0.45$ panel (c)
and $\lambda = 0.50, p=0.40, q=0.50$ panel (d). In
all panels point A represents a double critical point, point B a
regular critical point, point C a critical end-point, point D a
double critical end-point.}
\end{center}
\end{figure}
In plotting the phase diagram or the order parameter profile, the
temperature is usually chosen as the independent variable;
however, this is not the only choice as any other variable,
suitably chosen, can be. In the present case the randomness
strength $h_{0}$ is considered to be the control parameter for
studying the variation of the non-zero positive magnetization
$M_{2} = \omega_{2}*(kT/(Jz))$ for an FOPT, Eq. (\ref{foptsols}),
by forming the respective magnetization profile as a function of
$h_{0}$ for specific values of $\lambda$, $p$ and $q$; the
negative solution $M_{3} = - M_{2}$ behaves analogously. This
study reveals a complicated structure for magnetization profiles
and we present some representative of them; they appear in
Fig.~\ref{figh} displaying a variety of structures and
characterized by critical points of several kinds. Apart from the
simple profiles, with or without a critical point,
there are also profiles forming closed loops (one or two), closed
miscibility gap, Fig.~\ref{figh}. In addition to the
usual critical points (indicated by the letter B in all graphs),
there are double critical points (A points), critical end-points
(C points) as well as double critical end-points (D points)
\cite{asymtrim,hadjievans}. This behavior can be considered as an
FOPT between the two coexisting $M_{2}$ magnetizations with
respect to randomness $h_{0}$ as long as this is now the control
parameter, with upper and lower critical temperatures.
We consider, now, the values of $p$ and $q$ for which the system
exhibits only an SOPT; Eq. (\ref{magnet1a}) takes the form for
$A=1$,
\begin{equation}
F_{6}\omega^{5} + 20F_{4}\omega^{3} + 60F_{3}\omega^{2} = 0
\label{sopt1}
\end{equation}
The value $\omega_{1} = 0$ is again a solution (two-fold) or,
equivalently, $M_{1} = 0$ (PM phase); the other three ones are the
solutions to the equation
\begin{equation}
F_{6}\omega^{3} + 20F_{4}\omega + 60F_{3} = 0 \label{sopt2}
\end{equation}
\noindent which, depending on the value of $\lambda, p, q$ and
$h_{0}$, can have either only one real non zero solution if
$\Delta = u^{3} + v^{2} \geq 0$ ($v = -30F_{3}/F_{6}$,
$u=(20F_{4})/(3F_{6})$), namely
\begin{equation}
\omega_{2}=\sqrt[3]{v+\sqrt{\Delta}} + \sqrt[3]{v-\sqrt{\Delta}}
\label{root2}
\end{equation}
\noindent or three real non zero solutions for $\Delta < 0$, which are
\begin{eqnarray}
\omega_{2} & =& 2 \sqrt[3]{\rho}\;\cos(\theta/3) \nonumber \\
\omega_{3} & = & -\sqrt[3]{\rho}\;[\cos(\theta/3) +
\sqrt{3}\sin(\theta/3)] \nonumber \\
\omega_{4} & = & -\sqrt[3]{\rho}\;[\cos(\theta/3) -
\sqrt{3}\sin(\theta/3)] \label{root34}
\end{eqnarray}
\noindent where $\rho = \sqrt{v^{2} - \Delta}$, $\theta = \arctan(
\sqrt{-\Delta}/v )$ and $M_{i} = \omega_{i}*(kT/(Jz))$, $i=2,3,4$.
As a consequence, the solutions for an SOPT are classified into
two groups, group 1 includes the zero-solution ($M_{1}=0$) and the
single nonzero one $M_{2}$ of Eq. (\ref{root2}), whereas group 2
includes again the zero solution and the nonzero ones $M_{2},
M_{3}, M_{4}$ of the Eq. (\ref{root34}). Depending on the values
of $\lambda, p, q$ and $h_{0}$, there can be transitions between
these two groups. For the $p$ values an FOPT is present, the
solutions to the SOPT Eq. (\ref{sopt2}) belong to group 1; for
small $h_{0}$'s the zero solution ($M_{1}=0$) is the stable,
whereas for larger $h_{0}$'s (but smaller than those corresponding
to the respective FOPT) the $M_{2}$ solution (\ref{root2}) is the
stable.
The investigation was also extended to the zero-temperature case,
$T=0$; in this case the free energy (\ref{mfafren}) reduces to,
\begin{eqnarray}
F \equiv \frac{1}{N}\langle F \rangle_{h} & = & \frac{1}{2}
zJM^{2} - \frac{1}{\beta} \langle \ln\{ 2 \cosh [\beta (z J M +
h_{i})] \}
\rangle_{h} \nonumber \\
& = & \frac{1}{2} zJM^{2} - \langle |z J M + h_{i}| \rangle_{h} \nonumber \\
& = & \frac{1}{2} zJM^{2} - p\, |z J M + h_{0}| -q\, |z J M - \lambda*h_{0}|-r\, |zJM| \label{ztfren}
\end{eqnarray}
\noindent the external potential was omitted. Applying the
equilibrium condition $dF/dM = 0$ to (\ref{ztfren}) we get,
\begin{equation}
M = p\, \frac{| zJM + h_{0} |}{zJM + h_{0}} + q\, \frac{| zJM -
\lambda*h_{0} |}{zJM - \lambda*h_{0}} + r\, \frac{| zJM|}{zJM}
\label{ztmagnet}
\end{equation}
Analyzing Eq. (\ref{ztmagnet}), we find that $M = 1$ is a stable
solution for $p+r > \lambda h_{0}/zJ$, whereas for $p+r <
\lambda h_{0}/zJ$ the stable one is $M = 1 - 2q$. Also, across the
boundary $ \lambda h_{0}/zJ = p+r$ a first-order phase transition
occurs between the two ordered phases with $M = 1$ and $M = 1 -
2q$. If we consider the symmetric trimodal PDF
($p=q=\frac{1-r}{2}, \lambda =1 $), the results found by
Sebastianes and Saxena \cite{saxena} are recovered, that is, the
former result ($M = 1$) is stable for $ \frac{1+r}{2}> h_{0}/zJ$,
whereas the latter ($M = r$) for $\frac{1+r}{2} < h_{0}/zJ$, using
the current notation. The physical explanation for the existence
of the above two ordered phases ia that they can be attributed to
the competition between the ordering tendency, due to the first
term in Eq. (\ref{rham}), and the disorder induced because of the
presence of the second term in the same equation. In the first
case, $M = 1$, the condition $(p + r) > (\lambda h_{0}/(zJ))$
implies that the exchange interaction $J$ is much stronger than
the randomness $h_{0}$, and their ratio is always smaller than
one, thus forcing the system's spins to order according to the
first term in (\ref{rham}). The alternative condition $(p + r) <
(\lambda h_{0}/(zJ))$ implies, now, that randomness is no longer
negligible but strong enough to influence significantly the spins
enforcing a $p$-fraction of them to point up and a $q$-fraction
down, to randomly align with the local fields, thus, practically,
it dominates, so to speak, over the first term in Eq. (\ref{rham})
so that $M = p-q+r = 1 - 2q$. In addition to the aforementioned
two solutions, there are more; the result $M=2(p+q)-1$ is stable
for $p-r>\lambda h_{0}/zJ$, $M=2p-1$ for $ p-r < \lambda
h_{0}/zJ$, $M=1-2p$ for $ r-p > \lambda h_{0}/zJ$, $M=1-2(p+q)$
for $ r-p < \lambda h_{0}/zJ$ and $M=-1$ for $p+r+\lambda
h_{0}/zJ>0$.
\vspace{-6mm}
\section{Conclusions and discussions}
\vspace{-6mm}
In the current treatment we have determined the phase diagram and
discussed some critical phenomena of the Ising model under the
influence of a trimodal random field, an extension of the bimodal
one to allow for the existence of non magnetic particles or
vacancies in the system, for arbitrary values of the probabilities
$p$ and $q$ and different strengths of the random field in the up
and down directions specified by the competition parameter
$\lambda$ via the Landau expansion. The competition between the
ordering effects and the randomness induces a rich phase diagram.
The system is strongly influenced by the random field, which
establishes a new competition favoring disorder; this is obvious
from the appearance of first order transitions and tricritical
points, in addition to the second order transitions for some
values of $\lambda, p$ and $q$; the tricritical point temperature
has various modes of variation as a function of $p$ and $q$ and
for some cases there are two such points. The trimodal
distribution induces re-entrant behavior for the appropriate range
of $p, q$ and random field $h_{0}$. For some values of $p$ and $q$
the system can be found either in the PM phase or in the FM phase
for low and medium temperatures and high random fields; a
significant result is that the part of the phase diagram allocated
to the FM phase is reduced significantly as $\lambda$ tends to
one. A direct consequence of the asymmetric and anisotropic PDF is
the existence of residual mean magnetic field in the system, a
result of $<h_{i}>_{h} = (p-\lambda \,q)h_{0}$, making the TCP non
zero magnetization $M_{2}$ to be the stable one in comparison to
the zero one, $M_{1}$. Both asymmetric PDFs, bimodal and trimodal,
confirm the existence of a TCP and, nevertheless, yield similar
magnetization profiles as well as re-entrance; however, the
trimodal one predicts also the existence of a second TCP.
Griffiths extended the notion of the critical point to the
so-called multicritical points, e.g., the tricritical point,
the critical-end-point, double critical-end-point, fourth-order point,
ordered critical point, etc. \cite{griffiths}; however, in order
to describe these points (except the first two) the expansion considered
for the free energy (\ref{mfafren2}) has to be extended
to higher-order terms \cite{trimodal,crok3,galambirman,milmanetal}
so that the stability criteria for such a point are satisfied, but
this is beyond the scope of the current research.
The Landau theory breaks down close to the critical point (the non
classical region) because as the transition temperature is
approached the fluctuations become important and non classical
behavior is observed. A relative criterion, called the Ginzburg
criterion, determines how closely to the transition temperature
the true critical behavior is revealed, or, in other words, it
governs the validity of the Landau theory close to a critical
point \cite{ginzburg}. This criterion can rely on any
thermodynamic quantity but the specific heat is usually considered
for determining the critical region around $T_{c}$ where the mean
field solution cannot correctly describe the phase transition. The
Landau theory is valid for lattice dimensionality greater than the
upper critical dimension $d_{u}=4$ in the case of the presence of
only thermal fluctuations. However, in the current case the
presence of random fields enhances fluctuations causing the
critical region to be wider than the one due only to the thermal
fluctuations \cite{kaufmankardar,nielsen} and the upper critical
dimension is increased by $2$ to $d_{u}=6$. Occasionally, the non
classical region for some physical systems is extremely narrow so
that the respective critical behavior expected from Landau theory
is observed for a wide range of temperatures because, in this
case, the fluctuation region is very narrow and hardly accessible
for experimental observation; such a system is the weak-coupling
superconductor in three dimensions for which the respective non
classical region is $|t_{CR}|\leq 10^{-16}$ ($t_{CR}$ is the
reduced temperature, $t_{CR}=(T-T_{c})/T_{c}$). However, on
reducing the space dimension as in the case of the weak-coupling
superconductor in two dimensions, the non classical region
expands, so that the critical exponents have their classical
values up to the interval $|t_{CR}|= 10^{-5}$, and thus the
reduction of the space dimensionality has serious consequences for
the critical behavior of the physical system; in contrast, there
are systems with a wide non classical interval as in the case of
the superfluid helium transition, for which the classical region
extends up to $|t_{CR}|= 1.0$ and so fluctuations are detectable
\cite{patapok,ivan,domb,goldenfeld}. In addition to
superconductivity, the extent of the non classical region for the
ferroelectric system triglycine sulfate (TGS) is relatively small
and its critical exponents have the respective classical values up
to $|t_{CR}|= 1.5\times 10^{-5}$ \cite{fe1,fe2,fe3}.
Our results indicate that on increasing the complexity of the
model system new phenomena can be revealed as in the current case
of including asymmetry in the PDF; this inclusion induces drastic
changes in the phase diagram, such as re-entrance and two TCPs,
thus confirming the necessity of treating the partial
probabilities $(p,q,r)$ of the PDF in the most general way to get
the complete phase diagram. A similar situation appears in the
model systems in Refs. \cite{galamaharony,galam1} wherein the
complexity considered has revealed a rich variety of phase
diagrams with known and new multicritical points. The results
obtained in the current investigation by using the MFA can provide
a basis for a comprehensive analysis as well as experimental
implementation. However, they are of no less importance, since
they nevertheless show the phenomena that expected to be observed.
\vspace{-9mm}
\ack{This research was supported by the Special Account for
Research Grants of the University of Athens ($E\Lambda KE$) under
Grant No. 70/4/4096.}
\newpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,537 |
GREEN DAY Finally Return To The Hollywood Palladium!
Tag: All Access, All Access Music Group, Concert Review, Green Day, Hollywood Palladium
On Monday, October 17th, the three leading members of Green Day proved that at ages 44 (Billie), 44 (Mike) and 43 (Tre) and with nearly 30 years together, they still know how to put on one incredible show.
For two and a half hours, the band entertained a rowdy and mighty crowd at the Hollywood Palladium in Hollywood, California. It's been 22 years since Green Day has played at this venue so they were certainly ready to remind people of what they've been missing all these years.
The rock band performed songs off most of their entire catalog including hits from "Dookie" (1994), "Nimrod" (1997), "Warning" (2000), "American Idiot" (2004), "21st Century Breakdown" (2009) and "Revolution Radio" (2016). Crowd favorites included "Basket Case", "When I Come Around", "American Idiot", "Jesus Of Suburbia", "Holiday", "Hitchin A Ride" and many others.
Crowd surfing, chanting and thunderous applause filled the arena for the better part of the show. Despite the heat inside the venue, the audience never let up and were hypnotized by Green Day front-man and all around rock icon, Billie Joe Armstrong. His energy was intense and constant from beginning to end. The crowd ate it up and he had them eating out of his hand. His politically charged statements fueled the audience with both disdain for Donald Trump but also positivity and a sense of unity towards each other.
Throughout the show, Armstrong continued to urge concert-goers to vote in this year's election and to always speak their mind. At one point, he even gave a shout-out to his mom who happened to be in the audience. He shared that she had bought him his first guitar at age 10 for $300.00 and he still plays it today.
Billie's Green Day band-mates Mike Dirnt (bass) and Tre Cool (drummer) continue to be impeccable musicians and received plenty of praise themselves.
Along with playing their greatest hits, towards the end of the night, the group also performed a medley of "Shout", "Hey Jude" and "(I Can't Get No) Satisfaction" and even a little "Careless Whisper" on the saxophone provided by the musician Jason Freese.
At the end of the concert, Armstrong shouted, "Thank you, thank you, thank you, thank you, thank you, thank you, thank you to all you fucking weird Californians."
No, Billie, thank you. Thanks to you and the rest of Green Day for showing just how timeless your music is and how much better with age you have all become.
An Interview With LA-Based Front-Man Kenny Becker On The Formation and Growth of His New Band, GOON!
An Interview With CMT's Nashville Actor and Singer-Songwriter MARK COLLIE! | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,388 |
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<parameterName>TOWN_SN</parameterName>
<parameterValue>128</parameterValue>
</parameter>
</location>
<location>
<lat>24.2587</lat>
<lon>120.5151</lon>
<locationName>梧棲</locationName>
<stationId>467770</stationId>
<time>
<obsTime>2016-05-22T13:40:00+08:00</obsTime>
</time>
<weatherElement>
<elementName>TIME</elementName>
<elementValue>
<value>1</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>ELEV</elementName>
<elementValue>
<value>7</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>WDIR</elementName>
<elementValue>
<value>340.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>WDSD</elementName>
<elementValue>
<value>3.1</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>TEMP</elementName>
<elementValue>
<value>27.5</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>HUMD</elementName>
<elementValue>
<value>0.83</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>PRES</elementName>
<elementValue>
<value>1005.4</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>24R</elementName>
<elementValue>
<value>6.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_FX</elementName>
<elementValue>
<value>4.5</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_XD</elementName>
<elementValue>
<value>150.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_FXT</elementName>
<elementValue>
<value>1244</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_F10</elementName>
<elementValue>
<value>2.5</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_10D</elementName>
<elementValue>
<value>170.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_F10T</elementName>
<elementValue>
<value>1242</value>
</elementValue>
</weatherElement>
<parameter>
<parameterName>CITY</parameterName>
<parameterValue>臺中市</parameterValue>
</parameter>
<parameter>
<parameterName>CITY_SN</parameterName>
<parameterValue>02</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN</parameterName>
<parameterValue>梧棲區</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN_SN</parameterName>
<parameterValue>118</parameterValue>
</parameter>
</location>
<location>
<lat>23.5672</lat>
<lon>119.5552</lon>
<locationName>澎湖</locationName>
<stationId>467350</stationId>
<time>
<obsTime>2016-05-22T13:40:00+08:00</obsTime>
</time>
<weatherElement>
<elementName>TIME</elementName>
<elementValue>
<value>0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>ELEV</elementName>
<elementValue>
<value>11</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>WDIR</elementName>
<elementValue>
<value>170.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>WDSD</elementName>
<elementValue>
<value>6.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>TEMP</elementName>
<elementValue>
<value>28.1</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>HUMD</elementName>
<elementValue>
<value>0.86</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>PRES</elementName>
<elementValue>
<value>1007.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>24R</elementName>
<elementValue>
<value>18.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_FX</elementName>
<elementValue>
<value>9.4</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_XD</elementName>
<elementValue>
<value>140.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_FXT</elementName>
<elementValue>
<value>1236</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_F10</elementName>
<elementValue>
<value>5.3</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_10D</elementName>
<elementValue>
<value>170.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_F10T</elementName>
<elementValue>
<value>1254</value>
</elementValue>
</weatherElement>
<parameter>
<parameterName>CITY</parameterName>
<parameterValue>澎湖縣</parameterValue>
</parameter>
<parameter>
<parameterName>CITY_SN</parameterName>
<parameterValue>23</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN</parameterName>
<parameterValue>馬公市</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN_SN</parameterName>
<parameterValue>212</parameterValue>
</parameter>
</location>
<location>
<lat>23.8830</lat>
<lon>120.8999</lon>
<locationName>日月潭</locationName>
<stationId>467650</stationId>
<time>
<obsTime>2016-05-22T13:40:00+08:00</obsTime>
</time>
<weatherElement>
<elementName>TIME</elementName>
<elementValue>
<value>0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>ELEV</elementName>
<elementValue>
<value>1015</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>WDIR</elementName>
<elementValue>
<value>200.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>WDSD</elementName>
<elementValue>
<value>0.8</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>TEMP</elementName>
<elementValue>
<value>23.7</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>HUMD</elementName>
<elementValue>
<value>0.76</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>PRES</elementName>
<elementValue>
<value>899.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>24R</elementName>
<elementValue>
<value>2.5</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_FX</elementName>
<elementValue>
<value>5.6</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_XD</elementName>
<elementValue>
<value>290.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_FXT</elementName>
<elementValue>
<value>1300</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_F10</elementName>
<elementValue>
<value>3.3</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_10D</elementName>
<elementValue>
<value>310.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_F10T</elementName>
<elementValue>
<value>1300</value>
</elementValue>
</weatherElement>
<parameter>
<parameterName>CITY</parameterName>
<parameterValue>南投縣</parameterValue>
</parameter>
<parameter>
<parameterName>CITY_SN</parameterName>
<parameterValue>13</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN</parameterName>
<parameterValue>魚池鄉</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN_SN</parameterName>
<parameterValue>164</parameterValue>
</parameter>
</location>
<location>
<lat>23.5104</lat>
<lon>120.8051</lon>
<locationName>阿里山</locationName>
<stationId>467530</stationId>
<time>
<obsTime>2016-05-22T13:40:00+08:00</obsTime>
</time>
<weatherElement>
<elementName>TIME</elementName>
<elementValue>
<value>0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>ELEV</elementName>
<elementValue>
<value>2413</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>WDIR</elementName>
<elementValue>
<value>250.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>WDSD</elementName>
<elementValue>
<value>1.2</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>TEMP</elementName>
<elementValue>
<value>15.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>HUMD</elementName>
<elementValue>
<value>0.91</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>PRES</elementName>
<elementValue>
<value>764.1</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>24R</elementName>
<elementValue>
<value>10.5</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_FX</elementName>
<elementValue>
<value>4.3</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_XD</elementName>
<elementValue>
<value>160.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_FXT</elementName>
<elementValue>
<value>1201</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_F10</elementName>
<elementValue>
<value>2.2</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_10D</elementName>
<elementValue>
<value>140.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_F10T</elementName>
<elementValue>
<value>1201</value>
</elementValue>
</weatherElement>
<parameter>
<parameterName>CITY</parameterName>
<parameterValue>嘉義縣</parameterValue>
</parameter>
<parameter>
<parameterName>CITY_SN</parameterName>
<parameterValue>17</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN</parameterName>
<parameterValue>阿里山鄉</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN_SN</parameterName>
<parameterValue>208</parameterValue>
</parameter>
</location>
<location>
<lat>23.4893</lat>
<lon>120.9517</lon>
<locationName>玉山</locationName>
<stationId>467550</stationId>
<time>
<obsTime>2016-05-22T13:40:00+08:00</obsTime>
</time>
<weatherElement>
<elementName>TIME</elementName>
<elementValue>
<value>1</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>ELEV</elementName>
<elementValue>
<value>3845</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>WDIR</elementName>
<elementValue>
<value>290.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>WDSD</elementName>
<elementValue>
<value>3.1</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>TEMP</elementName>
<elementValue>
<value>8.1</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>HUMD</elementName>
<elementValue>
<value>0.93</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>PRES</elementName>
<elementValue>
<value>643.9</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>24R</elementName>
<elementValue>
<value>13.5</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_FX</elementName>
<elementValue>
<value>7.6</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_XD</elementName>
<elementValue>
<value>280.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_FXT</elementName>
<elementValue>
<value>1227</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_F10</elementName>
<elementValue>
<value>3.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_10D</elementName>
<elementValue>
<value>260.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_F10T</elementName>
<elementValue>
<value>1230</value>
</elementValue>
</weatherElement>
<parameter>
<parameterName>CITY</parameterName>
<parameterValue>南投縣</parameterValue>
</parameter>
<parameter>
<parameterName>CITY_SN</parameterName>
<parameterValue>13</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN</parameterName>
<parameterValue>信義鄉</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN_SN</parameterName>
<parameterValue>175</parameterValue>
</parameter>
</location>
<location>
<lat>23.4977</lat>
<lon>120.4245</lon>
<locationName>嘉義</locationName>
<stationId>467480</stationId>
<time>
<obsTime>2016-05-22T13:40:00+08:00</obsTime>
</time>
<weatherElement>
<elementName>TIME</elementName>
<elementValue>
<value>0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>ELEV</elementName>
<elementValue>
<value>27</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>WDIR</elementName>
<elementValue>
<value>230.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>WDSD</elementName>
<elementValue>
<value>4.8</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>TEMP</elementName>
<elementValue>
<value>28.7</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>HUMD</elementName>
<elementValue>
<value>0.82</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>PRES</elementName>
<elementValue>
<value>1004.9</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>24R</elementName>
<elementValue>
<value>0.5</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_FX</elementName>
<elementValue>
<value>5.3</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_XD</elementName>
<elementValue>
<value>230.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_FXT</elementName>
<elementValue>
<value>1207</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_F10</elementName>
<elementValue>
<value>3.4</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_10D</elementName>
<elementValue>
<value>230.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_F10T</elementName>
<elementValue>
<value>1208</value>
</elementValue>
</weatherElement>
<parameter>
<parameterName>CITY</parameterName>
<parameterValue>嘉義市</parameterValue>
</parameter>
<parameter>
<parameterName>CITY_SN</parameterName>
<parameterValue>09</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN</parameterName>
<parameterValue>西區</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN_SN</parameterName>
<parameterValue>219</parameterValue>
</parameter>
</location>
<location>
<lat>22.5679</lat>
<lon>120.3080</lon>
<locationName>高雄</locationName>
<stationId>467440</stationId>
<time>
<obsTime>2016-05-22T13:40:00+08:00</obsTime>
</time>
<weatherElement>
<elementName>TIME</elementName>
<elementValue>
<value>0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>ELEV</elementName>
<elementValue>
<value>2</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>WDIR</elementName>
<elementValue>
<value>150.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>WDSD</elementName>
<elementValue>
<value>1.6</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>TEMP</elementName>
<elementValue>
<value>29.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>HUMD</elementName>
<elementValue>
<value>0.83</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>PRES</elementName>
<elementValue>
<value>1008.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>24R</elementName>
<elementValue>
<value>3.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_FX</elementName>
<elementValue>
<value>6.8</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_XD</elementName>
<elementValue>
<value>150.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_FXT</elementName>
<elementValue>
<value>1225</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_F10</elementName>
<elementValue>
<value>3.9</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_10D</elementName>
<elementValue>
<value>170.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_F10T</elementName>
<elementValue>
<value>1228</value>
</elementValue>
</weatherElement>
<parameter>
<parameterName>CITY</parameterName>
<parameterValue>高雄市</parameterValue>
</parameter>
<parameter>
<parameterName>CITY_SN</parameterName>
<parameterValue>05</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN</parameterName>
<parameterValue>前鎮區</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN_SN</parameterName>
<parameterValue>328</parameterValue>
</parameter>
</location>
<location>
<lat>22.0054</lat>
<lon>120.7381</lon>
<locationName>恆春</locationName>
<stationId>467590</stationId>
<time>
<obsTime>2016-05-22T13:40:00+08:00</obsTime>
</time>
<weatherElement>
<elementName>TIME</elementName>
<elementValue>
<value>1</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>ELEV</elementName>
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<value>22</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>WDIR</elementName>
<elementValue>
<value>150.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>WDSD</elementName>
<elementValue>
<value>0.9</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>TEMP</elementName>
<elementValue>
<value>26.9</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>HUMD</elementName>
<elementValue>
<value>0.95</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>PRES</elementName>
<elementValue>
<value>1006.9</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>24R</elementName>
<elementValue>
<value>9.5</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_FX</elementName>
<elementValue>
<value>8.2</value>
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</weatherElement>
<weatherElement>
<elementName>H_XD</elementName>
<elementValue>
<value>240.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_FXT</elementName>
<elementValue>
<value>1242</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_F10</elementName>
<elementValue>
<value>3.5</value>
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</weatherElement>
<weatherElement>
<elementName>H_10D</elementName>
<elementValue>
<value>230.0</value>
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</weatherElement>
<weatherElement>
<elementName>H_F10T</elementName>
<elementValue>
<value>1246</value>
</elementValue>
</weatherElement>
<parameter>
<parameterName>CITY</parameterName>
<parameterValue>屏東縣</parameterValue>
</parameter>
<parameter>
<parameterName>CITY_SN</parameterName>
<parameterValue>20</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN</parameterName>
<parameterValue>恆春鎮</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN_SN</parameterName>
<parameterValue>350</parameterValue>
</parameter>
</location>
<location>
<lat>24.7656</lat>
<lon>121.7479</lon>
<locationName>宜蘭</locationName>
<stationId>467080</stationId>
<time>
<obsTime>2016-05-22T13:40:00+08:00</obsTime>
</time>
<weatherElement>
<elementName>TIME</elementName>
<elementValue>
<value>0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>ELEV</elementName>
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<value>7</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>WDIR</elementName>
<elementValue>
<value>100.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>WDSD</elementName>
<elementValue>
<value>2.6</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>TEMP</elementName>
<elementValue>
<value>27.4</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>HUMD</elementName>
<elementValue>
<value>0.82</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>PRES</elementName>
<elementValue>
<value>1007.6</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>24R</elementName>
<elementValue>
<value>0.5</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_FX</elementName>
<elementValue>
<value>4.5</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_XD</elementName>
<elementValue>
<value>100.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_FXT</elementName>
<elementValue>
<value>1226</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_F10</elementName>
<elementValue>
<value>3.2</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_10D</elementName>
<elementValue>
<value>90.0</value>
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</weatherElement>
<weatherElement>
<elementName>H_F10T</elementName>
<elementValue>
<value>1235</value>
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</weatherElement>
<parameter>
<parameterName>CITY</parameterName>
<parameterValue>宜蘭縣</parameterValue>
</parameter>
<parameter>
<parameterName>CITY_SN</parameterName>
<parameterValue>07</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN</parameterName>
<parameterValue>宜蘭市</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN_SN</parameterName>
<parameterValue>075</parameterValue>
</parameter>
</location>
<location>
<lat>24.6017</lat>
<lon>121.8644</lon>
<locationName>蘇澳</locationName>
<stationId>467060</stationId>
<time>
<obsTime>2016-05-22T13:40:00+08:00</obsTime>
</time>
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<elementName>TIME</elementName>
<elementValue>
<value>0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>ELEV</elementName>
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<value>25</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>WDIR</elementName>
<elementValue>
<value>140.0</value>
</elementValue>
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<weatherElement>
<elementName>WDSD</elementName>
<elementValue>
<value>2.6</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>TEMP</elementName>
<elementValue>
<value>26.2</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>HUMD</elementName>
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<value>0.88</value>
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</weatherElement>
<weatherElement>
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<value>1005.9</value>
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</weatherElement>
<weatherElement>
<elementName>24R</elementName>
<elementValue>
<value>1.0</value>
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</weatherElement>
<weatherElement>
<elementName>H_FX</elementName>
<elementValue>
<value>6.9</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_XD</elementName>
<elementValue>
<value>110.0</value>
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</weatherElement>
<weatherElement>
<elementName>H_FXT</elementName>
<elementValue>
<value>1250</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_F10</elementName>
<elementValue>
<value>4.7</value>
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</weatherElement>
<weatherElement>
<elementName>H_10D</elementName>
<elementValue>
<value>110.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_F10T</elementName>
<elementValue>
<value>1256</value>
</elementValue>
</weatherElement>
<parameter>
<parameterName>CITY</parameterName>
<parameterValue>宜蘭縣</parameterValue>
</parameter>
<parameter>
<parameterName>CITY_SN</parameterName>
<parameterValue>07</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN</parameterName>
<parameterValue>蘇澳鎮</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN_SN</parameterName>
<parameterValue>354</parameterValue>
</parameter>
</location>
<location>
<lat>23.9770</lat>
<lon>121.6050</lon>
<locationName>花蓮</locationName>
<stationId>466990</stationId>
<time>
<obsTime>2016-05-22T13:40:00+08:00</obsTime>
</time>
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<elementName>TIME</elementName>
<elementValue>
<value>0</value>
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<weatherElement>
<elementName>ELEV</elementName>
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<value>16</value>
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</weatherElement>
<weatherElement>
<elementName>WDIR</elementName>
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<value>150.0</value>
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<weatherElement>
<elementName>WDSD</elementName>
<elementValue>
<value>2.8</value>
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<weatherElement>
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<value>26.8</value>
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</weatherElement>
<weatherElement>
<elementName>HUMD</elementName>
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<value>0.83</value>
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</weatherElement>
<weatherElement>
<elementName>PRES</elementName>
<elementValue>
<value>1006.2</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>24R</elementName>
<elementValue>
<value>0.5</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_FX</elementName>
<elementValue>
<value>7.1</value>
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<weatherElement>
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<elementValue>
<value>140.0</value>
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<weatherElement>
<elementName>H_FXT</elementName>
<elementValue>
<value>1239</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_F10</elementName>
<elementValue>
<value>5.6</value>
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</weatherElement>
<weatherElement>
<elementName>H_10D</elementName>
<elementValue>
<value>140.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_F10T</elementName>
<elementValue>
<value>1241</value>
</elementValue>
</weatherElement>
<parameter>
<parameterName>CITY</parameterName>
<parameterValue>花蓮縣</parameterValue>
</parameter>
<parameter>
<parameterName>CITY_SN</parameterName>
<parameterValue>21</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN</parameterName>
<parameterValue>花蓮市</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN_SN</parameterName>
<parameterValue>150</parameterValue>
</parameter>
</location>
<location>
<lat>23.0992</lat>
<lon>121.3654</lon>
<locationName>成功</locationName>
<stationId>467610</stationId>
<time>
<obsTime>2016-05-22T13:40:00+08:00</obsTime>
</time>
<weatherElement>
<elementName>TIME</elementName>
<elementValue>
<value>0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>ELEV</elementName>
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<value>34</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>WDIR</elementName>
<elementValue>
<value>180.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>WDSD</elementName>
<elementValue>
<value>3.8</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>TEMP</elementName>
<elementValue>
<value>27.1</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>HUMD</elementName>
<elementValue>
<value>0.83</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>PRES</elementName>
<elementValue>
<value>1003.6</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>24R</elementName>
<elementValue>
<value>0.5</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_FX</elementName>
<elementValue>
<value>4.6</value>
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</weatherElement>
<weatherElement>
<elementName>H_XD</elementName>
<elementValue>
<value>200.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_FXT</elementName>
<elementValue>
<value>1259</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_F10</elementName>
<elementValue>
<value>2.8</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_10D</elementName>
<elementValue>
<value>190.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_F10T</elementName>
<elementValue>
<value>1259</value>
</elementValue>
</weatherElement>
<parameter>
<parameterName>CITY</parameterName>
<parameterValue>臺東縣</parameterValue>
</parameter>
<parameter>
<parameterName>CITY_SN</parameterName>
<parameterValue>22</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN</parameterName>
<parameterValue>成功鎮</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN_SN</parameterName>
<parameterValue>258</parameterValue>
</parameter>
</location>
<location>
<lat>22.7540</lat>
<lon>121.1465</lon>
<locationName>臺東</locationName>
<stationId>467660</stationId>
<time>
<obsTime>2016-05-22T13:40:00+08:00</obsTime>
</time>
<weatherElement>
<elementName>TIME</elementName>
<elementValue>
<value>0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>ELEV</elementName>
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<value>9</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>WDIR</elementName>
<elementValue>
<value>200.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>WDSD</elementName>
<elementValue>
<value>3.3</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>TEMP</elementName>
<elementValue>
<value>29.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>HUMD</elementName>
<elementValue>
<value>0.79</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>PRES</elementName>
<elementValue>
<value>1006.9</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>24R</elementName>
<elementValue>
<value>0.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_FX</elementName>
<elementValue>
<value>6.1</value>
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</weatherElement>
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<elementName>H_XD</elementName>
<elementValue>
<value>210.0</value>
</elementValue>
</weatherElement>
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<elementName>H_FXT</elementName>
<elementValue>
<value>1300</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_F10</elementName>
<elementValue>
<value>2.8</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_10D</elementName>
<elementValue>
<value>180.0</value>
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</weatherElement>
<weatherElement>
<elementName>H_F10T</elementName>
<elementValue>
<value>1239</value>
</elementValue>
</weatherElement>
<parameter>
<parameterName>CITY</parameterName>
<parameterValue>臺東縣</parameterValue>
</parameter>
<parameter>
<parameterName>CITY_SN</parameterName>
<parameterValue>22</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN</parameterName>
<parameterValue>臺東市</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN_SN</parameterName>
<parameterValue>297</parameterValue>
</parameter>
</location>
<location>
<lat>22.3576</lat>
<lon>120.8957</lon>
<locationName>大武</locationName>
<stationId>467540</stationId>
<time>
<obsTime>2016-05-22T13:40:00+08:00</obsTime>
</time>
<weatherElement>
<elementName>TIME</elementName>
<elementValue>
<value>0</value>
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</weatherElement>
<weatherElement>
<elementName>ELEV</elementName>
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<value>8</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>WDIR</elementName>
<elementValue>
<value>180.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>WDSD</elementName>
<elementValue>
<value>3.1</value>
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</weatherElement>
<weatherElement>
<elementName>TEMP</elementName>
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<value>27.5</value>
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</weatherElement>
<weatherElement>
<elementName>HUMD</elementName>
<elementValue>
<value>0.87</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>PRES</elementName>
<elementValue>
<value>1007.6</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>24R</elementName>
<elementValue>
<value>2.0</value>
</elementValue>
</weatherElement>
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<elementName>H_FX</elementName>
<elementValue>
<value>8.9</value>
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</weatherElement>
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<elementName>H_XD</elementName>
<elementValue>
<value>190.0</value>
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</weatherElement>
<weatherElement>
<elementName>H_FXT</elementName>
<elementValue>
<value>1207</value>
</elementValue>
</weatherElement>
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<elementName>H_F10</elementName>
<elementValue>
<value>5.3</value>
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<elementValue>
<value>180.0</value>
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</weatherElement>
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<elementName>H_F10T</elementName>
<elementValue>
<value>1213</value>
</elementValue>
</weatherElement>
<parameter>
<parameterName>CITY</parameterName>
<parameterValue>臺東縣</parameterValue>
</parameter>
<parameter>
<parameterName>CITY_SN</parameterName>
<parameterValue>22</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN</parameterName>
<parameterValue>大武鄉</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN_SN</parameterName>
<parameterValue>342</parameterValue>
</parameter>
</location>
<location>
<lat>22.0387</lat>
<lon>121.5506</lon>
<locationName>蘭嶼</locationName>
<stationId>467620</stationId>
<time>
<obsTime>2016-05-22T13:40:00+08:00</obsTime>
</time>
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<elementName>TIME</elementName>
<elementValue>
<value>0</value>
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<weatherElement>
<elementName>ELEV</elementName>
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<value>324</value>
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</weatherElement>
<weatherElement>
<elementName>WDIR</elementName>
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<value>240.0</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>WDSD</elementName>
<elementValue>
<value>11.7</value>
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<weatherElement>
<elementName>TEMP</elementName>
<elementValue>
<value>24.2</value>
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</weatherElement>
<weatherElement>
<elementName>HUMD</elementName>
<elementValue>
<value>0.87</value>
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</weatherElement>
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<elementName>PRES</elementName>
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<value>972.5</value>
</elementValue>
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<weatherElement>
<elementName>24R</elementName>
<elementValue>
<value>6.5</value>
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</weatherElement>
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<elementName>H_FX</elementName>
<elementValue>
<value>14.9</value>
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<elementValue>
<value>230.0</value>
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</weatherElement>
<weatherElement>
<elementName>H_FXT</elementName>
<elementValue>
<value>1254</value>
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</weatherElement>
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<elementName>H_F10</elementName>
<elementValue>
<value>10.8</value>
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<weatherElement>
<elementName>H_10D</elementName>
<elementValue>
<value>230.0</value>
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</weatherElement>
<weatherElement>
<elementName>H_F10T</elementName>
<elementValue>
<value>1259</value>
</elementValue>
</weatherElement>
<parameter>
<parameterName>CITY</parameterName>
<parameterValue>臺東縣</parameterValue>
</parameter>
<parameter>
<parameterName>CITY_SN</parameterName>
<parameterValue>22</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN</parameterName>
<parameterValue>蘭嶼鄉</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN_SN</parameterName>
<parameterValue>351</parameterValue>
</parameter>
</location>
<location>
<lat>25.6294</lat>
<lon>122.0713</lon>
<locationName>彭佳嶼</locationName>
<stationId>466950</stationId>
<time>
<obsTime>2016-05-22T13:40:00+08:00</obsTime>
</time>
<weatherElement>
<elementName>TIME</elementName>
<elementValue>
<value>1</value>
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<weatherElement>
<elementName>ELEV</elementName>
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<value>102</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>WDIR</elementName>
<elementValue>
<value>70.0</value>
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<weatherElement>
<elementName>WDSD</elementName>
<elementValue>
<value>4.4</value>
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<weatherElement>
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<value>24.4</value>
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</weatherElement>
<weatherElement>
<elementName>HUMD</elementName>
<elementValue>
<value>0.88</value>
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</weatherElement>
<weatherElement>
<elementName>PRES</elementName>
<elementValue>
<value>998.0</value>
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</weatherElement>
<weatherElement>
<elementName>24R</elementName>
<elementValue>
<value>4.5</value>
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</weatherElement>
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<elementName>H_FX</elementName>
<elementValue>
<value>6.2</value>
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<elementValue>
<value>60.0</value>
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</weatherElement>
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<elementName>H_FXT</elementName>
<elementValue>
<value>1248</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>H_F10</elementName>
<elementValue>
<value>5.0</value>
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<weatherElement>
<elementName>H_10D</elementName>
<elementValue>
<value>50.0</value>
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<weatherElement>
<elementName>H_F10T</elementName>
<elementValue>
<value>1230</value>
</elementValue>
</weatherElement>
<parameter>
<parameterName>CITY</parameterName>
<parameterValue>基隆市</parameterValue>
</parameter>
<parameter>
<parameterName>CITY_SN</parameterName>
<parameterValue>99</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN</parameterName>
<parameterValue>中正區</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN_SN</parameterName>
<parameterValue>999</parameterValue>
</parameter>
</location>
<location>
<lat>23.2590</lat>
<lon>119.6596</lon>
<locationName>東吉島</locationName>
<stationId>467300</stationId>
<time>
<obsTime>2016-05-22T13:40:00+08:00</obsTime>
</time>
<weatherElement>
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<value>0</value>
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<weatherElement>
<elementName>ELEV</elementName>
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<value>43</value>
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<weatherElement>
<elementName>WDIR</elementName>
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<value>190.0</value>
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<weatherElement>
<elementName>WDSD</elementName>
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<value>9.4</value>
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<weatherElement>
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<value>27.5</value>
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</weatherElement>
<weatherElement>
<elementName>HUMD</elementName>
<elementValue>
<value>0.84</value>
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</weatherElement>
<weatherElement>
<elementName>PRES</elementName>
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<value>1003.3</value>
</elementValue>
</weatherElement>
<weatherElement>
<elementName>24R</elementName>
<elementValue>
<value>2.5</value>
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</weatherElement>
<weatherElement>
<elementName>H_FX</elementName>
<elementValue>
<value>14.1</value>
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<elementName>H_XD</elementName>
<elementValue>
<value>150.0</value>
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</weatherElement>
<weatherElement>
<elementName>H_FXT</elementName>
<elementValue>
<value>1242</value>
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</weatherElement>
<weatherElement>
<elementName>H_F10</elementName>
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<value>10.4</value>
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<elementName>H_10D</elementName>
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<value>160.0</value>
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</weatherElement>
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<elementName>H_F10T</elementName>
<elementValue>
<value>1244</value>
</elementValue>
</weatherElement>
<parameter>
<parameterName>CITY</parameterName>
<parameterValue>澎湖縣</parameterValue>
</parameter>
<parameter>
<parameterName>CITY_SN</parameterName>
<parameterValue>23</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN</parameterName>
<parameterValue>望安鄉</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN_SN</parameterName>
<parameterValue>231</parameterValue>
</parameter>
</location>
<location>
<lat>24.9608</lat>
<lon>121.5165</lon>
<locationName>新店</locationName>
<stationId>A0A9M0</stationId>
<time>
<obsTime>2016-05-22T13:40:00+08:00</obsTime>
</time>
<weatherElement>
<elementName>TIME</elementName>
<elementValue>
<value>0</value>
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<weatherElement>
<elementName>ELEV</elementName>
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<value>24</value>
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</weatherElement>
<weatherElement>
<elementName>WDIR</elementName>
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<value>60.0</value>
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</weatherElement>
<weatherElement>
<elementName>WDSD</elementName>
<elementValue>
<value>1.8</value>
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<weatherElement>
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<value>28.9</value>
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<weatherElement>
<elementName>HUMD</elementName>
<elementValue>
<value>0.76</value>
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</weatherElement>
<weatherElement>
<elementName>PRES</elementName>
<elementValue>
<value>1005.2</value>
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</weatherElement>
<weatherElement>
<elementName>24R</elementName>
<elementValue>
<value>0.5</value>
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</weatherElement>
<weatherElement>
<elementName>H_FX</elementName>
<elementValue>
<value>2.9</value>
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<elementValue>
<value>90.0</value>
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<elementName>H_FXT</elementName>
<elementValue>
<value>1300</value>
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</weatherElement>
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<elementName>H_F10</elementName>
<elementValue>
<value>1.3</value>
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</weatherElement>
<weatherElement>
<elementName>H_10D</elementName>
<elementValue>
<value>230.0</value>
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<elementName>H_F10T</elementName>
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<value>1219</value>
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</weatherElement>
<parameter>
<parameterName>CITY</parameterName>
<parameterValue>新北市</parameterValue>
</parameter>
<parameter>
<parameterName>CITY_SN</parameterName>
<parameterValue>06</parameterValue>
</parameter>
<parameter>
<parameterName>TOWN</parameterName>
<parameterValue>新店區</parameterValue>
</parameter>
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\section{Introduction}
\subsection{Motivation}
As the traffic safety has become of utmost importance, much attention is given to intelligent transportation systems (ITSs), and more specifically to vehicular communications (VCs) \cite{arif2019survey,mekki2017vehicular}. VCs offer a wide range of applications such as, traffic state, autonomous driving, and safety \cite{sam2016vehicle,singh2019multipath,boquet2018adaptive}. According to World Health organization, over 1.25 million people die each year on the roads, and road traffic crashes are the number one cause of death among young people \cite{world2015global}. Moreover, 50 \% of all crashes are in junction areas (intersections) including fatal crashes, injury crashes and property damage crashes \cite{traficsafety}. This makes intersections critical areas not only for vehicles, but also for pedestrians and cyclists.
VCs have numerous applications to prevent accidents, or alert vehicles when accidents happen in their surroundings.
Thus, low latency and high reliability communications are mandatory in safety-based VCs.
To increase spectral efficiency and data rate \cite{ding2017application} in the fifth generation (5G) of wireless communication systems, non-orthogonal multiple access (NOMA) is an promising candidate as a multiple access scheme.
NOMA, unlike orthogonal multiple access (OMA), allows several users to use the same resource with several power allocation levels.
Also, cooperative transmissions have been show to increase the reliability of the transmission link \cite{wang2018secure,nguyen2019performance,altieri2019performance}. On the other hand, co-channel interference is one of the major impairments that can degrade a transmission in VCs \cite{tripp2018comparison,mourad2017performance,campolo2016modeling}.
Hence, in this paper we propose to study the impact of interference in cooperative VCs at intersections using NOMA.
\subsection{Related Works}
\subsubsection{NOMA Works}
The performance of NOMA has been well studied in the literature (see \cite{dai2015non,islam2017power,ding2014performance} and the references therein).
As far as the impact of interference on NOMA is concerned, several papers have studied its effect \cite{zhang2016stochastic}. The authors in \cite{zhang2017uplink} analysed the impact of interference on a NOMA uplink transmission. The authors also analyzed the performance of a NOMA downlink transmission with a selection based pairing in \cite{zhang2017downlink}.
The improvement of using cooperative transmissions in NOMA have been also well investigated \cite{ding2016relay,liu2016cooperative,timotheou2015fairness,ding2014performance}.
A scenario involving $M$ number of randomly deployed users was investigated in \cite{ding2014performance}. The authors also evaluated the ergodic rate and outage performance in \cite{timotheou2015fairness}.
In \cite{ding2016relay}, the authors studied the impact of relay selection on cooperative NOMA, and showed that the two-stage scheme can achieve the optimal diversity gain and the minimal outage probability.
However, the impact of implementing NOMA into VCs has been lacking in the literature.
\subsubsection{VCs Works}
The performance of VCs in the presence of interference has attracted a lot of attention \cite{rakhshan2017improving,jiang2016information,farooq2016stochastic}.
Mainly, there are two types of scenarios in VCs, highways scenarios and intersections scenarios. Considering highway scenarios, the authors in \cite{jiang2016information} investigated the performance of RTS/CTS protocol considering Nakagami-$m$ channels fading. In \cite{rakhshan2017improving}, the authors studied how the interference affects the safety of vehicles in a VCs. The authors also derived the packet success probability for two different traffic models in VCs \cite{rakhshan2016packet}.
The authors in \cite{farooq2016stochastic} investigated the performance of carrier sense multiple access (CSMA) protocols, and derived the expressions of packet success probability. In \cite{tassi2017modeling}, the authors derived the outage probability and rate coverage probability when a line of sight path to the base station is absent.
Considering intersection scenarios, a success probability expression of a simple intersection scenario was derived in \cite{steinmetz2015stochastic}. The authors in \cite{abdulla2016vehicle} extended the work of \cite{steinmetz2015stochastic} and derived the success probability considering limited road segments with different path loss models. The authors of \cite{abdulla2016vehicle} also studied the average and the fine-grained reliability in an interference-limited vehicle to vehicle (V2V) communications with the aid of the meta distribution in \cite{abdulla2017fine}. The authors \cite{jeyaraj2017reliability} in investigated the performance of V2V communications for orthogonal streets. The authors also studied V2V communications at intersections and showed that, the performance of the ALOHA protocol can be considered as lower bound of performance of the CSMA protocols \cite{jeyaraj2018nearest}. The effect of vehicles mobility and interference dependence has been investigated in \cite{J2}. The authors also, studied the performance of three transmission schemes at intersection in line of sight scenario and non light of sight scenario considering Nakagami-$m$ fading channels in \cite{J1,belmekki2019outagearxiv}.
However, the performance of NOMA in VCs is lacking in the literature. The first to tackle this issue are the authors of the paper at hand. They computed the outage probability and average achievable rate of NOMA at intersection roads considering direct transmissions \cite{C1,J3} and cooperative transmissions \cite{belmekki2019outage,J4}. They also investigated the performance of NOMA in millimeter wave vehicular communications in \cite{Cmm1,Cmm2}. In \cite{bprotocol}, the authors proposed an adaptive NOMA protocol in VCs.
In this paper, the authors study the feasibility and improvement in performance by implementing both NOMA and maximum ratio combining (MRC) in VCs. Hence, we compare the proposed scheme with the classical OMA, and the classical cooperative NOMA, and see if the improvements justify and outweigh the complexity of implementing MRC and NOMA in VCs.
\subsection{Contributions}
The contributions of this paper are as follows:
\begin{itemize}
\item We establish a framework for performance analysis of VCs under Rayleigh fading and two perpendicular roads containing one-dimensional Poisson field of interference. We analyze the performance of implementing MRC in cooperative VCs transmission schemes using NOMA at intersections in terms of outage probability. We obtained closed form outage probability
expressions. We further extend the derivations when $K$ destination nodes are involved, and to a realistic intersection scenario involving multiple lanes.
\item We compare the performance of MRC cooperative NOMA with a classical cooperative NOMA \cite{J4}, and show that implementing MRC in cooperative NOMA transmission offers a significant improvement over the classical cooperative NOMA in terms of outage probability. We also compare the performance of MRC cooperative NOMA with MRC cooperative OMA \cite{J2}, and show that NOMA offers a better performance than OMA. It is shown that the outage probability increases when the vehicles are closer to the road intersection, and that using MRC considering NOMA improves significantly the performance in this context.
\item
The relationships between system performance
and different network parameters such as NOMA power allocation coefficient, date rates,
channel access probability, intensity of potential interfering vehicles, relay position, noise power levels, successive interference cancellation (SIC) coefficient
are discussed. The
results clearly demonstrate the advantages of implementing MRC into NOMA
the performance in VCs, even at the cost of implementation complexity.
\item We show that as we increases the data rate of $D_2$, MRC transmission using NOMA offers a better performance than MRC transmission using OMA. Whereas for $D_1 $, low data rates are suitable, since there is a condition imposed to its data rate. We also show how the imperfect SIC process can degrade the performance of NOMA. We also show that MRC transmission using NOMA outperforms cooperative NOMA.
Finally, we investigate the best relay position, and show that the optimal relay position for $D_1$ and $D_2$ is near the destination nodes.
\item To confirm the correctness of our theoretical derivations, extensive Monte-Carlo simulations are carried out.
\end{itemize}
\subsection{Organization}
The rest of this paper is organized as follows. Section \ref{Section2} presents the system model. In Section \ref{Section3}, outage analytical expressions are derived. The Laplace transform expressions are presented in Section \ref{Section4}. Extension to multiple lanes scenario is investigated in Section \ref{Section5}. Simulations and discussions are in Section \ref{Section6}. Finally, we conclude the paper in Section \ref{Section7}.
\section{System Model}\label{Section2}
\begin{figure}[]
\centering
\includegraphics[scale=0.65]{Fig1}
\caption{Cooperative NOMA system model for vehicular communications involving two destination nodes and a relay node. For this example, $S$ is a vehicle, $R$ is an infrastructure, $D_1$ is a vehicle, and $D_2$ is an infrastructure. }
\label{Figure1}
\end{figure}
\subsection{Intersection Scenario}
We consider a cooperative transmission using NOMA between a source $S$ and two destinations $D_1$ and $D_2$, with the aid of a relay $R$ as shown in Fig.\ref{Figure1}. As both V2V and V2I communications are of interest\footnote{The Doppler shift and time-varying effect of V2V and V2I channels is beyond the scope of this paper.}, the nodes $S$,$R$, $D_1$ and $D_2$ can be on the roads (as vehicles), or outside the roads (as infrastructures). For instance in Fig.\ref{Figure1}, the configuration is as follows: $S$ and $D_1$ are vehicles, whereas $R$ and $D_2$ are infrastructures. For the sake of notation simplicity, we denote by $M$ the receiving node, and by $m$ the distance between the node $M$ and the intersection, where $M \in \{R,D_1,D_2\}$ and $m \in \{r,d_1,d_2\}$, as shown in Fig.\ref{Figure1}. Also, the term $\theta_M $ denotes the angle between the node $M$ and the $X$ road.
In this paper, we study the performance at an intersection. The intersection has two two perpendicular roads, an horizontal road denoted by $X$, and a vertical road denoted by $Y$. We extend the analysis to the case when the intersection involves multiple lanes in Section \ref{Section5}.
The set of nodes $\lbrace{S, R, D_1, D_2}\rbrace$ is subject to interference originated from transmitting vehicles located on the roads.
The set of interfering vehicles located on the $Z$ road where $Z\in \{X,Y\}$, denoted by $\Phi_{Z}$ are modeled as a one-dimensional homogeneous Poisson point process (1D-HPPP), that is, $\Phi_{Z}\sim\textrm{1D-HPPP}(\lambda_{Z},z)$, where $z\in \{x,y\}$ and $\lambda_{Z}$ are the position of interfering vehicles and their intensity on the $Z$ road, respectively. This implies that the number of potential interfering vehicles within any closed and bounded set $\mathcal{B}\subseteq\mathbb{R}$ is a Poisson random variable with parameter $\lambda \vert \mathcal{B}\vert $.
\begin{figure}
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\begin{tikzpicture}
\tikzstyle{S}=[circle,very thick,draw=black, fill=red, minimum size=.7cm]
\tikzstyle{R}=[circle,very thick,draw=black, fill=orange, minimum size=.7cm]
\tikzstyle{D1}=[circle,very thick,draw=black, fill=blue, minimum size=.7cm]
\tikzstyle{D2}=[circle,very thick,draw=black, fill=green, minimum size=.7cm]
\node[S,label=below :{$S$}] (S) at (0,0) {};
\node[R,label=below :{$R$}] (R) at (2, 0) {};
\node[D1,label=below :{$D_1$}] (D1) at (4, 2) {};
\node[D2,label=below :{$D_2$}] (D2) at (4, -2) {};
\draw [->, very thick] (S) -- (R); \draw [->, very thick](S) -- (D1) ;\draw [->, very thick](S) -- (D2) ;
\node[draw,text width=3cm] at (2,4) {\textbf{First phase}};
\end{tikzpicture}
}
\hspace{1cm}
\fbox{%
\begin{tikzpicture}
\tikzstyle{S}=[circle,very thick,draw=black, fill=red, minimum size=.7cm]
\tikzstyle{R}=[circle,very thick,draw=black, fill=orange, minimum size=.7cm]
\tikzstyle{D1}=[circle,very thick,draw=black, fill=blue, minimum size=.7cm]
\tikzstyle{D2}=[circle,very thick,draw=black, fill=green, minimum size=.7cm]
\node[S,label=below :{$S$}] (S) at (6,0) {};
\node[R,label=below :{$R$}] (R) at (8, 0) {};
\node[D1,label=below :{$D_1$}] (D1) at (10, 2) {};
\node[D2,label=below :{$D_2$}] (D2) at (10, -2) {};
\draw [->, very thick](R) -- (D1); \draw [->, very thick](R) -- (D2);
\node[draw,text width=3cm] at (8,4) {\textbf{Second phase}};
\end{tikzpicture}
}
}
\caption{Transmission scheme using MRC and NOMA.} \label{fig:M1}
\end{figure}
\subsection{MRC and Cooperative Protocol}
In this paper, we use Decode and Forward (DF) cooperative protocol \cite{feteiha2016decode,altieri2014outage}. The transmission occurs in two phases, the duration of each phase is one time slot. Finally we consider we use MRC in NOMA setup as shown in Fig.\ref{fig:M1}. In the first phase, $S$ broadcasts the message, and the nodes $R$, $D_1$ and $D_2$ try to decode the message. In the second phase, if $R$ decodes $S$ message, it broadcasts the message to $D_1$ and $D_2$. Then, $D_1$ and $D_2$ add the power received in the first phase from $S$ and (if $R$ decodes $S$ message) the power received from $R$ during the second phase to decode the message.
\subsection{NOMA Scenario and Assumptions}
In NOMA, there are two main ways to order the users. The first one is to order the nodes according to their channel stats. Hence, the user with the weakest channel state comes first in the decoding order (see \cite{ding2014performance,ding2015cooperative} and references therein). The second one is that, the users are sorted according to their quality of service (QoS) priorities. Hence, a user with the higher priority comes first in the seconding order. It has been show in \cite{ding2016relay,ding2016mimo}, that ordering users according to their QoS is more realistic and reasonable assumption, since in practice, it is very likely that users who want to participate in NOMA have similar channel conditions. Without loss of generality, we study the case in which node $D_1$ has to be served immediately with a low data rate. For example, $D_1$ can be a vehicle that needs to receive safety information containing a few bytes, such as a road flood warning or incident avoidance alert message. Whereas node $D_2$ requires relatively high data rate but can be served later. For instance $D_2$ can be a user that accesses the internet connection.
\subsection{Channel Model}
We consider slotted ALOHA protocol with parameter $p$, i.e., every node can access the medium with a probability $p$. This performs an independent thinning the parent 1D-HPPP by probability $p$. Hence, the set of interfering vehicles at a given time slot also follow a a 1D-HPPP with intensity $p\lambda$.
The transmission between a node $a$ and $b$ experience a path loss given by $l_{ab}= (A r_{ab})^{-\alpha}$
, where $A$ is a constant depending on the antenna characteristics, $ r_{ab}=\Vert a-b\Vert$, and $\alpha$ is the path loss exponent. All the node transmit with power $P$.
The signal transmitted by $S$, denoted $ \chi_{S}$ is a mixture of the message intended to $D_1$ and $D_2$. This can be expressed as
\begin{equation}
\chi_{S}=\sqrt{a_1}\chi_{D1}+\sqrt{a_2}\chi_{D2}, \nonumber
\end{equation}
where $a_i$ is the power coefficients allocated to $D_i$, and $\chi_{Di}$ is the message intended to $D_i$, where $i \in \{1,2\}$. Since $D_1$ has higher power than $D_2$, that is $a_1 \ge a_2$, then $D_1$ comes first in the decoding order. Note that, $a_1+a_2=1$.\\
The signal received at $R$ and $D_i$, denoted respectively by $\mathcal{Y}_{R}$ and $\mathcal{Y}_{D_i}$, during the first time slot are expressed as
\begin{flalign}
&\mathcal{Y}_{R}=\underbrace{h_{SR}\sqrt{P l_{SR}}\:\chi_{S}}_\text{The signal of interest that contains $D_1$ message and $D_2$ message}+\nonumber \\
&\underbrace{\sum_{x\in \Phi_{X_{R}}}h_{Rx}\sqrt{P l_{Rx}}\:\chi_{x}}_\text{Aggregate interference form the $Y$ road at $R$}
+\underbrace{\sum_{y\in \Phi_{Y_{R}}}h_{Ry}\sqrt{P l_{Ry}}\:\chi_{y}}_\text{Aggregate interference form the $Y$ road at $R$}+\underbrace{\sigma^2}_\text{Noise related term}, \nonumber
\end{flalign}
and
\begin{flalign}
& \mathcal{Y}_{D_i}=h_{SD_i}\sqrt{P l_{SD_i}}\:\chi_{S}+ \nonumber \\
&\underbrace{\sum_{x\in \Phi_{X_{D_i}}}h_{D_ix}\sqrt{P l_{D_ix}}\:\chi_{x}}_\text{Aggregate interference form the $X$ road at $D_i$}
+ \underbrace{\sum_{y\in \Phi_{Y_{D_i}}}h_{D_iy}\sqrt{P l_{D_iy}}\:\chi_{y}}_\text{Aggregate interference form the $Y$ road at $D_i$}+\sigma^2. \nonumber
\end{flalign}
The signal received at $D_i$, denoted by $\mathcal{Y}_{D_i}$, during the second time slot is expressed as
\begin{flalign}
&\mathcal{Y}_{D_i}=h_{RD_i}\sqrt{P l_{RD_i}}\:\chi_{R}+ \nonumber \\
&\underbrace{\sum_{x\in \Phi_{X_{D_i}}}h_{D_ix}\sqrt{P l_{D_ix}}\:\chi_{x}}_\text{Aggregate interference form the $Y$ road at $D_i$}
+\underbrace{\sum_{y\in \Phi_{Y_{D_i}}}h_{D_iy}\sqrt{P l_{D_iy}}\:\chi_{y}}_\text{Aggregate interference form the $Y$ road at $D_i$}+\sigma^2, \nonumber
\end{flalign}
where signals transmitted by the interfering vehicles $x$ and $y$, are denoted by $ \chi_x$ and $\chi_y $, respectively. The term $h_{ab}$ denotes the fading coefficient between node $a$ and $b$, and it is modeled as $\mathcal{CN}(0,1)$ \cite{halimi2017wavelet1,halimi2017unsupervised2,halimi2017statistical3}, hence $|h_{ab}|^2 \sim \exp(1)$. The aggregate interference is defined as
\begin{equation}\label{eqation.1}
I_{Z_{M}}=\sum_{z\in \Phi_{Z_{M}}}P\vert h_{Mz}\vert^{2}l_{Mz} ,
\end{equation}
where $I_{Z_{M}} $ denotes the aggregate interference from the $Z$ road at $M$, $\Phi_{Z_{M}}$ denotes the set of the interfering vehicles from the $Z$ road at $M$.
\section{Outage Analytical Derivations}\label{Section3}
\subsection{Outage Events}
We define an outage event at the receiving node when the signal-to-interference plus noise ratio (SINR) at the receiver is below a given threshold. According to SIC \cite{hasna2003performance}, $D_1$ is decoded first since it has the higher power allocation, and $D_2$ message is considered as interference. The outage event at $R$ to not decode $D_1$, denoted $\mathcal{A}_{R_1}(\Theta_1)$, is defined as
\begin{equation}
\mathcal{A}_{R_1}(\Theta_1)\triangleq \frac{P\vert h_{SR}\vert^{2}l_{SR}\,a_1}{P\vert h_{SR}\vert^{2}l_{SR}a_2+I_{X_{R}}+I_{Y_{R}}+\sigma^2} < \Theta_1,
\end{equation}
where $\Theta_1=2^{2\mathcal{R}_1}-1$, and $\mathcal{R}_1$ is the target data rate of $D_1$.
Since $D_2$ has a lower power allocation, $R$ has to decode $D_1$ message, then decode $D_2$ message. The outage event at $R$ to not decode $D_2$ message, denoted $\mathcal{A}_{R_2}(\Theta_2)$, is defined as
\begin{equation}
\mathcal{A}_{R_2}(\Theta_2)\triangleq\frac{P\vert h_{SR}\vert^{2}l_{SR}\,a_2}{\beta P\vert h_{SR}\vert^{2}l_{SR}\,a_1+I_{X_{R}}+I_{Y_{R}}+\sigma^2}< \Theta_2,
\end{equation}
where $\Theta_2=2^{2\mathcal{R}_2}-1$, and $\mathcal{R}_2$ is the target data rate of $D_2$.
Similarly, the outage event at $D_1$ to not decode its intended message in the first phase ($S \rightarrow D_1$), denoted $\mathcal{B}_{D_{1\rightarrow 1}}(\Theta_1)$, is given by
\begin{equation}
\mathcal{B}_{D_{1\rightarrow 1}}(\Theta_1)\triangleq\frac{P\vert h_{SD_1}\vert^{2}l_{SD_1}\,a_1}{P\vert h_{SD_1}\vert^{2}l_{SD_1}a_2+I_{X_{D_1}}+I_{Y_{D_1}}+\sigma^2} < \Theta_1.
\end{equation}
Finally, in order for $D_2$ to decode its intended message, it has to decode $D_1$ message. The outage event at $D_2$ to not decode $D_1$ message in the first phase ($S \rightarrow D_2$), denoted $\mathcal{B}_{D_{2\rightarrow 1}}(\Theta_1)$, and the outage event at $D_2$ to not decode its intended message, denoted $\mathcal{B}_{D_{2\rightarrow 2}}(\Theta_2)$, are respectively given by
\begin{equation}
\mathcal{B}_{D_{2\rightarrow 1}}(\Theta_1)\triangleq\frac{P\vert h_{SD_2}\vert^{2}l_{SD_2}\,a_1}{P\vert h_{SD_2}\vert^{2}l_{SD_2}a_2+I_{X_{D_2}}+I_{Y_{D_2}}+\sigma^2}< \Theta_1,
\end{equation}
and
\begin{equation}
\mathcal{B}_{D_{2\rightarrow 2}}(\Theta_2)\triangleq\frac{P\vert h_{SD_2}\vert^{2}l_{SD_2}\,a_2}{\beta P\vert h_{SD_2}\vert^{2}l_{SD_2}\,a_1+I_{X_{D_2}}+I_{Y_{D_2}}+\sigma^2}< \Theta_2.
\end{equation}
During the second phase, $D_1$ adds the power received from $S$ and from $R$. Hence, the outage event at $D_1$ to not decode its message in the second phase, denoted $\mathcal{C}_{D_{1\rightarrow 1}}(\Theta_1)$, is expressed as
\begin{equation}
\mathcal{C}_{D_{1\rightarrow 1}}(\Theta_1)\triangleq \frac{P\sum_{[SD_1,RD_1]}(\vert h\vert^{2},l)\,a_1}{P\sum_{[SD_1,RD_1]}(\vert h\vert^{2},l)\,a_2+I_{X_{D_1}}+I_{Y_{D_1}}+\sigma^2} < \Theta_1,
\end{equation}
where $$\sum_{[SD_i,RD_i]}(\vert h\vert^{2},l)=\vert h_{SD_i}\vert^{2}l_{SD_i}+\vert h_{RD_i}\vert^{2}l_{RD_2}.$$
In the same way, in the second phase, $D_2$ adds the power received from $S$ and from $R$. Hence, the outage event at $D_2$ to not decode $D_1$ message, denoted $\mathcal{C}_{D_{2\rightarrow1}}(\Theta_1)$, and the outage event at $D_2$ to not decode its message, denoted $\mathcal{C}_{D_{2\rightarrow2}}(\Theta_2)$, are respectively expressed as
\begin{equation}
\mathcal{C}_{D_{2\rightarrow 1}}(\Theta_1)\triangleq \frac{P\sum_{[SD_2,RD_2]}(\vert h\vert^{2},l)\,a_1}{P\sum_{[SD_2,RD_2]}(\vert h\vert^{2},l)\,a_2+I_{X_{D_2}}+I_{Y_{D_2}}+\sigma^2} < \Theta_1,
\end{equation}
and
\begin{equation}
\mathcal{C}_{D_{2\rightarrow 2}}(\Theta_2)\triangleq \frac{P\sum_{[SD_2,RD_2]}(\vert h\vert^{2},l)\,a_2}{\beta P\sum_{[SD_2,RD_2]}(\vert h\vert^{2},l)\,a_1+I_{X_{D_2}}+I_{Y_{D_2}}+\sigma^2} < \Theta_2.
\end{equation}
The overall outage event related to $D_1$, denoted $\textit{O}_{(1)}$, is given by
\begin{equation}
\textit{O}_{(1)}\triangleq \Big[ \mathcal{B}_{D_{1\rightarrow 1}}(\Theta_1) \cap \mathcal{A}_{R_1}(\Theta_1) \Big]\cup\Big[\mathcal{A}_{R_1}^C(\Theta_1) \cap \mathcal{C}_{D_{1\rightarrow 1}}(\Theta_1) \Big].
\end{equation}
Finally, the overall outage event related to $D_2$, denoted $\textit{O}_{(2)}$, is given by
\begin{align}
\textit{O}_{(2)}\triangleq &\left[\Bigg\{\bigcup_{i=1}^{2} \mathcal{B}_{D_{2\rightarrow i}}(\Theta_i)\Bigg\} \cap \Bigg\{\bigcup_{i=1}^{2} \mathcal{A}_{R_i}(\Theta_i)\Bigg\}\right]
\nonumber\\&\bigcup \left[\Bigg\{\bigcap_{i=1}^{2} \mathcal{A}_{R_{i}}^C(\Theta_i)\Bigg\} \cap \Bigg\{\bigcup_{i=1}^{2} \mathcal{C}_{D_{2\rightarrow i}}(\Theta_i)\Bigg\}\right].
\end{align}
\subsection{Outage Probability Expressions}
In the following, we will express the outage probability $\textit{O}_{(1)}$ and $\textit{O}_{(2)}$. The probability $\mathbb{P}(\textit{O}_{(1)})$, when $\Theta_1 < a_1/ a_2$, is given by
\begin{align}
&\mathbb{P}(\textit{O}_{(1)})=1- \mathcal{J}_{(D_1)}\big(\frac{G_{1}}{l_{SD_1}}\big)-
\mathcal{J}_{(R)}\big(\frac{G_{1}}{l_{SR}}\big)+
\mathcal{J}_{(D_1)}\big(\frac{G_{1}}{l_{SD_1}}\big)\mathcal{J}_{(R)}\big(\frac{G_{1}}{l_{SR}}\big)
\nonumber\\
&+\mathcal{J}_{(R)}\big(\frac{G_{1}}{l_{SR}}\big)-
\frac{l_{RD_1}\mathcal{J}_{(R)}\big(\frac{G_{1}}{l_{SR}}\big)\mathcal{J}_{(D_1)}\big(\frac{G_{1}}{l_{RD_1}}\big)-l_{SD_1}\mathcal{J}_{(R)}\big(\frac{G_{1}}{l_{SR}}\big)\mathcal{J}_{(D_1)}\big(\frac{G_{1}}{l_{SD_1}}\big)}{l_{RD_1}-l_{SD_1}},
\end{align}
where $G_{1}=\Theta_1/(a_1-\Theta_1 a_2)$, and $\mathcal{J}_{(M)}\Big(\frac{A}{B}\Big)$ is expressed as \\
\begin{equation}
\mathcal{J}_{(M)}\Big(\frac{A}{B}\Big)=\mathcal{L}_{I_{X_{M}}}\Big(\frac{A}{B}\Big)\mathcal{L}_{I_{Y_{M}}}\Big(\frac{A}{B}\Big)\exp\Big(-\frac{\sigma^2 A}{P B}\Big).
\end{equation}
The probability $ \mathbb{P}(\textit{O}_{(2)})$, when $\Theta_1 < a_1/ a_2$ and $\Theta_2 < a_2/ \beta a_1$, is given by\\
\begin{align}
&\mathbb{P}(\textit{O}_{(2)})=1- \mathcal{J}_{(D_2)}\big(\frac{G_{\mathrm{max}}}{l_{SD_2}}\big)-
\mathcal{J}_{(R)}\big(\frac{G_{\mathrm{max}}}{l_{SR}}\big)+
\mathcal{J}_{(D_2)}\big(\frac{G_{\mathrm{max}}}{l_{SD_2}}\big)\mathcal{J}_{(R)}\big(\frac{G_{\mathrm{max}}}{l_{SR}}\big)\nonumber\\
&+\mathcal{J}_{(R)}\big(\frac{G_{\mathrm{max}}}{l_{SR}}\big)-
\frac{l_{RD_2}\mathcal{J}_{(R)}\big(\frac{G_{\mathrm{max}}}{l_{SR}}\big)\mathcal{J}_{(D_2)}\big(\frac{G_{\mathrm{max}}}{l_{RD_2}}\big)-l_{SD_2}\mathcal{J}_{(R)}\big(\frac{G_{\mathrm{max}}}{l_{SR}}\big)\mathcal{J}_{(D_2)}\big(\frac{G_{\mathrm{max}}}{l_{SD_2}}\big)}{l_{RD_2}-l_{SD_2}},
\end{align}
where $G_{\mathrm{max}}=\mathrm{max}(G_1,G_2)$, and $G_2=\Theta_2 /(a_2-\Theta_2 \beta a_1)$.
\textit{Proof}: See Appendix A.\hfill $ \blacksquare $ \\
\subsection{NOMA With $K$-Destinations}
\begin{figure}
\centering
\fbox{%
\fbox{%
\begin{tikzpicture}
\tikzstyle{S}=[circle,very thick,draw=black, fill=red, minimum size=.7cm]
\tikzstyle{R}=[circle,very thick,draw=black, fill=orange, minimum size=.7cm]
\tikzstyle{D1}=[circle,very thick,draw=black, fill=blue, minimum size=.7cm]
\tikzstyle{D2}=[circle,very thick,draw=black, fill=cyan, minimum size=.7cm]
\tikzstyle{DK}=[circle,very thick,draw=black, fill=green, minimum size=.7cm]
\node[S,label=below :{$S$}] (S) at (0,0) {};
\node[R,label=below :{$R$}] (R) at (2, -0.1) {};
\node[D1,label=below :{$D_1$}] (D1) at (4, 2) {};
\node[DK,label=below :{$D_K$}] (DK) at (4, -2) {};
\node[D2,label=below :{$D_2$}] (D2) at (4, 0.7) {};
\node at ($(D2)!.5!(DK)$) {\vdots};
\draw [->, very thick] (S) -- (R); \draw [->, very thick](S) -- (D1) ;\draw [->, very thick](S) -- (D2) ;
\draw [->, very thick] (S) -- (DK);
\node[draw,text width=3cm] at (2,4) {\textbf{First phase}};
\end{tikzpicture}
}
\hspace{1cm}
\fbox{%
\begin{tikzpicture}
\tikzstyle{S}=[circle,very thick,draw=black, fill=red, minimum size=.7cm]
\tikzstyle{R}=[circle,very thick,draw=black, fill=orange, minimum size=.7cm]
\tikzstyle{D1}=[circle,very thick,draw=black, fill=blue, minimum size=.7cm]
\tikzstyle{D2}=[circle,very thick,draw=black, fill=cyan, minimum size=.7cm]
\tikzstyle{DK}=[circle,very thick,draw=black, fill=green, minimum size=.7cm]
\node[S,label=below :{$S$}] (S) at (6,0) {};
\node[R,label=below :{$R$}] (R) at (8, 0) {};
\node[D1,label=below :{$D_1$}] (D1) at (10, 2) {};
\node[D2,label=below :{$D_2$}] (D2) at (10, 0.7) {};
\node[DK,label=below :{$D_K$}] (DK) at (10, -2) {};
\node at ($(D2)!.5!(DK)$) {\vdots};
\draw [->, very thick](R) -- (D1); \draw [->, very thick](R) -- (D2);
\draw [->, very thick] (R) -- (DK);
\node[draw,text width=3cm] at (8,4) {\textbf{Second phase}};
\end{tikzpicture}
}
}
\caption{Transmission scheme using MRC and NOMA considering multiple destinations.} \label{fig:M2}
\end{figure}
We extend the results of NOMA to $K$-destinations as depicted in Fig.\ref{fig:M2}.
We generalize the following events to $K$ destination nodes $D_K$ as
\begin{equation}
\mathcal{A}_{R_i}(\Theta_i)\triangleq \frac{P\vert h_{SR}\vert^{2}l_{SR}\,a_i}{P\vert h_{SR}\vert^{2}l_{SR}\, \big[\beta\sum_{h=1}^{i-1} a_h + \sum_{n=i+1}^{K} a_n \big]+I_{X_{R}}+I_{Y_{R}}+\sigma^2} < \Theta_i,
\end{equation}
\begin{equation}
\mathcal{B}_{D_{i\rightarrow t}}(\Theta_t)\triangleq\frac{P\vert h_{SD_i}\vert^{2}l_{SD_i}\,a_t}{P\vert h_{SD_i}\vert^{2}l_{SD_i}\, \big[\beta\sum_{h=1}^{t-1} a_h + \sum_{n=t+1}^{K} a_n \big]+I_{X_{D_i}}+I_{Y_{D_i}}+\sigma^2} < \Theta_t,
\end{equation}
and
\begin{align}
&\mathcal{C}_{D_{i\rightarrow t}}(\Theta_1)\triangleq \nonumber\\
&\frac{P\sum_{[SD_i,RD_i]}(\vert h\vert^{2},l)\,a_t}{\big[\beta\sum_{h=1}^{t-1} a_h + \sum_{n=t+1}^{K} a_n \big]P\sum_{[SD_i,RD_i]}(\vert h\vert^{2},l)+I_{X_{D_1}}+I_{Y_{D_1}}+\sigma^2} < \Theta_t.
\end{align}
Note that, when $h>t-1$, then $\sum_{h=1}^{t-1} a_h=0$, and when $n>K$, then $\sum_{n=t+1}^{K} a_n=0$.\\
The outage event at the $i$th destination node, denoted $\textit{O}_{(i)}$, is given by
\begin{align}
\textit{O}_{(i)}\triangleq &\left[\Bigg\{\bigcup_{m=K-i+1}^{K} \mathcal{B}_{D_{i\rightarrow i-(K-m)}}(\Theta_{i-(K-m)})\Bigg\} \cap \Bigg\{\bigcup_{m=K-i+1}^{K} \mathcal{A}_{R_{i-(K-m)}}(\Theta_{i-(K-m)})\Bigg\}\right] \nonumber\\
\bigcup &\left[\Bigg\{\bigcap_{m=K-i+1}^{K} \mathcal{A}_{R_{i-(K-m)}}^C(\Theta_{i-(K-m)})\Bigg\} \cap \Bigg\{\bigcup_{m=K-i+1}^{K} \mathcal{C}_{D_{i\rightarrow i-(K-m)}}(\Theta_{i-(K-m)})\Bigg\}\right].\nonumber
\end{align}
Finally, the outage probability of $D_i$ when $\bigcup\limits_{t=1}^{i} \frac{\,a_t}{\beta \, \sum_{h=1}^{t-1} a_h + \sum_{n=t+1}^{K} a_n}\leq \Theta_{t}$, is expressed by
\begin{align}
&\mathbb{P}(\textit{O}_{(i)})=\nonumber\\
&1- \mathcal{J}_{(D_i)}\big(\frac{G_{(i){\textrm{max}}}}{l_{SD_i}}\big)-
\mathcal{J}_{(R)}\big(\frac{G_{(i){\textrm{max}}}}{l_{SR}}\big)+
\mathcal{J}_{(D_i)}\big(\frac{G_{(i){\textrm{max}}}}{l_{SD_i}}\big)\mathcal{J}_{(R)}\big(\frac{G_{\mathrm{max}}}{l_{SR}}\big)
+\mathcal{J}_{(R)}\big(\frac{G_{(i){\textrm{max}}}}{l_{SR}}\big)\nonumber\\
&-
\frac{l_{RD_i}\mathcal{J}_{(R)}\big(\frac{G_{(i){\textrm{max}}}}{l_{SR}}\big)\mathcal{J}_{(D_i)}\big(\frac{G_{(i){\textrm{max}}}}{l_{RD_i}}\big)-l_{SD_i}\mathcal{J}_{(R)}\big(\frac{G_{(i){\textrm{max}}}}{l_{SR}}\big)\mathcal{J}_{(D_i)}\big(\frac{G_{(i){\textrm{max}}}}{l_{SD_i}}\big)}{l_{RD_i}-l_{SD_i}},\nonumber
\end{align}
and $G_{(i)\textrm{max}}$ is given by
\begin{align}\label{eq:23}
G_{(i){\textrm{max}}}=&\textrm{max} \Bigg\{ \frac{ \Theta_{i-(K-1)}}{a_{i-(K-1)}- \Theta_{i-(K-1)}[\beta \, \sum_{h=1}^{{i-(K-1)-1}} a_h + \sum_{n=i-(K-1)+1}^{K} a_n]},\nonumber \\
&\quad\quad\quad\frac{ \Theta_{i-(K-2)}}{a_{i-(K-2)}- \Theta_{i-(K-2)}[\beta \, \sum_{h=1}^{{i-(K-2)-1}} a_h + \sum_{n=i-(K-2)+1}^{K} a_n]},...,\nonumber\\
&\quad\quad\quad\frac{ \Theta_{i-(K-l)}}{a_{i-(K-l)}- \Theta_{i-(K-l)}[\beta \, \sum_{h=1}^{{i-(K-l)-1}} a_h + \sum_{n=i-(K-l)+1}^{K} a_n]} \Bigg\},\nonumber
\end{align}
where $l\in\{1,2,...,K\}$, $\Theta_t=2^{2\mathcal{R}_t}-1$, and $\mathcal{R}_t$ is target data rate of $D_t$. We impose the condition that $l>K-i$.
\section{Laplace Transform Expressions}\label{Section4}
The Laplace transform of the interference originated from the $X$ road at the received node denoted $M$, is expressed as \cite{J4}
\begin{equation}\label{eq27}
\mathcal{L}_{I_{X_M}}(s)=\exp\Bigg(-\emph{p}\lambda_{X}\int_\mathbb{R}\dfrac{1}{1+\big(A\Vert \textit{x}-M \Vert^\alpha\big)/sP}dx\Bigg),
\end{equation}
where
\begin{equation}\label{eq28}
\Vert \textit{x}-M \Vert=\sqrt{\big(m\sin(\theta_{M})\big)^2+\big(x-m \cos(\theta_M) \big)^2 }.
\end{equation}
The Laplace transform of the interference originated from the $Y$ road is given by
\begin{equation}\label{eq29}
\mathcal{L}_{I_{Y_M}}(s)=\exp\Bigg(-\emph{p}\lambda_{Y}\int_\mathbb{R}\dfrac{1}{1+\big(A\Vert \textit{y}-M \Vert^\alpha\big)/sP}dy\Bigg),
\end{equation}
where
\begin{equation}\label{eq30}
\Vert \textit{y}-M \Vert=\sqrt{\big(m\cos(\theta_{M})\big)^2+\big(y-m \sin(\theta_M) \big)^2 }.
\end{equation}
The Laplace transform expressions of the interference when $\alpha=2$ are given by
\begin{equation}\label{eq31}
\mathcal{L}_{I_{X_M}}(s)=\exp\Bigg(-\emph{p}\lambda_{X}\dfrac{sP}{A^{2}}\dfrac{\pi}{\sqrt{\big(m\sin(\theta_{M})\big)^2+sP/A^{2}}}\Bigg),
\end{equation}
and
\begin{equation}\label{eq32}
\mathcal{L}_{I_{Y_M}}(s)=\exp\Bigg(-\emph{p}\lambda_{Y}\dfrac{sP}{A^{2}}\dfrac{\pi}{\sqrt{\big(m\cos(\theta_{M})\big)^2+sP/A^{2}}}\Bigg).
\end{equation}
\begin{figure}
\begin{subfigure}{12cm}
\centering\includegraphics[height=8cm,width=9cm]{one_lane.pdf}
\caption{}
\label{Figurea(a)}
\end{subfigure}
\begin{subfigure}{12cm}
\centering\includegraphics[height=8cm,width=9cm]{two-lanes.pdf}
\caption{}
\label{Figureb(b)}
\end{subfigure}
\caption{Multiple lane modeling (a) one lane scenario.(b) two lanes scenario.}
\label{Figure.1}
\end{figure}
\section{Multi Lanes Scenario} \label{Section5}
Regarding lanes modeling, there are two main approaches to model vehicles on multi-lane roads. The first approach, is the single lane abstraction model or simply the line abstraction model shown in Fig.\ref{Figurea(a)} in which all the traffic lanes are merged into a single lane with the aggregated traffic intensity (see Appendix.C in \cite{rakhshan2016packet}). The second approach is to consider that the traffic is restricted into individual lanes separated by a fixed inter-lane distance, as illustrated in Fig.\ref{Figureb(b)}.
We will derive the outage probability for the two road scenario, then generalize the results for multiple lanes.
\subsection{Two-lanes case scenario}
We address the case where vehicles can drive in two opposite directions, on the horizontal roads and the vertical roads, and further on extend the analysis to $Nb_{lanes}$ number of roads. We refer to the case when we have two roads in the horizontal, and two roads in the vertical as the two-way road case (two lanes on each road). In this case, the horizontal road on which vehicles drive from left to right (resp. right to left) is denoted $X_1$ (resp. $X_2$). The same modification holds for the vertical road on which, vehicles drive from bottom up (resp. top down) is denoted $Y_1$ (resp. $Y_2$). For $\alpha=2$, the expressions of the Laplace transform from the $X_1$ road and the $Y_1$ road at the receiving node $M$ denoted respectively $ \mathcal{L}_{I_{{X_{1}}_M}}(s)$ and $\mathcal{L}_{I_{{Y_{1}}_M}}(s)$, are given by (\ref{49a}) and (\ref{50a}). The expressions of the Laplace transform from the $X_2$ road and from the $Y_2$ road at $M$ are given respectively by
\begin{equation}\label{49a}
\mathcal{L}_{I_{{X_{2}}_M}}(s)=\exp\Bigg(-\emph{p}\lambda_{X_2}\dfrac{s\pi}{\sqrt{(m \sin(\theta_{M})-d_{Y_{Road}})^2+s}}\Bigg),
\end{equation}
and
\begin{equation}\label{50a}
\mathcal{L}_{I_{{Y_{2}}_M}}(s)=\exp\Bigg(-\emph{p}\lambda_{Y_2}\dfrac{s \pi}{\sqrt{(m \cos(\theta_{M})-d_{X_{Road}})^2+s}}\Bigg),
\end{equation}
where $\lambda_{X_2}$ and $\lambda_{Y_2}$ are the intensities of the interferer nodes on the $X_2$ road and $Y_2$ road respectively, and $d_{X_{Road}}$ and $d_{Y_{Road}}$ are distance between $X_1$ and $X_2$, and between $Y_1$ and $Y_2$ respectively.
\textit{proof}: See \ref{chapter3:App4}.\hfill $ \blacksquare $\\
In the case when there are two roads on the vertical and two roads on the horizontal, the interference are generated from four roads, the outage probability of $D_1$ and $D_2$ become respectively
\begin{align}
&\mathbb{P}(\textit{O}_{(1)})=\nonumber\\
&1- \mathcal{J}^{(2)}_{(D_1)}\big(\frac{G_{1}}{l_{SD_1}}\big)-
\mathcal{J}^{(2)}_{(R)}\big(\frac{G_{1}}{l_{SR}}\big)+
\mathcal{J}^{(2)}_{(D_1)}\big(\frac{G_{1}}{l_{SD_1}}\big)\mathcal{J}^{(2)}_{(R)}\big(\frac{G_{1}}{l_{SR}}\big)
\nonumber\\
&+\mathcal{J}^{(2)}_{(R)}\big(\frac{G_{1}}{l_{SR}}\big)-
\frac{l_{RD_1}\mathcal{J}^{(2)}_{(R)}\big(\frac{G_{1}}{l_{SR}}\big)\mathcal{J}^{(2)}_{(D_1)}\big(\frac{G_{1}}{l_{RD_1}}\big)-l_{SD_1}\mathcal{J}^{(2)}_{(R)}\big(\frac{G_{1}}{l_{SR}}\big)\mathcal{J}^{(2)}_{(D_1)}\big(\frac{G_{1}}{l_{SD_1}}\big)}{l_{RD_1}-l_{SD_1}}, \nonumber
\end{align}
and
\begin{align}
&\mathbb{P}(\textit{O}_{(2)})=\nonumber\\
&1- \mathcal{J}^{(2)}_{(D_2)}\big(\frac{G_{\mathrm{max}}}{l_{SD_2}}\big)-
\mathcal{J}^{(2)}_{(R)}\big(\frac{G_{\mathrm{max}}}{l_{SR}}\big)+
\mathcal{J}^{(2)}_{(D_2)}\big(\frac{G_{\mathrm{max}}}{l_{SD_2}}\big)\mathcal{J}^{(2)}_{(R)}\big(\frac{G_{\mathrm{max}}}{l_{SR}}\big)\nonumber\\
&+\mathcal{J}^{(2)}_{(R)}\big(\frac{G_{\mathrm{max}}}{l_{SR}}\big)-
\frac{l_{RD_2}\mathcal{J}^{(2)}_{(R)}\big(\frac{G_{\mathrm{max}}}{l_{SR}}\big)\mathcal{J}^{(2)}_{(D_2)}\big(\frac{G_{\mathrm{max}}}{l_{RD_2}}\big)-l_{SD_2}\mathcal{J}^{(2)}_{(R)}\big(\frac{G_{\mathrm{max}}}{l_{SR}}\big)\mathcal{J}^{(2)}_{(D_2)}\big(\frac{G_{\mathrm{max}}}{l_{SD_2}}\big)}{l_{RD_2}-l_{SD_2}}.\nonumber
\end{align}
where the function is given by
\begin{equation}
\mathcal{J}^{(2)}_{(M)}\Big(\frac{A}{B}\Big)=\mathcal{L}_{I_{X_{M}}}\Big(\frac{A}{B}\Big)\mathcal{L}_{I_{Y_{M}}}\Big(\frac{A}{B}\Big)\mathcal{L}_{I_{{X_2}_{M}}}\Big(\frac{A}{B}\Big)\mathcal{L}_{I_{{Y_2}_{M}}}\Big(\frac{A}{B}\Big)\exp\Big(-\frac{\sigma^2 A}{P B}\Big)^2.\nonumber
\end{equation}
\subsection{Multi-lanes case scenario}
To generalize the above expressions form $Nb_{lanes}$ roads, we calculate the Laplace transform for the interference for $i^{th} X$ road, and $i^{th} Y$ road when $\alpha=2$ is respectively given by:
\begin{equation}\label{51}
\mathcal{L}_{I_{{X_{i}}_{M}}}(s)=\exp\Bigg(-\emph{p}\lambda_{X_i}\dfrac{s\pi}{\sqrt{(m \sin(\theta_{{M}})-\sum_{i=1}^{Nb_{lanes}-1}(i-1)d_{Y_{Road}})^2+s}}\Bigg)
\end{equation}
\begin{equation}\label{52}
\mathcal{L}_{I_{{Y_{i}}_{M}}}(s)=\exp\Bigg(-\emph{p}\lambda_{Y_i}\dfrac{s\pi}{\sqrt{(m \cos(\theta_{{M}})-\sum_{i=1}^{Nb_{lanes}-1}(i-1)d_{X_{Road}})^2+s}}\Bigg)
\end{equation}
where $\lambda_{X_i}$ and $\lambda_{Y_i}$ are the intensities of the interferer nodes on the $X_i$ road and $Y_i$ road respectively. Hence the outage probability of $D_1$ and $D_2$ are respectively given by
\begin{align}
&\mathbb{P}(\textit{O}_{(1)})=\nonumber\\
&1- \mathcal{J}^{(Nb_{lanes})}_{(D_1)}\big(\frac{G_{1}}{l_{SD_1}}\big)-
\mathcal{J}^{(Nb_{lanes})}_{(R)}\big(\frac{G_{1}}{l_{SR}}\big)+
\mathcal{J}^{(Nb_{lanes})}_{(D_1)}\big(\frac{G_{1}}{l_{SD_1}}\big)\mathcal{J}^{(Nb_{lanes})}_{(R)}\big(\frac{G_{1}}{l_{SR}}\big)
\nonumber\\
&+\mathcal{J}^{(Nb_{lanes})}_{(R)}\big(\frac{G_{1}}{l_{SR}}\big)\nonumber\\
&-\frac{l_{RD_1}\mathcal{J}^{(Nb_{lanes})}_{(R)}\big(\frac{G_{1}}{l_{SR}}\big)\mathcal{J}^{(Nb_{lanes})}_{(D_1)}\big(\frac{G_{1}}{l_{RD_1}}\big)-l_{SD_1}\mathcal{J}^{(Nb_{lanes})}_{(R)}\big(\frac{G_{1}}{l_{SR}}\big)\mathcal{J}^{(Nb_{lanes})}_{(D_1)}\big(\frac{G_{1}}{l_{SD_1}}\big)}{l_{RD_1}-l_{SD_1}},
\end{align}
and
\begin{align}
&\mathbb{P}(\textit{O}_{(2)})=\nonumber\\
&1- \mathcal{J}^{(Nb_{lanes})}_{(D_2)}\big(\frac{G_{\mathrm{max}}}{l_{SD_2}}\big)-
\mathcal{J}^{(Nb_{lanes})}_{(R)}\big(\frac{G_{\mathrm{max}}}{l_{SR}}\big)+
\mathcal{J}^{(Nb_{lanes})}_{(D_2)}\big(\frac{G_{\mathrm{max}}}{l_{SD_2}}\big)\mathcal{J}^{(Nb_{lanes})}_{(R)}\big(\frac{G_{\mathrm{max}}}{l_{SR}}\big)\nonumber\\
&+\mathcal{J}^{(Nb_{lanes})}_{(R)}\big(\frac{G_{\mathrm{max}}}{l_{SR}}\big)\nonumber\\
&-
\frac{l_{RD_2}\mathcal{J}^{(Nb_{lanes})}_{(R)}\big(\frac{G_{\mathrm{max}}}{l_{SR}}\big)\mathcal{J}^{(Nb_{lanes})}_{(D_2)}\big(\frac{G_{\mathrm{max}}}{l_{RD_2}}\big)-l_{SD_2}\mathcal{J}^{(Nb_{lanes})}_{(R)}\big(\frac{G_{\mathrm{max}}}{l_{SR}}\big)\mathcal{J}^{(Nb_{lanes})}_{(D_2)}\big(\frac{G_{\mathrm{max}}}{l_{SD_2}}\big)}{l_{RD_2}-l_{SD_2}}.
\end{align}
where
\begin{equation}
\mathcal{J}^{(Nb_{lanes})}_{(M)}\Big(\frac{A}{B}\Big)=\exp\Big(-\frac{\sigma^2 A}{P B}\Big)^{Nb_{lanes}}\times\prod_{i=1}^{{Nb_{lanes}}}\mathcal{L}_{I_{{X_i}_{M}}}\Big(\frac{A}{B}\Big)\mathcal{L}_{I_{{Y_i}_{M}}}\Big(\frac{A}{B}\Big).
\end{equation}
\begin{figure}[]
\centering
\includegraphics[height=8cm,width=9cm]{Fig2.pdf}
\caption{Outage probability as a function of $a_1$ considering NOMA and OMA.}
\label{Fig2}
\end{figure}
\section{Simulations and Discussions}\label{Section6}
In this section, we evaluate the performance of MRC with NOMA at road intersections. Monte-Carlo simulation are carried out by generating samples (which correspond to the interfering vehicles) according to a PPP, and we average over $50,000$ iterations of Rayleigh fading channel coefficients. The Monte-Carlo simulations match the theoretical analysis, which confirm the accuracy of our results.
Unless stated otherwise, $\beta=0$, $S=[100,0]$, $R=[50,0]$, $D_1=[0,0]$ and $D_2=[0,-10]$. We set, without loss of generality, $\lambda_X = \lambda_Y = \lambda$.
\begin{figure}[]
\centering
\includegraphics[height=8cm,width=9cm]{Fig3.pdf}
\caption{Outage probability as a function of the distance from the intersection considering NOMA and OMA.}
\label{Fig3}
\end{figure}
Fig.\ref{Fig2} compares the outage probability as a function of $a_1$, considering a NOMA relay transmission \cite{J4}, NOMA MRC transmission (the proposed method), OMA relay transmission, and OMA MRC transmission \cite{J2}.
The figure shows that implementing MRC with NOMA offers a significant improvement on the performance compared to the relay transmission. This improvement is event greater for $D_2$. To quantify this improvement, we notice that, MRC with NOMA offers decrease of the outage probability of $34\%$ compared to the relay transmission with NOMA. Whereas the improvement of MRC with OMA is $2\%$ compared to the relay transmission with OMA. We can also notice that there is an improvement of $60\%$ in terms of outage probability when using MRC with NOMA compared to MRC with OMA.
Fig.\ref{Fig3} depicts the outage probability as a function of the distance between the nodes and the intersection.
We can see, from Fig.\ref{Fig3}, that the outage probability has a peak at the intersection. This can be explained by the fact that the interfering vehicles from both $X$ and $Y$ road contribute to the aggregate interference. Whereas only one road contribute to the aggregate interference when the
nodes are far from the intersection. We also see that implementing MRC with NOMA offers a better performance than MRC with OMA for $D_1$ and $D_2$.
\begin{figure}[]
\centering
\includegraphics[height=8cm,width=9cm]{Fig4.pdf}
\caption{Outage probability as a function of $\lambda$ considering NOMA and OMA.}
\label{Fig4}
\end{figure}
Fig.\ref{Fig4} plots the outage probability as a function of the vehicles density $\lambda$.
We notice that as the intensity of the interfering vehicles increases, the outage probability increases.
The reason is that as the number of vehicles increases, the aggregate of interference increases at the receiver
node, which decreases the SIR and increases the outage probability. Note that the value of $a_1$ has to be chosen carefully, since when $a_1=0.6$, MRC with NOMA offers a better performance than MRC with OMA for $D_1$ and $D_2$. which is not the case when $a_1=0.8$
\begin{figure}[]
\centering
\includegraphics[height=8cm,width=9cm]{Fig5.pdf}
\caption{Outage probability as a function of $\lambda$ considering NOMA using different transmission schemes. }
\label{Fig5}
\end{figure}
Fig.\ref{Fig5} shows the outage probability as a function of $\lambda$ considering NOMA using different transmission schemes. We can clearly see that the MRC using NOMA outperforms the classical relay transmission using NOMA. This holds true for both $D_1$ and $D_2$. This result is intuitive since in the relay transmission using NOMA, $D_1$ and $D_2$ decode the message transmitted by the relay. However, in the MRC transmission scheme using NOMA, $D_1$ and $D_2$ combine the signal from the source, and from the relay, which increases the power at the $D_1$ and $D_2$, and consequently increases the SINR.
\begin{figure}
\begin{subfigure}{12cm}
\centering\includegraphics[height=8cm,width=9cm]{Fig6a.pdf}
\caption{}
\label{Fig6a}
\end{subfigure}
\begin{subfigure}{12cm}
\centering\includegraphics[height=8cm,width=9cm]{Fig6b.pdf}
\caption{}
\label{Fig6b}
\end{subfigure}
\caption{Outage probability as a function of the relay position. (a) $\alpha=2$. (a) $\alpha=4$.}
\label{Fig6}
\end{figure}
Fig.\ref{Fig6} depicts the outage probability as a function of the relay position,
using a relay transmission and MRC transmission considering NOMA. We set, without loss of generality,
$\Vert S-D_1 \Vert = \Vert S-D_2 \Vert=100$m.
We can notice from Fig.\ref{Fig6a} that when $\alpha=2$, the optimal position for the relay using
a relay transmission is near the destinations, $D_1$ and $D_2$, whereas for MRC, the optimal relay
position is when the relay is close to the destination nodes.
When $\alpha=4$, we can see, from Fig.\ref{Fig6b}, that the best position for the relay is at mid-distance
between $S$ and the destination nodes when using the relay transmission. But, when using MRC, the best relay position is when the relay is near the destination nodes.
This is because, when the relay is near the destination, the channel coefficients between and $S$
and the destination, and between $R$ and the destination are decorrelated, which increases the diversity gain.
\begin{figure}[]
\centering
\includegraphics[height=8cm,width=9cm]{Fig7.pdf}
\caption{Outage probability as a function of $\lambda$ for several noise power values.}
\label{Fig8}
\end{figure}
We can see form the Fig.\ref{Fig8} that the noise power greatly impact the performance
only for low values of $\lambda$. However, as the value of $\lambda$ increases,
the performance when considering noise power and without noise power tends the
same values. This because for high value of $\lambda$, the power of noise
become negligible compared to the power of interference.
Fig.\ref{Fig7} shows the impact of $\beta$ on the performance in terms of outage probability.
We can see from Fig.\ref{Fig7a} that for low values of $\beta$ the outage probability considering
NOMA is lower than OMA when using MRC transmission. However, as the value of $\beta$ increases, the outage probability
of NOMA increases. We can also see that as $a_1$ decreases, the values of the effect of $\beta$ becomes less dominant. This because as we allocate more power to $D_2$, it increases the SINR at $D_2$
hence decreasing the outage probability.
We can also see from Fig.\ref{Fig7b} that the MRC outperforms the relay transmission for both NOMA
and OMA. However, we can see that the value of $\beta$ when OMA outperforms NOMA
is the same for MRC and the relay transmission.
\begin{figure}
\begin{subfigure}{12cm}
\centering\includegraphics[height=8cm,width=9cm]{Fig8a.pdf}
\caption{}
\label{Fig7a}
\end{subfigure}
\begin{subfigure}{12cm}
\centering\includegraphics[height=8cm,width=9cm]{Fig8b.pdf}
\caption{}
\label{Fig7b}
\end{subfigure}
\caption{Outage probability of $D_2$ as a function of $\beta$ considering NOMA and OMA. (a) NOMA and OMA considering MRC transmission. (b) NOMA and OMA considering MRC transmission and relay transmission.}
\label{Fig7}
\end{figure}
\begin{figure}
\begin{subfigure}{12cm}
\centering\includegraphics[height=8cm,width=9cm]{Fig9a.pdf}
\caption{}
\label{Fig9a}
\end{subfigure}
\begin{subfigure}{12cm}
\centering\includegraphics[height=8cm,width=9cm]{Fig9b.pdf}
\caption{}
\label{Fig9b}
\end{subfigure}
\caption{Outage probability as a function of data rate considering NOMA and OMA. (a) Outage probability as a function of $\mathcal{R}_1$. (b) Outage probability as a function of $\mathcal{R}_2$.}
\label{}
\end{figure}
\begin{figure}
\begin{subfigure}{12cm}
\centering\includegraphics[height=8cm,width=9cm]{13a.pdf}
\caption{}
\label{Figure13(a)}
\end{subfigure}
\begin{subfigure}{12cm}
\centering\includegraphics[height=8cm,width=9cm]{13b.pdf}
\caption{}
\label{Figure13(b)}
\end{subfigure}
\caption{ Multiple lanes modeling considering MRC NOMA. (a) outage probability as a function of $\lambda$ for $Nb_{lane}=\{2,4,6,8\}$. (b) outage probability as a function of $p$ for the single lane model and the multiple lane model for several values of $Nb_{lane}$. }
\label{Figure.13}
\end{figure}
Finally, we investigate the impact of the data rates $\mathcal{R}_1$ and $\mathcal{R}_2$
on the performance considering NOMA and OMA using MRC and the relay transmission. We can see from Fig.\ref{Fig9a} that as $\mathcal{R}_1$ increases, the outage probability of $D_1$
increases. This is intuitive since increasing the data rate increases the decoding threshold
which increases the outage probability. We can also see that NOMA offers better performance than OMA. However, as $\mathcal{R}_1$ increases, OMA outperforms NOMA for both MRC transmission and relay transmission.
Also, we can see from Fig.\ref{Fig9b} that from small values of $\mathcal{R}_2$, that is, $\mathcal{R}_2<0.5$ bit/s, OMA offers better performance than NOMA in terms of outage probability. This is because, unlike the vehicle $D_1$, the vehicle $D_2$ has to decode $D_1$ message first, and then decode its own message. Hence, $\mathbb{P}(D_2)$ depends solely on $\mathcal{R}_1$ for small values of $\mathcal{R}_2$.
We also notice that, for large values of $\mathcal{R}_2$ $(\mathcal{R}_2>2 \textrm{bit/s})$, NOMA has better performance in terms of outage probability than OMA. This because for large values of $\mathcal{R}_2$, the decoding threshold of OMA increases linearly since it is multiplied by a factor of 4. This proves that cooperative NOMA has a better outage performance for high data rates. Finally, we can see that MRC transmission outperforms cooperative transmission for both NOMA and OMA.
Fig.\ref{Figure13(a)} plots NOMA outage probability as a function of $\lambda$ for considering MRC for several values of $Nb_{lane}$. We can see an increases in the outage probability as the number of lanes increases. This results is intuitive because when the number of lanes increases the interfering vehicles density increases as well, hence increasing the outage probability.
Fig.\ref{Figure13(b)} shows NOMA outage probability as a function of $p$ using NOMA and considering the 1D-HPPP with a single lanes model, and the 1D-HPPP with multiple lanes. We can see from the Fig.\ref{Figure13(b)} that the single lane model matches perfectly the multiple lanes model.
\section{Conclusion}\label{Section7}
In this paper, we implemented MRC using NOMA in VCs at road intersections.
We derived closed form expressions of the outage probability for a setup involving two destinations. Then we extended the analysis for a scenario involving $K$ destinations.
We also analyzed the performance for several road lanes.
We noticed that implementing MRC using NOMA in vehicles improvements significantly the performance.
compared to the standard cooperative transmission using NOMA.
We also noticed that MRC using NOMA significantly outperforms MRC using OMA.
From our results we concluded that it is always beneficial to use MRC and NOMA even at the cost of implementation complexity.
Finally, we demonstrated that the outage probability has a peak when the vehicles are at the intersection, and that using MRC considering NOMA offers a great improvement in this context.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,071 |
Megaselia juli är en tvåvingeart som först beskrevs av Charles Thomas Brues 1908. Megaselia juli ingår i släktet Megaselia och familjen puckelflugor. Inga underarter finns listade i Catalogue of Life.
Källor
Externa länkar
Puckelflugor
juli | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,568 |
Пуерто-Лумбрерас () — муніципалітет в Іспанії, у складі автономної спільноти Мурсія. Населення — осіб (2010).
Муніципалітет розташований на відстані близько 360 км на південний схід від Мадрида, 75 км на південний захід від Мурсії.
На території муніципалітету розташовані такі населені пункти: (дані про населення за 2010 рік)
Кабесо-де-ла-Хара: 15 осіб
Ерміта: 149 осіб
Еспаррагаль: 2260 осіб
Естасьйон: 494 особи
Гоньяр: 70 осіб
Пуерто-Адентро: 134 особи
Лас-Касікас: 47 осіб
Пуерто-Лумбрерас: 10951 особа
Демографія
Галерея зображень
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Муніципальна рада
Примітки
Муніципалітети Мурсії | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,493 |
Does Florida Permit Lawsuits for "Tortious Interference" with an Expected Inheritance?
There is a special type of claim that occasionally comes up in probate law known as "tortious interference." Let's say Lewis makes a will leaving his estate to his daughter Sarah. But Lewis subsequently makes a new will disinheriting Sarah. Sarah believes her father's change-of-heart was due to the fact he was under the undue influence of a third party. After Lewis dies, Sarah could file a contest to the new will on undue influence grounds. But she might also attempt to sue the third party directly for tortious interference with her "expected inheritance rights."
Florida Follows "Probate First" Rule in Tortious Interference Cases
At least, that is the legal theory behind tortious interference claims. In practice, such claims are difficult to prove, and in many states they are not even allowed. For instance, the South Dakota Supreme Court recently said it would not "recognize a cause of action for tortious interference with inheritance or expectancy of inheritance." The South Dakota court acted in response to a question from a federal judge overseeing a tortious interference claim brought by a man who alleged his sister used undue influence to coerce their mother into amending her trust.
In reviewing how other state supreme courts have addressed this subject, the South Dakota decision noted Florida has "adopted the tort but required plaintiffs to first exhaust probate remedies." Indeed, the Florida Supreme Court was also asked to answer a question regarding the existence of the tortious interference claim by a federal court, in this case the Fifth Circuit Court of Appeals, which previously had federal appellate jurisdiction over Florida.
In a 1981 decision, DeWitt v. Duce, the Florida court noted that "[a]lthough a cause of action for wrongful interference with a testamentary expectancy has been recognized in this state," it could not be used to mount a "collateral attack" on the original probate proceedings. In plain English, if an heir feels they were cheated out of an inheritance due to an act of undue influence, they should contest the will in probate court. Only if the alleged interference somehow prevents the probate court from dealing with the situation is it appropriate for the heir to file a separate lawsuit for tortious interference.
For example, if the alleged undue influence or fraud was not discovered until after the administration of the probate estate was completed, the Florida court said an affected heir "is allowed to bring a later action for damages since relief in probate was impossible."
Speak with a Florida Estate Litigation Lawyer Today
Whether we are talking about tortious interference or undue influence, it is important to understand the person making the challenge has the burden of proof. As a general rule, people are free to change their wills as they wish and leave their property to whomever they choose. The mere fact you "expected" an inheritance that never came to pass is not, in and of itself, proof of undue influence or tortious interference on the part of a third party.
If you need legal advice or assistance in this area from an experienced Fort Myers estate litigation attorney, call the Kuhn Law Firm, P.A., at 239-333-4529 to schedule a free consultation.
scholar.google.com/scholar_case?case=8538889066514639124 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,048 |
Beacon Views, Aleshia Howell: Merging of home and workspaces should be taken cautiously
Aleshia Howell /
Beacon Magazine
When we think about the future of work, it's useful to think in terms of technologies that impact our bodies, minds and communities.
Robotics promises to replace manual labor and elevate workers from drudgery. Artificial intelligence takes on lower-level tasks of the mind, like speech and pattern recognition, and is now able to teach itself via machine learning without direct human input. And the growing gig economy is changing how and when we work, allowing us to build our jobs around our lives, not the other way around.
However, when we thought about the future of work at the dawn of the new decade, we were not thinking that our homes would play such a prominent role.
Home is where the heart is, and pandemic conditions have dictated that it is also where the office is. This was a difficult transition for those called upon to learn new technologies quickly, divide limited workspace between partners working at home together, and develop the necessary discipline for a healthy work-life balance on the fly. But for seasoned remote workers like Kevin Lawver, it's just another work day.
Lawver, a local tech community organizer and self-proclaimed nerd, has been working remotely for seven of the 12 years that he and his family have lived in Savannah. He currently serves as chief technology officer of the New York-based startup Planted, which connects companies with technical talent.
His strategy for effective remote work is threefold. He first builds a "commute" to move from home to work mode. His 20-minute morning routine includes making coffee ("I make really good coffee," he reassures) and creates structure to the beginning of his day.
Similarly, he is an advocate of taking regular breaks and strongly recommends that those new to remote work try the Pomodoro Technique, a time management system which breaks the day into 25-minute chunks separated by five-minute breaks.
Finally, he stresses the importance of communicating with colleagues. His work team uses the communication platform Slack to keep in touch throughout the day.
"Slack is the water cooler where no water cooler exists," Lawver said. "You're not getting up and having a 15-minute conversation with people you bump into in the hallway, so you have to be intentional about checking in and reaching out."
There are many others like Kevin Lawver in Savannah's tech community who work from home and "third spaces," like coffee shops and co-working offices; in fact, a tagline of TechSAV, the organization Lawver helps to lead, is: "Live here, work anywhere."
"People want to live in a walkable city, a bikeable city. People want arts and culture and not to have bad weather all the time," said Jennifer Bonnett, vice president of innovation and entrepreneurship at the Savannah Economic Development Authority. "All the things that made Savannah interesting to me are why others choose Savannah."
Bonnett, who relocated to Savannah from Atlanta in 2018, has orchestrated a new relocation incentive for remote technology workers who want to live in Savannah. The program offers reimbursement of up to $2,000 in moving expenses for qualified individuals in exchange for a two-year commitment to living and working in Chatham County. There is an additional $500 incentive for locals who refer remote tech workers.
Though the incentive was not a product of COVID-19, the pandemic allowed the approval process to be expedited.
"If you love Savannah and have always dreamed of living here, then why not choose to do so now?" Bonnett asks. She hopes the incentive will draw 15 to 20 remote tech workers to the area in 2020 and to double that number in 2021.
But will workers already in Savannah return to their offices post-COVID-19? Bonnett isn't so sure.
"It will be hard to go back," she said, citing that the pandemic has proven many jobs can be done remotely, and many may prefer the safety and convenience of home. "Every situation is different and personal, and people are going to react differently as Savannah reopens."
Lawver speaks more cautiously.
"When remote work is forced, unplanned and doesn't go as well as hoped, there's a great reason to think it doesn't work," he said.
Lawver also warns that, while large Silicon Valley tech companies are setting a precedent for remote work en masse, there will be ethical questions to answer about pay scales and management trust as well as consideration for unintended consequences.
"Remote work impacts the economies of downtowns," he said. "It will fundamentally disrupt a lot of stuff." | {
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Next on the list today is the ASUS 7800GT TOP Silent graphics cards which offers high-end performance without the noise!
Heat pipe graphics cards have never really seemed to kick in the market. The main reason is probably due to the fact that they have been more concentrated on mid-range cards that generally generate less heat and can run well under a heat pipe situation.
Today we will be looking at the highest-end heat pipe graphics card we have seen before, the GeForce 7800GT, but not only that we have got ourselves a pair to see just what happens we put them together in SLI mode.
ASUS have stepped outside the square and gone for what looks like a really efficient heat pipe setup. It's the type of setup that could (and probably will) give those people who want silent systems, the frame rates they thought they would never get. We also hope to find out if we can overclock at all or is stock speeds maxing out the powerful 7800GT core for all its worth with silent cooling.
Let's begin and take a close look! | {
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Keep It Steady är en singel av Songs: Ohia, utgiven 2002 på Secretly Canadian
Låtlista
A-sida
"Keep It Steady" - 4:43
B-sida
"United or Lost Alone" - 3:18
Referenser
Musiksinglar 2002 | {
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} | 9,538 |
{"url":"https:\/\/cseducators.stackexchange.com\/tags\/python\/hot","text":"# Tag Info\n\n44\n\nI'm not as familiar with Python as I am with other languages, but I'm sure your students have played Minecraft. If you haven't, I suggest taking a few minutes to find some introductory \"Lets Play\" videos on YouTube first. Let's talk Blocks. Minecraft has dozens of blocks. Dirt, some, water, colored wool... All blocks can be broken, picked up, placed, ...\n\n19\n\nA few years ago, the answer to this would have been \"stick with Python 2; the libraries aren't ready for Python 3 yet\". In many cases, that would have been a deal-breaker, because many of the older libraries were pretty useful, particularly for scientific computation. However, the story's a bit different now, in 2017. There isn't much of a reason to stick ...\n\n19\n\nTL;DR Those two aren't your only options. The main concern is cognitive load: learning to program is difficult enough without adding incidental complexity. 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It isn't that \"things\" change type. They don't. Objects and other values have a well determined type when they are created and that type never changes. Names never have a type to be ...\n\n9\n\nThe real question is this: do you want to teach your students what is actually going on, or teach them which magic buttons to press in an IDE? Of course for professional programming work nobody would NOT use the most functional IDE they could find for the task they were doing. But if you throw a complete beginner into the deep end of a tool like MS Visual ...\n\n9\n\nUnfortunately, GUI programming is sufficiently different from algorithmic programming that if you start with it students can get the wrong idea about what a program should look like. For example, when I write an algorithmic program, using good OO techniques, a method of five lines is starting to be too long. The granularity of a good OO program can be very ...\n\n9\n\nActually, the code is terrible, but I don't think its purpose is to illustrate a stack so much as to illustrate in a very rudimentary way how heap allocation works. (Worse than \"terrible\", it isn't \"pythonic\"). But you are wrong about the efficiency. Only the initialize function is O(n). Push an pop are O(1) as should be obvious. But no serious code ...\n\n7\n\nMy coding school gave one particular (weeks-long) project that I felt nailed the concept of inheritance, and why it could be useful: Simulating a circuit board with logic gates. The framework of the exercise can be adjusted, but here's a short example: A circuit board is composed of circuit inputs, logic gates and circuit outputs; each of these ...\n\n7\n\nIf you are tutoring her, it is wonderful that you are trying to motivate the material in a practical way, but don't beat yourself up too much if you aren't that successful at persuading her. Some people just get themselves into a sort of myopic headspace where the only things that they need to learn are the exact operations that they will perform later on in ...\n\n7\n\nYou certainly don't need a list longer than this one. If you do even half of this you will have learned enough to know pretty much what should be next. Having a complete list now gives you very little. What you really need is practice and feedback. For practice, find some significant problem and build a program to solve it. Use the best methodology you ...\n\n6\n\nI agree with Peter. It needs to be fun, and games really help. It doesn't take much to exercise basic programming concepts such as variables, loops, selection, input, and output. I would start with simple games like Tic-Tac-Toe and Number Guessing Game. Implementing a simple AI would be fun. I've found that in-class programming contests work well too. For ...\n\n5\n\nI like to ease people in. You have mentioned that this is for first year. 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Dirt Cheap Photo Guide to Grand Teton National Park
Jeff R. Clow
Copyright 2011 Jeff R. Clow
Smashwords Edition
Smashwords Edition, License Notes
This eBook is licensed for your personal enjoyment only. This eBook may not be re-sold
or given away to other people. If you would like to share this book with another person,
please purchase an additional copy for each recipient. If you are reading this book and did
not purchase it, or it was not purchased for your use only, then please return to
Smashwords.com and purchase your own copy. Thank you for respecting the hard work
of this author.
Chapter One - The Photographer's Quest
Oxbow Bend. The Snake River Overlook. Moulton Barn. Legendary landscape locations in Grand Teton National Park that have beckoned photographers for decades. You've heard of them and you've seen countless shots from each of these spots, but do you know what's the best time to visit them during the day? Do you know the alternative angles that produce images that aren't the standard postcard view of the place?
And what about off the beaten path locations such as Buffalo Fork Ridge or Hedrick Pond Overlook? Have you heard of them? Probably not. They are accessible but without a knowledge of the back roads that cross the main highways, they don't get many visitors. They are equally impressive when compared to their more famous brethren in the park, but they don't show up on any maps. Do you know how to get to them - and is it safe to do so in a rental car?
That's where this inexpensive photo guide comes in - a written recap of the area in and around Jackson Hole, Wyoming that is focused on one thing. Photography. Not on where to stay or where to eat, but where to stand and when to do so.
I've been visiting the area for ten years and have been leading photo tours for the last three years to this landscape and nature photographer's paradise. And the first time I visited Jackson Hole and Grand Teton National Park, I missed out on dozens of different photo spots because I didn't know who to ask and I didn't know my way around the area.
I remember asking myself repeatedly - is this dirt road on my left private or public? Is the fence in front of the barn a barrier to me or to the bison and can I go around it without getting in trouble? And then I would visit some of the well known galleries in the town of Jackson, Wyoming and see a shot that I'd love to take, but I had no idea how to get there. Or I would have two days to take photos and I wasn't sure where to devote that first hour just before sunrise. Should I go to the barns or go to Oxbow? Thus, my first couple of visits to the area were equal parts reward and frustration.
But I was lucky. My step-daughter and her husband had just moved to the area and so my wife and I had a reason to visit. We loved the place and we could visit without needing to shell out hundreds for hotel rooms. And so I began my quest - my quest to find the best places in Jackson Hole for a photographer and my quest to understand when the light was going to be best at each of those locations.
And slowly, but surely, I began running into local photographers at some of the sites. I would strike up a conversation and ask what other spots they would recommend. Some were very willing to tell me and others acted like that I was asking them for the secret formula for a well known soft drink. They clammed up and basically gave me the cold shoulder. But after years of asking and exploring every back road I could find in the park, I began to get a real sense of the place. A sense of the "vibe" that you sometimes hear the locals talk about.
Fast forward to today. Many of those photos this self taught landscape and wildlife photographer took ended up catching the eye of editors at Getty Stock Images. They are now being used by companies all over the world in advertising and promotion campaigns. My own portfolio on Flickr.com has been visited over four million times and I've won lots of accolades. But the truth is that so much of good photography is simply based on one thing - and one thing only.
If you want to get better and better at photography **\- you've got to stand in front of better things**.
Then things like light and composition and depth of field come into play. But - other than repeated visits to the area - how does a photographer on a quest figure out where and how to spend their precious time when they visit Grand Teton National Park? Do they grab a rental car and head north on Highway 89 out of Jackson? Well, that will work. Some of the most iconic locations are literally on the main road. But the problem becomes one of prioritizing and knowledge - or lack thereof. Oxbow Bend is easy to find, but if you wait until noon to hit this magical spot, the famous mirror reflection of Mount Moran in the bend of the river is likely going to be gone.
Why? Simple weather patterns and the fact that on most days the winds pick up after 9:00 a.m. just enough to ripple the water and disrupt the reflection. Will you still get a decent shot of the place? Sure. But the iconic shot that you've seen in all the guidebooks may be out of your grasp.
Thus, you need a guide. You can employ one of the many local photographers who offer guided tours of the area for visiting photographers. I personally know a couple of people who do a terrific job and will take you exactly where you need to go and when. The downside? They are expensive - usually in the range of $500 to $900 per day per person. But you'll get the shots that you've always wanted.
Or you could join a photo tour or photo workshop focused on the area. During the spring, summer and fall months there are workshops almost every weekend. Many led by well known photographers whose work has appeared in many national publications. My small company - Dirt Cheap Photo Tours - offers inexpensive guided tours for small groups several times a year. But once again, there are some limitations. You have to be in the area when the workshop or tour is held. And you have to travel with six to ten other people - a fun experience but maybe not what you want to do if you're visiting the area with your family and the photography part of your trip is only part of the overall experience.
Or you could use this book as your personal guide to the area that will allow you to travel at your own pace and visit the sites that resonate with you personally.
As my knowledge of the Jackson Hole area became well known, I started getting emails from strangers almost every week. They would invariably begin like this:
_"Hi, Jeff. I saw your photos of Grand Teton National Park online and I'm visiting the area next month and was hoping you could give me some pointers. I'm going to be traveling with my kids and I only will_ be in _Jackson Hole for a couple of days because we're spending most of our time in Yellowstone. Can you tell me where I should devote the morning hours to during my visit?"_
Over the course of the last several years, I've received countless requests similar to this and I've tried my best to be responsive. But the harsh fact is that there is too much too see - and photograph - in the park to be able to give a definitive answer. If you're a landscape photographer, then there are a half dozen places that are "must visit" in my opinion.
But if you also want to have the best chance to encounter - and photograph - wildlife, then there are other places that are better to visit if time is of the essence. You won't get a killer landscape photo at these locales, but you'll be much more likely to run into a deer or a moose or sometimes a bear.
So, let's say you are one of those who would trade a sunrise at a landscape location for a chance to photograph a moose. Then the place you need to be standing is on the shoreline of one of the three ponds that are adjacent to the dirt road on Moose-Wilson Road.
Moose-Wilson Road
Contrary to what you might be thinking, Moose-Wilson Road was not named thusly because you see moose often on this stretch of roadway. It was named this because it starts in the town of Moose, Wyoming and runs southwest until it ends in the town of Wilson, Wyoming.
Throughout this book you'll see many places that will become your own personal target list as you prepare for your visit to the area. Each time you come across one of these that you want to add to your "must see" sites - you'll want to make a note of it and develop your own shooting list for each day you are at the park. You can then make adjustments if the weather changes on you or if you encounter unexpected wildlife where you least expect it.
Now, back to our quest for a wildlife shot during a brief stay in the park. This - and all subsequent directions - assumes you are staying overnight in the town of Jackson, Wyoming. If you are staying inside the park or nearby elsewhere, you'll still be able to find each spot because I'll use well known roads as directional start points.
First, determine when sunrise is going to occur for the day you plan to head out. In the summer months, this may be an hour or two earlier than in the spring or fall. I would recommend using one of the many well known apps that show sunrise and sunset times for any location if you have a smart phone. If you don't , check the local forecast on the television the night before.
Leaving your hotel in downtown Jackson at least 45 minutes prior to sunrise - head north on Highway 89 out of town. At the Moose junction, turn left (west) and drive across the bridge that spans the Snake River. As you cross the bridge, look to your right and left near the banks of the river - often moose congregate here to feed on the willows that line the banks.
And of course, during your drive to this location, watch for groupings of vehicles pulled off the road. Many times this will be your first indication that there are wildlife close by - and you should slow down or stop and check it out.
After crossing the bridge and immediately past the visitors center, you'll see a road veering left that is marked "Moose-Wilson Road". Take that road - which is paved at this point - and begin driving south/southwest. On your right and left immediately after you turn on to the road are stands of aspen trees. Watch closely because often you'll catch a deer or an elk in the early morning hours walking through the trees here.
As you continue south/southwest, you'll come to a paved overlook on your left side. Park here and look down and you'll see a meadow that is often frequented in the early morning by elk and moose and the occasional bear. Your best course of action is to stand on the ridge for several moments quietly and listen to the sounds of nature - and often you'll hear a rustle or see a movement that indicates wildlife.
After you've worked this area, continue driving south/southwest and the road will narrow. On your left will be a series of shallow ponds that are home to moose, bear, owls and deer. Drive slowly and you'll see a couple of small turnouts where a few cars can park. If there is wildlife present, there will likely be others who've already spotted them and have pulled off the road. Sometimes, if there's a major event - like a bear spotting - there may even be a park ranger or two present. Pull your vehicle off the road and grab your tripod and camera and walk towards the activity.
Tripod? Yes, if you have one you'll want to carry it. I know - it's a heavy burden and it slows you down, but if you want crisp nature and wildlife shots, you need to use it. Vibration reduction and image stabilization lenses do work, but the majority of great wildlife shots you'll ever take will be with a tripod or monopod.
If there is not activity, don't despair. You may be in luck and be the first in the area. Walk slowly toward the shore of one of the small ponds and listen intently. If you're alone, put all your senses on full alert. There have been several times in this area when I almost walked on top of a moose before I saw it. And remember that noise can startle a wild animal and make them bolt, so tread lightly.
The reason that there's a good chance you'll find wildlife here is because there is vegetation on the bottom of the pond that moose find irresistible to eat. There's also an abundance of willow trees along the road that are also sought out by moose.
If you detect a moose, keep your distance. Park rangers have for years told me that a moose is much more dangerous than almost any other wild animal. They are prickly and may charge you if they think you are invading their space - so give them plenty of breathing room. That said, don't be too scared as they are fairly docile if you stay far enough way.
For this particular area, I'd recommend using a zoom lens that is at least a 200mm or longer. Ideally a 300mm - perhaps with a teleconverter - will get you a good shot of the animal at a comfortable distance. You'll need to set your ISO up high - probably in the 800 to 1200 range - to keep enough light on your subject during the early morning moments before and just after dawn. If you're new to photography, just set your camera on the program mode and it will figure out ISO and shutter speed for you.
The most important thing to do is to use a tripod or a monopod and aim your lens so that your central focus point is the moose's eye - or head. And remember, digital film is cheap - so take lots of shots at different exposures so you'll be sure to have a few keepers after all is said and done.
Once you have gotten your shots at the ponds, continue your drive south/southwest. Approximately a half mile south after the ponds you'll come across a series of willow trees alongside the left hand side of the road with water at their base. Not really ponds, per se - but water that keeps the willow growing well throughout the summer droughts. To the east you'll see a meadow beyond the willow trees. And as you clear the area, on your left will be a dirt road. If you haven't seen anything up to this point, you may want to turn on to the dirt road and park on the area to the right and get out of your vehicle and do a search of the meadow to your north.
Here's a shot I took of the meadow with the willow trees behind the bull moose. If you look closely, you can see that he's standing guard over his doe who is laying in the grass on the lower right hand side of the shot. Only her ears are sticking out of the grass.
One of the things about this open space is that it gives you a chance to see the wildlife in the context of the larger landscape around them. So don't forget to shoot some frames with a wider angle lens that will help others "see" what you saw when you had your encounter.
After you leave this area, you can continue south as the paved road turns into dirt. You'll be surrounded on both sides by dense forest, but in the early morning you're apt to come across elk or deer in the forest. They are tough to photograph because there is so much dense brush, but if you're lucky, you may spot a clearing along the way that will give you an open shot. This land was once the private reserve of the Rockefeller family, who deeded it to the Park Service several years ago.
Or you can turn around at this juncture - since you have the room to do so - and head back north in search of another stellar site that is less than thirty minutes away and will have perfect light for its location as you arrive there. The location? The magical Schwabacher Landing.
Chapter Two - Schwabacher Landing - Upper, Lower and the Beaver Pond
Schwabacher Landing - if you're a fan of landscape images, you've surely seen hundreds of images taken at this iconic location. It probably - along with Oxbow Bend and Moulton Barn - is the most photographed site in Grand Teton National Park. Yet, many newcomers to the area never venture down the dirt road that leads you to the multitude of photo opportunities that are present along the branch of the Snake River that is accessible via the landing.
At this point, we should set our bearings. The vast majority of great locations in Grand Teton National Park are off of only three main roads - Hwy 89 that runs north from Jackson to the Moran junction turn, and then veers left (west) and ultimately leads to the southern entrance to Yellowstone National Park.
The second road that you'll need to become familiar with is Teton Park Road that runs primarily north and south at the base of the Tetons and is mostly parallel with Highway 89, although much further west. You can access it by turning left at the Moose Junction turn and driving past the visitors center headed west/northwest.
The third road is Highway 287 that branches off headed east from the Moran Junction. This will lead you out of the park, but offers some spectacular panoramic views that we'll review later in this book.
Headed north on Hwy 89 out of downtown Jackson, you'll pass by the airport and then Moose Junction - both on your left. You'll continue north past the Blacktail Ponds Overlook and the Glacier View turnout. You'll probably want to stop a half dozen times during your journey because of the magnificent scenes all around you, but stay focused and continue northward.
Approximately a quarter mile past Glacier View, watch for a dirt road on your left with a small sign that says "Schwabacher Landing". It is such a small sign that it is quite easy to miss.
The road is rutted, but you can safely take a rental car down it if you drive slowly. There are actually three distinct stops that you'll want to make, but initially drive the mile or so until you come to the gravel parking lot at the end of the road. You'll know you have made it because you'll see toilet facilities erected by the Park Service at the lot. And a word to the wise - never pass a toilet without taking advantage of it, since they are few and far between within the park.
As you pull into the lot, before you will be a spectacular scene - a slow moving branch of the Snake River and the Teton mountain range hovering over the river. Since you're getting here early - let's call it an hour or so after sunrise - the winds should be quiet and you should see a mirror reflection of the Tetons in the water.
The scene in front of you is breathtaking, and you'll want to take out your tripod and wide angle lens to capture the panoramic view. I would recommend a lens in the 16mm to 28mm range. Running alongside the eastern side of the river is a dirt path that you'll want to use later on to get to the beaver pond to the north.
But first, you'll want to work the scene before you - and if you have mobility issues, you can literally get out of the car and shoot from the parking lot. But, if you're able, then you'll want to head down to the bank of the river. You'll need to experiment with lots of different angles - shooting from the top extension of your tripod, shooting from a level at your knees and then ultimately as low as you can get against the ground. The reflection changes dramatically as you take these different shots and the lower you get, the more of the foreground interest you can get. Foreground interest is the stuff that will give your photo depth - the flowers on the bank, the rocks in the river. Since you are using a tripod, you'll be able to set your f/stop to f/18 or even f/22, which will give you crisp clarity throughout the image. If you have a remote shutter release, you'll want to use it to avoid the miniscule shutter shake that occurs when your finger touches the shutter. If you don't have a remote, use the timer function to shoot the shot. The clarity of these methods will give you the sharpness that most photographers treasure in their images.
Another thing to consider is the inclusion of people in your shot to give the viewer some sense of the scale of the place. I've been at Schwabacher often when there is an artist sketching the scene - and I often ask if they mind if I take their photo while they work with the Tetons in front of them.
Another option is to take a bride with you to enhance the scene.
If you don't have a willing helper to assist you, then you might want to consider setting up your tripod behind you and taking a self portrait of yourself sitting on the banks of the river and gazing up at those magnificent Tetons in the distance.
Many landscape photographers will disagree with the placement of a person in a scene, but others will tell you that without scale, it is very hard for a person viewing the photo to gain an understanding of just how massive the mountains are in the Tetons as they jut up from the valley floor.
The shot above was taken from the parking lot and you can see the dirt path that the two photographers are standing upon as they take in the magnificent scene before them. I'd urge you to not spend all of your time with a lens in front of your face, but to stop for a few minutes and breathe in the crisp air and let your eyes feast on nature's wonder.
Once you've worked this scene from parking lot to the bank of the river, then you'll want to head north along the dirt path. Many photographers stop at the lot and don't realize that an even more pristine location is just a ten minute walk up a fairly level path to the beaver ponds.
Okay, so you may be saying at this point that you've got a tight schedule and you want to hit one of the dozen other sites that beckon you. You're not sure you want to invest another 45 minutes at this site as the light begins to climb in the sky. Don't be tempted to leave. Walk the pathway and you'll never regret it.
The path leads you to a stream that has branched off the main river and has been dammed up by beavers. Unfortunately, you'll see that there has been some infestation alongside the west side of the stream that has killed off a lot of the trees. But there will be a couple of spots as you venture north that will give you a chance to capture a different view of the Teton range with trees reflected in the still water alongside the path.
Once again you'll want to experiment with your shooting height as you traverse the pathway. The shot above was taken from a knee high height, which allowed me to get both the tree tops and their reflections in the water. Clouds are a bonus, but you'll find that they can be both a gift and a hindrance as they are capable of obscuring the peaks as the day progresses. The same can be said of residual smoke from area forest fires - which has a tendency to hang at the base of the mountain range until winds blow through the area.
But that's the luck of the draw, and something you can't control during your visit. What you can control is your ability to make the best of the situation in front of you.
As you move on down the path, you'll find the stream widening as the beaver ponds appear. If you're alone as you walk down the path, move gently and you may be fortunate to see wildlife at the water's edge. But most of the time you'll be there alongside others - especially on the weekends during the summer and fall months when Schwabacher sees its largest crowds.
Your ultimate destination is easy to find - the park service has erected a log bench at the conclusion of the dirt path that points west. Sit down before you set up your tripod and take in one of the most magical nature scenes in North America. What you are looking at has been the subject of countless cover photos since the turn of last century.
What makes this scene so special is the trees that frame the Teton range and the fact that you can get such a perfect mirror reflection in the still waters of the beaver pond. There are often ducks on the pond that will swim in your scene, but be aware that their ripples may break up the reflection. You will also want to move alongside the eastern bank of the pond - there are trees on your left that you can get under if you crouch low enough and that perspective will create a completely different shot for you.
Because of the number of things in front of you and the changing nature of the sky and the light, I'd urge you to bracket a series of shots as you photograph this scene. If you haven't bracketed, it means simply taking a shot that the camera tells you is exposed correctly and then taking another that is under exposed and another over exposed. I normally do this via the aperture setting that most cameras possess. And I normally use a tripod and start out with an aperture setting of f/8 or f/10 and then experiment with settings all the way up to f/22 - all at different exposures after getting the standard exposure that the camera and lens have calculated. When you get home and start processing your shots, you'll often find that a shot other than the correct exposure may actually be a better image.
Once you've gotten your fill of this great location, you could move northward along a much smaller trail that runs alongside the water. However, just north of the bench, the view from the east side becomes one of trees alone as you lose the view of the Teton mountain range. If you're looking for something different, you can experiment with shots in this vicinity.
But if you're pressed for time - and wanting to save the good light for best use - return to your car in the parking lot and drive back on the dirt road headed south for a quarter mile or so until you see a second parking lot on your right (west) side. This is the third photo spot that I referenced at the start of this chapter. Park here and take a moment to check out this alternative scene in front of you. This is known as lower Schwabacher, which is counterintuitive since it is at a level higher than the parking lot at the landing.
Although you're not as likely to spend a ton of time here, there are some great spots alongside the meandering Snake River below that will give you a view of water and the Tetons as the river bends west towards the mountain range. You can shoot the scene from the parking area, but the better shots are taken when you walk down to the side of the river bank and follow it west.
Chapter Three - Hedrick Pond Overlook and the Triangle X Ranch
Our journey continues north as we head up Highway 89 towards a spot that is a photographer's nirvana, but one that doesn't show up on any maps of the area. The Hedrick Pond overlook. This is one of those places that took on a special meaning for me personally because I had only seen one photo from this location during repeated visits to all the photo galleries in downtown Jackson. But what a photo it was - a view from a high point that overlooks the valley and gives one a panoramic view for miles with a beautiful pond as a focal point in the center of the valley.
First things first, however. To get to this hidden overlook will require a drive on a deeply rutted road that is not conducive to a normal rental car. If you have a vehicle that has high clearance - like a large SUV or a jeep, then you're in good shape and you can literally drive to the point where you setup your tripod. Otherwise, you'll need to hike in when the road gets too difficult for a regular car to handle. I made that mistake the first time I visited this spot and now I always get a rental vehicle with high clearance.
Although this particular photo location could also be a sunrise spot, since you'll be facing west towards the Tetons when you shoot, I would recommend it for mid-morning. Two reasons - safety and the ability to have enough light on all of the panorama in front of you. Because of the narrow, rutted road I would not recommend trying to find your way in near darkness. And although a sunrise shot would show good light on the mountains, part of the valley below you would still be in the shadows. So, if you can get there around 9:30 a.m. or 10:00 a.m., I think you'll get a marvelous shot.
If you're headed north from Schwabacher Landing or up Highway 89 from downtown Jackson, you'll simply want to drive until you come to the Teton Point turnout on the left hand side of the road. That will tell you that you're getting close. Although it also has a beautiful view of the Snake River and part of the valley, I'd recommended you continue northward and watch for a gravel and dirt road on your right that is just before the well known Snake River Overlook. This gravel and dirt road leads to the Lost Creek Ranch and you may see a sign to that effect - although your best bet is to look ahead and see the Snake River Overlook on your left in about a quarter of a mile. You'll know you've found the right road.
Head east on this road and you'll travel approximately a mile and then on your left you'll see a much less traveled dirt road that splits off the main dirt road. You'll want to turn here and head north. And if you're in a rental that won't get you too far, then you can park on the plateau to your left almost immediately and still get a very nice view of the Tetons with a multitude of trees in the foreground. This is also a great place to stop if you're traveling during the fall season when the trees are turning colorful in front of the Tetons.
As you can see above, the view is pretty darn special here and if you're inclined, you can work this scene for a good half hour as you check out the various angles available looking west and northwest.
But I would recommend stopping only briefly and then continuing your drive east along the ridge. As you drive, you may want to stop at a point a half mile or so from your last stop and shoot the scene to your rear that shows the dirt road atop the ridge and the Tetons in the distance.
Drive slowly and watch for another deeply rutted road that will veer left (west) in approximately 1.1 miles. You should turn here and _slowly_ follow this road north and then west as it leads you to an area that has picnic benches and has a flat spot you can park and also turn around.
You've made it and as soon as you get out you'll see that it was worth the considerable effort. You're standing at a spot that probably less than 1 in 10,000 visitors to Grand Teton National Park ever visit, and you've got a wide ranging view of the Tetons and the area that is called Jackson Hole. In fact, I would be surprised if the first mountain men didn't stand right here and marvel at the sight that lay before them.
As an interesting aside, the pond below you - Hedrick Pond - was the site of a movie called Spencer's Mountain back in the last century. It is a credit to the film crew that they returned the pond to its pristine state when they finished the film.
When you take out your tripod, consider using a wide angle lens to capture the scene - and even then I would suggest taking a series of shots from left to right that you can later stitch into a panorama. And surprisingly, I have gotten decent cell phone coverage standing on this ridge in the past, so you may be able to take a shot or two with your cell phone camera and send it out to your friends and family to create "instant" envy.
Remember to work the scene from left to right and you'll have to make some decisions about whether to include the tree that stands at the edge of the ridge as a framing device in your composition. Also, if you're able to climb down a bit, you'll find that the angles below the ridge offer some different viewpoints than can be had while standing on top.
I might also suggest you add a person in one or more of your shots, or include yourself in the scene by using the self timer on your camera. As I have mentioned before, the inclusion of a person allows that great thing known as scale - or perspective - to enter into the image. And don't hesitate to take a shot of a person's back as they gaze upon the valley. Future viewers will imagine themselves as that person standing on the ridge.
In terms of camera settings, it is always good to use the standard landscape aperture settings that range from f/16 to f/22 if you are shooting with a tripod and you've got good light. Experiment a bit and even try a radical different aperture setting of f/2.8 or f/4.0 if you're shooting a portrait on the ridge You'll have good focus on the person's face or body and then the limited depth of field behind them will blur the scene as the distance spreads out behind them.
After you've shot to your heart's content and you begin to take your gear back to your vehicle, I hope you'll stop and enjoy this world class view one more time without a lens in front of your face. If you are a lover of nature and landscapes, it is not going to get much better than this in North America.
Once you are back in the driver's seat, retrace the dirt road headed east until you come to the main dirt road and then turn back right which will have you headed south along the ridge for a mile or so. You will then run into the gravel road that first led you to the ridge and you'll want to turn right headed west and drive back down to Highway 89.
My next recommended stop is the wooden rail fence that sits directly across from the road that leads to the Triangle X Ranch - a distance of approximately four miles headed north on Highway 89. You'll probably be tempted to turn out for the Snake River Overlook that is just on your left as you rejoin the main highway, but I'd suggest you bypass this spot today and leave it for a morning dawn shot later during your visit.
As you drive north, you'll notice that you are making a fairly sharp directional turn to the east and then returning in a northward direction. If you look up the hill to your right as the S curve straightens out, you'll see the ridge you were standing on that overlooks Hedrick Pond. Hedrick Pond is not visible from Highway 89 and thus is missed as a photo opportunity by so many photographers as they search out photo spots along the highway.
On your left the valley floor will begin to spread out before you and you'll see lots of brush and the beginnings of a wooden rail fence. Often there will be bison grazing near this spot - and if they are, you'll want to stop and take a few shots. With the right light and correct angle, you may even be able to get them standing in front of the Teton range. But if they aren't present, I'd suggest you continue to drive until you see a large wooden gate on your left and a road to Triangle X on your right. You can pull off on the east side of the road as there is a parking area there that is open to the public.
Full disclosure - like so many locales in this book, this site is also another great sunrise location. However, I'm going on the assumption that you only have a few sunrises in your trip and this wouldn't be my first choice.
That said, there is one circumstance that would make it my first choice - and that's when the wranglers at the Triangle X Ranch move their horse herd from the pasture on the west to the corrals on the east side of Highway 89 at first light. Unfortunately, they don't do this every day or even every week - it all depends on many factors - including weather and Park Service permission. In my ten years of visiting the area frequently, I've only had three separate times when it all came together.
But, when it does, you can get an iconic shot of a herd of horses in front of the Tetons as the first light illuminates the mountain range.
The best way to ascertain if there is an opportunity for this shot is to call the ranch directly at 307.733.2183 a day before and ask them if they will be moving the horses from the western pasture to the corrals in the morning. If you're lucky, they will be and you can join the handful of other photographers who know about this great photo opportunity at first light the next day.
But let's assume that your luck wasn't perfect this time around. You haven't missed out completely because there are rolling hills covered with an old time wooden fence just across the road and they are beckoning you and your camera and tripod.
Depending on the time of the year of your visit, you may encounter the various shades of green and brown that dot the landscape via the sagebrush that grows wild in the area. I'd suggest you take some panoramic views and then walk up and down the fence rail as you line up your shots. I've also found that getting very low to the ground and shooting up towards the fence and the mountain range can produce good results.
This is also a good spot to practice the use of bracketing as the many colors of the landscape are so hard to capture with just one frame. Taking several frames that are under and over exposed along with the correct exposure will allow you to blend these when you return home. Of course, many of you know that there are software programs that will do this type of high dynamic range (HDR) blending for you and will help you capture the nuances of color and light that were present when you took your shots.
As you wrap up your shooting at this location, be sure and shoot east as well. It won't be as magnificent as the Teton range, but there are some good shots looking northeast across the road.
We'll continue our photo journey together in the next chapter as we head back south and then east for two places that the afternoon sun will help instead of hindering our photo work.
Chapter Four - Afternoon trek to the Red Rocks and Teton National Forest
If you've followed my recommendations so far, then you've used up most - if not all - of your morning light on some really good locations. Good for you. If you're like so many others before you, you've probably been marveling at all the beauty around you and the magical scenes in front of your camera. And if it is your first visit to Jackson Hole, then I would guess you've already filled up a couple of memory cards with hundreds of images.
Realistically, you may not be following the road - and the text - from chapter to chapter. You may have spent more time at one of the previous locations, or run into wildlife along the way. You may have turned off at one of the many photo ops that you passed on the way to the locations we've covered so far.
No problem. This guide is supposed to help you with your journey and it is written with a couple of things in mind. Some people want to know about a route that will maximize their limited time in Grand Teton National Park. And some people want to know about the hidden and not so hidden places but want to travel at their own pace and make their own route as they go.
Whether you fall in one of those camps or some variation of the two, this chapter is designed to take you to some photo spots that benefit from mid-day and afternoon light. So if you're carrying this text along with you as your travel through the park, you'll want to wait till afternoon to visit the locations in this chapter.
Assuming you're on Highway 89 at the Triangle X at this point, I'd suggest you turn back south and head back past the Snake River Overlook and down to the Moose Junction. You can then turn right (west) and grab a bite to eat at Dornan's - which is just on your right down a short road after you turn off Highway 89. During the summer they have an outdoor grill that makes a mean buffalo burger and they also have a deli that serves sandwiches, and also a pizza restaurant that has a second floor dining area that gives you a stellar view of the Tetons as you eat your lunch.
There are also restrooms near the big teepee that you'll see on the western side of the parking lot.
Once you've refueled your body - and your vehicle at the gas pumps if necessary - it's time to head back to Highway 89 and continue south past the Airport and watch for the Gros Ventre Road intersection that runs east and west. You'll want to head east towards the town of Kelly when you get to this major artery that will take you out of Grand Teton National Park and in to the Teton National Forest.
As you drive east, you'll see the Gros Ventre River running parallel to the road. There are a couple of pullouts on your right and it is worth stopping briefly to see if there are any moose feeding on the willow trees that line the river. And of course if you see a lot of cars pulled over, then that will be a strong indicator that wildlife are present.
Continuing east you'll drive through the small town of Kelly, Wyoming and then the road will turn left (north) and you'll see pasture land on both sides of the road. Often there are bison grazing here and so you may want to stop if the herd is in this vicinity and get some shots.
Of course you'll want to use a long zoom (300mm) or prime lens (400mm) if you have one to get a good close-up of a bison. Bison are docile most of the time if you keep your distance, but if they sense you are too close, they'll definitely charge. So stay safe and stay a good distance away from them. In fact, a Park Ranger told me that there have been more deaths in National Parks from bison attacks than from bear attacks and whenever I find myself moving toward the bison to get a better shot, I recall his story and it reminds me to keep lots of space between me and the 1400 lb animal I'm photographing.
If you don't run in to bison here, don't worry - there is a large herd that stays within the confines of the park and you'll be bound to run into them at some point as they graze in the open fields that traverse the Jackson Hole Valley.
As you head north out of Kelly, watch for a road on your right and turn east when you come to it - it is called Gros Ventre Road not surprisingly. Immediately after you make your turn, you'll notice a parking area and that's there because there is a prairie dog village on the south side of the road. Although prairie dogs (or ground squirrels) don't have the allure of a bison, if you're traveling with kids then it would make sense to stop for a few minutes and stay quiet and you'll see the little fellows emerge from their holes.
Continuing east on Gros Ventre Road you'll soon encounter a couple of broken down wooden cabins on the left (north ) side of the road. Although they are in poor shape, they do have some interesting background and they might be worth a stop if you're so inclined. This is the old set where they shot all of the ranch scenes for the famous western movie called Shane. And yes, it's okay to explore the cabins, as this is still public land. If you do decide to get out and look around a bit, you'll find that you can get inside the remains of the cabins and shoot through the open windows looking west. There you'll just be able to make out the peaks of the Teton mountain range. If you climb the ridge to the north of the cabins, you'll have a better shot of the mountains.
When you commence your drive east you'll notice that you'll be leaving Grand Teton National Park and entering the Bridger-Teton National Forest. There is a parking lot near the sign that announces your arrival to this new area and if you stop, you can get a couple of good shots of the rolling hills to your north that are different than the landscape you've encountered so far.
I am personally always fascinated by a lone tree on a ridge and I think it makes for a good landscape photo opportunity. And since you are shooting northward the bright sun above is actually much more helpful than if you were trying to shoot west.
As you continue your drive east you will notice that the road goes from being paved to being simple gravel. If you are in a rental car, don't worry because the road is well maintained and you're not going to run in to any major ruts as you drive. But you'll want to adjust your speed, especially when there are cars coming towards you from the east that will kick up lots of dust that will obscure your view for a few seconds.
About five miles in and halfway to your ultimate destination, you'll see a large lake looming on your right (south) side - this is Lower Slide Lake. The road you are on will take you high above this lake and there are some stunning views if you'd like to stop and grab a few shots.
The lake is also home to many different types of birds - including eagles - that can often be seen perched on the dead trees in the center of the lake as they hunt for a meal. Unless you have a long lens (400mm or longer), you'll not be able to get a decent shot of them, unfortunately.
There is a campground called Atherton Creek on your left as the road descends and it's tempting to turn down the road there and take some shots of the lake looking westward. I'd recommend that you don't stop, however, as you'll be frustrated by the light since you've most likely driven this way in the afternoon.
Your best opportunity to get some great shots is just beyond the lake and you're now about a fifteen or twenty minute drive from the less than well known Red Hills.
The road narrows again and you'll see on your left (north) as you head east that the hills are beginning to take on a red hue that is different from anything you've seen so far. The mineral composition in the soil here turns the hills into varying shades of red and you'll be tempted to stop as soon as they begin to appear, but I'd urge you to continue forward and the road will rise in front of you. At the top of the rise you'll see a parking area on your right that you can park in and take in the full magnificence of the Red Hills spread out in front of you. Here you'll see a magical panorama with a valley full of horse farms and those amazing Red Hills jutting from the valley and begging you to set up your tripod.
For this scene, you'll want to use a wide angle lens if you have one and you may also want to take several shots in succession from left to right so you can stitch them together when you return home.
Although not necessary, this spot is a good place to use a neutral density filter or a circular polarizer on your lens to cut down on the bright light above the hills. I normally use a warming circular polarizer that does double duty and is an effective tool for a landscape photographer. They come in almost all lens sizes and there are many different brands and prices for all budgets.
You'll probably want to take a lot of different angles from your high vantage point and I would once again urge you to include the road in some of your shots. The horses on the right will definitely give the scene the scale you are wanting, but I find that the vanishing point effect that is created by the road running seemingly into the hills is an effective visual bonus for the viewer.
After you've taken plenty of shots up high, I'd suggest you continue your drive east and you'll come to a dirt road on your left that drops down rapidly headed south. I usually will pull my vehicle over here and take my gear and hike down a few hundred feet to the southeast to get a shot of the Red Hills looming over me as I shoot from the valley with the horse farm and fences in front of the lens. If you're lucky you're apt to have a few horses come over to the fence and investigate if you have any treats for them and you can snap off a shot or two with them in front of the Red Hills.
If you haven't already done so, look over your shoulder behind you as you return to the car and the road above and you'll see some beautiful photo opportunities. Spread out in front of you will be the Gros Ventre River and the heavily forested hills.
Once you're back in the car, head down to the eastern edge of the Red Hills and shoot some photos northward that will give you a completely different perspective of this amazing geological formation.
If you started out for this spot at noon, you're probably pushing the 3:00 p.m. hour by now and I'd suggest you retrace your route headed west this time until you rejoin Highway 89 at the Gros Ventre Junction in Grand Teton National Park. It will probably take you the better part of an hour to get back to the main highway.
Once you get to Highway 89, head to town or the place you're staying while in the area and rest a bit or review your shots from today. Then grab an early dinner so you can head back to Moose-Wilson Road for some wildlife viewing as the light begins to fade during the early evening hours. Wildlife tend to come out to feed during the early light and twilight hours and you'll never know what you might find.
Chapter Five - The Barns and Mormon Row
We'll be covering one of the most famous photo locations in the world in this chapter, but one that won't show up on your official National Park Service map of Grand Teton National Park. And that's one of the reasons that this book exists, because of the simple fact that the iconic Moulton Barn in front of the Tetons literally isn't on the map that everyone gets when they visit inside the park on their first visit.
I can't tell you how many emails I get on a regular basis from fans of the Tetons telling me that they visited the park and never saw the barns. It has probably been mentioned to me a thousand times or more. And the sad thing is that they were so close when they traveled through the main arteries of the park.
I'm assuming that you're headed out at least an hour before first light from your hotel in downtown Jackson. You'll be driving north on Highway 89 and the trip is approximately 27 miles from downtown to the two barns. But you'll need to allow yourself more than 30 minutes because the speed limit is only 45 or 50 miles per hour throughout the park. And as you head north you might easily encounter elk or moose or bison feeding near the road in the moments before dawn - so watch carefully as you make your trek.
You'll be driving past the airport junction and Moose junction and you'll be looking for a sign that says "Antelope Flats Road" on your right about a mile north of the Moose Junction turnoff. Here's a hint - almost directly opposite of this road on the west side of Highway 89 is the Blacktail Ponds Overlook \- and I'd suggest you make a quick detour to see if there are any moose grazing in the meadow just below the parking lot. I have found that the abundance of willow trees in this area makes this a magnet for moose during the early morning hours.
If you see a moose, great - set up you tripod and begin shooting from the relative safety of the ridge. But if no moose are present, I'd urge you to jump back in your vehicle and head east on Antelope Flats Road right away as you will want to be all set up as the first light hits the southern Moulton Barn.
To access the barn, you'll want to turn right when you encounter the dirt road that heads south about two miles east from the intersection of Highway 89 and Antelope Flats Road. This road is known as Mormon Row because it was where the first Mormon settlers set up their homes and ranches in the early 1900's. And to say they knew a little about the old real estate adage of "location, location, location" would be an understatement.
As you head east approximately a quarter of a mile you'll see the outline of Thomas Moulton's barn on your right. If you are visiting during a holiday weekend or during the fall season, you will also probably see a half dozen cars parked ahead and at least that many photographers set up alongside the fence in front of the barn.
Don't worry, it's not always like the scene you see above and I've been there many a morning when there were only a couple of other photographers present. And the key thing is that you are now personally standing in front of what a couple of magazines have called "the most photographed barn in the world".
That said, I'm going to tell you my own personal bias. I've been to Grand Teton National Park many, many times and I would never miss the opportunity to shoot at least one morning at the barn. The light is always different, the clouds and weather are always different and the crowd is always different. There is an electricity in the air as the first light begins to illuminate the peaks in front of you and then more great vibe when the weathered old barn starts to light up at sunrise.
One of the questions first timers ask often is if the barn is on private property. And the answer is no, it is on land that is part of Grand Teton National Park and thus you are free to roam around and cross over to get a different perspective than the one you get from shooting behind the fence. What I find truly amazing is that in the early 1970's the Park Service was considering an option to tear down the barn as an "eyesore" in the park. Luckily, a handful of local photographers got together and worked to preserve the barn and today it is truly the most iconic symbol of the area - and the old west - in the western United States.
I personally have three separate shots that I take when I visit and all offer the photographer a chance to get something other than the traditional head on shot with the barn's peak mirroring the peak of Grand Teton behind it. I'd recommend you once again use a tripod and a remote shutter to reduce camera shake \- which is especially important in the early morning when you will be shooting with little light present. And if you don't have a tripod, you can set your camera on one of the wooden fence posts to stabilize it.
My first shooting setup is to go over to the trees on the left hand side of the barn and position myself and the tripod low to the ground. If there has been rain in the area recently, there will usually be a bit of water present that you can use to lead the viewer's eyes back to the barn and the mountains.
Most people tend to try to shoot through the trees and their composition does not include the trees, but I'm a contrarian and think that the trees add some balance to the barn and the mountain range. This is especially true if there aren't a lot of clouds present when you are shooting. I would recommend you try several different zoom lengths from panoramic to focusing solely on the barn and the mountain peaks behind it.
And remember to stay low. It seems like most photographers like to shoot at comfortable levels like at their chest or in front of their face while they are standing. That makes sense from a comfort viewpoint, but it definitely hampers your creative ability. If you get down low, you'll probably have a creak or two in your joints, but my guess is that you will be quite happy with the results when you view them later.
After working this scene, I would move back to your right (north) and set up a shot that has the second stream or gully in the foreground. This one is on the right side of the barn and is another item that is often overlooked by the first time visitor who is content to stand behind the fence and shoot away.
This is another place where I would once again urge you to get low and stay low as you line up your shot. If you are present at the end of the summer or in the early fall, you will have the opportunity to use the brown and tan sagebrush in front of you to create some depth in your image. This will lead to the magic layering effect of foreground, mid ground and background that landscape photographers strive for in their compositions.
And here's a radical thought - what about leaving the barn completely out of the scene and just use the trees across the meadow from you to frame the Teton range? That's actually my third favorite spot when I visit the southern barn and try to shoot something different that takes advantage of the truly stellar landscape that is spread out in front of you.
In terms of aperture settings, I would suggest f/16 if the light is good at this spot and assuming that sunrise is behind you. You'll have to adjust depending on your light and weather conditions, but you'll want to try to get sharp detail from the front to the back of your shot.
Next, I'd urge you to jump in your car and drive north about a quarter of a mile to the second Moulton Barn on Mormon Row. Yes, there are actually two barns built by two brothers that dot the landscape here and the second one has some real character all of its own.
This barn was built by homesteader John Moulton around 1911 or 1912 and is often referred to by photographers as the northern barn. What makes this barn different than the barn you've just left is that this barn has a corral and a different backdrop from its more famous sibling to the south.
And like so many things in the photography world, the amount of clouds and light you have will make a huge difference in how you want to set up your composition at this barn. Many photographers head up close, but I prefer to walk out in the field covered with sagebrush and use that brush for foreground interest in my shots.
You'll probably want to experiment with different compositions - with and without the tree on the right is always a big question. I personally like it as a framing device for the mountains, but you might find that a zoom in shot with the barn and the corral sans tree is more your cup of tea. But either way - stay low. I know I sound like a broken record on this subject, but I can assure you that it is one of the things you should consider on every landscape shot you take during your visit to Jackson Hole.
Another thing to consider is whether you want to include any of the out buildings that are just to the left of the barn behind a row of trees. I have seen some panoramas done that included these and that's a matter of taste as well. I prefer the barn alone because it really gives an old time western feel to the landscape. And to think that at one time that was a new barn just completed by the Moulton brothers and there wasn't a photographer anywhere to be seen.
If you do decide to move in closer to the barn, be considerate of the fact that there will likely be several other photographers shooting at the same time you are setting up you shot. When I'm there, I always announce to anyone within earshot that I'm going in for a couple of shots but I'll be back out within a few minutes so as not to spoil their shots.
But don't be surprised if others aren't quite as neighborly. I've had many a good shot spoiled by another photographer walking blissfully unaware right in to my shot as he or she pondered their own shot. It's a fact of life at the barns and don't let it get you down. It is still early in the morning and you've got lots of good light left.
Once you have filled up one of your digital film cards - and you probably will - then I'd suggest you return to your vehicle and head back to Highway 89 by driving west a couple of miles. While there are photo opportunities all around you, I'd suggest we turn left at the highway and head south to the Moose Junction and then turn right as we head for the interior artery within the park - Teton Park Road.
We'll cover Teton Park Road in detail in the next chapter.
Chapter Six - Teton Park Road
Teton Park Road is the second most traveled road within Grand Teton National Park after Highway 89. It runs near the base of the Teton mountain range for almost 20 miles. It begins its journey at the Moose Junction, heads north/northwest and it ends at the Jackson Lake Junction on the eastern shoreline of Jackson Lake. During those miles, you'll experience a whole different perspective because of your proximity to the mountains and there are almost a dozen pullouts that are worth exploring.
However, we're going to concentrate on the handful that you can shoot in a three hour period before the noon sun gets a bit too harsh for good landscape photo work. We'll assume that you spent your sunrise hour at the barns on Mormon Row and you are ready for some different views in front of your lens.
As you turn west on Teton Road from the Moose Junction off Highway 89, you'll cross the Snake River within the first half mile. Once again, you'll want to drive slowly and watch both sides of the road for a moose spotting. Because of the abundance of willow trees, this is a location that has more than its fair share of moose and they can often be seen alongside the river. Of course, you probably won't have to look and listen too carefully, because if there are moose present there will be a moose jam with cars lining the road.
As you cross the river, you'll continue west past the visitors center which is a great place to visit when the afternoon light makes photography less than stellar. You will then come to the ranger entrance where you'll have to pay your fee to enter this part of the park. The fee covers you and your vehicle so if you have more than one person in your party, you'll pay one set price.
Immediately after the entrance, you'll want to take an immediate right turn on the paved road that leads to the Chapel of the Transfiguration. You may want to get a shot of the chapel with the Tetons in the background, but I prefer the scene on the western side of the chapel. So, when you see the chapel on your left side as you head east, I'd suggest you pull over and take your tripod to the field just behind the chapel.
There you will see a grove of aspen trees that are on a slight ridge in front of the mountains and this makes a very compelling image - especially during the autumn season when the aspens will turn various shades of color.
After you have worked this scene from the meadow, I'd suggest going beyond the aspens to the west where the view without the trees is also a good one of the Teton range.
Then you can either walk or jump back in your car and head east to the parking lot that sits on the north side of the entrance to the chapel. If you drive, park there and get out and walk on the path headed east to the ferry landing. The big opportunity here is the two stores that sit on the path. On the backside of these stores are windows that will reflect the Tetons perfectly if you can get the right light and angle to set up your shot. This is a shot that will separate your images from those of others because it is virtually unknown. I stumbled across the idea when talking to an old timer at the pond on Moose-Wilson road during one of my visits, and he said that he thought it was one of the unique shots in the park. I agree.
You'll want to experiment with your lens and tripod because it is a tricky shot. If you get too far in front of the window, you'll be in the shot and if you get too far to the left or the right, you will lose the reflection. You will also want to take some different aperture and focus point shots - some with the window frame in clear focus and some with the mountains in focus and the window frame out of focus.
If you then walk about one hundred feet west from this location to the service road that runs north, you can follow it for a short distance and you'll have another nice view of the Tetons that you can shoot with the trees framing the shot in the foreground.
Then it is time to get back in your vehicle and drive west until you run back into Teton Park Road where you'll turn right and head north. At the Windy Point turnout, I'd suggest you make a quick stop by parking on the lot on your right and taking your camera and lens to the west side of Teton Park Road. Be careful about the traffic, but if you can pull it off I'd suggest you get a shot of the road as it heads north headed directly in to the base of the Tetons.
Many, if not most, photographers want to get a shot without the road but I'd once again suggest that you need some scale to let the ultimate viewer see just how massive the mountain range is versus the road below it. For those new to photography, this type of shot is called a vanishing point image as the road appears to vanish as it heads into the distant mountains and landscape. There are several spots in the park that you can pull off this type of shot safely, and in the mid morning light you can usually get a fine shot at this particular turnout.
The next major stop is a breathtakingly beautiful scene where you will have an opportunity to catch one or more horses feeding in a meadow in front of the Tetons. There is a parking lot on the east side of the road and you can park there and head across the street to set up your shot.
One of the major decisions you'll have to make at this location is whether you want to include the fence or not in your landscape photo. I would recommend you take some shots right at the fence line and some farther back to include the old wooden rail fence. I personally think the fence adds to the old west feel that the scene portrays. And you'll want to use your tripod, a relatively wide angle lens such as a 18 to 24mm and an aperture setting of f/16 if it's a sunny day. And of course if you're relatively new to photography, you can get a pretty darn good shot by using the auto setting on your camera.
The horses are park service animals and they are normally present in the meadow during the morning hours. However, you may drive by when they are absent and then I'd suggest you return to this locale when you seem them in the field on one of your many excursions up and down Teton Park Road. It is a pretty scene without the horses, but the horses are the proverbial cherry on top of the sundae that you want to include if at all possible.
One other hint that most photographers don't know about is that if you walk to the north, the fence begins to turn westward and you can get a nice shot of the fence and the trees with the mountain range.
Once you hop back in your vehicle, we'll continue north/northwest for a few miles and you'll see a concrete bike path that has been built on the west side that parallels the road. If you are so inclined, pull off when you see a rider or group of riders and get a shot or two from the east side of the road with the bikes at the base of the Tetons. Once again you'll get that scale that we're always wanting to include in a scene.
As you drive north you'll see Jenny Lake and the visitor center on your left and I'd suggest you pass it by this morning and continue north. We'll cover Jenny Lake and Leigh Lake in another chapter, but to maximize your photo productivity this morning, your best bet is to keep moving north. There are several turnouts that exist on both sides of the road and I know it will be hard to keep on driving. But I think you can get your most impressive shots past the North Jenny Lake Junction which will be about four miles north of the first Jenny Lake turnout.
The reason I suggest the effort to head north is that you get some of the best perspectives in the area between the Mount Moran Turnout and the Potholes Turnout. Stopping at these points or pulling off the road anywhere between them will give you a terrific view of the Teton mountain range with a vast prairie below it. I think that when most people see this vista for the first time, it takes their breath away and it is a great place for taking wide angle landscape photos.
I personally like shots from the Mount Moran turnout if you walk about 75 feet west/southwest of the parking lot and set up your tripod in the vast field of scrub brush and sage. And remember to take a few shots with the tripod at its lowest setting so you can get the foreground interest of the brush inside the frame of your shot.
I would then suggest going down the road and pulling off when you have some room and waiting for a clear moment on the road when there aren't any cars present. You've got another fine opportunity for a vanishing point shot and the landscape makes this area perfect for this type of image.
What you are looking at from this location is called the Cathedral Group because the first visitors to the area thought that the mountains and their peaks looked like the vast spires of the old churches found in Europe.
As the morning light winds down, you've got another hour or so and I'd suggest you continue your drive north and head to a location just past the Signal Mountain Lodge where you can pull over and get a superb shot of the mountains reflected in the lake if the winds aren't too strong. The parking is on the west side of the road and the view will be off to the west/southwest. You will want to work this scene
from both the road and then hike down to the shoreline of the lake for a different perspective. And as you've used so often this morning, a wide angle lens will come in very handy here as well. If you don't have a wide angle, just remember to take a series of shots with the lens you do have while panning from left to right using your tripod. You can then stitch them together later in a post processing step that will give you the magnitude of the scene that appears in front of you.
If the wind has picked up too much to get that mirror reflection that you're looking for in this scene, I'd suggest you make a mental note of this location and plan to return here at another time when you can nail the shot that you want. But if you're pressed for time and you may not pass this way again, then you'll want to get a few shots here because it gives you the best scenic view of Mount Moran from the Teton Park Road. This is the same Mount Moran that will show up again when you visit Oxbow Bend, but this particular view isn't as well known as the iconic view at Oxbow.
And as we close out this chapter, I'd suggest that you take some shots of this scene with the thought of converting them to black and white or infrared using post processing. Because of the elements of this scene, it works pretty darn well and creates a sort of timeless appeal that speaks to many photographers - and fans - of Jackson Hole, Wyoming.
Chapter Seven - Afternoon side trips to Two Ocean Lake and Pacific Creek Road
Well, if you've been following along on a day by day basis, then this chapter will find you on the second afternoon you've been in the park. You've captured some stunning views and you've got lots of great locations left, but the light is overhead now and most shots taken looking west will have that notorious flat light from this point forward. If you're skipping around amongst the various chapters, file this one in your mental bank as an afternoon recommendation for one of your days at Grand Teton National Park.
So, assuming you're just finishing up at the shore of Jackson Lake you've got a couple of options. You can head back to downtown Jackson for some food, fuel and rest or you can head north and east a bit and take in two little locations that are once again off the proverbial beaten path. I shoot as much as I can when I'm in Jackson Hole, so I always carry water and protein bars with me so I can push forward with more photo work. But if you didn't bring any of that with you, you might want to go back to the Signal Mountain Lodge and grab a sandwich before you head out northward.
You'll follow Teton Park Road across the Jackson Lake Dam and you'll come to a junction - you'll want to turn right (east) at this junction and you'll find yourself on Highway 89 and Highway 287 as it passes by Oxbow Bend on your right in less than a mile. Now the temptation will be overwhelming to stop here because you'll be seeing one of the most famous locales in North America on your south side.
And I'd suggest you do so for a few brief minutes. Not necessarily to take any shots just right now because the overhead sunlight will be harsh, but to get a sense of the layout of the shooting location. You'll be headed back here during the pre-dawn hour on your next morning trip from your hotel or campground and in the midday light you can get a sense of where you will want to setup your tripod. We'll cover Oxbow Bend in an upcoming chapter, but since you are driving past it you'll want to see what is it all about.
And you'll also want to look south over this bend in the Snake River because often you'll see moose in the distance grazing on the shoreline. One of my favorite shots occurred here around noon when I happened to glance over and see a moose feeding quietly in the distant water. No one else seemed to have spotted him, and I didn't have a really long lens with me so I used what I had and got a nice shot of the bull silhouetted within the context of his surroundings.
Once again a combination of luck and knowing where to look paid off and I'd urge you to be constantly aware of the possibility that you may see wildlife at anytime in the park.
After checking out Oxbow Bend, you'll be heading east again and you'll want to watch for a small sign on your left that will say Pacific Creek Road. This road will lead you north to a split in the road that will allow you to veer left or right. I'd suggest you veer left first and drive the gravel dirt road north/northwest to its completion where it dead ends at Two Ocean Lake.
You are now on the banks of an unusual lake that drains to both the oceans east and west because it lies directly on the Continental Divide. Although you cannot see the Teton mountains from here, you are in an area that many of the local horseback riding outfitters utilize for trail rides. It is easy to understand why they ride here when you consider the scenic beauty and its remoteness from the crowds that can invade the park in the summer months.
And because there is much less population density in this spot, you'll have a much better chance of spotting wildlife along the road as you travel to the lake, so drive slowly and keep your eyes and ears open.
Once you get to the southern shoreline of the lake, you'll see a picnic area with restrooms and a small parking lot on your left. Park here and take your gear and head down to the shoreline and you'll see a trail that meanders around the eastern side of the lake. If your timing is right, you'll see riders on the trail.
One caution - this location has a lot of insects and so you'll need to come equipped with bug spray to keep the flies at bay. You can walk on the trail as it curves northward, but some of the best shots are simply shooting north from the campground and shoreline of the lake when the sun is illuminating the distant tree covered hills. And don't wander too far off the trail because there are a lot of bears that inhabit this area and bear sightings occur here on a regular basis during the summer and fall months.
Hopefully you'll see a bear or a deer or a moose at a far enough distance to get a good shot without your adrenaline pumping too much. And I do hope that when you visit here, you'll put down your camera for awhile and just take in the mountain air and the view around you. You may find yourself catching your breath occasionally since you are near 7000 feet in elevation at this point.
Once you've worked the location, you should hop in your vehicle and retrace your route headed south until you come to the Pacific Creek Road again. This time, turn left and head eastward. The road ultimately dead ends at the far eastern edge of Grand Teton National Park. You'll see a creek on your right as you weave in and out of the woods and you'll have a terrific opportunity to view wildlife since this portion of the park is seldom visited by summer tourists. There are several turnouts that will allow you to park your car and get out and do some exploring - which I would encourage you to do as long as you are aware that wildlife is all around you although you may not be able to see it initially. Walk slowly and listen intently and be sure an look up because since there is water nearby, you may be able to spot osprey or even eagles as they fly overhead in search of their next meal.
Once you get near to the end of the road, you'll want to park and walk through the trees on your right and head east/southeast to the creek's edge. As you look back to the west, you'll see glimpses of the Tetons through the trees.
After exploring this section, you are in good position to get some landscape shots with the river flowing eastward while the sun continues to give you good light in that direction. Although you may want to catch the Tetons to the west, the best shots are actually of the natural landscape, trees and river looking south and east.
After awhile here, you'll want to head back west on Pacific Creek Road and you'll see some great views of the Tetons as you drive along the gravel road. Although the sun won't be in your favor, you will probably want to snap a couple of shots from this perspective as it is another view that very few people see - or photograph - within the park.
This is especially true if you are visiting Jackson Hole in the autumn season when the abundance of aspens alongside the road will allow you to get some seasonal shots. Hopefully, the photos you take will help convey the beauty to others when they see your images taken along Pacific Creek Road.
As you rejoin the main highway, you'll want to turn right on Highway 89 and drive east to the Moran Junction. From there you'll want to turn right and head south on the highway and watch for a dirt road on your left not too far after you cross the Buffalo Fork of the Snake River. It is unmarked, but it is public property still and you want to turn and head east on this road.
As you drive, you'll notice that you'll come up on a clearing that has both fences and some old buildings. You can continue to drive east and don't be surprised if you run into bison alongside the dirt road. This is one of their favorite spots within the park and I've seen bison here many, many times. As long as you're careful, you can get some really close up shots of them by staying in your car and using it as a photo blind. Just roll down your window and put your lens on the windowsill to serve as a temporary tripod and shoot away.
You'll want to try lots of different exposures because the bison are so dark that it is hard to pick up details of their massive bodies.
If there's no bison to shoot, you've still good one good option left before headed back to downtown Jackson.
Just east of the old buildings, you will see a stream and some small ponds that are on both sides of the rutted road. You can drive further east, but unless you have a four wheel drive, it may be tough. However, if the weather and clouds are cooperating when you arrive here, you can get some great reflection shots of the eastern and southern low lying mountains. Just be sure and get down low and experiment with some different angles.
Okay, it's not the Tetons - but if you're looking for a way to use the afternoon light, then this is definitely an option versus shopping at the Square in Jackson.
Because the road gets iffy from this point forward, I'd suggest you turn around here and head west back towards the main highway. As you drive in that direction, you'll notice that you've got an unusual viewpoint of the top of Mount Moran with a small ridge and some trees in the foreground. Although you'll be shooting into the sun, if you bracket your exposures (one exposed correctly, one underexposed and one over exposed) while using your tripod, you can get a series of shots that you can blend later into a high dynamic range image that won't be too bad in spite of the less than stellar light conditions present in the afternoon.
Once you rejoin Highway 89, you'll want to turn left and drive south back to Jackson. But you will utilized your afternoon wisely and you'll end up with some photos that most photographers never had the chance to take because they didn't drive off the main highways.
Chapter Eight - Morning at Oxbow Bend
Well, if you've been following these chapters sequentially you've probably already asked yourself a couple of times a very salient question.
"What about Oxbow Bend?" or "Is he ever going to write about Oxbow Bend?"
Whether you've been to Grand Teton National Park a dozen times or this is your first, you probably already know that Oxbow Bend is one of the truly great landscape photo locations in North America. It regularly appears on the cover of travel, nature and photography magazines and it is a magnet for even the most casual photographer.
I believe that you can get the best shots of this location by being there early - as in at least a half hour before official sunrise. So, wherever you are staying, plan to head out with sufficient time to get there before first light. If you are staying in downtown Jackson, then allow for approximately 45 minutes driving time when you head north on Highway 89 and then continue west when the highway turns left at the Moran Junction.
If you have any mobility issues, this shouldn't hamper you one bit as you can actually shoot Oxbow Bend from inside a car. There is a parking lot where the majority of shots are taken, but there is a pullout area just west of the parking lot on the south side of the road that would allow you to compose a pretty decent shot without leaving the comfort of your vehicle.
I wouldn't suggest that be your only shot, but it's definitely an option.
One of the most fascinating aspects of Oxbow Bend is that it contains so many elements that landscape photographers love in a place - a natural beauty coupled with foreground interest, a strong middle point and a magical reflection that captivates a viewer.
If you've not been to Oxbow Bend, some clarification may be in order for you. You are actually looking at a bend in the Snake River that occurred naturally a long time ago and created a body of still water that is perfect for creating a mirror effect. Now that mirror effect is not always present because the winds pick up often in mid-morning, but if you are there early for first light, then your likelihood of getting the reflection is much improved.
You'll want to set up as you have before - with a tripod and a remote release shutter trigger if you have that type of gear. If you don't have the remote, just use your timer to take the shot as opposed to pressing the shutter button with your finger. And you'll want to use an aperture setting of f/8 or better, and thus the need for a tripod in the low light of sunrise becomes really important. Don't be surprised if your camera's internal light meter ends up taking a slow shutter speed shot to compensate for the lack of full light.
I would also recommended that you start at the well worn ridge just west of the parking lot for your first series of shots and that you bracket once again with correct, over and under exposures that will allow you to work with high dynamic range post processing. You'll want to take a minimum of three shots with this method, but using a five photo bracket would also work well here. In fact, I know of several landscape photographers who have taken as many as nine bracketed shots at this spot.
This is one of the places where you can also get a sweeping panoramic shot by using your tripod and shifting the view from left to right. Taking multiple shots with an overlap of about a third of the image between each frame you take will give you a great opportunity to merge them all into one magnificent panorama for your home or office.
And don't hesitate to rotate quite a bit to your left when you set up your panorama as the trees on the southern bank of the Snake River are a good anchor for the image. Too many first timers try to zoom in on the distant Mount Moran and its reflection - which is a fine shot in itself - but they neglect to include a broader sweeping view of the area.
One of the things that you'll always want to remember at a place like this is that you want to capture not only the shot you see in all the postcards of the area, but a shot that you haven't seen anywhere else.
Okay, let's say that you've just pulled into the parking lot and there are heavy clouds present that cause you to sigh a deep sigh of disappointment. I've been there and I've heaved the same sigh a number of times. But I have also learned that weather is constantly changing and the presence of clouds can end up being a beautiful gift for a photographer.
The clouds may not break as the sun rises over the eastern hills, but you won't know until you stay for awhile and see what type of nature luck occurs this particular morning. I was once set up on the ridge with my tripod and I saw dozens of photographers stop, decide it wasn't worth the effort, and drive on.
But a few short minutes later, there was a break in the clouds and a bolt of light shot through that enabled me to take a shot that has appeared in publications all over the world and is a constant best seller . I recall being the only person on the ridge that moment and I was very glad that I had not given in to my thought that maybe I wasn't being very smart standing here when all these other photographers had bailed out.
As we discussed in Chapter One, a great way to improve your photography is to stand in front of better things, and at Oxbow Bend you are standing in front of one of the best scenes you'll ever have in front of your camera. So don't worry so much about the clouds, the weather, or the other photographers. Concentrate on creating an image that reflects the scene right now while you are standing on the ridge and every time you look at the shot, you'll remember the moment.
The next point you'll want to relocate to would be to follow the well worn trail to the water below the ridge. Be careful, since it's a fairly sharp decline, but it is well worth the effort to get to the shoreline.
When you get to the northern shore of the river, you'll notice that you're not the first one to have ventured down here. It is probably safe to say that you are walking in the footsteps of hundreds of thousands of other photographers who come to Oxbow Bend year after year. That said, there are only two people to consider when lining up your shot. Yourself and what visually appeals to your eyes and your potential viewer and how you can capture something that will appeal to them. They will be looking at the photo on a computer screen or as a print and so you'll want to draw them in and through the complete frame of your shot, so think about getting down low. When you go low, you can add interest to your composition by including the floating log or the white pelican that is paddling by you at this very moment.
And consider if you want to have everything in focus. If you do, you'll want to stick with an aperture setting of f/8 or higher. But think about using f/4.5 and having the log in focus and Mount Moran and the distant shoreline a bit out of focus. This limited depth of field will make for a different shot that may end up being one of your favorites. The scene will still be instantly recognizable because almost everyone has seen a photo of Oxbow Bend, but they may never have seen it from the surface of the river.
And here's a radical idea. Don't shoot Mount Moran at all for a few of your frames. Point your camera at the shoreline just south of where you are now standing and you'll likely have a beautiful mirror reflection of the evergreens in the Snake River.
It took me several visits to realize that focusing in on Mount Moran completely was causing me to miss the beauty that is all around Oxbow Bend. This is particularly true during the last week of September when the trees that line the western shoreline light up with the turning of their leaves. Leaving Mount Moran out of your composition allows you to show the viewer nature's seasonal beauty just as effectively.
I would urge you to walk up and down the bank and try out lots of different compositions and angles. Working the scene may allow you to serendipitously "find" a shot that you didn't see initially and this happens often in a place like Oxbow. Experiment with different focal lengths and pull out your wide angle if you've got one. Zoom in on Mount Moran and check out how that view might look in one of your compositions. In essence, act like you did when you first got a camera and started to experiment with photography. Remember how you were shooting everything in sight? This would be a perfect place to recreate that approach.
Once you've filled up a memory card or two, I'd suggest you return to your car and drive about a quarter of a mile east and pull into the parking lot on your right. This is a different vantage point that is often overlooked by first time visitors and it allows you to see the scene from a completely different perspective. True, you do lose the magical reflection - but what you gain is a more sweeping vista that works very well as a landscape photo.
And since you are here early in the morning, there is a good chance that you will see the herd of elk that frequents the meadow directly south of this location. They should be grazing in the dawn hour and with a long zoom you should be able to get some decent shots with your lens and tripod.
You may also want to hike down to the left on the trees that are directly to the west of the parking lot because when you clear the trees you'll get another vista of the mountain range looking west/northwest that is equally impressive. And then if you've got enough energy, head across the parking lot and cross Highway 89 and climb about halfway up the hill in front of you for yet another stellar photo location site. This last position allows you to once again see the bend in the river and the panoramic view of the Mount Moran with the valley spread out for miles and miles. As you hike up the hillside with your equipment, you'll probably be saying "I sure hope this is worth it" - and it is, I assure you.
Although it will be hard to leave this magical place, you'll need to move on since you've still got good light this morning and you'll want to put it to good use. So hop in your car and head west as we trek over to Jackson Lake Lodge for some morning coffee and some absolutely stunning views.
Chapter Nine - Jackson Lake Lodge and Colter Bay
As you leave the parking lot near Oxbow Bend, there's another relatively unknown location that 98% of visitors miss during their trip to the park. Approximately one quarter of a mile west of the main Oxbow Bend parking lot is an unmarked dirt road on your left. This is known to locals as Cattleman's Bridge, although the bridge was torn down years ago by the Park Service. Turn left onto this road.
Except during heavy rains, this road is bumpy but you can drive a rental car on it all the way to the end. But what you'll want to do is drive very slowly because this is a very popular destination for moose, deer and elk. There are dense trees and meadows and about a quarter mile in you will come to a large meadow on your right. Pull over and you can get a different landscape shot of just Mount Moran as it rises majestically out of the west. There are a couple of large trees in the meadow and you can decide whether you would like to add them as a framing device or just have an open shot of the scene.
As you drive to the end of the road, you'll see that you've come to a spot that is popular with the locals for trout fishing and kayaking. There is a small parking area where you can get out and explore a bit on the level ground. Be sure and spend some time watching the Snake River to your south as often you can see river otters and white pelicans in the area.
Although you don't have a view of the Tetons from this point, you do get a chance to experience the quiet and solitude of the moment and you may luck out and get a decent wildlife shot while you're in the area.
Then you'll want to hop back in your vehicle and retrace your drive in until you rejoin the main highway. Turn left and proceed past the road on your left (which is Teton Park Road) and veer north. Approximately a half mile on your left after the junction, you'll see a parking lot. Pull into the lot and you'll get a grand view of the meadows below framed by aspen trees.
This area is also frequented by grizzlies in the spring as they prey on newborn elk, so be on the lookout and don't wander too far west in search of a perfect shot. In fact, the whole area around this meadow is prime bear country and you'll often see signs posted by the Park Service saying "no access beyond this point". I'd urge you to follow the advice of the Park Rangers who post these signs. I was standing in this same parking lot a couple of years ago in mid-morning when a lady drove into the lot and asked me if I had seen the grizzly. I looked at her quizzically and she pointed right across the road where a grizzly was moving up the hill there. To say I was surprised would be an understatement as I was so focused on setting up my landscape shot in front of me that I didn't look around.
But I've learned from that and I hope you will too. Even if your intent in to shoot landscape and nature shots only, remember that you're standing in an area that has the largest concentration of wildlife in North America. That's why I now travel with two camera bodies - each equipped with a different lens. On my full frame DSLR camera, I have a relatively wide angle zoom lens that goes from 24mm to 70mm. On my other camera, which has the DX sensor - which is smaller - I have a long zoom lens that goes from 70mm to 300mm. And because it has the digital DX sensor, the equivalent range goes up to 450mm - which is usually effective for wildlife that is not too far away.
The view you see west from this lot - and the lot a quarter of a mile further north - is a grand panorama of the area known as Willow Flats. You also see the complete Teton mountain range with Jackson Lake at its base. You'll definitely want to pull out your tripod and do a series of shots from left to right so you can use your panorama software when you return home. And if you don't have a wide angle lens at this point, you'll need to figure out just how you'd like to frame up the shot.
A slightly different - but similar - viewpoint awaits you when you visit Jackson Lake Lodge, so if you are pressed for time, you can avoid this stop and head on up to the lodge.
Assuming you do have some time, I'd suggest you take a series of shots - some wide and some with one particular focal point. I think the long zoom shot of Lake Jackson in the foreground with the peak of Grand Teton centered in the frame is effective as is the shot of Mount Moran zoomed in with just a part of Willow Flats within the frame.
Once you are happy with your work, then hop back in your car and head north about a half a mile. You'll cross a bridge just before you get to the Jackson Lake Lodge turn. You will want to scan the area left and right when you are on the bridge because often you can spot wildlife from this vantage point. If you do see wildlife present, you can pull over on the right immediately after the bridge as there is room for a car on that side.
But if you don't see any wildlife, just proceed north and you'll see the sign for the Lodge almost immediately. You will want to turn left and head west. Once you make the turn, on your left in about a quarter of a mile is another road that you can turn on if you need gasoline. There is a station there that also has bottled water, snacks and cold drinks.
But if your gas tank is still full, head west until the road dead ends. You can park anywhere you can find a spot as it is free parking. And then grab your gear and walk right in to the main doors of the Lodge for a visual treat that amazes most people.
As I was composing this book, I thought about inserting a photo here of the view you see when you walk up the main stairs of the lodge and enter the main room. It is spectacular \- to say the least - and I decided that this is one of those places where a photo doesn't do the experience the justice it deserves. So, be ready to be surprised.
And here's a quick history lesson about this very spot in which you are now standing. This is the famous location that Yellowstone Park Superintendant Horace Albright brought the Rockefeller family back in the 1930s. At that time, this area was not a National Park and it was being taken over by commercial interests. He brought the family here and told them that this view was going to be lost to future generations forever and the government didn't have the means to do anything about it.
They enjoyed a picnic here and returned home. A few days later, Horace Albright received a telegram from Rockefeller. It said simply "Buy it all". And the Rockefeller family did just that over the next decade and ultimately gifted the land to the public. The Lodge was owned by them for decades and serves as a reminder that Horace Albright and the Rockefeller family were visionaries that saved this magnificent area for all of us to enjoy.
In fact, the very spot where they had their picnic on that fateful day is up a steep hill to your right when you exit west from the main lobby of the Lodge. This is still known today as "Picnic Hill" and if you're a history buff, then you'll want to make the strenuous hike upwards to see what they saw exactly. But for the rest of the world who want to capture the view, the area directly behind the Lodge is magnificent. I'd urge you to shoot away and then after you've shot all that can be shot, I'd suggest that you sit down on one of the outside bar stools and have a soft drink or a cocktail. Then you can just gaze at the wide expanse of nature that I believe will fill your eyes with wonder.
If you're feeling adventurous, you can leave the large patio area and walk south to where the patio ends and you can explore the southern area that has both cabins and hotel accommodations. Although there are good views here as well, I personally think the best views are right from the patio area. And be sure and look closely at the meadows below you. Because of the abundance of willow trees, there are often one or more moose grazing below. Because it is such a large area, you'll need to look closely as the moose or elk or bear may just appear as a small dark figure amidst the greenery.
Before you leave, you'll want to consider grabbing an early lunch or sandwich at one of the two restaurants within the lodge. I personally am a big fan of the bison chili in the Mural Room - which also happens to have spectacular views from every table. And because you are in a National Park facility, the prices aren't too pricey. And don't forget to utilize the restrooms that are at the top on the stairs before you leave.
After you've had your relaxation moment, then you'll want to jump back into your vehicle and return to the main highway. When you reach the highway, you'll want to turn left and head north to Colter Bay. It is a bit of a drive - almost five miles - but the scenery is spectacular along the way and there are several turnouts that you'll no doubt want to pull out into for a shot or two.
When you reach the Colter Bay Junction, you'll want to turn left and head west. This area is named for John Colter, who left the Lewis and Clark expedition and is considered the first white explorer to discover this area - as well as Yellowstone to the north.
You are entering in to an area that is pretty populated during the summer months with visitors from all around the world. This is the largest concentration of cabins and tent sites within the park. If you have some extra time on one of your afternoons, you might want to explore this area by driving the many short roads that head north and south through the woods.
But we are headed west until the parking lots end. You'll then want to turn right and drive north until that parking lot there also ends. Park somewhere in this vicinity and grab your gear and head down to the shoreline of Jackson Lake.
As you walk the shoreline, you will note that it is covered with millions of small rocks. Be sure and get down low with your tripod and use the rocks as foreground interest. And of course if there is a log or piece of driftwood present, then that would make a good foreground anchor that will help lead your viewer's eyes through the frame of the photo. Then they will see the magnificence of the Teton range that seems to arise directly from the waters of the lake and dominates the western bank.
Walk up and down the shoreline and experiment with wide and short angles and with large and small depth of field. This can be accomplished by using different aperture settings - from f/4.0 to f/22. And because you may be here as the midday light begins to get harsh, I'd suggest you take some bracketed under and over exposed shots as well that you can blend when you get home.
Let's assume that the afternoon light is now upon you as you finish up at Colter Bay. You are a scant few miles from the southern entrance to Yellowstone and so if you want to take a quick side trip up to that park for an afternoon, this would be the closest place to do so. Otherwise, you'll want to head south and back to town or your campsite for a few hours . As you have done before you can get some rest before you head out again in the early evening for some more wildlife viewing.
Chapter Ten - Morning at the Two Overlooks \- Snake River and Blacktail Ponds
As you prepare to head out before dawn once again, be sure and consume an energy bar or two this morning since you've got a full morning of shooting ahead of you. Since by now you are an old hand at checking for today's sunrise time, I'd suggest you leave at least 45 minutes prior to sunrise. As in the past, this will give you time to set up your tripod properly and it allows a cushion in the event that you encounter wildlife on your journey to your next landscape location.
Heading north on Highway 89 out of downtown Jackson, you are going to travel approximately 20 miles to the location made famous by Ansel Adams - the Snake River Overlook. Almost everyone in North America has seen his iconic shot of the bending river below the ridge and his timeless interpretation of the scene.
When you arrive, you'll turn left off the highway and you'll notice a long parking lot with a turnaround at the end. If you are early enough, the lot may be deserted - but if you're late, you might scramble to find a spot. This is what I call I magnet location - it draws everyone on vacation who has a camera.
As you get out of your car and approach the overlook, you'll see a four foot stone wall that has been built out of native boulders and that extends for a hundred feet or more. You'll want to set up as far to the north as possible where you still have a clear view of the river below. And although the wall is there to keep the crowds from walking off the overlook and down the hillside, you can easily jump up on the wall and get on the other side if you think that's a better setup location.
One thing you will note almost immediately is that this is not the exact scene before you that you saw in Ansel Adam's outstanding photo. Well, that's absolutely true. The trees in front of you have grown in the last 50 years and they do block more of the scene than they did when Ansel was standing with his tripod where you are right now.
But you can still get that great turn in the Snake River below as it shifts its course west from its southerly flow. And if you are here before sunrise and the weather cooperates, you should be able to get that famous alpenglow light that turns the mountain range a beautiful shade of pink and red when the first light illuminates the mountains.
Many photographers feel that they can't get high enough to shoot over the trees like they would like - but it's the harsh reality of the location today. You can try standing on the wall and setting up your tripod on the wall itself, but that's pretty iffy as a stable platform.
Another thing to look and hope for when you visit this location is the early morning fog that will hang just above the river when it is cool in the morning. Now if you are visiting in the middle of summer, it may be too hot for this to occur for you. But if you are here in the early spring or late autumn time period, it is likely to happen. And so you'll obviously want to incorporate this nature phenomenon in your shot - hopefully where you can still see the river below peeking through the fog.
Because it will be very early when you visit, you'll be dealing with uneven light as the first light touches the mountains. I would suggest you crank up your ISO setting as far as possible without causing too much noise to appear. On most newer DSLR cameras, you can safely go to 1000 ISO and still retain good clarity. I would then set my aperture setting at f/8 (if cloud cover is extensive) or f/16 (if little cloud cover is present) and use your remote to trigger the shutter while your camera in on the tripod. In the early morning light, this may cause a relatively long exposure, but it will allow you to capture the nuances of color and light that appear at this magical site. And remember, if you don't have a remote trigger, just use the built in timer that virtually every camera made it the last ten years has available.
And if you happen to not have gotten up as early as you planned, or if you decided to have breakfast with the family before you headed out, don't despair. Although this is one of the four locations that I think merit a specific sunrise trip (Oxbow Bend, Moulton Barn and Triangle X Ranch are the others) - it doesn't mean you can't take good shot at any point in the morning. Because of the location, you can reasonably shoot here until at least noon, but you'll find that the afternoon sun and haze may make it untenable past that time.
You may also want to experiment with setting up your tripod a little bit south of the standard location where everyone sets up. You have to move left past the trees in the foreground and you'll come to another opening in the scene. You cannot capture the bend in the river as well, but it is still a fine shot and allows the depth that so many landscape photographers crave when shooting this type of photo.
One other thing you might want to consider is to drop back by here for a few brief minutes in mid-morning. The reason is that you can then use a circular polarizer or a warming filter much more effectively when you have good light. This will help you control the exposure and should result in a couple of images that you'll really like as a contrast to your early morning shots.
Once you have completed working the scene at the Snake River Overlook, I'd suggest you hop in your vehicle and head approximately seven miles south to our next stop - Blacktail Ponds Overlook.
You may feel like you are backtracking a bit \- and you are - but I wanted to put you at the best spot first during the very early morning light. Because of its proximity, Blacktail Ponds Overlook is a great second destination after the sun has risen high enough on the eastern horizon to illuminate the meadow below the mountain.
As you turn west off Highway 89 into the roadway that leads to the Blacktail Ponds parking lot, you'll notice that the ridge you are approaching is covered with sagebrush. Be sure and consider this as an appropriate foreground subject when you are composing your shots at this location. Although I'm the first to suggest a tripod when you are doing landscape photo work, I also think it's important to try getting low and seeing how this vantage point can change a composition. When I lead a photo tour and share this advice, invariably by the end of the tour half the group is laying on the ground and shooting landscapes and they often tell me that their favorites were the shots taken from the ground.
The meadow that lies below the ridge at Blacktail Ponds Overlook is full of willow trees and so you may happen upon a moose or two grazing below you. Most people spend all their time shooting off the ridge, but I find that if you are the adventurous type, you can get some killer shots by hiking down the ridge and shooting from the meadow itself. Because of the way that the trees frame the bottom of the Teton mountain range, you can get some layers within the frame of your composition. And of course, it also is predicated on when you are visiting as the colors before you change as the season progresses and the hot sun turns the grasses from green to brown.
The scene before you at this overlook provides a multitude of different angles and composition options. There is of course the straight ahead shot with the meadow below. But after having visited this site many times, I've come to appreciate the subtle differences that you can make in your landscape photo work here by pivoting a bit from straight on and shooting in a west/southwest angle. You can accomplish this by walking to your right and head north for a hundred feet or so until you get a view of the small branch of the Snake River that is headed towards the mountains.
In terms of lens and camera setup, you've got lots and lots of options here as well. The cloud cover - or lack thereof - will help you determine what f/stop to go with and if the day is sunny, you'd be in good shape at f/16 as long as you are shooting from a tripod. I'd suggest a relatively wide angle lens here to capture the expanse that lays below you, or you could rotate the tripod and capture a series of side by side shots to stitch together as a panorama later.
But what should you do if you've got extensive cloud cover when you visit Blacktail Ponds? Well, you can mark it as a place you'd like to return to on a different day, but if you are pressed for time then you'll have to make the best of a mediocre situation.
I'd suggest you put the cloud cover to good use by taking some darker exposures by going to manual exposure and overriding the "correct" exposure setting. You can experiment and see how the exposure will change the scene, but if you're willing to shoot several different exposures, I think you'll be amazed at how the clouds can become ominous and foreboding. I have found that always shooting on the "A" setting (Aperture Priority) is normally a good default but that there are many other options available that will depend specifically on the light and the scene.
Another option is to experiment with filters if you have them with you. A polarizer or a warming filter can do wonders to change the scene into something different . There are also a lot of post processing filters on the market that will do this in the comfort of your own home if you didn't have a filter when you took the original shot.
After you have had a chance to work this scene, you'll probably notice that the sun is continuing its inevitable rise and the light and color nuances are changing. With that in mind, I'd suggest you pack up your gear and head back to Highway 89 and turn north for the next destination on our photography journey - Elk Ranch Flats.
Chapter Eleven - Elk Ranch Flats and Buffalo Fork Ridge
As you head north once again on Highway 89, you'll be passing a lot of photo opportunities on your left and, as always, you'll want to keep your eyes peeled for wildlife. Normally mid morning is not prime wildlife viewing, but I've come across bears and moose at all times of the day. What is more likely is that you may encounter the bison herd that frequents the Jackson Hole valley that you are now driving through. And if they happen to be on the plains to your left with a shot of the Tetons as a backdrop, you'll certainly want to stop and set up a photo.
But your main objective for this drive is the turnoff at Elk Ranch Flats which is approximately 12 miles north of the Blacktail Ponds Overlook. And what we are in search of is that classic shot that you've seen a few dozen times of horses grazing in a large pasture with the Teton Range in the distance. Now if you have been lucky earlier in your visit, you've already gotten a similar shot on the inner Teton Park Road. But this shot is actually significantly different because of the different perspective - as well as the much longer distance that you are from the base of the mountains.
One of the unknown factors in this location is the presence of the horse herd. This is one of the locations that you will pass on the way to Oxbow Bend and Jackson Lake Lodge and so you may have an opportunity to see horses in the pasture when you least expect it. By itself, it is a classic panorama view of the Tetons spread as far as the eye can see. But with the horses, it takes on a wholly different look and feel and the old time wooden post fence adds to the appeal.
And that old wooden fence also has an added bonus of being a makeshift tripod if you just want to grab a quick shot on your way to another location. Another thing to be aware of is the fact that this location works equally well as a vertical or horizontal shot - sometimes referred to as portrait or landscape orientation. If you decide to shoot vertically with your camera on its side, you won't grab the panorama, but you will be able to create a sense of depth between the fence, the horses, the trees and the distant mountains.
I've only seen the horses come all the way to the fence a couple of times - they mostly keep their distance and stay together as a herd in the center of the pasture. But what you'll want to look for is a handful of horses that are doing something visually interesting - whether it be standing still in tandem or fighting each other with kicks and teeth baring. Horses do this all the time, but you have to watch for it and have a fast enough shutter speed to be able to capture the action in the split second in which it occurs. You may also want to single out one horse and use them as foreground interest even though they may be surrounded by a dozen other horses. And it is usually easiest to do this when you shoot vertically.
Okay, you may be reading this in the comfort of your favorite chair before you actually make your trip to Jackson Hole. I do the same whenever I visit a new location. But I hope you plan to take this book with you to the park as a shooting guide, or simply to trigger your decision of where you want to shoot if you are on a tight schedule. This is always particularly true if you are traveling with others who aren't really into photography.
I mention this here because the one thing that we've discussed previously that is out of anyone's control is the weather. The light may not be perfect, or you may be at this great pasture location in the evening instead of late morning - or there may simply be haze on the horizon because it is late afternoon. But instead of bemoaning your luck or lack of luck, I'd suggest you stop and look around and figure out what you can do with the conditions that present themselves to you. The fence isn't going away, nor is that great pasture in front of you and those mountains still are magnificent. Maybe it's time to get on your knees and shoot through the fence with the fence in detail at an aperture like f/4.5 and then the mountains will blur in the background. They will still be identifiable because of their unique nature and shape, but you'll have a shot that is different than almost anyone else.
You can also drive just a bit farther north from this location and you'll see a band of trees that are much closer to the road. This gives you the opportunity to use them to frame up a shot that works pretty well even in the harsh afternoon light.
And another thing to watch for is the fact that the bison herd is well known for using the pastures on both sides of the road for grazing. Although the view to the east is not as spectacular, you can often get some good shots of the bison herd at this location. And if you have a longer zoom lens, you are likely to see antelope grazing in the eastern field alongside the bison and cattle that frequent Elk Ranch Flats.
There are also gaps in the wooden fences on both sides - this has been done to allow the wildlife to travel from one pasture to the other. If you are fortunate enough to come by here when the bison are relocating, be sure and set up your camera and tripod at the gap in the fence line. You'll have to look for these gaps, but they are definitely there and can end up helping you get a terrific shot of the herd going through them single file.
After you have finished working this location with your camera and lens, I'd suggest you load up your vehicle and continue north on Highway 89 to the Moran Junction. At this junction, you'll want to veer right and continue on Highway 26/287. As you begin to drive up a small hill just after the junction, you'll notice that the Buffalo Fork of the Snake River is paralleling the road. Be sure and keep a quick eye out on your right as often there are wildlife present near the banks of the river.
As you continue your climb, you'll see a dirt road that turns off the highway on your right at the crest of the ridge. You'll also notice that there is no sign here, but simply a dirt road that leads off into the distance. Turn on this dirt road and follow it and veer right when there is an opportunity to do so. You will find yourself on a plateau overlooking the Jackson Hole valley with a grove of aspen trees just below you.
Although this location doesn't show up on any maps that I've seen of the area, I've named it Buffalo Fork Ridge simply because of its location over the river. It is also another one of those spots that first timers to Grand Teton National Park seldom encounter because it is relatively unknown and unmarked. But it is the eastern boundary of the park and you are welcome to drive to it and shoot panoramic shots to your heart's content.
I'd suggest using the widest angle lens in your bag and once again dragging out your tripod. You probably think I sound like a broken record, but I made the mistake of not using my tripod when I first visited this location and the photos turned out to be just a little soft. When you are doing panorama shooting, you'll want the crisp detail throughout the frame that an aperture setting of f/22 will give you. And you'll want to take into consideration the fact that there is so much real estate in front of you that you are not going to be able to capture it all with a single frame.
I'd personally take the first shot directly from where the ridge drops off the plateau and get as much of the aspens and Teton range in the frame as you can. I would also highly recommend that you take a series of shots from left to right for a panorama stitch when you return home. And you really ought to try the bracketing technique mentioned earlier - one at exposure, one at overexposure and one at underexposure - so that you can blend them together and bring out all the color that this scene presents you.
After shooting the straight on shot from a couple of different spots along the ridgeline, I'd urge you to walk towards the wooden fence line that is south of you. The fence allows you a different type of foreground interest if you get down low and shoot the fence as it trails off west and down the ridge. Try several different compositions to see what strikes your fancy, but remember that using both ends of the aperture spectrum, - let's say f/4.5 first and f/24 next - produces some strikingly different outcomes when shooting at the same location.
It really boils down to perspective and how much of the scene you want to have in focus. When I first visited the Tetons, I shot at the classic landscape settings of f/8 and f/16 - depending on whether I had sun or heavy clouds. But as I have progressed as a photographer, I've learned that having everything in perfect focus is not always the ideal technique. Sometimes that creamy blur that results from a setting like f/4.5 is actually better than absolute clarity throughout the image. This blur - that the Japanese named "bokeh" - is something you'll want to experiment with during your visit to Grand Teton National Park. Because there is little cost to compact flash or sd cards these days, you can experiment freely with these different aperture settings and discard them later if you decide you don't like the effect. But you may find that some of your favorites will incorporate bokeh as part of your overall visual presentation of what you saw during your visit.
If you are following this book's timeline, you'll find yourself on the Buffalo Fork Ridge as the morning light turns to noon light. I'd suggest you finish up with this location and return to the main highway and head back south/southwest for either a quick lunch at one of the park's lodges or a return to the town of Jackson for some afternoon rest and relaxation with your family or friends
Chapter Twelve - Final Thoughts
As we wind down our whirlwind photo visit to the Jackson Hole area, I hope you have enjoyed the journey as well as the images you've set up and produced. I've attempted here to describe the best times to visit the majority of Grand Teton National Park's best landscape and wildlife locations.
However, there are other locations that are worth exploring if you have more than four or five days in the area. Most visitors and photographers don't have the extra time and thus I've had to make some tough choices. And throughout this book, I've stayed away from recommending places that require a hike - since many photographers don't have the time nor the inclination to hike a long distance for one landscape shot. But I wanted to make you aware that there are two more hidden spots worth seeking out. They definitely require more effort - and time - before you can setup your tripod and shoot. But they are special shots that the dedicated landscape shooter might add to their shooting list when visiting Jackson Hole.
The first place is Heron Pond at the far north end of the park in Colter Bay. This requires an hour hike over relatively flat ground to get to the site, but the end result is a beautiful pond with more magical reflections of the Tetons on the pond's surface. You'll want to be sure you have a good day weather wise before you attempt this hike, as you'll need relatively clear skies and little wind to be sure your efforts pay off.
There is a well marked trail just off the south end of the most western parking lot at Colter Bay. This is the lot that ends at the edge of the lake when you drive in to Colter Bay from the main highway. You'll need to head west off HIghway 89 at the Colter Bay intersection and you'll come to the western lots. Then turn left (south) and park as close as you can to the end of the lot. Take some water with you and your tripod and follow the trail to Heron Pond. You can then set up along the shore of the pond and get the reflection shot that eludes almost every visitor to Jackson Hole.
There is another hidden location that requires a similar roundtrip hike of about an hour over a series of small hills. This is the path to Old Patriarch - a famous solitary pine that is a great foreground item for a wonderful panoramic shot of the Teton mountain range. To access this location, you'll want to drive south from the northern Jenny Lake junction approximately .6 miles and park on the side of the road. Then take your gear and hike directly east over a series of ridges. You'll ultimately come to a small gathering of trees and you'll be able to see the lone pine below you. The hike over should take you about 25 minutes and you'll want to set up east of the Old Patriarch so you can shoot the mountains behind the tree. I'd recommend a wide angle shot and also a series of zoom shots at different aperture settings - some with the tree in crisp detail and some with the mountains in focus and the tree less focused in the foreground.
Some of my other favorite spots revolve around the Jenny Lake and it's lodge, so if you do decide to hike to Old Patriarch - be sure and also drive west at the northern junction and take some shots of the towering mountains from the porch of this rustic lodge in the woods. You can't miss it because the road west from the Jenny Lake northern junction will take you right by the front door.
And speaking of Jenny Lake, there are also some fine photo opportunities at the two lakes just north of Jenny Lake. String Lake and Leigh Lake are favorites for kayak and canoe trips and you can often catch these on the water at the foot of the mountains. They are both accessed by turning west off the Teton Park Road at the northern Jenny Lake Junction and veering right into their parking lots when the road splits just a half mile before you will encounter Jenny Lake Lodge.
One of the dilemmas of these two lakes is that you are so close to the foot of the mountains that it is sometimes hard to get a wide enough angle to encompass the entire scene that you can see so clearly with your own eyes. But if you have extra time, it still is a lot of fun to try. There is a pathway from the parking lot at String Lake headed north that will take you to the bank of Leigh Lake and there is a decent reflection shot available if the wind is cooperating when you visit.
Depending on the time of year of your visit, you may want to follow the shoreline of the lake south to get some shots with the reeds in the shallow water of the lake. You are most likely to see them when you encounter a small bridge crossing that veers slightly east from the shoreline. This is also an area that wildlife has been known to frequent, so once again be on the lookout for their presence.
As we've seen so often in our journey through the park together, there is beauty all around you that is worth a stop. Although I've spent little time describing the photo opportunities from Teton Park Road, it is not because they are not there - I just wanted to be sure that with limited time you didn't miss the truly must see locations. And it took me several visits to realize that I had to prioritize my stops in order to use the light and the time most effectively.
But if you are truly without a time limit crunch, then I'd highly recommend that you take a turn down any of the other dirt roads that you encounter on your visit. Some of them are dead ends and some of them are little visited. However, unless it is specifically marked as private, you can explore to your heart's content throughout the Jackson Hole area. That's how I discovered many of my favorite spots within Grand Teton Park and that's part of the wonder and joy of this magical national park.
I wanted to also invite you to visit my website - www.jeffclow.com - when you have a chance. I'm easy to reach via this site and I plan to always have a new tip or two to share with my friends. Since you've taken the time to read this book and accompany me on a written photo journey, I hope you'll consider me a friend as well.
In closing, I wish you good light and great skies on your visit to Grand Teton National Park. It has been a pleasure to share with you some of my favorite places in all of North America, and I'm looking forward to hearing from you if you discover a new spot that I can include in a future edition of this book.
Safe travels.
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"redpajama_set_name": "RedPajamaBook"
} | 563 |
Max Payne for Minion Masters ?
I'll pass on Walking Dead but I'll do Minon Masters for Max Payne if you want. Thanks!
Anything here for Hexcells Complete Pack?
Hey! Thanks for asking but sorry I don't see anything.
Alright, adde you on steam.
hi. i want 'the way' + 'chroma squad'. please see if you want anything here.
Hi, Walking Dead Season 1 or something here for ESO pet, minion masters, Wargame: Red Dragon?
Hi! Something here for Outlast 2, ABZU, GOD EATER 2, Minion Masters, DoW III, Wargame & KF2?
Sorry for the late reply. My daughter has been sick. Anyway great list man. I no longer have Minion Masters but I'm interested in Dead Rising 4, Portal Knights, and Ruiner. If they are still available. Thanks!
NP at all, I hope she's okay now. Great list too. Which of these games (Outlast 2, ABZU, GE2, DoW III, Wargame & KF2) would you do for Dead Rising 4, Portal Knights & RUINER?
ABZU, Outlast 2, and GE2. Does that sound good?
Would you throw in Wargame or DoW III instead of GE2?
Sure, contact me whenever you can!
I'm going to sleep now, but I'll send you my games tomorrow. When you activate all, you can send me your games. Talk to you tomorrow!
Sorry I'm already doing that exact trade today. Also I already have everything I want from your list. Thanks for the reply!
anything for Wargame: Read Dragon?
1,5 euro Market Transfer for Quake Champions Early Access humble link?
Then any game from the store for 2 dollar?
Do you have CSGO keys?
I'm interested in Mother Russia Bleeds, and maybe some other games.
Would you like trade for somehting here?
Hi, sorry for the late reply. Still interested?
Hey! Sorry for the wait but I was actually able to pick up Staxel in a bundle. Thanks!
Can you find Something here for Mad Gaes Tycoon and/or Furi ?
Hey, really late reply but are you still interested?
Hello Dear, i would like to buy Bristlegut Piglet Pet Code for Elder Scrolls and i can pay 1x CS:GO or TF2 case key on it. If u dont like case key currency also i can offer an equivalent game from my list: https://www.steamtrades.com/trade/khDrd/selling-and-trading-some-games. U can add me to fast trade. Good Day !
Hey! So sorry for the late reply. Are you still interested?
Hi! I'm interested in The Banner Saga 2.
Hey! Sorry for the late reply. Are you still interested?
I'm also interested in The Beginner's Guide (giftlink).
Regency Solitaire for postworld or play with me.
He yo! Something here for Dawn of War III?
Something in here https://www.steamtrades.com/trade/cW7UF/make-your-offers for STRAFE?
anything here for Lost Castle, Slime-san and Okhlos?
Are you interested in trade your Seasons after Fall for something from my list?
Sorry for the late response. I have everything I want from your list. Cheers!
Hi, some of these for Replica ?
Never dealt with gems before. How much is a sack worth?
I'll pass this time. Thanks for the offer!
Nothing I could see. You can make me an offer if you want. To be honest I didn't look at most of them since they aren't in alphabetical order.
Hey there! Not really interested in Hello Neighbor. I looked at your trades and saw Hollow Knight. I know it's a long shot but do I have anything for that? Thanks!
Your Passpartout: The Starving Artist for my Interplanetary: Enhanced Edition?
Sorry I'm dumb haha I thought you were asking if I wanted Solstice. I forgot I had it :) I looked at your list and I like NBA Playgrounds and Running with Rifles.
That's fine. I'm fine with doing a 1-1 trade if you're interested.
Would you like to trade Four Sided Fantasy for Action Henk?
How much for Ashes of the Singularity: Escalation? Thanks. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,299 |
Experiences with sharks
Activities family
Games and printables
For infant schools
Discover L'Aquarium
The Oceanarium
Mediterranean aquariums
Themed aquariums
Jewels of the sea
Explora! Children´s area
Tropical aquariums
Events and venue hire
V @en
Violet sea urchin
Scientific name: Sphaerechinus granularis
The violet sea urchin is a sea urchin flattened in the centre. It is dark purple with porous whitish fields; the needles are almost always violet with white points.
Its shell is a maximum 15 cm in diameter.
BIOLOGY:
It is a solitary animal.
It feeds on algae and occasionally sessile or very slow animals, and also on carcasses, therefore it is an omnivorous species.
Its reproduction is oviparous and there is always separation between sexes, both the sperm and the ovules being laid freely in the water. There are not many cases of sexual dimorphism, that is, there is no external difference between males and females.
They live preferably on sandy beds with or without marine grass, or on coral, rocky beds at depths of 1 to 20 m.
They are found in the Mediterranean, the eastern Atlantic from the English Channel to the gulf of Guinea and the Azores.
Species not evaluated (according to the red list of threatened species).
CURIOSITIES:
Among their needles, they have some that have turned into a kind of clamp, part of which have poisonous glands that serve as a defence against enemies.
Do you know why they have remains of molluscs and other invertebrates on their needles? They do this to protect themselves from the light, which they flee from.
[contact-form-7 id="899″ title="newsletter-en"] | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 263 |
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